mikhailg.katz vladimirkanovei tahlnowiku.cs.biu.ac.il/~katzmik/egreg826old3.pdf · 28.10. de rham...

True infinitesimal differential

geometry

Mikhail G. Katz∗

Vladimir Kanovei

Tahl Nowik

M. Katz, Department of Mathematics, Bar Ilan Univer-

sity, Ramat Gan 52900 Israel

E-mail address : [email protected]

V. Kanovei, IPPI, Moscow, and MIIT, Moscow, Russia


T. Nowik, Department of Mathematics, Bar Ilan Uni-

versity, Ramat Gan 52900 Israel


∗Supported by the Israel Science Foundation grant 1517/12.

Abstract. This text is based on two courses taught at Bar IlanUniversity. One is the undergraduate course 89132 taught to agroup of about 120 freshmen. This course used infinitesimals tointroduce all the basic concepts of the calculus, mainly followingKeisler’s textbook. The other is the graduate course 88826 taughtto about 10 graduate students.

The graduate course developed two viewpoints on infinitesimalgenerators of flows on manifolds. Much of what we say here isdeeply affected by the pedagogical experience of teaching the twogroups.

The classical notion of an infinitesimal generator of a flow ona manifold is a (classical) vector field whose relation to the actualflow is expressed via integration. A true infinitesimal frameworkallows one to re-develop the foundations of differential geometryin this area. The True Infinitesimal Differential Geometry (TIDG)framework enables a more transparent relation between the infini-tesimal generator and the flow.

Namely, we work with a combinatorial object called a hyper-real walk constructed by hyperfinite iteration. We then deduce thecontinuous flow as the real shadow of the said walk.

Namely, a vector field is defined via an infinitesimal displace-ment in the manifold itself. Then the walk is obtained by iterationrather than integration. We introduce synthetic combinatorial con-ditions Dk for the regularity of a vector field, replacing the classicalanalytic conditions of Ck type, that guarantee the usual theoremssuch as uniqueness and existence of solution of ODE locally, Frobe-nius theorem, Lie bracket, small oscillations of the pendulum, andother concepts.

Here we cover vector fields, infinitesimal generator of flow,Frobenius theorem, hyperreals, infinitesimals, transfer principle,ultrafilters, ultrapower construction, hyperfinite partitions, micro-continuity, internal sets, halo, prevector, prevector field, regularityconditions for prevector fields, flow of a prevector field, relation toclassical flow.

Contents

Chapter 1. Preface 11

Part 1. True Infinitesimal Differential Geometry 13

Chapter 2. Introduction 152.1. Breakdown by chapters 162.2. Historical remarks 172.3. Preview of flows and infinitesimal generators 18

Chapter 3. A hyperreal view 193.1. Successive extensions N, Z, Q, R, ∗R 193.2. Motivating discussion for infinitesimals 203.3. Introduction to the transfer principle 213.4. Transfer principle 243.5. Orders of magnitude 253.6. Standard part principle 273.7. Differentiation 27

Chapter 4. Ultrapower construction and transfer principle 314.1. Introduction to filters 314.2. Examples of Filters 324.3. Properties of filters 334.4. Real number field as quotient of sequences of rationals 344.5. Extension of Q 344.6. Examples of infinitesimals 344.7. Ultrapower construction of the hyperreal field 35

Chapter 5. Microcontinuity, EVT, internal sets 375.1. Construction via equivalence relation 375.2. Extending the ordered field language 375.3. Existential and Universal Transfer 395.4. Examples of first order statements 395.5. Galaxies 415.6. Microcontinuity 435.7. Uniform continuity in terms of microcontinuity 45

3

4 CONTENTS

5.8. Hyperreal extreme value theorem 465.9. Intermediate value theorem 475.10. Ultrapower construction applied to P(R); internal sets 485.11. The set of infinite hypernaturals is not internal 495.12. Transfering the well-ordering of N; Henkin semantics 505.13. From hyperrationals to reals via ultrapower 515.14. Repeating the construction 53

Chapter 6. Application: small oscillations of a pendulum 556.1. Small oscillations of a pendulum 556.2. Vector fields, walks, and flows 566.3. The hyperreal walk and the real flow 576.4. Adequality pq 586.5. Infinitesimal oscillations 596.6. Linearisation 596.7. Adjusting linear prevector field 606.8. Conclusion 60

Chapter 7. Differentiable manifolds 637.1. Definition of differentiable manifold 637.2. Open submanifolds, Cartesian products 647.3. Tori 657.4. Projective spaces 667.5. Definition of a vector field 677.6. Zeros of vector fields 687.7. 1-parameter groups of transformations of a manifold 697.8. Invariance of infinitesimal generator under flow 707.9. Lie derivative and Lie bracket 71

Chapter 8. Theorem of Frobenius 738.1. Invariance under diffeomorphism 738.2. Frobenius thm, commutation of vector fields and flows 758.3. Proof of Frobenius 768.4. A-track and B-track approaches 788.5. Relations of ≺ and ≺≺ 788.6. Smooth manifolds; notion nearstandardness 798.7. Results for general culture on topology 80

Chapter 9. Gradients, Taylor, prevectorfields 839.1. Properties of the natural extension 839.2. Remarks on gradient 839.3. Prevectors 859.4. Tangent space to a manifold via prevectors 86

CONTENTS 5

9.5. Action of a prevector on smooth functions 879.6. Maps acting on prevectors 88

Chapter 10. Prevector fields, classes D1 and D2 9110.1. Prevector fields 9110.2. Realizing classical vector fields 9210.3. Regularity class D1 for prevectorfields 9410.4. Second differences, class D2 9410.5. Secord-order mean value theorem 9510.6. The condition C2 implies D2 9610.7. Bounds for D1 prevector fields 97

Chapter 11. Prevectorfield bounds via underspill, walks, flows 9911.1. Operations on prevector fields 9911.2. Sharpening the bounds via underspill 10011.3. Further bounds via underspill 10111.4. Global prevector fields, walks, and flows 10211.5. Internal induction 10411.6. Dependence on initial condition and prevector field 10411.7. Dependence on initial conditions for local prevector field 10611.8. Inducing a real flow 10811.9. Geodesic flow 109

Chapter 12. Universes, transfer 11112.1. Universes 11112.2. The reals as the base set 11212.3. More sets in universes 11312.4. Membership relation 11412.5. Ultrapower construction of nonstandard universes 11412.6. Los theorem 11712.7. Elementary embedding 11912.8. Definability 12212.9. Conservativity 123

Chapter 13. More regularity 12713.1. D2 implies D1 12713.2. Injectivity 12813.3. Time reversal 129

Part 2. 500 yrs of infinitesimal math: Stevin to Nelson 131

Chapter 14. 16th century 13314.1. Simon Stevin 13314.2. Decimals from Stevin to Newton and Euler 135

6 CONTENTS

14.3. Stevin’s cubic and the IVT 136

Chapter 15. 17th century 13915.1. Pierre de Fermat 13915.2. James Gregory 13915.3. Gottfried Wilhelm von Leibniz 139

Chapter 16. 18th century 14116.1. Leonhard Euler 141

Chapter 17. 19th century 14317.1. Augustin-Louis Cauchy 143

Chapter 18. 20th century 14518.1. Thoralf Skolem 14518.2. Edwin Hewitt 14518.3. Jerzy Los 14518.4. Abraham Robinson 14518.5. Edward Nelson 145

Bibliography 147

Chapter 19. Theorema Egregium 15319.1. Preliminaries to the Theorema Egregium of Gauss 15319.2. The theorema egregium of Gauss 15519.3. Intrinsic versus extrinsic 15619.4. Gaussian curvature and Laplace-Beltrami operator 15719.5. Duality in linear algebra 15819.6. Polar, cylindrical, and spherical coordinates 15919.7. Duality in calculus; derivations 161

Chapter 20. Derivations, tori 16320.1. Space of derivations 16320.2. Constructing bilinear forms out of 1-forms 16520.3. First fundamental form 16520.4. Expressing the metric in terms of 1-forms 16620.5. Hyperbolic metric 16720.6. Surface of revolution 16720.7. Coordinate change 16920.8. Conformal equivalence 17020.9. Uniformisation theorem for tori 17020.10. Isothermal coordinates 17120.11. Tori of revolution 172

Chapter 21. Conformal parameter 175

CONTENTS 7

21.1. Conformal parameter τ of standard tori of revolution 17521.2. Comformal parameter of standard torus of rev 17621.3. Curvature expressed in terms of θ(s) 17821.4. Total curvature of a convex Jordan curve 17921.5. Signed curvature kα of Jordan curves in the plane 18021.6. Coordinate patch defined by family of normal vectors 18321.7. Total Gaussian curvature and Gauss–Bonnet theorem 185

Chapter 22. Addendum on a Bernoullian framework 18722.1. Ultrapower construction 18722.2. Saturation and topology 187

Chapter 23. More on Gauss-Bonnet 18923.1. Second proof of Gauss–Bonnet theorem 18923.2. Euler characteristic 18923.3. Exercise on index notation 190

Chapter 24. Bundles 19124.1. Trivial bundle (E,B, π), function as a section (mikta) 19124.2. Mobius band as a first example of a nontrivial bundle 19224.3. Tangent bundle 19224.4. Hairy ball theorem 19324.5. Cotangent bundle 19324.6. Exterior derivative of a function 19424.7. Extending gradient and exterior derivative 19424.8. Plan for building de Rham cohomology 19524.9. Exterior algebra, area, determinant, rank 19624.10. Formal definition, alternating property 19624.11. Example: rank 1 19724.12. Areas in the plane 19724.13. Vector and triple products 19924.14. Anticommutativity of the wedge product 20024.15. The k-exterior power; simple (decomposable) multivectors20024.16. Basis and dimension of exterior algebra 20124.17. Rank of a multivector 201

Chapter 25. Multilinear forms; exterior differential complex 20325.1. Formal construction of the exterior algebra 20325.2. Exterior differential complex 20425.3. Alternating multilinear forms 20525.4. Case of 2-forms 20625.5. Norm on 1-forms 20625.6. Polar coordinates and dual norms 207

8 CONTENTS

25.7. Dual bases and dual lattices in a Euclidean space 208

Chapter 26. Comass norm, Hermitian product, symplectic form 21126.1. Λ2(V ) is isomorphic to the standard model Λ2

0(Rn) 211

26.2. Euclidean norm on k-multivectors and k-forms 21226.3. Comass norm of a form 21226.4. Symplectic form from a complex viewpoint 21326.5. Hermitian product 21426.6. Orthogonal diagonalisation 215

Chapter 27. Wirtinger inequality, projective spaces 21727.1. Wirtinger inequality 21727.2. Wirtinger inequality for an arbitrary 2-form 21927.3. Real projective space 22027.4. Complex projective line CP1 221

Chapter 28. Complex projective spaces, de Rham cohomology 22328.1. A clarification 22328.2. Complex projective space CPn 22328.3. Real projective space as a quotient 22428.4. Unitary group 22428.5. Hopf bundle 22428.6. Flags and cell decomposition of CPn 22528.7. g, A, and J in a complex vector space 22628.8. Fubini-Study 2-form on CP1 22728.9. Exterior differential complex 22828.10. De Rham cocycles and coboundaries 22928.11. De Rham cohomology ring of M , Betti numbers 229

Chapter 29. de Rham cohomology continued 23129.1. Comparison of topology and de Rham cohomology 23129.2. de Rham cohomology of a circle 23129.3. 2-dimensional cohomology of CP1 23229.4. 2-dimensional cohomology of the torus 233

Chapter 30. Elements of homology theory 23530.1. Relation between Fubini-Study 2-form and volume form on CPn23530.2. Cup product 23630.3. Cohomology of complex projective space 23630.4. Abelianisation in group theory 23730.5. The fundamental group π1(M) 23730.6. The second homotopy group π2(M) 23830.7. First singular homology group 238

CONTENTS 9

Chapter 31. Stable norm in homology 24131.1. Length, area, volume of cycles 24131.2. Length in 1-dimensional homology of surfaces 24131.3. Stable norm for higher-dimensional manifolds 24231.4. Stable 1-systole 24331.5. 2-dimensional homology and systole 24431.6. Stable norm in two-dimensional homology 24431.7. Pu’s inequality 24531.8. Integration of a 1-form over a circle 24631.9. Volume form of a surface 24731.10. Duality of homology and de Rham cohomology 248

Chapter 32. Orientation and fundamental class 25132.1. Orientation and induced orientation 25132.2. Fundamental class in homology 25232.3. Duality of comass and stable norm 25332.4. Fubini-Study metric on complex projective line 25432.5. Fubini-Study metric on complex projective space 25532.6. Gromov’s inequality for complex projective space 256

Chapter 33. Proof of Gromov’s theorem 25733.1. Homology class α and cohomology class ω 25733.2. Comass and application of Wirtinger inequality 25733.3. Volume calculation 258

Chapter 34. Eiseinstein, Hermite, Loewner, Pu, and Guth 26134.1. Eisenstein integers and Hermite constant 26134.2. Standard fundamental domain and Eisenstein integers 26234.3. Loewner’s torus inequality 26334.4. Loewner’s inequality with defect 26334.5. Computational formula for the variance (shonut) 26434.6. An application 26434.7. Fundamental domain and Loewner’s torus inequality 26534.8. Boundary case of equality 26634.9. Hopf fibration h 26734.10. Tangent map 26734.11. Riemannian submersions 26834.12. Hamilton quaternions 26834.13. Complex structures on the algebra H 269

Chapter 35. From surface tension to Loewner’s inequality 27135.1. Surface tension and shape of a water drop 27135.2. Isoperimetric inequality in the plane 271

10 CONTENTS

35.3. Central symmetry 27135.4. Property of a centrally symmetric polyhedron in 3-space 27235.5. Notion of systole 27235.6. The real projective plane RP2 27335.7. Pu’s inequality 27335.8. Loewner’s torus inequality 27435.9. Bonnesen’s inequality 274

Chapter 36. Pu’s inequality 27536.1. From complex structure to fibration 27536.2. Lie groups 27636.3. A special isomorphism 27636.4. Sphere as homogeneous space 27736.5. Dual real projective plane 27836.6. A pair of fibrations 27836.7. Double fibration of SO(3) and integral geometry on S2 27936.8. An inequality 28036.9. Proof of Pu’s inequality 281

Chapter 37. Gromov’s essential inequality 28337.1. Essential manifolds 28337.2. Gromov’s inequality for essential manifolds 28337.3. Filling radius of a loop in the plane 28337.4. The sup-norm 28537.5. Riemannian manifolds as spaces with a distance function28537.6. Kuratowski imbedding 28537.7. Homology theory for groups 28537.8. Relative filling radius 28637.9. Absolute filling radius 28637.10. further 286

CHAPTER 1

Preface

Terence1 Tao has recently published a number of monographs wherehe exploits ultrapowers in general, and Robinson’s infinitesimals in par-ticular, as a fundamental tool; see e.g., [Tao 2014], [Tao & Vu 2016].In the present text, we apply such an approach to some basic topics indifferential geometry.

Differential geometry has its roots in the study of curves and sur-faces by methods of mathematical analysis in the 17th century. Ac-cordingly it inherited the infinitesimal methods typical of these earlystudies in analysis. Weierstrassian effort at the end of the 19th centurysolidified mathematical foundations by breaking with the infinitesimalmathematics of Leibniz, Euler, and Cauchy, but in many ways the babywas thrown out with the water, as well. This is because mathematics,including differential geometry, was stripped of the intuitive appeal andclarity of the earlier studies.

Shortly afterwards, the construction of a non-Archimedean totallyordered field by Levi-Civita in 1890s paved the way for a mathemat-ically correct re-introduction of infinitesimals. Further progress even-tually lead to Abraham Robinson’s hyperreal framework, a modernfoundational theory which introduces and treats infinitesimals to greateffect with full mathematical rigor. Since then, Robinson’s frameworkhas been used in a foundational role both in mathematical analysis[Keisler 1986] and other branches of mathematics, such as differ-ential equations [Benoit 1997], measure, probability and stochasticanalysis [Herzberg 2013], [Albeverio et al. 2009], asymptotic se-ries [Van den Berg 1987], mathematical economics [Sun 2000], etc.These developments have led both to new mathematical results and tonew insights and simplified proofs of old results.

The goal of the present monograph is to restore the original infini-tesimal approach in differential geometry, on the basis of true infinitesi-mals in Robinson’s framework. In the context of vector fields and flows,the idea is to work with a combinatorial object called a hyperreal walk

1Issues in note 7 (chapter 5), note 2 (chapter11), note 10 (chapter 11), (chap-ter 12), (chapter 12).

11

12 1. PREFACE

constructed by hyperfinite iteration. We then deduce the continuousflow as the real shadow of the said walk.

We address the book to a reader willing to benefit from both the in-tuitive clarity and mathematical rigor of this modern approach. Read-ers with prior experience in differential geometry will find new (orrather old but somewhat neglected since Weierstrass) insights on knownconcepts and arguments, while readers with experience in applicationsof the hyperreal framework will find a new and fascinating applicationin a thriving area of modern mathematics.

Part 1

True Infinitesimal DifferentialGeometry

CHAPTER 2

Introduction

The classical notion of an infinitesimal generator of a flow on amanifold is a (classical) vector field whose relation to the actual flowis expressed via integration. A true infinitesimal framework allows oneto re-develop the foundations of differential geometry in this area insuch a way that the relation between the infinitesimal generator andthe flow becomes more transparent.

The idea is to work with a combinatorial object called a hyper-real walk constructed by hyperfinite iteration. We then deduce thecontinuous flow as the real shadow of the said walk.

Thus, a vector field is defined via an infinitesimal displacementdefined by a self-map F of the manifold itself. The flow is then obtainedby iteration F N (rather than integration), as in Euler’s method. As aresult, the proof of the invariance of a vector field under a flow becomesa consequence of basic set-theoretic facts such as the associativity ofcomposition, or more precisely the commutation relation F F N =F N F .

Furthermore, there are synthetic, or combinatorial, conditions Dk

for the regularity of a (pre)vector field, replacing the usual analytic con-ditions of Ck type, that guarantee the usual theorems such as unique-ness and existence of solution of ODE locally, Frobenius theorem, Liebracket, and other concepts.

A pioneering work in applying true infinitesimals to treat differen-tial geometry of curves and surfaces is [Stroyan 1977]. Our approachis different because Stroyan’s point of departure is the analytic class C1

of vector fields, whereas we use purely combinatorial synthetic condi-tions D1 (and D2) in place of C1 (and C2). Similarly, the Ck conditionswere taken as the point of departure in [Stroyan & Luxemburg 1976],[Lutz & Goze 1981], and [Almeida, Neves & Stroyan 2014].

To illustrate the advantages of this approach, we provide a proofof a case of the theorem of Frobenius on commuting flows. We alsotreat small oscillations of a pendulum (see Section 6.1), where the ideathat the period of oscillations with infinitesimal amplitude should beindependent of the amplitude finds precise mathematical expression.

15

16 2. INTRODUCTION

One advantage of the hyperreal approach to solving a differentialequation is that the hyperreal walk exists for all time, being definedcombinatorially by iteration of a selfmap of the manifold. The focustherefore shifts away from proving the existence of a solution, to estab-lishing the properties of a solution.

Thus, our estimates show that given a uniform Lipschitz bound onthe vector field, the hyperreal walk for all finite time stays in the finitepart of the ∗M . If M is complete then the finite part is nearstandard.Our estimates they imply that the hyperreal walk for all finite timedescends to a real flow on M .

2.1. Breakdown by chapters

In Chapters 3 and 4, we introduce the basic notions of an extendednumber system while avoiding excursions into mathematical logic thatare not always accessible to students trained in today’s undergraduateand graduate programs. We start with a syntactic account of a numbersystem containing infinitesimals and infinite numbers, and progress tothe relevant concepts such as filter, ultrapower construction, the exten-sion principle (from sets and functions over the reals to their naturalextensions over the hyperreals), and the transfer principle.

In Chapter 5, we present some typical notions and results fromundergraduate calculus and analysis, such as continuity and uniformcontinuity, extreme value theorem, etc., from the point of view of theextended number system, introduce the notion of internal set gener-alizing that of a natural extention of a real set, and give a relatedconstruction of the reals out of hyperrationals.

In Chapter 6.1 we apply the above to treat infinitesimal oscillationsof the pendulum.

In Chapter 7.1, we present the traditional approach to differentiablemanifolds, vector fields, flows, and 1-parameter families of transforma-tions.

In Chapter 7.8, we present the traditional “A-track” (see section 2.2)approach to the invariance of a vector field under its flow and to thetheorem Frobenius on the commutation of flows.

In Chapters 8.4 and 9.6, we present an approach to vectors and vec-tor fields as infinitesimal displacements, and obtain bounds necessaryfor proving the existence of the flow locally.

In Chapters 10 and 11 we approach flows as iteration of the (pre)vectorfield.

Chapters 12 and following contain more advanced material on uni-verses.

2.2. HISTORICAL REMARKS 17

2.2. Historical remarks

Many histories of analysis are based on a default Weierstrassianfoundation taken as a primary point of reference.

In contrast, the article [Bair et al. 2013] developed an approachto the history of analysis as evolving along separate, and sometimescompeting, tracks.

These are:

• the A-track, based upon an Archimedean continuum; and• the B-track, based upon what we refer to as a Bernoullian (i.e.,

infinitesimal-enriched) continuum.1

Historians often view the work in analysis from the 17th to the mid-dle of the 19th century as rooted in a background notion of continuumthat is not punctiform. This necessarily creates a tension with modern,punctiform theories of the continuum, be it the A-type set-theoreticcontinuum as developed by Cantor, Dedekind, Weierstrass, and oth-ers, or B-type continua as developed by [Hewitt 1948], [ Los 1955],[Robinson 1966], and others. How can one escape a trap of pre-sentism in interpreting the past from the viewpoint of set-theoreticfoundations commonly accepted today, whether of type A or B?

A possible answer to this query resides in a distinction betweenprocedure and ontology. In analyzing the work of Fermat, Leibniz,Euler, Cauchy, and other great mathematicians of the past, one mustbe careful to distinguish between

(1) its practical aspects, i.e., actual mathematical practice involv-ing procedures and inferential moves, and,

(2) semantic aspects related to the actual construction of the enti-ties such as points of the continuum, i.e., issues of the ontologyof mathematical entities such as numbers or points.

In Chapter 14 we provide historical comments so as to connect thenotions of infinitesimal analysis and differential geometry with theirhistorical counterparts, with the proviso that the connection is meantin the procedural sense as explained above.

1Scholars attribute the first systematic use of infinitesimals as a foundationalconcept to Johann Bernoulli. While Leibniz exploited both infinitesimal methodsand “exhaustion” methods usually interpreted in the context of an Archimedeancontinuum, Bernoulli never wavered from the infinitesimal methodology. To notethe fact of such systematic use by Bernoulli is not to say that Bernoulli’s foundationis adequate, or that that it could distinguish between manipulations with infinites-imals that produce only true results and those manipulations that can yield falseresults.

18 2. INTRODUCTION

2.3. Preview of flows and infinitesimal generators

As discussed in Section 7.7, the infinitesimal generator X of a flowθt(p) = θ(t, p) : R × M → M on a differentiable manifold M is a(classical) vector field defined by the relation

Xp(f) = lim∆t→0

1

∆t

(f(θ∆t(p)) − f(p)

)

satisfied by all functions f on M . Even though the term “infinitesimalgenerator” uses the adjective “infinitesimal”, this traditional notiondoes not actually exploit any infinitesimals. The formalism is somewhatinvolved, as can be sensed already in the proof of the invariance of theinfinitesimal generator under the flow it generates, as well as the proofof the special case of the Frobenius theorem on commuting flows inSection 8.2.

We will develop an alternative formalism where the infinitesimalgenerator of a flow is an actual infinitesimal (pre)vector field. As anapplication, we will present more transparent proofs of both the invari-ance of a vector field under its flow and the Frobenius theorem in thiscase.

CHAPTER 3

A hyperreal view

3.1. Successive extensions N, Z, Q, R, ∗R

Our reference for true infinitesimal calculus is Keisler’s textbook[Keisler 1986], downloadable athttp://www.math.wisc.edu/~keisler/calc.html

We start by motivating the familiar sequence of extensions of num-ber systems

N → Z → Q → R

in terms of their applications in arithmetic, algebra, and geometry.Each successive extension is introduced for the purpose of solving prob-lems, rather than enlarging the number system for its own sake. Thus,the extension Q ⊆ R enables one to express the length of the diagonalof the unit square and the area of the unit disc in our number system.

The familiar continuum R is an Archimedean continuum, in thesense that it satisfies the following Archimedean property:

(∀ǫ > 0)(∃n ∈ N) [nǫ > 1] .

We therefore provisionally denote this continuum A where “A” standsfor Archimedean. Thus we obtain a chain of extensions

N → Z → Q → A,

as above. In each case one needs an enhanced ordered1 number systemto solve an ever broader range of problems from algebra or geometry.Some historical remarks on Simon Stevin and his real numbers can befound in Section 14.1.

The next stage is the extension

N → Z → Q → A → B,

where B is a Bernoullian continuum containing infinitesimals, definedas follows.

Definition 3.1.1. A Bernoullian extension of R is any proper ex-tension which is an ordered field.

1sadur

19

http://www.math.wisc.edu/~keisler/calc.html

20 3. A HYPERREAL VIEW

Any Bernoullian extension allows us to define infinitesimals and dointeresting things with those. But things become really interesting ifwe assume the Transfer Principle (Section 3.4), and work in a truehyperreal field, defined as in Definition 3.3.1 below. We will providesome motivating comments for the transfer principle in Section 3.3.

3.2. Motivating discussion for infinitesimals

Infinitesimals can be motivated from three different angles: geo-metric, algebraic, and arithmetic/analytic.

θ

Figure 3.2.1. Horn angle θ is smaller than every recti-linear angle

Here is another figure

(1) Geometric (horn angles): Some students have expressedthe sentiment that they did not understand infinitesimals untilthey heard a geometric explanation of them in terms of whatwas classically known as horn angles. A horn angle is thecrivice between a circle and its tangent line at the point oftangency. If one thinks of this crevice as a quantity, it is easy toconvince oneself that it should be smaller than every rectilinearangle (see Figure 3.2.1). This is because a sufficiently smallarc of the circle will be contained in the convex region cutout by the rectilinear angle no matter how small. When onerenders this in terms of analysis and arithmetic, one gets apositive quantity smaller than every positive real number. Wecite this example merely as intuitive motivation (our actualconstruction is different).

3.3. INTRODUCTION TO THE TRANSFER PRINCIPLE 21

(2) Algebraic (passage from ring to field): The idea is torepresent an infinitesimal by a sequence tending to zero. Onecan get something in this direction without reliance on anyform of the axiom of choice. Namely, take the ring S of allsequences of real numbers, with arithmetic operations definedterm-by-term. Now quotient the ring S by the equivalencerelation that declares two sequences to be equivalent if theydiffer only on a finite set of indices. The resulting object S/Kis a proper ring extension of R, where R is embedded by meansof the constant sequences. However, this object is not a field.For example, it has zero divisors. But quotienting it furtherin such a way as to get a field, by extending the kernel K to amaximal ideal K ′, produces a field S/K ′, namely a hyperrealfield.

(3) Analytic/arithmetic: One can mimick the construction ofthe reals out of the rationals as the set of equivalence classesof Cauchy sequences, and construct the hyperreals as equiva-lence classes of sequences of real numbers under an appropriateequivalence relation. This viewpoint is detailed in Section 4.5.

Some more technical comments on the definability of a hyperrealline can be found in Section 12.8.

3.3. Introduction to the transfer principle

The transfer principle is a type of theorem that, depending on thecontext, asserts that rules, laws or procedures valid for a certain num-ber system, still apply (i.e., are “transfered”) to an extended numbersystem. Thus, the familiar extension Q ⊆ R preserves the propertyof being an ordered field. To give a negative example, the exten-sion R ⊆ R∪±∞ of the real numbers to the so-called extended realsdoes not preserve the property of being an ordered field. The hyperrealextension R ⊆ ∗R (defined below) preserves all first-order properties(i.e., properties involving quantification over elements but not over sets;see Section 5.4 for a fuller discussion) of ordered fields. For example,the formula sin2 x+ cos2 x = 1, true over R for all real x, remains validover ∗R for all hyperreal x, including infinitesimal and infinite valuesof x ∈ ∗R.

Thus the transfer principle for the extension R ⊆ ∗R is a theoremasserting that any statement true over R is similarly true over ∗R,


and vice versa. Historically, the transfer principle has its roots in theprocedures involving Leibniz’s Law of continuity.2

We will explain the transfer principle in several stages of increasingdegree of abstraction. More details can be found in Sections 3.4, 5.2,and 5.3.

Definition 3.3.1. An ordered field B, properly including the fieldA = R of real numbers (so that A $ B) and satisfying the TransferPrinciple, is called a hyperreal field. If such an extended field B is fixedthen elements of B are called hyperreal numbers,3 while the extendedfield itself is usually denoted ∗R.

Theorem 3.3.2. Hyperreal fields exist.

For example, a hyperreal field can be constructed as the quotient ofthe ring RN of sequences of real numbers, by an appropriate maximalideal. This will be shown in Section 4.7.

Definition 3.3.3. A positive infinitesimal is a positive hyperrealnumber ǫ such that

(∀n ∈ N) [nǫ < 1]

More generally, we have the following.

Definition 3.3.4. A hyperreal number ε is said to be infinitelysmall or infinitesimal if

−a < ε < a

for every positive real number a.

In particular, one has ε < 12, ε < 1

3, ε < 1

4, ε < 1

5, etc. If ε > 0

is infinitesimal then N = 1ε

is positive infinite, i.e., greater than everyreal number.

A hyperreal number that is not an infinite number are called finite.Sometimes the term limited is used in place of finite.

Keisler’s textbook exploits the technique of representing the hyper-real line graphically by means of dots indicating the separation betweenthe finite realm and the infinite realm. One can view infinitesimals withmicroscopes as in Figure 3.6.1. One can also view infinite numbers withtelescopes as in Figure 3.3.1. We have an important subset

finite hyperreals ⊆ ∗R

2Leibniz’s theoretical strategy in dealing with infinitesimals was analyzedin detailed studies [Katz & Sherry 2012] as well as [Katz & Sherry 2013],[Sherry & Katz 2014], [Bair et al. 2013]. We found Leibniz’ strategy to bemore robust than George Berkeley’s flawed critique thereof.

3Similar terminology is used with regard to integers and hyperintegers; seeSection 5.5.

3.3. INTRODUCTION TO THE TRANSFER PRINCIPLE 23

Finite

−2 −1 0 1 2

Positive

infinite

Infinite

telescope1ǫ− 1

1/ǫ

1ǫ

+ 1

Figure 3.3.1. Keisler’s telescope

which is the domain of the function called the standard part function(also known as the shadow) which rounds off each finite hyperreal tothe nearest real number (for details see Section 3.6).

Example 3.3.5. Slope calculation of y = x2 at x0. Here we usestandard part function (also called the shadow)

st : finite hyperreals → R

(for details concerning the shadow see Section 3.6). If a curve is definedby y = x2 we wish to find the slope at the point x0. To this end weuse an infinitesimal x-increment ∆x and compute the corresponding y-increment

∆y = (x0 + ∆x)2−x20 = (x0 + ∆x+x0)(x+ ∆x−x0) = (2x0 + ∆x)∆x.

The corresponding “average” slope is therefore

∆y

∆x= 2x0 + ∆x

which is infinitely close to 2x, and we are naturally led to the definitionof the slope at x0 as the shadow of ∆y

∆x, namely st

(∆y∆x

)= 2x0.


The extension principle expresses the idea that all real objects havenatural hyperreal counterparts. We will be mainly interested in sets,functions and relations. We then have the following extension principle.

Extension principle. The order relation on the hyperreals con-tains the order relation on the reals. There exists a hyperreal numbergreater than zero but smaller than every positive real. Every set D ⊆ Rhas a natural extension ∗D ⊆ ∗R. Every real function f with domain Dhas a natural hyperreal extension ∗f with domain ∗D.4

Here the naturality of the extension alludes to the fact that suchan extension is unique, and the coherence refers to the fact that thedomain of the natural extension of a function is the natural extensionof its domain.

A positive infinitesimal is a positive hyperreal smaller than everypositive real. A negative infinitesimal is a negative hyperreal greaterthan every negative real. An arbitrary infinitesimal is either a positiveinfinitesimal, a negative infinitesimal, or zero.

Ultimately it turns out counterproductive to employ asterisks forhyperreal functions (in fact we already dropped it in equation (3.4.1)).This issue will be dealt with in (5.4.3).

3.4. Transfer principle

Definition 3.4.1. The Transfer Principle asserts that every first-order statement true over R is similarly true over ∗R, and vice versa.

Here the adjective first-order alludes to the limitation on quantifi-cation to elements as opposed to sets, as discussed in more detail inSection 5.4.

Listed below are a few examples of first-order statements. A de-tailed account in terms of a first-order limitation can be found in Sec-tion 5.4.

Example 3.4.2. The commutativity rule for addition x+y = y+xis valid for all hyperreal x, y by the transfer principle.

Example 3.4.3. The formula

sin2 x + cos2 x = 1 (3.4.1)

is valid for all hyperreal x by the transfer principle.

4Here the noun principle (in extension principle) means that we are going toassume that there is a function ∗f : B → B which satisfies certain properties.It is a separate problem to define B which admits a coherent definition of ∗f forall f : A → A, to be solved below.

3.5. ORDERS OF MAGNITUDE 25

Example 3.4.4. The statement

0 < x < y =⇒ 0 <1

y<

1

x(3.4.2)

holds for all hyperreal x, y.

Example 3.4.5. The characteristic function χQ of the rationalnumbers equals 1 on rational inputs and 0 on irrational inputs. Bythe transfer principle, its natural extension ∗χQ = χ∗Q will be 1 onhyperrational numbers ∗Q and 0 on hyperirrational numbers (namely,numbers in the complement ∗R \ ∗Q).

To give additional examples of real statements to which transferapplies, note that all ordered field -statements are subject to Trans-fer. As we will see below, it is possible to extend Transfer to a muchbroader category of statements, such as those containing the functionsymbols exp or sin or those that involve infinite sequences of reals.

A hyperreal number x is finite if there exists a real number r suchthat |x| < r. A hyperreal number is called positive infinite if it isgreater than every real number, and negative infinite if it is smallerthan every real number.

3.5. Orders of magnitude

Hyperreal numbers come in three orders of magnitude: infinitesi-mal, appreciable, and infinite. A number is appreciable if it is finitebut not infinitesimal. In this section we will outline the rules for ma-nipulating hyperreal numbers.

To give a typical proof, consider the rule that if ǫ is positive infini-tesimal then 1

ǫis positive infinite. Indeed, for every positive real r we

have 0 < ǫ < r. It follows from (3.4.2) by transfer that that 1ǫ

is greaterthan every positive real, i.e., that 1

ǫis infinite.

Let ǫ, δ denote arbitrary infinitesimals. Let b, c denote arbitraryappreciable numbers. Let H,K denote arbitrary infinite numbers. Wehave the following theorem.

Theorem 3.5.1. We have the following rules for addition:

• ǫ + δ is infinitesimal;• b+ ǫ is appreciable;• b+ c is finite (possibly infinitesimal);• H + ǫ and H + b are infinite.

We have the following rules for products.

• ǫδ and bǫ are infinitesimal;• bc is appreciable;


• Hb and HK are infinite.

We have the following rules for quotients.

• ǫb, ǫH, bH

are infinitesimal;

• bcis appreciable;

• bǫ, Hǫ, Hbare infinite provided ǫ 6= 0.

We have the following rules for roots, where n is a standard naturalnumber.

• if ǫ > 0 then n√ǫ is infinitesimal;

• if b > 0 thenn√b is appreciable;

• if H > 0 then n√H is infinite.

Note that the traditional topic of the so-called “indeterminate forms”can be treated without introducing any ad-hoc terminology by meansof the following remark.

Remark 3.5.2. We have no rules in certain cases, such as ǫδ, HK

, Hǫ,and H +K.

Theorem 3.5.3. Arithmetic operations on the hyperreal numbersare governed by the following rules.

(1) every hyperreal number between two infinitesimals is infinites-imal.

(2) Every hyperreal number which is between two finite hyperrealnumbers, is finite.

(3) Every hyperreal number which is greater than some positiveinfinite number, is positive infinite.

(4) Every hyperreal number which is less than some negative infi-nite number, is negative infinite.

Example 3.5.4. The difference√H + 1 −

√H − 1 (where H is

infinite) is infinitesimal. Namely,

√H + 1 −

√H − 1 =

(√H + 1 −

√H − 1

) (√H + 1 +

√H − 1

)(√

H + 1 +√H − 1

)

=H + 1 − (H − 1)(√H + 1 +

√H − 1

)

=2√

H + 1 +√H − 1

is infinitesimal. Once we introduce limits, this example can be refor-mulated as follows: limn→∞

(√n + 1 −

√n− 1

)= 0.

3.7. DIFFERENTIATION 27

Definition 3.5.5. Two hyperreal numbers a, b are said to be in-finitely close, written

a ≈ b,

if their difference a− b is infinitesimal.

It is convenient also to introduce the following terminology andnotation.

Definition 3.5.6. Two nonzero hyperreal numbers a, b are said tobe adequal, written

a pq b,

if either ab≈ 1 or a = b = 0.

Note that the relation sin x ≈ x for infinitesimal x is immediatefrom the continuity of sine at the origin (in fact both sides are infin-itely close to 0), whereas the relation sin x pq x is a subtler relationequivalent to the computation of the first order Taylor approximationof sine.

3.6. Standard part principle

Theorem 3.6.1 (Standard Part Principle). Every finite hyperrealnumber x is infinitely close to an appropriate real number.

Proof. The result is generally true for an arbitrary proper orderedfield extension E of R. Indeed, if x ∈ E is finite, then x induces aDedekind cut on the subfield Q ⊆ R ⊆ E via the total order of E. Thereal number corresponding to the Dedekind cut is then infinitely closeto x.

The real number infinitely close to x is called the standard part, orshadow, denoted st(x), of x.

We will use the notation ∆x, ∆y for infinitesimals.

Remark 3.6.2. There are three consecutive stages in a typical cal-culation: (1) calculations with hyperreal numbers, (2) calculation withstandard part, (3) calculation with real numbers.

3.7. Differentiation

An infinitesimal increment ∆x can be visualized graphically bymeans of a microscope as in the Figure 3.7.1.

The slope s of a function f at a real point a is defined by setting

s = st

(f(a+ ∆x) − f(a)

∆x

)


r r + β r + γr + α

st

−1

−1

0

0

1

1

2

2

3

3

4

4r

Figure 3.6.1. The standard part function, st, “roundsoff” a finite hyperreal to the nearest real number. The func-tion st is here represented by a vertical projection. Keisler’s“infinitesimal microscope” is used to view an infinitesimalneighborhood of a standard real number r, where α, β,and γ represent typical infinitesimals. Courtesy ofWikipedia.

whenever the shadow exists (i.e., the ratio is finite) and is the samefor each nonzero infinitesimal ∆x. The construction is illustrated inFigure 3.7.2.

Definition 3.7.1. Let f be a real function of one real variable.The derivative of f is the new function f ′ whose value at a real x isthe slope of f at x. In symbols,

f ′(x) = st

(f(x + ∆x) − f(x)

∆x

)

whenever the slope exists.

Equivalently, we can write f ′(x) ≈ f(x+∆x)−f(x)∆x

. When y = f(x)we define a new dependent variable ∆y by settting

∆y = f(x + ∆x) − f(x)

called the y-increment, so we can write the derivative as st(∆y∆x

).

3.7. DIFFERENTIATION 29

a

a a+ ∆x

Figure 3.7.1. Infinitesimal increment ∆x under the microscope

Example 3.7.2. If f(x) = x2 we obtain the derivative of y = f(x)by the following direct calculation:

f ′(x) ≈ ∆y

∆x

=(x + ∆x)2 − x2

∆x

=(x + ∆x− x)(x + ∆x + x)

∆x

=∆x(2x + ∆x)

∆x= 2x+ ∆x

≈ 2x.


a

|f(a+ ∆x) − f(a)|

a a+ ∆x

Figure 3.7.2. Defining slope of f at a

Given a function y = f(x) one defines the dependent variable ∆y =f(x+∆x)−f(x) as above. One also defines a new dependent variable dyby setting dy = f ′(x)∆x at a point where f is differentiable, and setsfor symmetry dx = ∆x. Note that we have

dy pq ∆y

as in Definition 3.5.6 whenever dy 6= 0. We then have Leibniz’s notation

f ′(x) =dy

dxand rules like the chain rule acquire an appealing form.

CHAPTER 4

Ultrapower construction and transfer principle

To motivate the material on filters contained in this chapter, wewill first outline a construction of a hyperreal field ∗R exploiting filters.A more detailed technical presentation of the construction appears inSection 4.7.

4.1. Introduction to filters

Let RN denote the ring of sequences of real numbers, with arithmeticoperations defined termwise. Then we have

∗R = RN/MAX (4.1.1)

where “MAX” is an appropriate maximal ideal. What we wish toemphasize is the formal analogy (4.1.1) and the construction of the realfield as the quotient field of the ring of Cauchy sequences of rationalnumbers. Note that in both cases, the subfield is embedded in thesuperfield by means of constant sequences.

We now describe a construction of such a maximal ideal. The idealMAX consists of all “negligible” sequences 〈un〉, i.e., sequences whichvanish for a set of indices of full measure 1 namely,

m(n ∈ N : un = 0

)= 1.

Here m : P(N) → 0, 1 (thus m takes only two values, 0 and 1)is a finitely additive measure taking the value 1 on each cofinite set,1

where P(N) is the set of subsets of N. The subset Fm ⊆ P(N) consistingof sets of full measure m is called a free ultrafilter. These originate with[Tarski 1930]. The construction of a Bernoullian continuum outlinedabove was therefore not available prior to that date. The construc-tion outlined above is known as an ultrapower construction. The firstconstruction of this type appeared in [Hewitt 1948], as did the termhyper-real.

1For each pair of complementary infinite subsets of N, such a measure m decidesin a coherent way which one is negligible (i.e., of measure 0) and which is dominant(measure 1).

31

32 4. ULTRAPOWER CONSTRUCTION AND TRANSFER PRINCIPLE

As already mentioned, to present an infinitesimal-enriched contin-uum of hyperreals, we need some preliminaries on filters. Let I be anonempty set (usually N). The power set of I is the set

P(I) = A : A ⊆ I

of all subsets of I.

Definition 4.1.1. A filter on I is a nonempty collection F ⊆ P(I)of subsets of I satisfying the following axioms:

• Intersections: if A,B ∈ F , then A ∩B ∈ F .• Supersets: if A ∈ F and A ⊆ B ⊆ I, then B ∈ F .

Thus to show B ∈ F , it suffices to show

A1 ∩ · · · ∩ An ⊆ B,

for some finite n and some A1, . . . , An ∈ F .The full power set P(I) is itself a filter.A filter F contains the empty set ∅ if and only if F = P(I).

Definition 4.1.2. We say that F is proper if ∅ 6∈ F .

Every filter contains I, and in fact the one-element set I is thesmallest filter on I.

Definition 4.1.3. An ultrafilter is a proper filter that satisfies thefollowing additional property:

• for any A ⊆ I, either A ∈ F or Ac ∈ F , where Ac = I \ A.

4.2. Examples of Filters

We present several examples of filters.

(1) Principal ultrafilter. Choose i ∈ I. Then the collection

F i = A ⊆ I : i ∈ A

is an ultrafilter, called the principal ultrafilter generated by i.If I is finite, then every ultrafilter on I is of the form F i forsome i ∈ I, and so is principal.

(2) Frechet filter. The filter FFre = A ⊆ I : I \A is finite is thecofinite, or Frechet, filter on I, and is proper2 if and only if Iis infinite. Note that FFre is not an ultrafilter.

2amiti

4.3. PROPERTIES OF FILTERS 33

(3) Filter generated by a collection. If ∅ 6= H ⊆ P(I), then thefilter generated by H, i.e., the smallest filter on I including H,is the collection

FH = A ⊆ I : A ⊃ B1 ∩ · · · ∩Bn for some n and some Bi ∈ HFor H = ∅ we put FH = I.

If H has a single member B, then FH = A ⊆ I : A ⊃ B,which is called the principal filter generated by B. The ultrafil-ter F i of Example (1) is the special case of this when B = i,namely a set containing a single element.

4.3. Properties of filters

Theorem 4.3.1. An ultrafilter F satisfies

A ∩B ∈ F iff A ∈ F and B ∈ F ,as well as

A ∪B ∈ F iff A ∈ F or B ∈ F ,and also

Ac ∈ F if and only if A 6∈ F .Theorem 4.3.2. Let F be an ultrafilter and A1, . . . , An a finite

collection of pairwise disjoint (Ai ∩ Aj = ∅) sets such that

A1 ∪ . . . ∪ An ∈ F .Then Ai ∈ F for exactly one i such that 1 ≤ i ≤ n.

Theorem 4.3.3. A nonprincipal (or free) ultrafilter must containeach cofinite set. Thus every free ultrafilter F includes the Frechetfilter FFre, namely

FFre ⊆ F .This is a crucial property used in the construction of infinitesimals

and infinitely large numbers.

Remark 4.3.4. A filter F is an ultrafilter on I if and only if it is amaximal proper filter on I, i.e., a proper filter that cannot be extendedto a larger proper filter on I.

Definition 4.3.5. A collection H ⊆ P(I) has the finite intersectionproperty if the intersection of every nonempty finite subcollection of His nonempty, i.e.,

B1 ∩ · · · ∩ Bn 6= ∅ for any n and any B1, . . . , Bn ∈ H.Note that the filter FH is proper if and only if the collection H has

the finite intersection property (see Definition 4.3.5).


4.4. Real number field as quotient of sequences of rationals

To motivate the construction of the hyperreal numbers, we will firstanalyze the construction of the real numbers via Cauchy sequences.

Let QN denote the ring of sequences of rational numbers. Let

QNC

denote the subring consisting of Cauchy sequences. The reals are bydefinition the quotient field

R := QNC

/MAX (4.4.1)

where MAX is the maximal ideal consisting of null sequences (i.e.,sequences tending to zero). Note that QN

C is only a ring, whereas thequotient (4.4.1) is a field.

The point we wish to emphasize is that a field extension is con-structed starting with the base field and using sequences of elementsin the base field.

4.5. Extension of Q

We will think of sets in the ultrafilter as dominant and their com-plements as negligible.3

We choose a nonprincipal ultrafilter F on N. We form an ideal MAX =MAXF ⊆ QN consisting of all sequences 〈un : n ∈ N〉 of rational num-bers such that

n ∈ N : un = 0 ∈ F .An infinitesimal-enriched field extension ∗Q of Q may be obtained viathe ultrapower construction by forming the quotient as follows:

∗Q = QN/MAX,

where we dropped the subscript F from MAX for simplicity.

Remark 4.5.1. This is not the same maximal ideal as the one usedin Section 4.4. We use the same notation to emphasize the similarityof the two constructions.

4.6. Examples of infinitesimals

With respect to the construction presented in Section 4.5, we nowgive some examples of infinitesimals.

Definition 4.6.1. The equivalence class of a sequence u = 〈un〉will be denoted [u] or alternatively [un].

3zaniach

4.7. ULTRAPOWER CONSTRUCTION OF THE HYPERREAL FIELD 35

Example 4.6.2. Let α =[1n

]. Then α is smaller than every positive

real number r > 0.

In more detail, α = [u] where u = 〈un : n ∈ N〉 is the null sequence(i.e., sequence tending to zero) un = 1

n. Consider the set

S = n ∈ N : un < r.Equivalently, we can write

S =n ∈ N : 1

r< n

.

Since 1r

is a (necessarily finite) real number, there are only finitely many

integers that fail to satisfy the condition 1r< n. Therefore the set S is

cofinite. Thus S is a member of the Frechet filter (see Section 4.2). Butthe Frechet filter is included in each nonprincipal ultrafilter F , and inparticular in the one used in the ultrapower construction. Hence S ∈ Fand therefore α < r.

Example 4.6.3. The sequence⟨

(−1)n

n

⟩(4.6.1)

represents a nonzero infinitesimal in ∗Q, whose sign depends on whetheror not the set 2N of even natural numbers is a member of the ultrafilter.

Indeed, given a fixed real ǫ > 0, there will be only finitely manyterms in the sequence such that un ≥ ǫ. Therefore the correspondingset of indices is a member of the Frechet filter. Therefore it is a memberof every ultrafilter and in particular the chosen ultrafilter F . If (thecollection of even natural numbers) 2N ∈ F then the sequence (4.6.1)is equivalent to

⟨1n

⟩and therefore generates a positive infinitesimal.

4.7. Ultrapower construction of the hyperreal field

To obtain a full hyperreal field model of a B-continuum, we re-place Q by R in the construction of Section 4.5, and form a similarquotient

∗R := RN/MAX (4.7.1)

as explained below. We wish to emphasize the analogy with for-mula (4.4.1) defining the A-continuum, and also a key difference: thebasic domain is RN rather than RN

C . In more detail, we proceed asfollows.

(1) We define RN to be a ring with componentwise operations.(2) Choose a nonprincipal ultrafilter F ⊆ P(N).(3) Define MAXF to be a subset of RN consisting of real sequences

〈un〉 such that n ∈ N : un = 0 ∈ F .


(4) We observe that MAXF is an ideal of RN.(5) We observe that the quotient RN /MAXF is a field and in fact

an ordered field (see Definition 4.7.1).(6) We denote it by RN /F to simplify notation.

Definition 4.7.1. A natural order relation <∗ on ∗R is defined bysetting

[un] <∗ [vn]

if and only of the relation < holds for a “dominant” set of indices,where “dominance” is determined by a fixed ultrafilter F :

[un] <∗ [vn] if and only if n ∈ N : un < vn ∈ F .The complement of a dominant set can be viewed as “negligible.”4

More details on the ultrapower construction can be found in [Davis 1977].

Theorem 4.7.2. The field RN /F satisfies the transfer principle.

We give a proof of the transfer principle in Chapter 12.See [Chang & Keisler 1990, Chapter 4], on a more general model

theoretic study of ultrapowers and their applications.

4zaniach

CHAPTER 5

Microcontinuity, EVT, internal sets

5.1. Construction via equivalence relation

From now on, we will work with the following version of the con-struction (4.7.1) formulated in terms of an equivalence relation, which ismore readily generalizable to other contexts (see Section 5.10). Namely,we set

∗R = RN /∼where ∼ is an equivalence relation defined as follows:

(〈un〉 ∼ 〈vn〉

)if and only if n ∈ N : un = vn ∈ F .

Since the relation ∼ depends on F , so does the quotient RN/∼. For thisreason the logicians, and they are definitely a special breed, decided towrite directly

RN /F .But we will see that this abuse of notation offers some insights.

Definition 5.1.1. Subsets ∗Q and ∗N of ∗R consist of F -classes ofsequences 〈un〉 with rational (respectively, natural) terms.

If X ⊆ R then the subset ∗X of ∗R consists of F -classes of se-quences 〈un〉 with terms un ∈ X for each n. In particular the subsets ∗Qand ∗N of ∗R consist of F -classes of sequences 〈un〉 with rational (re-spectively, natural) terms.

Definition 5.1.2. If X ⊆ R and f : X → R, then the map ∗f :∗X → ∗R sends each F -class of a sequence 〈un〉 (un ∈ X) to the F -classof the sequence 〈vn〉 where vn = f(un) for each n.

It is easy to check that this definition ∗f is independent of choicesmade in the construction.

As noted in Definition 4.7.1, the order on ∗R is defined by set-ting [u] <∗ [v] if and only if n ∈ N : un < vn ∈ F .

5.2. Extending the ordered field language

Let the ordered field language be defined starting with a pair ofoperations · and +, and enhanced as follows.

37

38 5. MICROCONTINUITY, EVT, INTERNAL SETS

(1) We allow expressions like (x + y) · z + (x− 2y) called terms ;(2) we further define formulas like T = T ′ and T < T ′ where T, T ′

are terms;(3) we allow more complex formulas by means of logical connec-

tives and quantifiers;(4) given a concrete ordered field F , we allow the replacement of

free variables in formulas by elements of F ;(5) this leads to the notion of a formula being true in F ;(6) this finally enables us to express and study various properties

of F in a formal and well defined way.

However, the ordered field language is not sufficient as a basis foreither calculus or analysis. For instance there is no way to express phe-nomena related to non-algebraic functions like exp, sin, etc. Thereforewe need to extend the ordered field language further.

Definition 5.2.1. We will use the term extended real number lan-guage to refer to the ordered field language extended further as follows:

(7) we can freely use expressions like f(x), where f : R → R isany particular function;

(8) we can freely use formulas like x ∈ D, where D ⊆ R is anyparticular set;

(9) we can freely use any particular real numbers as parameters.

Theorem 5.2.2 (Transfer revisited). Let A be a sentence of theextended real number language, and let ∗A be obtained from A by sub-stituting ∗f and ∗D for each f or D which occur in A. Then A is trueover R if and only if ∗A is true over ∗R.

In Chapter 12 we will prove a generalisation of Theorem 5.2.2.In Remark 5.4.3 we will explain our approach to dropping the as-

terisks on functions.

Remark 5.2.3. In the statement of the theorem we wrote trueover R, or over ∗R, rather than the usual true in R, to emphasizethe fact that the sentences involved refer not only to real numbersthemselves but also to sets of reals as well as real functions, namely,objects usually thought of as elements of a superstructure over R; seeChapter 12 for more details.

Further analysis of the transfer principle appears in Section 5.3. Acomplete proof of the transfer principle appears in Chapter 12.

5.4. EXAMPLES OF FIRST ORDER STATEMENTS 39

5.3. Existential and Universal Transfer

Some motivating comments for the transfer principle already ap-peared in Section 3.3. Recall that by Theorem 4.7.2 we have an exten-sion R ⊆ ∗R of an Archimedean continuum by a Bernoullian one whichfurthermore satisfies transfer. To define ∗R, we fix a nonprincipal ultra-filter F over N and let ∗R = RN /F . By construction, elements [u] of ∗Rare represented by sequences u = 〈un : n ∈ N〉 of real numbers, u ∈ RN,with an appropriate equivalence relation defined in terms of F as inSection 4.7, or in symbols

∗R := RN/F .Here an order is defined as follows. We have [u] > 0 if and only if theset n ∈ N : un > 0 belongs to F .

Definition 5.3.1. The inclusion R → ∗R is defined by sending areal number r ∈ R to the constant sequence un = r, i.e.,

〈r, r, r, . . .〉.Since we view R as a subset of ∗R, we will denote the resulting (stan-dard) hyperreal by the same symbol r.

The transfer principle was formulated in Theorem 5.2.2. To summa-rize, it asserts that truth over R is equivalent truth over ∗R. Thereforethere are two directions to transfer in.

Definition 5.3.2. Transfer of statements in the direction

R ∗R

is called universal,1 whereas transfer in the opposite direction∗R R

is called existential.2

5.4. Examples of first order statements

In this section we will illustrate the application of the transfer prin-ciple by several examples of transfer, applied to sentences familiar fromcalculus. The statements transfer applies to are first-order quantifiedformulas, namely formulas involving quantification over field elementsonly. Quantification over subsets of the field is disallowed. Thus, thecompleteness property of the reals is not transferable because its for-mulation involves quantification over all subsets of the field.

1haavara colelet2haavara yeshit


Example 5.4.1. Rational numbers q > 0 satisfy the following:

(∀q ∈ Q+)(∃n,m ∈ N)[q =

n

m

],

where Q+ is the set of positive rationals. By universal transfer, weobtain the following statement, satisfied by all hyperrational numbers:

(∀q ∈ ∗Q+)(∃n,m ∈ ∗N)[q =

n

m

].

Here of course n,m may be infinite. Hypernatural numbers will bediscussed in more detail in Section 5.5.

Example 5.4.2. Recall that a function f is continuous at a realpoint c ∈ R if the following condition is satisfied:

(∀ǫ ∈ R+)(∃δ ∈ R+)(∀x ∈ R)[|x− c| < δ ⇒ |f(x) − f(c)| < ǫ

].

By universal transfer, a continuous function f similarly satisfies thefollowing formula over ∗R:

(∀ǫ ∈ ∗R+)(∃δ ∈ ∗R+)(∀x ∈ ∗R)[|x− c| < δ ⇒ | ∗f(x) − ∗f(c)| < ǫ

].

Note that the formula is satisfied in particular for each positive infini-tesimal ǫ ∈ ∗R. We remark that Transfer is applicable to functions, aswell (in addition to the ordered field formulas).

Remark 5.4.3. In practice most calculations exploit the hyperrealextension of a function f , rather than f itself. We will therefore con-tinue on occasion to denote by f the extended hyperreal function, aswell.

Example 5.4.4. A real function f is continuous in a domain D ⊆ Rif the following condition is satisfied:

(∀x ∈ D)(∀ǫ ∈ R+)(∃δ ∈ R+)(∀x′ ∈ D)[|x′ − x| < δ ⇒ |f(x′) − f(x)| < ǫ

].

or equivalently

(∀ǫ ∈ R+)(∀x ∈ D)(∃δ ∈ R+)(∀x′ ∈ D)[|x′ − x| < δ ⇒ |f(x′) − f(x)| < ǫ

].

(5.4.1)

By universal transfer, the hyperreal extension ∗f of such a functionwill similarly satisfy the following formula over ∗R:

(∀ǫ ∈ ∗R+)(∀x ∈ ∗D)(∃δ ∈ ∗R+)(∀x′ ∈ ∗D)[|x′ − x| < δ ⇒ |∗f(x′) − ∗f(x)| < ǫ

],

(5.4.2)

where ∗D ⊆ ∗R is the natural extension of D ⊆ R (see Section 3.1).Note that transfer is applicable to all such sets D.

5.5. GALAXIES 41

Example 5.4.5. A real function f is uniformly continuous in adomain D ⊆ R if the following condition is satisfied:

(∀ǫ ∈ R+)(∃δ ∈ R+)(∀x ∈ D)(∀x′ ∈ D)[|x′ − x| < δ ⇒ |f(x′) − f(x)| < ǫ

].

By universal transfer, the hyperreal extension ∗f of such a function willsimilarly satisfy the following formula over ∗R:

(∀ǫ ∈ ∗R+)(∃δ ∈ ∗R+)(∀x ∈ ∗D)(∀x′ ∈ ∗D)[|x′ − x| < δ ⇒ |∗f(x′) − ∗f(x)| < ǫ

].

An alternative characterisation of uniform continuity, of reducedquantifier complexity, is presented in Section 5.7.

Example 5.4.6. To give an example of existential transfer, considerthe problem of showing that if f is differentiable at c ∈ R and f ′(c) > 0then there is a point x ∈ R, x > c such that f(x) > f(c). Indeed, for

infinitesimal α > 0 we have st(f(c+α)−f(c)

α

)> 0 and hence f(c+ α) >

f(c). Setting x = c + α, we see that the following formula is satisfiedover ∗R:

(∃x)[f(x) > f(c)].

Therefore by existential transfer, the same formula holds over the realfield R. Thus there is a real x such that f(x) > f(c).

Remark 5.4.7. Another example of existential transfer will begiven in Section 5.6 following formula (5.6.3).

5.5. Galaxies

Let us consider the hypernatural numbers ∗N (positive hyperinte-gers) in more detail.

Definition 5.5.1. A hyperreal number x is called finite if |x| isless than some real number. Equivalently, x is finite if |x| is less thansome natural number. A number is infinite if it is not finite. Thus aninfinite positive hypereal is a hyperreal number bigger than each realnumber.

Now consider the extension N ⊆ ∗N constructed by an ultrapoweras in Section 4.7. Recall that a sequence u ∈ NN given by

u = 〈un : n ∈ N〉is equivalent to a sequence v ∈ NN given by v = 〈vn : n ∈ N〉 if andonly if the set of indices

n ∈ N : un = vn


is a member of a fixed ultrafilter F ⊆ P(N).We identify N with its image in ∗N under the natural embedding

generated by n 7→ 〈n, n, n, . . .〉.Theorem 5.5.2. Every finite hypernatural number is a natural num-

ber.

Proof. Let [u] ∈ ∗N where u = 〈un : n ∈ N〉 and suppose [u] isfinite. Then there exists a real number

r > 0

such that [u] < r. This means that the set of indices

n ∈ N : un < ris a member of the fixed ultrafilter F . Modifying the sequence u on anegligible set of terms does not affect its equivalence class [u]. Thereforewe can replace the remaining members of the sequence by 0 so that thecondition un < r is now satisfied for all n ∈ N. Now consider the finitecollection of integers

0, 1, 2, . . . ⌊r⌋.

Then for each n ∈ N we have either un = 0, or un = 1, or un = 2,. . . ,or un = ⌊r⌋. In other words, we represent N as a disjoint union offinitely many sets, each of type Si = n ∈ N : un = i. We now applythe defining property of an ultrafilter. It follows that exactly one ofthese sets, say

Si0 = n ∈ N : un = i0,is dominant, i.e., it is a member of F . Therefore [u] = i0 and so [u] isa natural number.

We provide an alternative proof exploiting the transfer principle.One observes that natural numbers satisfy the formula

(∀n ∈ N)[n < r ⇒ n = 0 ∨ n = 1 ∨ n = 2 ∨ . . . ∨ n = ⌊r⌋

].

Therefore by transfer, hypernatural numbers satisfy

(∀n ∈ ∗N)[n < r ⇒ n = 0 ∨ n = 1 ∨ n = 2 ∨ . . . ∨ n = ⌊r⌋

].

Therefore the hypernatural number [u] must necessarily be one of thenatural numbers 0, 1, 2, . . . , ⌊r⌋.

Corollary 5.5.3. Every member of ∗N is either a natural numberor an infinite number.

Proof. This follows from Theorem 5.5.2 by the law of excludedmiddle relied upon in classical logic.

5.6. MICROCONTINUITY 43

Definition 5.5.4. An equivalence relation ∼g on ∗N is defined asfollows: for n,m ∈ ∗N, we have

n ∼g m if and only if |n−m| is finite.

Definition 5.5.5. An equivalence class of the relation ∼g is calleda galaxy. The galaxy of an element x will be denoted gal(x).3

By Corollary 5.5.3, there is a unique galaxy containing finite num-bers, namely the galaxy

gal(0) = 0, 1, 2, . . . = N .

Each galaxy containing an infinite number such as H is of the form

gal(H) = . . . , H − 3, H − 2, H − 1, H,H + 1, H + 2, H + 3, . . .and is therefore order-isomorphic to Z (rather than N).

Corollary 5.5.6. The ordered set (∗N, <) is not well-ordered.4

5.6. Microcontinuity

Let D = Df ⊆ R be the domain of a real function f , where thenotation was discussed in Remark 5.4.3.

Definition 5.6.1. We say that f is microcontinuous at x if when-ever x′ ≈ x, one also has f(x′) ≈ f(x) for all x′ is in the domain ∗Df

of ∗f . Here x′ ≈ x means that x′ − x is infinitesimal.

Remark 5.6.2. It is a good exercise to prove that ∗(Df) = D∗f .

Remark 5.6.3. This definition can be applied not only at a realpoint x ∈ Df but also at a hyperreal point x ∈ ∗Df ; see Example 5.7.3.

Microcontinuity at all points in a segment [x0, X ] provides a usefulmodern proxy for Cauchy’s definition of continuity in his 1821 textCours d’Analyse:

the function f(x) is continuous with respect to x be-tween the given limits if, between these limits, an in-finitely small increment in the variable always pro-duces an infinitely small increment in the functionitself. (translation by [Bradley & Sandifer 2009,p. 26]).

3Sometimes galaxies are defined in the context of larger number systems, suchas ∗R. Our main interest in galaxies lies in illustrating the concept of a non-internalset; see Section 5.11.

4Sadur tov


In the current cultural climate it needs to be pointed out that Cauchywas obviously not familiar with our particular model of a B-continuum.On the other hand, his procedures and inferential moves find closerproxies in the context of modern infinitesimal-enriched continua thanin the context of modern Archimedean continua.

Theorem 5.6.4. Let c ∈ R. A real function f is continuous inthe ǫ, δ sense at c if and only if ∗f is microcontinuous at c.

Proof. We are following [Goldblatt 1998, p. 76]. Recall that areal function f is continuous at c in the ǫ, δ sense if

(∀ǫ ∈ R+)(∃δ ∈ R+)(∀x ∈ Df )[|x− c| < δ ⇒ |f(x) − f(c)| < ǫ

].

(5.6.1)Assume that ∗f is microcontinuous at c, so that

x ≈ c ⇒ ∗f(x) ≈ ∗f(c). (5.6.2)

Let us prove that f is continuous in the sense of (5.6.1). Choose areal ǫ > 0 as in the leftmost quantifier in (5.6.1). Let d > 0 be infini-tesimal. If |x−c| < d then in particular x ≈ c. Thus by microcontinuityof ∗f we have

∗f(x) ≈ ∗f(c),

and therefore |∗f(x) − ∗f(c)| < ǫ. Assigning the value d to the quanti-fied variable δ in (5.6.3) below we therefore obtain the following trueformula:

(∃δ ∈ ∗R+)(∀x ∈ ∗Df)[|x− c| < δ ⇒ |∗f(x) − ∗f(c)| < ǫ

]. (5.6.3)

More precisely, the bound |f(x) − f(c)| < ǫ is satisfied if we substi-tute δ = d, proving (5.6.3).

We now apply existential transfer (see Section 5.3 for the definition)to obtain

(∃δ ∈ R+)(∀x ∈ Df )[|x− c| < δ ⇒ |f(x) − f(c)| < ǫ

].

We conclude that there exists a real δ > 0 as required.Conversely, assume that the ǫ, δ condition (5.6.1) holds. Let ǫ be a

positive real. By (5.6.1) there is an appropriate real δ > 0 such thatthe following sentence is true:

(∀x ∈ Df)[|x− c| < δ ⇒ |f(x) − f(c)| < ǫ

]. (5.6.4)

Applying universal transfer to formula (5.6.4), we obtain

(∀x ∈ ∗Df)[|x− c| < δ ⇒ |∗f(x) − ∗f(c)| < ǫ

].

But now if we choose x such that x ≈ c, the condition |x − c| < δis automatically satisfied since x − c is infinitesimal. Therefore the

5.7. UNIFORM CONTINUITY IN TERMS OF MICROCONTINUITY 45

inequality |∗f(x) − ∗f(c)| < ǫ holds for arbitrary real ǫ > 0. It fol-lows that ∗f(x) ≈ ∗f(c), proving the relation (5.6.2) and the oppositeimplication.

5.7. Uniform continuity in terms of microcontinuity

The ǫ, δ definition of uniform continuity was reviewed in Section 5.3.We now introduce an equivalent definition in terms of microcontinuity.

Definition 5.7.1 (Alternative definition). A real function f is uni-formly continuous in its domain D = Df if and only if

(∀x ∈ ∗Df)(∀x′ ∈ ∗Df ) [x ≈ x′ ⇒ ∗f(x) ≈ ∗f(x′)] .

Remark 5.7.2. This definition of uniform continuity of a real func-tion f can be reformulated in terms of its natural extension ∗f to thehyperreals as follows:

for all x ∈ ∗Df ,∗f is microcontinuous at x.

This sounds startlingly similar to the definition of continuity it-self. What is the difference between the two definitions? The point isthat microcontinuity is now required at every point of the Bernoulliancontinuum rather than merely at the points of the Archimedean con-tinuum, i.e., in the domain of ∗f which is the natural extension of thereal domain of f .

Example 5.7.3. The function f(x) = x2 fails to be uniformly con-tinuous on its domain D = R because of the failure of microcontinuityof its natural extension ∗f at any single infinite hyperreal H ∈ ∗Df =∗R. The failure of microcontinuity at H is checked as follows. Considerthe infinitesimal e = 1

H, and the point H + e infinitely close to H . To

show that ∗f is not microcontinuous at H , we calculate∗f(H + e) = (H + e)2 = H2 + 2He+ e2 = H2 + 2 + e2 ≈ H2 + 2.

This value is not infinitely close to ∗f(H) = H2:∗f(H + e) − ∗f(H) = 2 + ǫ 6≈ 0.

Therefore microcontinuity fails at the point H ∈ ∗R. Thus the squaringfunction f(x) = x2 is not uniformly continuous on R.

The term “microcontinuity” is exploited in [Davis 1977] as wellas [Gordon et al. 2002]. It reflects the existence of two definitions ofcontinuity, one using infinitesimals, and one using epsilons. The formeris what we refer to as microcontinuity. It is given a special name to dis-tinguish it from the traditional definition of continuity. Note that mi-crocontinuity at a non-standard hyperreal does not correspond to any


notion available in the epsilontic framework limited to an Archimedeancontinuum.

Example 5.7.4. Consider the function f given by f(x) = 1x

on theopen interval D = (0, 1) ⊂ R. Then ∗f fails to be microcontinuousat a positive infinitesimal. To see this, choose an infinite H > 0 andlet x = 1

Hand x′ = 1

H+1. Clearly x ≈ x′. Both of these points are in

the extended domain ∗D = ∗(0, 1). Then∗f(x′) − ∗f(x) = H + 1 −H = 1 6≈ 0.

It follows that the real function f is not uniformly continuous on theopen interval (0, 1) ⊆ R.

5.8. Hyperreal extreme value theorem

Example 5.8.1. The unit interval [0, 1] ⊆ R has a natural extension∗[0, 1] ⊆ ∗R. This contains all positive infinitesimals as well as allnumbers smaller than 1 and infinitely close to 1.

Consider the extreme value theorem (EVT). The classical proof ofthe EVT usually proceeds in two or more stages.

(1) Typically, one first shows that the function is bounded.(2) Then one proceeds to construct an extremum by one or an-

other procedure involving choices of sequences.

The hyperreal approach is both more economical (there is no need toprove boundedness first) and less technical.

Furthermore, in the hyperreal approach one can actually write downa succinct formula for a maximum (rather than merely prove its exis-tence); see formula (5.8.3).

Theorem 5.8.2. A continuous function f on [0, 1] ⊂ R has a max-imum.

Proof. LetH ∈ ∗N \N

be an infinite hypernatural number.5 The real interval [0, 1] has anatural hyperreal extension

∗[0, 1] = x ∈ ∗R : 0 ≤ x ≤ 1.Consider its partition intoH subintervals of equal infinitesimal length 1

H,

with partition points

xi = iH, i = 0, . . . , H.

5For instance, the one represented by the sequence 〈1, 2, 3, . . .〉 with respect tothe ultrapower construction outlined in Section 4.7.

5.9. INTERMEDIATE VALUE THEOREM 47

The existence of such a partition follows by universal transfer (seeSection 5.3) applied to the first order formula

(∀n ∈ N) (∀x ∈ [0, 1]) (∃i < n)(in≤ x < i+1

n

).

The function f has a natural extension ∗f defined on the hyperrealsbetween 0 and 1. Note that in the real setting (when the number ofpartition points is finite), a point with the maximal value of f amongthe partition points xi can always be chosen by induction. We have thefollowing first order property expressing the existence of a maximumof f over a finite collection:

(∀n ∈ N) (∃i0 ≤ n) (∀i ≤ n)(f( i0

n) ≥ f( i

n)).

We now apply the transfer principle to obtain

(∀n ∈ ∗N) (∃i0 ≤ n) (∀i ≤ n)(∗f( i0

n) ≥ ∗f( i

n)), (5.8.1)

where ∗N is the collection of hypernatural numbers. Formula (5.8.1) istrue in particular for a specific infinite hypernatural value of n givenby H ∈ ∗N \N. Thus, there is a hypernatural i0 such that 0 ≤ i0 ≤ Hand

(∀i ∈ ∗N)[i ≤ H ⇒ ∗f(xi0) ≥ ∗f(xi)

]. (5.8.2)

Consider the real point

c = st(xi0) (5.8.3)

where “st” is the standard part function. By continuity of f at c ∈ R,we have ∗f(xi0) ≈ ∗f(c), and therefore

st(∗f(xi0)) = ∗f(st(xi0)) = f(c).

An arbitrary real point x lies in an appropriate sub-interval of thepartition, namely x ∈ [xi, xi+1], so that st(xi) = x, or xi ≈ x. Applying“st” to the inequality in the formula (5.8.2), we obtain

st(∗f(xi0)) ≥ st(∗f(xi)).

Hence f(c) ≥ f(x), for all real x, proving c to be a maximum of f (andby transfer, of ∗f as well).

5.9. Intermediate value theorem

Theorem 5.9.1. Let f be a continuous real function on [0, 1] andassume f(0) < 0 and f(1) > 0. Then there is a point c ∈ [0, 1] suchthat f(c) = 0.


Proof. Let H ∈ ∗ N \N and consider the corresponding partitionof ∗[0, 1] with partition points xi = i

H, i = 0, . . . , H .

Consider the set of all partition points xi such that for all pointsbefore xi, the function is nonpositive. By transfer this set has a lastpoint xi0 . By hypothesis

f(xi0) ≤ 0 < f(xi0+1).

Applying standard part we obtain

st(f(xi0)) ≤ 0 ≤ st(f(xi0+1)).

But by microcontinuity st(f(xi0)) = st(f(xi0+1)). Therefore the pointc = st(xi0) is a zero of the function.

5.10. Ultrapower construction applied to P(R); internal sets

Consider the set of subsets of R, denoted P(R). Thus A is containedin P(R), i.e., A ∈ P(R) means that A is included in R, i.e., A ⊆ R. Bythe extension principle (see Section 3.3) we have the corresponding sub-set ∗A ⊆ ∗R called the natural extension of A. We would like to applythe ultrapower construction to P(R). For convenience, we will use theshortened notation P = P(R). Then the natural extension ∗P of P isconstructed as before as a quotient of PN by an appropriate equivalencerelation defined in terms of an ultrafilter using the dominant/negligibledichotomy.

The natural extensions of subsets of R constitute a useful familyof subsets. We will now construct an important larger class of subsetsof ∗R called internal sets. These are the members of ∗P.

Definition 5.10.1. An α ∈ ∗P is an equivalence class α = [A]where A is a sequence A = 〈An ∈ P : n ∈ N〉 of elements of P (i.e.,subsets of R), where A ∼ B if and only if n ∈ N : An = Bn ∈ F .

In more detail, we have the following definition.

Definition 5.10.2. Consider a sequence A = 〈An ∈ P(R) : n ∈ N〉of subsets of R. We use it to define a set α = [A] ⊆ ∗R as follows. Ahyperreal [u] = [un] is an element of the set α = [A] if and only if onehas

n ∈ N : un ∈ An ∈ F .Such sets α ⊆ ∗R are called internal.

Here as usual F ⊆ P(N) is the fixed free ultrafilter used in theconstruction of the hyperreal field.

5.11. THE SET OF INFINITE HYPERNATURALS IS NOT INTERNAL 49

Example 5.10.3. An example of a subset of ∗R which is internal butis not a natural extension of any real set is the interval [0, H ] where His an infinite hyperreal. To show that it is internal, represent H by asequence u = 〈un : n ∈ N〉, so that H = [u]. Then the set α = [0, H ] isthe equivalence class α = [A] of the sequence A = 〈An : n ∈ N〉 of realintervals An = [0, un] ⊆ R.

5.11. The set of infinite hypernaturals is not internal

Does the construction of Section 5.10 give all possible subsets of ∗R?The answer turns out to be negative even in the case of the hypernat-urals, as we show in Theorem 5.11.4.

Theorem 5.11.1. Each internal subset of ∗N has a least element.

Remark 5.11.2. In other words, ∗N is internally well-ordered inthe sense that the well-ordering property is satisfied if we only dealwith internal sets.

Proof. Assume that an internal α ia given by α = [A] where A =〈An〉. Here each An can be assumed nonempty by Lemma 5.11.3 below.Since N is well-ordered we can choose a minimal element un ∈ An foreach n. Consider the sequence u = 〈un : n ∈ N〉. Its equivalenceclass [u] is the minimal element in the set α.

Lemma 5.11.3. A nonempty set α = [A] where A = 〈An〉 can alwaysbe represented by a sequence where each of the sets An in the sequenceis nonempty.

Proof. The set of indices n for which An = ∅ is negligible sinceotherwise α would be the empty set itself. For each n from this neg-ligible set of indices n, we can replace the corresponding set An = ∅by An = N without affecting the equivalence class α = [A]. In the newsequence all the sets An are nonempty.

Theorem 5.11.4. The set of infinite hypernaturals, ∗N \ N, is notan internal subset of ∗N.

Proof. If the set ∗N \ N were internal it would have a least ele-ment by Theorem 5.11.1. But there is no minimal infinite hypernaturalbecause if H is an infinite hypernatural then H − 1 is another infinitehypernatural.

Recall that every galaxy of infinite hypernaturals is order-isomorphicto Z. For the structure of the galaxies see Section 5.5.

Thus the set of infinite hypernaturals is external. It follows thatevery internal set including ∗N \ N will necessarily contain also some


elements that are finite natural numbers. This principle is called un-derspill.

This property is expploited in Section 11.2.

5.12. Transfering the well-ordering of N; Henkin semantics

Is it possible to transfer the well-ordering of the natural numbers?All the examples of transfer given in Section 5.3 deal with quantifi-

cation over numbers, such as natural, rational, or real numbers.As already pointed out, quantification over sets cannot be encom-

passed by the transfer principle because the least upper bound propertyfor bounded sets in R fails when interpreted literally over ∗R.

Example 5.12.1. The set of all infinitesimals in ∗R does not admita least upper bound. Indeed, if C > 0 were such a bound, then either Cis infinitesimal or it is appreciable. If C were infinitesimal, then theinfinitesimal 2C is greater than C, contradicting the claim that C is al.u.b. If C is appreciable, then C/2 is also appreciable and thereforea smaller upper bound than C. The contradiction shows that there isno l.u.b.

On the other hand we showed in Section 5.11 that, while the set ofinfinite hypernaturals is not well-ordered with respect to the naturalordering induced from ∗N, nevertheless internal subsets of ∗N do satisfythe well-ordering condition.

In fact, quantification over sets in N can be transfered on conditionof being interpreted as quantification over elements of P = P(N), asfollows. The condition of well-ordering can be stated as follows. Tosimplify the formula we will use the symbol ∀′ to denote quantificationover nonempty sets only:

(∀′A ⊆ N)(∃u ∈ A)(∀x ∈ A)[u ≤ x

].

To make this transferable, we reformulate the condition as follows:

(∀A ∈ P \ ∅)(∃u ∈ A)(∀x ∈ A)[u ≤ x

],

where P = P(N). At this point transfer can be applied, resulting inthe following sentence:

(∀A ∈ ∗P \ ∅

)(∃u ∈ A)(∀x ∈ A)

[u ≤ x

].

Here quantification is over elements of ∗P. In other words, quantifica-tion is over internal subsets of ∗N, resulting in a correct sentence. Thepoint is that

∗P(N) $ P(∗N).

5.13. FROM HYPERRATIONALS TO REALS VIA ULTRAPOWER 51

Remark 5.12.2. Robinson refers to this approach to quantificationas Henkin semantics.

All sentences involving quantification over subsets of

A ⊆ R (5.12.1)

can similarly be transfered provided we interpret the quantification asranging over

A ∈ P(R) (5.12.2)

and transfering formula (5.12.2) instead of (5.12.1), to obtain

A ∈ ∗P(R),

entailing quantification over internal subsets. For example, the leastupper bound property for bounded sets holds over the hyperreals, pro-vided we interpret it as applying to internal subsets only.

5.13. From hyperrationals to reals via ultrapower

This section supplements the material of Section 4.7 on the ultra-power construction.

We used the ultrapower construction to build the hyperreal fieldout of the real field. But the ultrapower construction can alo be usedto give an alternative construction of the real number system startingwith the rationals.

Let Q be the field of rational numbers. The star-transform of Qgives an extension

Q ⊆ ∗Qto the hyperrationals. Here ∗Q = QN /MAX (see Section 4.5 for de-tails).

Consider the subring F ⊆ ∗Q consisting of all finite hyperrationals.More precisely, we have the following definition.

Definition 5.13.1. We set

F =x ∈ ∗Q : (∃r ∈ Q)

[|x| < r

].

We also define the ideal I ⊆ F of infinitesimal elements by setting I =x ∈ F : x < Q+

, or more precisely

I =x ∈ F : (∀r ∈ Q)

[r > 0 ⇒ |x| < r

].

Here the absolute value bars denote the natural extension of theusual absolute value function on the rationals (as usual the star super-script is suppressed). Here F is not a field because an infinitesimal isnot invertible in F . We will now show that the ring F admits a naturalquotient which is isomorphic to R.


Theorem 5.13.2. The ideal I ⊆ F is maximal, so that the quotientfield

ρ = F/I

is naturally isomorphic to R.

Proof. We will indicate the isomorphisms in each direction, asfollows: φ : ρ→ R and ψ : R → ρ.

Given an element x ∈ ρ, choose a representative x ∈ F . Since x isfinite, its standard part is well-defined, and we set

φ(x) = st(x),

where “st” is the standard part. This is independent of the choice ofthe representative because the standard part function suppresses allinfinitesimal differences. We therefore obtain a map φ : ρ→ R.

To construct a homomorphism in the opposite direction, considera positive real number y ∈ R. The procedure will be similar for anegative number but we describe it for a positive number so as to fixideas. Let

K = ⌊10Hy⌋so that

0 ≤ 10Hy −K < 1.

Dividing by 10H we obtain

0 ≤ y − K

10H<

1

10H

so that y ≈ K10H

. In other words, we are considering the decimal ap-proximation of y truncated6 at rank H , resulting in a hyperrational

K

10H∈ F ⊆ ∗Q.

Then the map ψ : R → ρ is defined by sending the real number y to theequivalence class of the finite hyperrational K

10Hin F/I, or in formulas

ψ(y) =K

10H+ I.

Then we have φ(ψ(y)) = st( K10H

) = y, proving the theorem.7

6katum (with kuf)? mekutzatz? google translate gives “esroni mekutzatz”7Should the following material be included in the main text?This can be elaborated as follows in terms of the extended decimal expansion.

Consider the decimal expansion of y: y = a.a1a2a3 . . . an . . . defined for each naturalindex n ∈ N. By transfer, the decimal digits an are defined for each hypernaturalrank, as well: y = a.a1a2a3 . . . an . . . ; . . . aH−1aHaH+1 . . . Here ranks to the leftof the semicolon are finite, while ranks to the right of the semicolon are infinite.Choose a infinite hypernatural H ∈ ∗N \N. The product 10Hy has the form 10Hy =

5.14. REPEATING THE CONSTRUCTION 53

5.14. Repeating the construction

What happens if we apply the F/I-type construction but nowwith R in place of Q? Namely, consider the star-transfer ∗R, and thering of finite hyperreals ∗RF ⊆ ∗R, as well as the ideal of infinitesimals

∗RI ⊆ ∗RF .

It turns out that we do not get anything new in this direction, asattested to by the following.

Theorem 5.14.1. The quotient ∗RF/∗RI is naturally isomorphic

to R itself.

The proof is essentially the same, with the main ingredient beingthe existence of the standard part function with range R:

st : ∗RF → R

with kernel precisely ∗RI .

Remark 5.14.2. Similar considerations apply to Rn, showing thatthe quotient ∗Rn

F/∗Rn

I is naturally isomorphic to Rn itself. We willreturn to this observation in Section 8.5.

a a1a2a3 . . . an . . . aH−1aH . aH+1 . . . with a decimal point after the digit aH . There-

fore its integer part is the hyperinteger K = ⌊10Hy⌋ = a a1a2a3 . . . an . . . aH−1aH ,

and we proceed as above.

CHAPTER 6

Application: small oscillations of a pendulum

The material in this chapter first appeared in Quantum Studies:Mathematics and Foundations [Kanovei, Katz & Nowik 2016].

6.1. Small oscillations of a pendulum

In his 1908 book Elementary Mathematics from an Advanced Stand-point, Felix Klein advocated the introduction of calculus into the high-school curriculum. One of his arguments was based on the problemof small oscillations of the pendulum.1 The problem had been treateduntil then using a somewhat mysterious superposition principle. Thelatter involves (a vertical plane projection of) a hypothetical circularmotion of the pendulum. Klein advocated what he felt was a bet-ter approach, involving the differential equation of the pendulum; see[Klein 1908, p. 187].

The classical problem of the pendulum translates into the secondorder nonlinear differential equation x = −g

ℓsin x for the variable an-

gle x with the vertical direction, where g is the constant of gravity and ℓis the length of the (massless) rod or string. The problem of small os-cillations deals with the case of small amplitude,2 i.e., x is small, sothat sin x is approximately x. Then the equation is boldly replacedby the linear one x = −g

ℓx, whose solution is harmonic motion with

period 2π√ℓ/g.

This suggests that the period of small oscillations should be in-dependent of their amplitude. The intuitive solution outlined abovemay be acceptable to a physicist, or at least to the mathematicians’proverbial physicist. The solution Klein outlined in his book does notgo beyond the physicist’s solution. The Hartman–Grobman theorem[Hartman 1960], [Grobman 1959] provides a criterion for the flowof the nonlinear system to be conjugate to that of the linearized sys-tem, under the hypothesis that the linearized matrix has no eigenvaluewith vanishing real part. However, the hypothesis is not satisfied forthe pendulum problem.

1Tnudat metutelet2misraat

55

56 6. APPLICATION: SMALL OSCILLATIONS OF A PENDULUM

To provide a rigorous mathematical treatment for the notion ofsmall oscillation, it is tempting to exploit a hyperreal framework fol-lowing [Nowik & Katz 2015]. Here the notion of small oscillationcan be given a precise sense, namely infinitesimal amplitude. Notehowever the following.

Remark 6.1.1. Even for infinitesimal x one cannot boldly replace sin xby x.

Therefore additional arguments are required.The linearisation of the pendulum is treated in [Stroyan 2015] us-

ing Dieners’ “Short Shadow” Theorem; see Theorem 5.3.3 and Exam-ple 5.3.4 there. This text can be viewed as a self-contained treatementof Stroyan’s Example 5.3.4.

The traditional setting in the context of the real continuum can onlymake sense of the claim that “the period of small oscillations is inde-pendent of the amplitude” by means of a paraphrase in terms of limits,rather than a specific oscillation. In the context of an infinitesimal-enriched continuum, such a claim can be formalized more literally; seeCorollary 6.8.1.

6.2. Vector fields, walks, and flows

The framework developed in [Robinson 1966] involves a properextension ∗R ⊇ R preserving the properties of R to a large extent dis-cussed in Remark 6.3.3. Elements of ∗R are called hyperreal numbers.A positive hyperreal number is called infinitesimal if it is smaller thanevery positive real number.

We choose a fixed positive infinitesimal λ ∈ ∗R (a restriction on thechoice of λ appears in Section 6.8).

Remark 6.2.1. We will work with flows in a two-dimensional con-text where the manifold can be conveniently identified with C. We willalso exploit the natural extension ∗C ⊇ C.

Definition 6.2.2. Given a classical vector field V = V (z) where z ∈C, one forms an infinitesimal displacement δF (of the intended prevec-torfield Φ) by setting δΦ = λV .

Thus the aim is to construct the corresponding flow Φt in the plane.Note that a zero of δΦ corresponds to a fixed point of the flow.

Definition 6.2.3. The infinitesimal generator of the flow is thefunction Φ : ∗C → ∗C, also called a prevector field, defined by

Φ(z) = z + δΦ(z), (6.2.1)

6.3. THE HYPERREAL WALK AND THE REAL FLOW 57

where δΦ(z) = λV (z) in the case of a displacement generated by aclassical vector field as above.

Remark 6.2.4. An infinitesimal generator, or prevector field, couldbe a more general internal function. See Section 5.10 and Chapter 8.4for details on internal sets, and Definition 10.3.3 for the D1 conditionfor prevectorfields.

6.3. The hyperreal walk and the real flow

We propose a concept of solution of differential equation based onEuler’s method with infinitesimal step, with well-posedness based on aproperty of adequality (see Section 6.4), as follows.

Definition 6.3.1. The hyperreal flow, or walk, Φt(z) is a t-paramet-rized map ∗C → ∗C defined whenever t is a hypernatural multiple t =Nλ of λ, by setting

Φt(z) = ΦNλ(z) = Φ Φ . . . Φ(z) = ΦN (z), (6.3.1)

where ΦN is the N -fold composition. For arbitrary hyperreal t thewalk can be defined by setting Φt = Φ⌊t/λ⌋λ.

Remark 6.3.2. The fact that the infinitesimal generator Φ givenby (6.2.1) is invariant under the flow Φt of (6.3.1) receives a transparentmeaning in this framework, expressed by the commutation relation

Φ ΦN = ΦN Φ

due to transfer of associativity of composition of maps.

Remark 6.3.3. As discussed in Section 3.3, the transfer principleis a type of theorem that, depending on the context, asserts that rules,laws or procedures valid for a certain number system, still apply (i.e.,are transfered) to an extended number system. Thus, the familiar ex-tension Q ⊆ R preserves the properties of an ordered field. To give anegative example, the extension R ⊆ R∪±∞ of the real numbersto the so-called extended reals does not preserve the properties of anordered field. The hyperreal extension R ⊆ ∗R preserves all first-orderproperties, including the identity sin2 x + cos2 x = 1 (valid for all hy-perreal x, including infinitesimal and infinite values of x ∈ ∗R). Thenatural numbers N ⊆ R ⊆ ∗R are naturally extended to the hypernat-urals ∗N ⊆ ∗R. For a more detailed discussion, see Chapter 12.

Definition 6.3.4. The real flow φt on C for t ∈ R when it exists isconstructed as the shadow (i.e., standard part) of the hyperreal walk Φt

by settingφt(z) = st

(ΦNλ(z)

)


where N =⌊tλ

⌋, while ⌊x⌋ rounds off the hyperreal number x to the

nearest integer no greater than x, and “st” (standard part or shadow)rounds off each finite hyperreal to its nearest real number.

For t sufficiently small, appropriate regularity conditions ensurethat the point ΦNλ(z) is finite so that the shadow is well-defined.

The usual relation of being infinitely close is denoted ≈. Thus forfinite (i.e., non-infinite) z, w we have z ≈ w if and only if st(z) = st(w).This relation is an additive one.

The appropriate relation for working with small prevector fields isa multiplicative rather than an additive one, as detailed in Section 6.4.

6.4. Adequality pq

We will use Leibniz’s notation pq . Leibniz actually used a symbolthat looks more like ⊓ but the latter is commonly used to denote aproduct. Leibniz used the symbol to denote a generalized notion ofequality “up to” (though he did not distinguish it from the usual sym-bol = which he also used in the same sense). A prototype of such arelation (though not the notation) appeared already in Fermat underthe name adequality. We will use it for a multiplicative relation among(pre)vectors.

Definition 6.4.1. Let z, w ∈ ∗C. We say that z and w are adequaland write z pq w if either one has z

w≈ 1 (i.e., z

w− 1 is infinitesimal)

or z = w = 0.

This implies in particular that the angle between z, w is infinites-imal (when they are nonzero), but the relation pq entails a strongercondition. If one of the numbers is appreciable, then so is the otherand the relation z pq w is equivalent to z ≈ w. If one of z, w is infin-itesimal then so is the other, and the difference |z − w| is not merelyinfinitesimal, but so small that the quotients |z − w|/z and |z − w|/ware infinitesimal, as well.

We are interested in the behavior of orbits in a neighborhood ofa fixed point 0, under the assumption that the infinitesimal displace-ment satisfies the Lipschitz condition. In such a situation, we have thefollowing theorem.

Theorem 6.4.2. Prevector fields defined by adequal infinitesimaldisplacements define hyperreal walks that are adequal at each finite time.

In particular, if δΦ pq δG then φt = gt where φt and gt are thecorresponding real flows. This was shown in [Nowik & Katz 2015,Example 5.12].

6.6. LINEARISATION 59

6.5. Infinitesimal oscillations

Let x denote the variable angle between an oscillating pendulumand the downward vertical direction. By considering the projection ofthe force of gravity in the direction of motion, one obtains the equationof motion mℓx = −mg sin x where m is the mass of the bob of thependulum, ℓ is the length of its massless rod, and g is the constant ofgravity. Thus we have a second order nonlinear differential equation

x = −gℓ

sin x. (6.5.1)

The initial condition of releasing the pendulum at angle a (for ampli-tude) is described by

x(0) = a

x(0) = 0

We replace (6.5.1) by the pair of first order equationsx =

√gℓy,

y = −√

gℓ

sin x,

and initial condition (x, y) = (a, 0). We identify (x, y) with x + iyand (a, 0) with a + i0 as in Section 6.2. The classical vector fieldcorresponding to this system is then

V (x, y) =√

gℓy −

(√gℓ

sin x)i. (6.5.2)

The corresponding prevector field (i.e., infinitesimal generator of theflow) Φ is defined by the infinitesimal displacement

δΦ = (λ√

gℓy) − (λ

√gℓ

sin x)i

so thatΦ(z) = z + δΦ(z).

We are interested in the flow of Φ, with initial condition a + 0i. Theflow is generated by hyperfinite iteration of Φ.

6.6. Linearisation

Consider also a prevector field E(z) = z + δE(z) defined by thedisplacement

δE(z) = λ√

gℓy − iλ

√gℓx

where as before z = x+ iy. We are interested in small oscillations, i.e.,the case of infinitesimal amplitude a. Since sin x pq x for infinitesimal xwe have

δE pq δΦ.


Due to the multiplicative nature of the relation of adequality, the rescal-ings of E and Φ by change of variable z = aZ remain adequal andtherefore define adequal hyperreal walks by Theorem 6.4.2.

6.7. Adjusting linear prevector field

We will compare E to another linear prevector field H defined by

H(x+ iy) = e−iλ√gℓ (x+ iy)

=(x cos λ

√gℓ

+ y sinλ√

gℓ

)+(−x sin λ

√gℓ

+ y cosλ√

gℓ

)i

given by clockwise rotation of the x, y plane by infinitesimal angle λ√

gℓ.

The corresponding hyperreal walk, defined by hyperfinite iteration ofH ,satisfies the exact equality

Ht(a, 0) =(a cos

√gℓt,−a sin

√gℓt)

(6.7.1)

whenever t is a hypernatural multiple of λ. In particular, we have theperiodicity property H 2π√

g/ℓ

(z) = z and hence

Ht+ 2π√g/ℓ

= Ht (6.7.2)

whenever both t and 2π√gℓ

are hypernatural multiples of λ. The usual

estimatescosλ

√gℓ− 1

λ≈ 0,

sinλ√

gℓ

λ√

gℓ

≈ 1. (6.7.3)

imply an adequality δE pq δH whenever x and y are finite. By Theo-rem 6.4.2 the hyperfinite walks of E and H satisfy

Et(a, 0) pq Ht(a, 0)

for each finite initial amplitude a and for all finite time t which is ahypernatural multiple of λ.

6.8. Conclusion

The advantage of the prevector field H is that its hyperreal walkis given by an explicit formula (6.7.1) and is therefore periodic withperiod precisely 2π√

gℓ

, provided we choose our base infinitesimal λ in

such a way that 2π

λ√gℓ

is hypernatural. We obtain the following con-

sequence of the periodicity property (6.7.2): modulo an appropriatechoice of a representing prevector field (namely, H) in the adequality

6.8. CONCLUSION 61

class, the hyperreal walk is periodic with period 2π√ℓ/g. This can be

summarized as follows.

Corollary 6.8.1. The period of infinitesimal oscillations of thependulum is independent of their amplitude.

If one rescales such an infinitesimal oscillation to appreciable sizeby a change of variable z = aZ where a is the amplitude, and takesstandard part, one obtains a standard harmonic oscillation with pe-riod 2π

√ℓ/g. The formulation contained in Corollary 6.8.1 has the

advantage of involving neither rescaling nor shadow-taking.

CHAPTER 7

Differentiable manifolds

7.1. Definition of differentiable manifold

A n-dimensional manifold M is a set covered by a collection of sub-sets (called coordinate charts or neighborhoods), typically denoted Aor B, and having the following properties. For each coordinate neigh-borhood A we have an injective map u : A → Rn whose image is anopen set in Rn. Thus, the coordinate neighborhood is a pair

(A, u).

The maps are required to satisfy the following compatibility condition.Let

u : A→ Rn, u = (ui)i=1,...,n,

and similarly

v : B → Rn, v = (vα)α=1,...,n.

Whenever the overlap A ∩ B is nonempty, it has a nonempty imagev(A ∩ B) in Euclidean space. Restricting to this subset, we obtain aone-to-one map

u v−1 : v(A ∩B) → Rn (7.1.1)

from an open set v(A ∩ B) ⊆ Rn to Rn.Similarly, the map v u−1 from the open set u(A∩B) ⊆ Rn to Rn

is one-to-one.

Definition 7.1.1. The defining condition of a smooth manifoldis that the map (7.1.1) is differentiable for all choices of coordinateneighborhoods A and B as above. The maps u v−1 are called thetransition maps.1 The collection of coordinate charts as above is calledan atlas.

The coordinate charts induce a topology on M by imposing theusual conditions (arbitrary unions of open sets are open; finite inter-sections of open sets are open).

1funktsiot maavar (nothing to do with ekron haavara)

63

64 7. DIFFERENTIABLE MANIFOLDS

Remark 7.1.2. We will usually assume that M is connected. Giventhe manifold structure as above, connectedness of M is equivalent topath-connectedness2 of M .

Remark 7.1.3 (Metrizability). There are some pathological non-Hausdorff examples like two copies of R glued along an open halfline.To rule out such examples, the simplest condition is that of metrizabil-ity; see e.g., Example 7.3.2, Theorem 7.4.2.

Identifying A ∩ B with a subset of Rn by means of the coordi-nates (ui), we can think of v as given by n functions vα(u1, . . . , un)as α runs from 1 to n.

Remark 7.1.4. The manifold condition can be stated as the re-quirement that the functions vα(ui) are all smooth.

Example 7.1.5. The usual hierarchy of smoothness denoted Ck

(or C∞) in Euclidean space generalizes to manifolds. Thus, for k = 1a manifold M is C1 if and only if all the partial derivatives

∂vα

∂ui, α = 1, . . . , n, i = 1, . . . , n

are continuous. Similarly, M is C2 if all second partials

∂2vα

∂ui∂uj

are continuous, etc.

7.2. Open submanifolds, Cartesian products

The notion of open and closed set in M is inherited from Euclideanspace via the coordinate charts. An open subset C ⊆ M of a mani-fold M is itself a submanifold, with differentiable structure obtainedby the restriction of the coordinate map u = (ui) of (A, u) or in formu-las uA∩C .

Let Matn,n(R) be the set of square matrices with real coefficients.This is identified with Euclidean space of dimension n2, and is thereforea manifold.

Theorem 7.2.1. Define a subset GL(n,R) ⊆ Matn,n(R) by setting

GL(n,R) = X ∈ Matn,n(R) : det(X) 6= 0.Then GL(n,R) is a submanifold.

2kshir-mesila

7.3. TORI 65

Proof. The determinant function is a polynomial in the entries xijof the matrix X . Therefore it is a continuous function of the entries,

which are the coordinates in Rn2

. Hence GL(n,R) is an open set, andtherefore a manifold.

Note that its complement is the closed set consisting of matrices ofzero determinant.

Theorem 7.2.2. Let M and N be two differentiable manifolds ofdimensions m and n. Then the Cartesian product M × N is a differ-entiable manifold of dimension m + n. The differentiable structureis defined by coordinate neighborhoods of the form (A × B, u × v),where (A, u) is a coordinate chart on M , while (B, v) is a coordinatechart on N . Here the function u× v on A× B is defined by

(u× v)(a, b) = (u(a), v(b))

for all a ∈ A, b ∈ B.

7.3. Tori

Example 7.3.1. The torus T 2 = S1 × S1 is a 2-dimensional man-ifold. More generally, the n-torus T n = S1 × · · · × S1 (product of ncopies of the circle) is an n-dimensional manifold.

Example 7.3.2. The circle S1 = (x, y) ∈ R2 : x2 + y2 = 1 itselfis a manifold. Let us give an explicit atlas for the circle. Let A+ ⊆ S1

be the open upper halfcircle

A+ = a = (x, y) ∈ S1 : y > 0.Consider the coordinate chart (A+, u), namely

u : A+ → R,

defined by setting u(x, y) = x. The open lower halfcircle A− also givesa coordinate chart (A−, u) where the coordinate u is defined by thesame formula. We similarly define the right halfcircle

B+ = a = (x, y) ∈ S1 : x > 0,yielding a coordinate chart (B+, v) where v(x, y) = y, and similarlyfor B−. The transition function between A+ and B+ is calculated bynoting that in the overlap A+ ∩ B+ one has both x > 0 and y > 0.

Let us calculate a typical transition function u v−1. The map v−1

sends y ∈ R1 to the point (√

1 − y2, y) ∈ S1, and then the coordinate

map u sends (√

1 − y2, y) to the first coordinate√

1 − y2 ∈ R1. Thusthe composed map u v−1 given by

y 7→√

1 − y2


is the transition function in this case. Since this is smooth for ally ∈ (0, 1), the circle is a C∞ manifold of dimension 1 (modulo spellingout the remaining transition functions). Setting d(p, q) = arccos〈p, q〉we see that S1 is metrizable.

7.4. Projective spaces

Definition 7.4.1. Let X = Rn+1 \0 be the collection of (n +1)-tuples x = (x0, . . . , xn). Define an equivalence relation ∼ amongtuples x, y ∈ X by setting x ∼ y if and only if there is a real number t 6=0 such that y = tx, i.e.,

yi = txi, i = 0, . . . , n.

The collection of equivalence classes is called the real projective spaceand denoted RPn.

Theorem 7.4.2. The space RPn is a smooth n-dimensional mani-fold.

Proof. We define coordinate neighborhoods Ai, where i = 0, . . . , nby setting

Ai = x : xi 6= 0. (7.4.1)

We will now define the coordinate pair (Ai, ui), where ui : Ai → Rn,namely the corresponding coordinate chart. We can set

ui(x) =

(x0

xi, . . . ,

xi−1

xi,xi+1

xi, . . . ,

xn

xi

)

since division by xi is allowed in Ai by (7.4.1). The coordinate ui iswell-defined because if x ∼ y then ui(x) = ui(y). To calculate thetransition map, let u = ui and v = uj, where we assume for simplicitythat i < j. We wish to calculate the map uv−1 associated with Ai∩Aj.Take a point z = (z0, . . . , zn−1) ∈ Rn. Since we work with xj 6= 0, wecan rescale the homogeneous coordinates so that xj = 1, so we canrepresent v−1(z) by

v−1(z) = (z0, . . . , zj−1, 1, zj, . . . , zn−1).

Now we apply u = ui, obtaining

u v−1(z) =

(z0

zi, . . . ,

zi−1

zi,zi+1

zi, . . .

zj−1

zi,

1

zi,zj+1

zi, . . . ,

zn−1

zi

)(7.4.2)

All transition functions appearing in (7.4.2) are rational functions andare therefore smooth. Thus RPn is a differentiable manifold. Set-ting d(p, q) = arccos |〈p, q〉| for unit p, q we see that RPn is metriz-able.

7.5. DEFINITION OF A VECTOR FIELD 67

7.5. Definition of a vector field

Recall that the tangent space TpM at a point p ∈ M is the collectionof all tangent vectors at the point p. Here a tangent vector can bedefined via derivations.3

With respect to a coordinate chart (A, u) where u = (ui)i=1,...,n, wehave a basis ( ∂

∂ui) for TpM . Therefore an arbitrary vector X at p is a

linear combination

X i ∂

∂ui,

for appropriate X i. Here we use the Einstein summation convention.Since the vectors

(∂∂ui

)form a basis for the tangent space at every

point p ∈ A, a choice of component functions X i(u1, . . . , un) in theneighborhood will define a vector field

X i(u1, . . . , un)∂

∂ui

in A. Here the components X i are required to be of an appropriatedifferentiability type. More precisely, we have the following definition.

Definition 7.5.1. (see [Boothby 1986, p. 117]) Let M be a C∞

manifold. A vector field X of class Cr on M is a function assigning toeach point p of M a vector Xp ∈ TpM whose components in any localcoordinate (A, u) are functions of class Cr.

Example 7.5.2. Let M be the Euclidean plane R2. Via obviousidentifications the Euclidean norm in the (x, y)-plane leads naturallyto a Euclidean norm | | on the tangent space (i.e., tangent plane) atevery point with respect to which both tangent vectors ∂

∂xand ∂

∂yare

orthogonal and have unit norm:∣∣∣∣∂

∂x

∣∣∣∣ =

∣∣∣∣∂

∂y

∣∣∣∣ = 1.

Therefore the dual basis (dx, dy) of 1-forms also satisfies |dx| = |dy| =1, and the two basis 1-forms are orthogonal.

Note that each of ∂∂x

and ∂∂y

defines a global vector field on R2

(defined at every point).

Example 7.5.3 (Vector fields defined by polar coordinates). In-teresting examples of vector fields are provided by polar coordinates.These may be undefined at the origin, i.e., apriori only defined in the

3This material was covered in the undergraduate course 88201.


open submanifold R2 \0. Here ∂∂θ

is represented by the path (r cos t, r sin t)with derivative (−r sin t, r cos t). Hence

∣∣∣∣∂

∂θ

∣∣∣∣ = r (7.5.1)

so that 1r∂∂θ

is of norm 1. Passing to the dual basis we obtain that rdθ

has norm 1 because ∂∂r

, ∂∂θ

are orthogonal.4

7.6. Zeros of vector fields

Example 7.6.1 (Zero of type source/sink). We mentioned that thevector field ∂

∂rin the plane is undefined at the origin, but r ∂

∂ris a

continuous (C0) vector field vanishing at the origin. Moreover it isequal to x ∂

∂x+ y ∂

∂dyand hence smooth.

The vector field in the plane defined by r ∂∂r

is sometimes called

a “source” while the vector field X = −r ∂∂r

is sometimes called a“sink”.5 Note that the integral curves of a source flow from the originand away from it, whereas the integral curves of a sink flow into theorigin, and converge to it at for large time. At a point p ∈ R2 withpolar coordinates (r, θ) this is given by

X(p) =

−r ∂

∂rif p 6= 0

0 if p = 0.

Example 7.6.2 (Zero of type “circulation”). The vector ∂∂θ

in theplane, viewed as a tangent vector at a point at distance r from theorigin, tends to zero as r tends to 0, as is evident from (7.5.1). Thereforethe vector field defined by ∂

∂θon R2 \0 extends by continuity to the

point p = 0. Moreover it equals −y ∂∂x

+ x ∂∂y

and hence smooth.

4Alternatively, since dr2 = dx2 + dy2 by Pythagoras, we have |dr| = 1 as well.

Meanwhile θ = arctan yx and therefore dθ = 1

1+(y/x)2d(y/x) = x2

y2+x2

xdy−ydxx2 =

xdy−ydxr2 . Hence

|dθ| =|xdy − ydx|

r2=

r

r2=

1

r. (7.5.2)

Thus rdθ is a unit covector. We therefore have an orthonormal basis (dr, rdθ) forthe cotangent space. Since dθ

(∂∂θ

)= 1, equation (7.5.2) implies (7.5.1). Therefore

1r

∂∂θ is a unit vector.

5Would these be makor and kior?

7.7. 1-PARAMETER GROUPS OF TRANSFORMATIONS OF A MANIFOLD 69

Thus we obtain a continuous vector field p 7→ X(p) = −y ∂∂x

+ x ∂∂y

on R2 which vanishes at the origin:

X(p) =

∂∂θ

if p 6= 0

0 if p = 0.

Such a vector field is sometimes described as having circulation6 aroundthe point 0.

Note that the integral curves of a circulation are circles around theorigin.

Example 7.6.3 (Zero of type “circulation”on the sphere). Sphericalcoordinates (ρ, θ, ϕ) in R3 restrict to the unit sphere S2 ⊆ R3 to givecoordinates (θ, ϕ) on S2. The north pole is defined by ϕ = 0. Herethe angle θ is undefined but the vector field ∂

∂θcan be extended by

continuity as in the plane, and we obtain a zero of type “circulation”.Similarly the south pole is defined by ϕ = π. Here ∂

∂θhas a zero

of type “circulation” but going clockwise (with respect to the naturalorientation on the 2-sphere).

7.7. 1-parameter groups of transformations of a manifold

Let θ : R×M → M be a smooth mapping (this θ is in generalunrelated to polar coordinates). Denote coordinates t ∈ R and p ∈ M .We will write θt(p) for θ(t, p). Assume θ satisifies the following twoconditions:

(1) θ0(p) = p for all p ∈M ;(2) θt θs(p) = θt+s(p) for all p ∈M and all s, t ∈ R.

Then θ is called a 1-parameter group of transformations, a flow, or anaction for short.

An action θ defines a vector field X on M called the infinitesimalgenerator of θ, as follows.

Definition 7.7.1. The infinitesimal generator X of the flow θt isdefined at a point p ∈M as a derivation Xp defined by setting for eachfunction f on M ,

Xp(f) = lim∆t→0

1

∆t

(f(θ∆t(p)) − f(p)

).

Remark 7.7.2. This is a generally accepted term defined as abovein a context where actual infinitesimals are not present.

6machzor, tzirkulatsia.


7.8. Invariance of infinitesimal generator under flow

In this section we follow [Boothby 1986]. Consider a flow θt on amanifold M as in Section 7.7, namely a map

θ : R×M →M,

where we write θt(p) for θ(t, p), such that θ0(p) = p for all p ∈M , andθt θs(p) = θt+s(p) for all p ∈M and all s, t ∈ R.

Then one obtains an induced action θt∗ on a vector field Y on M ,producing a new vector field θt∗(Y ) as follows. We view Y as a deriva-tion acting on functions f on M . Given a function f on M , we need tospecify the manner in which the new vector field θt∗(Y ) differentiatesthe function f .

Definition 7.8.1. The induced action on the vector field Y of theflow θ of the manifold M is defined by setting

θt∗

(Yp)f = Yp(f θt)so that the vector θt∗(Yp) resides at the point θt(p) ∈M .

In more detail, let q = θt(p). If a function f is defined in a neigh-borhood of q ∈ M then the composed function f θt is defined in aneighorhood of p and therefore it makes sense apply the derivation Ypto the function f θt.

Recall that the infinitesimal generator X of a flow θ is the vectorfield satisfying

Xp(f) = lim∆t→0

1

∆t

(f(θ∆t(p)) − f(p)

).

What happens when we apply the induced action of the flow to thevector field which is the infinitesimal generator of the flow itself? Ananswer is provided by the following theorem.

Theorem 7.8.2 (Boothby p. 124). The infinitesimal generator Xof an action θt is invariant under the flow:

θt∗(Xp) = Xθt(p).

Proof. The proof is a direct computation, obtained by testing thefield on a function f ∈ Dq where q = θt(p) (and Dq is the space ofsmooth functions defined in a neighborhood of q) as follows:

θt∗(Xp)f = Xp(f θt)

= lim∆t→0

1

∆t

(f θt (θ∆t(p)) − f θt(p)

).

7.9. LIE DERIVATIVE AND LIE BRACKET 71

Since R is abelian, we have θt θ∆t = θt+∆t = θ∆t θt (by definition ofa flow). Hence

θt∗(Xp)f = lim∆t→0

1

∆t

(f θ∆t(θt(p)) − f(θt(p))

).

= Xθt(p)f,

proving the theorem.

Note that while theorem 7.8.2 is intuitively “obvious”, the receivedformalism involved in proving it is bulky. We will develop a more directapproach in Section 9.6.

7.9. Lie derivative and Lie bracket

In this section as in the previous one we follow Boothby.

Definition 7.9.1 (Boothby p. 154). Let θt be a flow on M withinfinitesimal generator X , and let Y be a vector field on M . The vectorfield LXY , called the Lie derivative of Y with respect to X , is definedby setting

LXY = limt→0

1

t

(θ−t∗(Yθ(t,p)) − Yp

)

Theorem 7.9.2. If X and Y are vector fields on M then

LXY = [X, Y ],

where [X, Y ]f = Xp(Y f) − Yp(Xf) is the Lie bracket.

Proof. This can be checked in coordinates using a Taylor formulawith remainder.

Example 7.9.3. The coordinate vector fields commute with respectto the Lie bracket, i.e., [

∂

∂ui,∂

∂uj

]= 0

for all i, j = 1, . . . , n. This is an equivalent way of stating the equalityof mixed partials (sometimes called Schwarz’s theorem or Clairaut’stheorem).

CHAPTER 8

Theorem of Frobenius

8.1. Invariance under diffeomorphism

In this section as in Section 7.9 we follow Boothby. We first reviewsome concepts from advanced calculus. If

σ : M → N

is a smooth map such that σ(p) = q ∈ N , we obtain an inducedmap σ∗ : TpM → TqN . Here σ∗ is the map induced on the tangentspace. This can be defined as follows.

Definition 8.1.1. Choose a representing curve α(t) so that X =α′(0), and define σ∗(X) to be the vector represented by the curve de-fined by the composition σ α on the image manifold N .

In coordinate-free notation, chain rule takes the following form.

Theorem 8.1.2 (Chain rule). for smooth maps σ, τ among mani-folds asserts that the composition σ τ satisfies

(σ τ)∗ = σ∗ τ∗,i.e., (σ τ)∗(X) = σ∗ τ∗(X) for all tangent vectors X.

Definition 8.1.3. A diffeomorphism of M is a bijective 1-1 map φ :M →M such that both φ and φ−1 are C∞ functions for all coordinatecharts.

Recall that we have the notion of a flow θt : M →M depending onparameter t ∈ R.

Definition 8.1.4. A vector field X is said to generate a flow θtif X is the infinitesimal generator of θt as in Definition 7.7.1.

Recall from Definition 7.7.1 that the vector field (which is the in-finitesimal generator of the flow) and the flow itself are related by

Xp(f) = lim∆t→0

1

∆t

(f(θ∆t(p)) − f(p)

). (8.1.1)

73

74 8. THEOREM OF FROBENIUS

Definition 8.1.5. An integral curve,1 or orbit, of a vector field Xthrough a fixed point p ∈M is a curve θ(t, p) satisfying θ(0, p) = p and∂θ∂t

= X where t ranges through an open interval containing 0.

Recall that a vector field X is said to be invariant under a smoothmap σ if σ∗(X) = X at every point. In more detail, if q = σ(p) ∈ Mthen we require that σ∗(Xp) = Xq.

Theorem 8.1.6 (Boothby p. 142). Let X be a vector field gener-ating a flow θ(t, p) on M and let F : M → M be a diffeomorphism.Then X is invariant under F if and only if F commutes with the flow,i.e., F (θ(t, p)) = θ(t, F (p)).

Proof of the easy direction. First we show the easy direc-tion. Suppose X is invariant under the map F , so that F∗(Xp) = XF (p)

for all p ∈M . Let σp(t) be an integral curve through p ∈M . Since wewish to use p as the index, we use the letter σ in place of θ to avoidclash of notation with θt introduced earlier (see (8.1.1)). By chain rulewe have for each t that

(F σp)∗ = F∗ σp∗.Thus the image curve F σp sends the vector d

dt(a tangent vector

to R) to F∗(X); see (8.1.3) for more details. Thus F takes the inte-gral curve σp of the vector field X to an integral curve of the vectorfield F∗(X). Since F∗(X) = X by hypothesis, the uniqueness of in-tegral curves implies that F (θ(t, p)) = θ(t, F (p)), proving the easydirection.

Proof of the opposite direction. To prove the opposite di-rection, suppose

F (θ(t, p)) = θ(t, F (p)) (8.1.2)

and let us prove that F∗(Xp) = XF (p).Consider an integral curve σp through p defined by σp : R → M

where

σp(t) = θ(t, p),

so that σp(0) = p ∈ M . Here σp may be defined on a smaller intervalthan R but we used R for simplicity. As before, we use the letter σinstead of θ so as to avoid a clash of notation with θt = θ(t, p) usedpreviously.

1Akuma-integralit (or akom-integrali). The odd term akumat-iskum foundat http://hebrew-terms.huji.ac.il/teva english result.asp? seems to be a canaan-ite innovation.

8.2. FROBENIUS THM, COMMUTATION OF VECTOR FIELDS AND FLOWS75

Let ddt

be the natural basis for the tangent line T0R at the origin.

Let σ = dσdt

. Then

Xp = σp(0) = σp∗

(d

dt

). (8.1.3)

We now apply the isomorphism F∗ : TpM → TF (p)M to formula (8.1.3).We obtain

F∗(Xp) = F∗ σp∗(d

dt

)= (F σp)∗

(d

dt

)= σF (p)∗

(d

dt

),

where the last equality follows from F σp(t) = σF (p)(t) as in (8.1.2).

Now the theorem results from the equality σF (p)∗(ddt

)= XF (p).

8.2. Frobenius thm, commutation of vector fields and flows

In this section as in the previous one we follow Boothby. Here weprove a special case of the Frobenius theorem on flows, in the case ofcommuting flows.

Recall from Definition 7.9.1 that if θt is a flow on M with infin-itesimal generator X , and Y is a vector field on M , then the vectorfield LXY , called the Lie derivative of Y with respect to X , is definedby setting

LXY = limt→0

1

t

(θ−t∗(Yθ(t,p)) − Yp

),

and furthermore LXY = [X, Y ].

Theorem 8.2.1 (Boothby p. 156). Let X and Y be C∞ vector fieldson a manifold M . Let θ = θt be the flow generated by X, and η = ηsbe the flow generated by Y . Then the following two conditions areequivalent:

(1) [X, Y ] = 0;(2) for each p ∈ M there exists a δp > 0 such that ηs θt(p) =

θt ηs(p) whenever |s| < δp and |t| < δp.

Proof. First we show the easy direction (2) =⇒ (1). We applyTheorem 8.1.6 with F = θt to conclude that the infinitesimal genera-tor Y of the flow ηs is invariant under θt for a fixed small parameter t,i.e., θt∗(Yq) = Yθ(t,q). Now consider the Lie derivative. We obtain2

[X, Y ]q = (LXY )q = limt→0

1

t

(θ−t∗(Yθ(t,q)) − Yq

)= lim

t→0

1

t(Yq − Yq) = 0.

The other direction will be shown in the next segment.

2The calculation in Boothby on page 157 contains some misprints involvingmisplaced parentheses.


Proof of the opposite direction. Now let us show the oppo-site direction (1) =⇒ (2) in Theorem 8.2.1. Assume that [X, Y ] = 0identically on M . Consider a point q ∈ M . Define a vector Zq(t) at qby setting

Zq(t) = θ−t∗(Yθ(t,q)

)∈ TqM. (8.2.1)

Let us show that the t-derivative Z(t) vanishes. Consider the point q′ =θ(t, q). Then

Zq(t) = lim∆t→0

1

∆t

(θ−(t+∆t)∗

(Yθ(∆t,q′)

)− θ−t∗(Yq′)

)(8.2.2)

Decomposing the action of θ−(t+∆t)∗ as composition

θ−(t+∆t)∗ = θ−t∗ θ−∆t∗,

we obtain from (8.2.2) and linearity of induced maps that

Zq(t) = θ−t∗

(lim∆t→0

1

∆t

(θ−∆t∗

(Yθ(∆t,q′)

)− Yq′

))(8.2.3)

Now we recognize the expression 1∆t

(θ−∆t∗

(Yθ(∆t,q′)

)− Yq′

)as the Lie

derivative LXY of Y at the point q′ = θ(t, q). Therefore we obtain

Zq(t) = θ−t∗ ((LXY )q′) = 0

since by hypothesis (1), the Lie bracket vanishes everywhere, includingat the point q′ = θ(t, q).

Therefore the vector Zq(t) of (8.2.1) is constant for |t| < δ, andtherefore equal to Yq. This means that Y is invariant under the flow θt.Therefore by Theorem 8.1.6, we have the commutation of flows ηs θt(q) = θt ηs(q).

8.3. Proof of Frobenius

Theorem 8.3.1 (Frobenius). Let U ⊆ Rn be open, and X1, . . . , Xk :U → Rn be k independent C1 classical vector fields such that [Xi, Xj]

′ =∑km=1C

mijXm for Cm

ij : U → R. (As before [·, ·]′ is the classical Lie

bracket.) Let git be the classical flow of Xi. Given p ∈ U , let r > 0be such that ϕ(t1, . . . , tk) = g1t1 g2t2 · · · gktk(p) is defined on (−r, r)kand ∂ϕ

∂t1, . . . , ∂ϕ

∂tkare independent there. Then span ∂ϕ

∂t1(a), . . . , ∂ϕ

∂tk(a) =

spanX1(ϕ(a)), . . . , Xk(ϕ(a)) for every a ∈ (−r, r)k.Proof. By definition of giti ,

∂∂ti

(giti · · · gktk)(p) = Xi. So to

show ∂ϕ∂ti

∈ spanX1, . . . , Xk one needs to show that

(g1t1 · · · gi−1ti−1

)∗(Xi) ∈ spanX1, . . . , Xk.

8.3. PROOF OF FROBENIUS 77

So one needs to show that (gjtj )∗(Xi) ∈ spanX1, . . . , Xk for every i, j,

and for convenience of notation say j = 1. Using the flow g1t we maychange coordinates so that X1 = ∂

∂x1, and so the flow of X1 is sim-

ply gt(x) = x+(t, 0, . . . , 0), and we need to show that spanX1, . . . , Xkis invariant under the flow gt, t ∈ (−r, r).

For fixed q let Xi(t) = Xi(q + (t, 0, . . . , 0)) then

d

dtXi(t) = [X1, Xi]

′ =

k∑

m=1

Cm1i (t)Xm(t).

Since this simple flow maps a vector to the “same” (i.e. parallel) vectorat the image point, spanX1, . . . , Xk being invariant under the flowmeans that spanX1(t), . . . , Xk(t) is the same subspace for all t ∈(−r, r).

Let us change basis in Rn such that X1(q), . . . , Xk(q) are the first kstandard basis vectors.

Let us denote the coordinates of Xi by Xji , j = 1, . . . , n, so we must

show that Xji (t) = 0 for all 1 ≤ i ≤ k, k + 1 ≤ j ≤ n and t ∈ (−r, r).

Let W ⊆ (−r, r) be the set of all t satisfying Xji (t) = 0 for all 1 ≤

i ≤ k, k + 1 ≤ j ≤ n. Then W is clearly closed and 0 ∈ W , so if weshow that W is also open then W = (−r, r) and we are done.

At this point we pass to the nonstandard extension. To showthat W is open we must show that for every w ∈ W and every s ≈ w,s ∈ ∗W . By transfer s ∈ ∗W means that Xj

i (s) = 0 for all 1 ≤ i ≤k, k + 1 ≤ j ≤ n. Say s > w.

Let A = max |Xji (u)|, w ≤ u ≤ s, 1 ≤ i ≤ k, k + 1 ≤ j ≤ n, and

assume this maximum is attained at u = u′, with i = a, j = b.Let B = max | d

dtXji (u)|, w ≤ u ≤ s, 1 ≤ i ≤ k, k + 1 ≤ j ≤ n, and

assume this maximum is attained at u = u′′, with i = c, j = d.Then since Xb

a(w) = 0 we have

A = |Xba(u

′)|= |Xb

a(u′) −Xb

a(w)| ≤ |u′ − w|B

= |u′ − w| · | ddtXdc (u′′)|

= |u′ − w| · |∑

m

Cm1c(u

′′)Xdm(u′′)|

≺ |u′ − w|A≺≺ A.


We have A ≺≺ A and so necessarily A = 0 and so Xji (s) = 0 for

all 1 ≤ i ≤ k, k + 1 ≤ j ≤ n.The notation ≺ is explained in Section 8.5.

8.4. A-track and B-track approaches

Here we follow [Nowik & Katz 2015]. In Chapter 7.8, we pre-sented the received A-track3 formalism for dealing with the infinitesi-mal generator of a flow on a manifold. Note that the adjective “infin-itesimal” in this context is a dead metaphor as it no longer refers totrue infinitesimals. We will now develop an infinitesimal formalism fordealing with the infinitesimal generator of a flow.

Of course, an infinitesimal (B-track) formalism enables a more in-tuitive approach not merely to the infinitesimal generator of a flow,but to many other topics in calculus and analysis. Thus, the conceptsof derivative, continuity, integral, and limit can all be defined via in-finitesimals. For example, in Section 5.8 we presented the infinitesimalapproach to the extreme value theorem.

8.5. Relations of ≺ and ≺≺We will work in the hyperreal extension R ⊆ ∗R.

Definition 8.5.1. For r, s ∈ ∗R, we will write

r ≺ s

if r = as for finite a. We will also write

r ≺≺ s

if r = as for infinitesimal a.

Thus r ≺ 1 means that r is finite, while

r ≺≺ 1

means that r is infinitesimal.More generally, suppose we are given a finite dimensional vector

space V over R. Then, given v ∈ ∗V , and s ∈ ∗R, we will write

v ≺ s

if ∗‖v‖ ≺ s for some norm ‖ · ‖ on V . We will generally omit the ∗ fromfunction symbols and so will simply write ‖v‖ ≺ s. This condition isindependent of the choice of norm since all norms on V are equivalent.Similarly we will write

v ≺≺ s

3nativ

8.6. SMOOTH MANIFOLDS; NOTION NEARSTANDARDNESS 79

if ‖v‖ ≺≺ s. We will also write

v ≈ w

when v−w ≺≺ 1. We choose a basis for V thus identifying it with Rn.

Lemma 8.5.2. An element x = (x1, . . . , xn) ∈ ∗Rn satisfies x ≺ sor x ≺≺ s if and only if for each i, the component xi satisfies thecorresponding relation.

This can be checked directly for the Euclidean norm on Rn.

Definition 8.5.3. Given 0 < s ∈ ∗R, consider the subspaces V sF =

v ∈ ∗V : v ≺ s and V sI = v ∈ ∗V : v ≺≺ s.

Then V sI ⊆ V s

F ⊆ ∗V are linear subspaces over R.

Theorem 8.5.4. We have a canonical isomorphism

V sF/V

sI∼= V.

Indeed, by Section 5.14 we have V 1F /V

1I

∼= V , and multiplicationby s maps V 1

F onto V sF and V 1

I onto V sI .

8.6. Smooth manifolds; notion nearstandardness

Our object of interest is a smooth manifold M together with anenlargement ∗M ⊇M .

Definition 8.6.1. For p ∈M , the halo of p, which we denote by

hal(p) = halM(p),

is the set of all points x in ∗M ⊇ M , for which there is a coordinateneighborhood U of p such that x ∈ ∗U and x ≈ p in the given coordi-nates.

Remark 8.6.2. By including the index M in the definition of thehalo, we emphasize its dependence on the ambient manifold. For ex-ample, the halo of the origin in ∗R consists of hyperreal infinitesimals,while the halo of the origin in ∗R2 consists of points both of whosecoordinates are infinitesimal.

In fact, the definition of hal(p) does not require coordinates, butrather depends only on the topology of M . The points of M are calledstandard.

Definition 8.6.3. A point which is in hal(p) for some p ∈ M iscalled nearstandard.4

4kestandartit


Definition 8.6.4. If a is nearstandard then st(a) is the unique p ∈M such that a ∈ hal(p).

Definition 8.6.5. For a subset A ⊆M , the halo hA of A is a subsetof ∗M defined by

hA =⋃

a∈AhalM(a).

In particular, hM is the set of all nearstandard points in ∗M .The following is proved in [Davis 1977, p. 90, Theorem 5.6]. Let X

be a metric space, e.g., a finite-dimensional smooth manifold equippedwith a Riemannian metric.

Theorem 8.6.6 (Davis). For a metric space X the following areequivalent:

(1) every bounded closed set in X is compact;(2) every finite point of ∗X is nearstandard.

In particular, in every finite-dimensional complete Riemannian man-ifold, finite and nearstandard are equivalent.

8.7. Results for general culture on topology

We have the following characterisations of open sets and compactsets. A good reference is [Davis 1977].

Theorem 8.7.1. A subset A is open in M if and only if hA ⊆ ∗A.

Example 8.7.2. The subset of M = R given by the closed inter-val A = [0, 1] is not open since ∗A fails to contain negative infinitesimalswhich are in the halo of A, i.e., halM(0) 6⊆ ∗A.

Theorem 8.7.3 (Davis p. 78, item 1.6). A set A ⊆ M is compactif and only if ∗A ⊆ hA.

Example 8.7.4. The subset of R given by the open interval A =(0, 1) is not compact since a positive infinitesimal ǫ is in ∗A but ǫ is notin the halo of A, since it is not infinitely close to any point of A becauseevery point of A is appreciable. Since every point of A is appreciable,so is every point of hA, but infinitesimals are not appreciable.

In particular, if A is compact then ∗A ⊆ hM . Recall that the haloboth of a point and of a subset are defined relative to the ambientmanifold M (see Remark 8.6.2).

Theorem 8.7.5 (Davis p. 78, item 1.7). The manifold M itself iscompact if and only if hM = ∗M .

8.7. RESULTS FOR GENERAL CULTURE ON TOPOLOGY 81

If M is noncompact then hM is an external set.5

Example 8.7.6. The 1-dimensional manifold M = R is not com-pact. The halo hR of R consists of all finite hyperreals. This is anexternal subset of ∗R. Recall that hR is the domain of the standardpart function:

st : hR → Rdefined in Section 3.6.

5A set is external if it is not internal. See Section 5.10.

CHAPTER 9

Gradients, Taylor, prevectorfields

9.1. Properties of the natural extension

Given an open W ⊆ Rn and a smooth function f : W → R wenote some properties of the extension ∗f : ∗W → ∗R, obtained bytransfer. When there is no risk of confusion, we will omit the ∗ fromthe function symbol ∗f and simply write f for both the original functionand its extension. Recall that hW ⊆ ∗W consists of all nearstandardpoints (see Section 8.6).

Lemma 9.1.1. For open W ⊆ Rn, let f : W → R be continuous.Then f(a) is finite for every a ∈ hW .

Proof. Given a ∈ hW , let U ⊆W be a neighborhood of st(a) suchthat U ⊆W and U is compact.

By compactness, there is a constant C ∈ R such that |f(x)| ≤ Cfor all x ∈ U . By transfer |f(x)| ≤ C for all x ∈ ∗U . In particular wehave |f(a)| ≤ C. Therefore f(a) is finite. (As for functions, we omitthe ∗ from relation symbols, writing ≤ in place of ∗≤.)

9.2. Remarks on gradient

Recall that hU is the set of nearstandard points of ∗U , where ∗Uis the star-transform of U . Given an open U ⊆ Rn and a smoothfunction f : U → R, we can compute the partial derivatives ∂f

∂uias

usual.

Definition 9.2.1. The partial derivatives of ∗f are by definitionthe functions

∗

(∂f

∂ui

),

i.e., the extensions of the partial derivatives of f .

Definition 9.2.2. Given f ∈ C1 we consider the row vector

Da

of partial derivatives at each point a ∈ ∗U .

83

84 9. GRADIENTS, TAYLOR, PREVECTORFIELDS

One similarly defines the higher partial derivatives at a point a ∈∗U .

Definition 9.2.3. Given f ∈ C2 we consider the Hessian matrix

Ha

of the second partial derivatives at each a ∈ ∗U .

By Lemma 9.1.1, Da and Ha are finite throughout hU . For fu-ture applications, we would like to consider the infinitesimal intervalbetween two infinitely close points in the halo of U .

Remark 9.2.4. Let a, b ∈ hU with a ≈ b, then the interval be-tween a and b is included in hU ⊆ ∗U .

Theorem 9.2.5. If f is C1 then f(b)−f(a) = Dx(b−a) for some xin the interval between a and b.

Proof. This results by transfer of the mean value theorem. Notethat here Dx(b− a) is a product of matrices (row vector times columnvector).

Equivalently, we can write

f(b) − f(a) −Dx(b− a) = 0

for this specific choice of x. Meanwhile, for an arbitrary point in placeof x we have the following weaker statement.

Theorem 9.2.6. Let f be C1 and let a, b ∈ hU be infinitely close.Then for every y in the interval between a and b, we have

f(b) − f(a) −Dy(b− a) ≺≺ ‖b− a‖.Proof. By hypothesis, the partial derivatives Dx are themselves

continuous functions of x. Then the characterization of continuityvia infinitesimals and microcontinuity in Section 5.6 implies that Dx−Dy ≺≺ 1 for any y ≈ a (e.g., for y = st(a)). Therefore we can write

f(b) − f(a) = Dy(b− a) + (Dx −Dy)(b− a).

By continuity of Dx at x0 = st(x) = st(y), the difference Dx − Dy isinfinitesimal, proving the theorem.

Remark 9.2.7. If the first partial derivatives are Lipschitz (e.g., if fis C2), and we are given an infinitesimal constant β such that ‖x−y‖ ≺β for all x in the interval between a and b, then we have the strongercondition

f(b) − f(a) −Dy(b− a) ≺ β‖b− a‖.

9.3. PREVECTORS 85

If f is C2 then by transfer of the Taylor approximation theorem wehave

f(b) − f(a) = Da(b− a) +1

2(b− a)tHx(b− a)

for some x in the interval between a and b, and remarks similar to thosewe have made regarding Dx −Dy apply to Hx −Hy.

If ϕ = (ϕi) : U → Rn then the n rows Dia corresponding to ϕi form

the Jacobian matrix Ja of ϕ at a. By applying the above considerationsto each ϕi we obtain that

ϕ(b) − ϕ(a) − Jy(b− a) ≺≺ ‖b− a‖.If all partial derivatives are Lipschitz (e.g., if ϕ is C2) then ϕ(b) −ϕ(a) − Jy(b− a) ≺ β‖b− a‖ with β as above.

9.3. Prevectors

We now choose a positive infinitesimal λ ∈ ∗R, and fix it once andfor all. We will now define the concept of a prevector.1

Definition 9.3.1. Given a ∈ hM , a prevector based at a is apair (a, x), where x ∈ hM , such that for every smooth function f :M → R, we have f(x) − f(a) ≺ λ.

Note that the hypothesis f(x) − f(a) ≺ λ is stronger than requir-ing f(x) and f(a) to be infinitely close.

We can avoid quantification over functions as in Definition 9.3.1by giving an equivalent condition of being a prevector in coordinatesas follows. Consider coordinates in a neighborhood W of st(a) in M ,whose image in Euclidean space is U ⊆ Rn. Let a, x ∈ ∗U be thecoordinates for a, x ∈ ∗W .

Theorem 9.3.2. The pair (a, x) is a prevector based at the point aif and only if

x− a ≺ λ,

where the difference x − a is defined using the linear structure of thevector space ∗Rn ⊇ ∗U .

Proof. Let us show that the two definitions are indeed equivalent.Assume the first definition, and let (u1, . . . , un) be the chosen coordi-nate functions. Since each xi is smooth, we get ui(x) − ui(a) ≺ λ foreach i, i.e., x− a ≺ λ.

Conversely, assume (a, x) satisfies the second definition. Let f :M → R be a smooth function. Then by the intermediate value the-orem, f(x) − f(a) = Dc(x − a) for some c in the interval between a

1kdam-vector (with a kuf).


and x. Now the components of Dc are finite by Lemma 9.1.1. There-fore f(x) − f(a) ≺ ‖x− a‖ ≺ λ.

9.4. Tangent space to a manifold via prevectors

Definition 9.4.1. We denote by Pa = Pa(M) the set of prevectorsbased at a.

This is an infinite-dimensional space. We will use Pa to definethe finite-dimensional tangent space of M . First we will define anequivalence relation ≡ on Pa as follows.

Definition 9.4.2. We write

(a, x) ≡ (a, y)

if one of the following two equivalent conditions is satisfied:

(1) f(y) − f(x) ≺≺ λ for every smooth function f : M → R;(2) in coordinates as above, y − x ≺≺ λ.

The equivalence of the two definitions follows by the same argumentas in Section 9.3.

Definition 9.4.3. We denote by

Ta = Ta(M)

the set of equivalence classes Pa/≡. In the spirit of physics notation,the equivalence class of (a, x) ∈ Pa will be denoted

−→ax ∈ Ta,

and referred to as an ivector (short for infinitesimal vector) so as todistinguish it from a classical vector.

Theorem 9.4.4. A choice of coordinates in a neighborhood of Mgives an identification of Ta with Rn for every nearstandard a.

Proof. Given a ∈ hM , let W ⊆ M be a coordinate neighborhoodof the point st(a) ∈M . Thus we have a map W → Rn with image U ⊆Rn. Here by definition x 7→ x for all x ∈ W . Recall that (Rn)λF is thespace of points in ∗Rn that are not infinite compared to λ. We definea map

Pa → (Rn)λFby sending (a, x) 7→ x − a. This induces an identification of thespace Ta = Pa/≡ with the space Rn = (Rn)λF/(R

n)λI . Under thisidentification, the space Ta inherits the structure of a vector spaceover R.

9.5. ACTION OF A PREVECTOR ON SMOOTH FUNCTIONS 87

Remark 9.4.5. If we choose a different coordinate patch in a neigh-borhood of st(a) with image U ′ ⊆ Rn, then if ϕ : U → U ′ is the changeof coordinates, then by Definition 9.2.1, we have

ϕ(x) − ϕ(a) − Jst(a)(x− a) ≺≺ ‖x− a‖ ≺ λ,

where J is the Jacobian matrix. Hence we obtain

ϕ(x) − ϕ(a) − Jst(a)(x− a) ≺≺ λ.

This means that the map Rn → Rn induced by the two identificationsof Ta with Rn provided by the two coordinate maps, is given by multi-plication by the matrix Jst(a), and therefore is linear. Thus it preservesthe linear space structure of the quotient.

We see that the vector space structure induced on Ta via coordinatesis independent of the choice of coordinates, and so we have a welldefined vector space structure on Ta over R. Note that the objectdenoted Ta is a mixture of standard and nonstandard notions. It is avector space over R, but defined at every nearstandard point a ∈ hM .2

9.5. Action of a prevector on smooth functions

Prevectors act on functions in a way that does not rely on eitherlimits or standard part.

Definition 9.5.1. A prevector (a, x) ∈ Pa acts on a smooth func-tion f : M → R as follows:

(a, x) f =1

λ

(f(x) − f(a)

).

The right hand side is finite by definition of prevector. This actioninduces a differentiation as follows.

Definition 9.5.2. The differentiation of f : M → R by an ivec-tor −→a x ∈ Ta is defined as follows:

−→a x f = st((a, x)f

).

2If a, b ∈ hM and a ≈ b, then given coordinates in a neighborhood of st(a),the identifications of Ta and Tb with Rn induced by these coordinates induces anidentification between Ta and Tb. Given a different choice of coordinates, the ma-trix Jst(a) used in the previous paragraph is the same matrix for a and b, and so theidentification of Ta with Tb is well defined, independent of a choice of coordinates.Thus when a ≈ b ∈ hM we may unambiguously add an ivector −→a x ∈ Ta and an

ivector−→b y ∈ Tb.


The action −→a xf is well defined by definition of the equivalence rela-tion ≡. Note again that the ivector −→a x is a nonstandard object basedat the nonstandard point a, but it assigns a standard real number tothe standard function f .

Theorem 9.5.3. The action (a, x)f satisfies the Leibniz rule up toinfinitesimals.

Proof. Indeed, given functions f and g we can compute the actionas follows:

(a, x)(fg) =1

λ

(f(x)g(x) − f(a)g(a)

)

=1

λ

(f(x)g(x) − f(x)g(a) + f(x)g(a) − f(a)g(a)

)

= f(x)1

λ

(g(x) − g(a)

)+

1

λ

(f(x) − f(a)

)g(a)

≈ f(a)1

λ

(g(x) − g(a)

)+

1

λ

(f(x) − f(a)

)g(a).

Here the final relation ≈ is justified by the continuity of f and finiteness

of the action 1λ

(g(x) − g(a)

).

Theorem 9.5.4. Differentiation by an ivector −→a x satisfies the fol-lowing version of the Leibniz rule:

−→a x(fg) = f(a0) · −→a xg + −→a xf · g(a0)

where a0 = st(a).

Proof. By Theorem 9.5.3, we obtain the following for the differ-entiation −→a x(fg):

−→a x(fg) = st(f(a)) · −→a xg + −→a xf · st(g(a))

= f(a0) · −→a xg + −→a xf · g(a0),

where the second equality is by continuity of f and g.

When a is real we obtain the ordinary Leibniz rule for differentia-tion.

9.6. Maps acting on prevectors

We continue to follow [Nowik & Katz 2015]. Recall that λ isa fixed infinitesimal, Pa denotes the space of prevectors (a, x) at anearstandard point a ∈ hM , i.e., x− a ≺ λ.

Let f : M → N be a smooth map between smooth manifolds.Let a ∈ hM be a nearstandard point.

9.6. MAPS ACTING ON PREVECTORS 89

Definition 9.6.1. We define the differential dfa of f ,

dfa : Pa(M) → Pf(a)(N),

by(a, x) 7→ (f(a), f(x)).

Definition 9.6.2. The differential dfa induces a tangent map

Tfa : Ta(M) → Tf(a)(N)

by choosing a prevector representing an ivector −→a x ∈ TaM and setting

Tfa (−→a x) =−−−−−−→f(a) f(x).

The relation between f and df seems more transparent here thanin the corresponding classical definition.

Theorem 9.6.3. For a standard point a ∈ M , the space Ta is nat-urally identified with the classical tangent space of M at a.

Proof. It suffices to show this for an open set U ⊆ Rn, wherethe classical tangent space at any point a is Rn itself. A classicalvector v ∈ Rn is then identified with

−→a x where x = a+ λ · v.Under this identification, our definitions of the differentiation −→a xf (seeSection 9.5) and the action of the differential Tfa(

−→a x) coincide withthe classical ones.

Remark 9.6.4. When the manifolds are open subsets of Euclideanspace and Ta is the classical tangent space, the map Tfa : Ta → Tf(a) isidentified with the Jacobian matrix Ja (at the point a) whose rows arethe gradients Da of all n components. Recall that Da v for a columnvector v is the matrix product yielding a scalar result.

CHAPTER 10

Prevector fields, classes D1 and D2

10.1. Prevector fields

The notion of an internal set was discussed in Section 5.10. Recallthat λ is a fixed infinitesimal. A prevector field is, intuitively, theassignment of an infinitesimal displacement (of size comparable to λ)at every point, given by an internal map.

Remark 10.1.1. Here, as usual, a map is called internal if it isa member of a star transform of a standard object. For example, amap f : M → N is defined by a subset of the Cartesian product M×N .Therefore f can be thought of as an element of P = P(M × N). Aninternal map from ∗M to ∗N is then an element of ∗P.

More precisely, we have the following definition. Recall that Pa isthe set of prevectors at a point a ∈ hM , i.e., pairs (a, x) such that incoordinates x− a ≺ λ.

Definition 10.1.2. A prevector field Φ on a smooth manifold Mis an internal map Φ : ∗M → ∗M such that we have (a,Φ(a)) ∈ Pa forevery a ∈ hM .

In coordinates where addition and subtraction are possible, thiscondition translates into Φ(a) − a ≺ λ for every a ∈ hM .

Remark 10.1.3. Even though the condition is imposed only atpoints of hM , we define Φ on all of ∗M since we need Φ (and thereforeboth its domain and range) to be an internal entity so as to be able toapply hyperfinite iteration which is the basis of the hyperreal walk.

The relation ≡ between prevectors (a, x) and (a, y) in Pa was definedin Section 9.4. Namely, in coordinates y − x ≺≺ λ.

Definition 10.1.4. Let Φ and G be prevector fields. We say Φ isequivalent to G and write Φ ≡ G if Φ(a) ≡ G(a) for every a ∈ hM , orin coordinates Φ(a) −G(a) ≺≺ λ.

Definition 10.1.5. A local prevector field on an open U ⊆M is aninternal map Φ : ∗U → ∗V satisfying the above condition, where V ⊇ Uis open.

91

92 10. PREVECTOR FIELDS, CLASSES D1 AND D

2

When the distinction is needed, we will call a prevector field definedon all of ∗M a global prevector field.

The reason for allowing the values of a local prevector field definedon ∗U to lie in a possibly larger range ∗V is in order to allow a prevectorfield Φ to be restricted to a smaller domain which is not invariantunder Φ.

Example 10.1.6. Let M = R. If we wish to restrict the prevectorfield Φ on M given by

Φ(a) = a + λ

to the domain ∗(0, 1) ⊆ ∗R, then we need the flexibility of allowing aslightly larger range for Φ. In the sequel we will usually not mentionthe slightly larger range V when describing a local prevector field, butwill tacitly assume that we have such a V when needed. A secondinstance where it may be required for the range to be slightly largerthan the domain is the following natural setting for defining a localprevector field.

Example 10.1.7. Let p ∈ M . Let V ⊆ M a coordinate neigh-borhood of p with images p ∈ V ′ ⊆ Rn. Let X be a classical vectorfield on V , given in coordinates by X ′ : V ′ → Rn.1 Then there is aneighborhood U ′ of p ∈ Rn, with U ′ ⊆ V ′, such that we can define alocal prevector field Φ′ : ∗U ′ → ∗V ′ by setting

Φ′(a) = a+ λ ·X ′(a). (10.1.1)

Indeed, it suffices to choose U ′ such that U ′ is compact and further-more U ′ ⊆ V ′. For the corresponding open set U ⊆ V in M , thisinduces a local prevector field Φ : ∗U → ∗V which realizes the classicalvector field X on U in the sense of Definition 10.2.1 below (cf. Theo-rem 9.6.3).

10.2. Realizing classical vector fields

Definition 10.2.1. We say that a local prevector field Φ on Urealizes a classical vector field X on U if for every smooth function h :U → R we have the following for the new function Xh:

Xh (a) =−−−−→a Φ(a) h

for all a ∈ U , where−−−−→a Φ(a) is the ivector represented by the prevec-

tor (a,Φ(a)).

1It is not entirely unambiguous how one is to choose a trivialized vector field X ′

starting with a vector field X on M. One may want to make this more precise.

10.2. REALIZING CLASSICAL VECTOR FIELDS 93

Remark 10.2.2. When realizing a vector field as in Example 10.1.7(namely, formula (10.1.1) above) it may be necessary to restrict to asmaller neighborhood U . For example, when M = V = V ′ = (0, 1)and X = 1 (i.e., X = 1 d

dxin classical notation), one needs to take U =

(0, r) for some r ∈ R, r < 1 in order for Φ(a) = a + λ to lie alwaysin ∗V .

Note that Definition 10.2.1 involves only standard points.Different coordinates for the same neighborhood U will induce equiv-

alent realizations in ∗U . More precisely, we show the following.

Remark 10.2.3. Let U ⊆ Rn. Let X : U → Rn be a classical vec-tor field. Let ϕ : U → W ⊆ Rn be a change of coordinates. Thus Tϕsends Ta to Tϕ(a). Let Y : W → Rn be the classical vector field corre-sponding to X under the coordinate change. Then

Y (ϕ(a)) = JaX(a), (10.2.1)

where Ja is the Jacobian matrix of ϕ at a.

Proposition 10.2.4. In the notation of Remark 10.2.3, let Φ, G bethe prevector fields given by

(1) Φ(a) = a+ λX(a),(2) G(a) = a + λY (a)

as in Example 10.1.7. Then Φ ≡ ϕ−1 G ϕ, or equivalently,

ϕ Φ(a) −G ϕ(a) ≺≺ λ

for all a ∈ hU .

Proof. Let ϕi, Y i, Gi be the ith component of ϕ, Y,G respectively,and let Di

a be the differential of ϕi at a, so that Dia is the ith row of Ja.

We apply the mean value theorem to ϕi together with (10.2.1) to obtainthe following for a suitable point c:

ϕi Φ(a) −Gi ϕ(a) = ϕi(a + λX(a)) −Gi ϕ(a)

= ϕi(a + λX(a)) − ϕi(a) − λY i(ϕ(a))

= DicλX(a) − λDi

aX(a)

= λ(Dic −Di

a)X(a)

≺≺ λ,

where Dic is the differential of ϕi at a suitable point c in the interval

between a and a + λX(a). Such a point c exists by Theorem 9.2.5.Since this is true for each component i, we obtain

ϕ Φ(a) −G ϕ(a) ≺≺ λ


2


10.3. Regularity class D1 for prevectorfields

Defining various operations on prevector fields, such as their flow,or Lie bracket, requires assuming some regularity properties. First werecall the following classical definition.

Definition 10.3.1. A classical vector field X : U → Rn is called K-Lipschitz, where K ∈ R, if

‖X(a) −X(b)‖ ≤ K‖a− b‖ for all a, b ∈ U. (10.3.1)

For the local prevector field Φ of Example 10.1.7, where Φ(a)−a =λX(a), this translates into

∥∥∥(

Φ(a) − a)−(

Φ(b) − b)∥∥∥ ≤ Kλ‖a− b‖ (10.3.2)

for a, b ∈ ∗U .

Remark 10.3.2. Note the presence of the factor of λ on the righthand side of (10.3.2) which is not in (10.3.1). This is due to the factthat (10.3.2) deals with infinitesimal prevectors rather than classicalvectors.

The bound (10.3.2) motivates the following definition.

Definition 10.3.3. A prevector field Φ on a smooth manifold Mis of class D1 if whenever a, b ∈ hM and (a, b) ∈ Pa then in coordinatesthe following condition is satisfied:

Φ(a) − a− Φ(b) + b ≺ λ‖a− b‖. (10.3.3)

As mentioned earlier, condition (10.3.3) is equivalent to the Lips-chitz condition when Φ is defined by a classical vector field.

10.4. Second differences, class D2

To work with Lie brackets a stronger condition than D1 is necessary.

Definition 10.4.1. A prevector field Φ on a smooth manifold Mis of class D2 if for any a ∈ hM , the following condition is satisfiedin coordinates: for each pair v, w ∈ ∗Rn with v, w ≺ λ, the seconddifference

∆2v,wΦ(a) = Φ(a) − Φ(a + v) − Φ(a+ w) + Φ(a + v + w)

is sufficiently small:

∆2v,wΦ(a) ≺ λ‖v‖‖w‖.

10.5. SECORD-ORDER MEAN VALUE THEOREM 95

In Proposition 10.6.1 we will show that the prevector field of Ex-ample 10.1.7, induced by a classical vector field of class Ck, is a Dk

prevector field (k = 1, 2). This is in fact the central motivation for ourdefinition of Dk.

Remark 10.4.2. Our Dk is in fact a weaker condition than Ck,e.g. Φ of Example 10.1.7 is D1 if X is (order 1) Lipschitz, which isweaker than C1.

A definition of D0 along the above lines would amount to requiringΦ(a)− a ≺ λ, i.e. Φ being a prevector field. Note, however, that in ourdefinitions above, being a prevector field is part of the definition of anelement of Dk.

10.5. Secord-order mean value theorem

We will need the result from [Rudin 1976, Theorem 9.40]. We willassume that f is C2 for simplicity (Rudin presents weaker conditions).

Theorem 10.5.1 (Rudin Theorem 9.40). Suppose a two-variablefunction f(u1, u2) is defined and of class C2 in an open set in theplane including a closed rectangle Q with sides parallel to the coordinateaxes, having (a, b) and (a + h, b+ k) as opposite vertices, where h > 0and k > 0. Set

∆2(f,Q) = f(a+ h, b+ k) − f(a+ h, b) − f(a, b+ k) + f(a, b).

Then there is a point (x, y) in the interior of Q such that

∆2(f,Q) = hk

(∂2f

∂u1∂u2

)(x, y).

Remark 10.5.2. Note the analogy with the mean value theorem.

Proof. Define a 1-variable function

u(t) = f(t, b+ k) − f(t, b) where t ∈ [a, a+ h].

Two applications of the mean value theorem will yield the result. Notethat the mean value theorem is applicable to the first partial of fsince f ∈ C2 by hypothesis. Namely there is an x between a and a+h,and there is a y between b and b+ k, such that

∆2(f,Q) = u(a+ h) − u(a)

= hu′(x)

= h

((∂f

∂u1

)(x, b+ k) −

(∂f

∂u1

)(x, b)

)

= hk

(∂2f

∂u1∂u2

)(x, y)


2


10.6. The condition C2 implies D2

Recall our “second difference” notation for a prevector field Φ at anearstandard point a:

∆2v,wΦ(a) = Φ(a) − Φ(a + v) − Φ(a + w) + Φ(a + v + w).

The condition D2 for prevector fields was defined in Section 10.4. Thisis the condition ∆2

v,wΦ(a) ≺ λ‖v‖‖w‖ for all v, w ≺ λ. This is infact related to [Zygmund 1945]. Given a classical vector field X , onedefines a prevector field Φ(a) = a + λX(a). We will show that if X isof class C2 then Φ is of class D2.

Proposition 10.6.1. Let W ⊆ Rn be open. Let X : W → Rn bea classical C2 vector field, and consider its natural extension to ∗W .Then for each a ∈ hW and each pair of infinitesimal v, w ∈ ∗Rn, wehave

∆2v,wX(a) ≺ ‖v‖‖w‖.

We obtain the following immediate corollary.

Corollary 10.6.2. Let X be of class C2. If Φ is the prevectorfield on ∗W of Example 10.1.7, i.e., Φ(a) = a + λX(a), then Φ is D2.

Proof of Proposition 10.6.1. Let U ⊆ W be a smaller neigh-borhood of st(a) for which all second partial derivatives of X arebounded by a constant Const ∈ R. Let (X1, . . . , Xn) be the com-ponents of X . Given p ∈ U and v1, v2 ∈ Rn such that we have p +s1v1 + s2v2 ∈ U for all 0 ≤ s1, s2 ≤ 1, we define functions ψi by

ψi(s1, s2) = X i(p+ s1v1 + s2v2).

Then by chain rule, the mixed second partial of ψ satisfies∣∣∣∣∂2ψi

∂s1∂s2

∣∣∣∣ ≤ Const|v1| |v2|

where Const depends on the second partials of X i.By Theorem 10.5.1 (second-order mean value theorem) there is a

point (t1, t2) ∈ [0, 1] × [0, 1] such that

∑

(e1,e2)∈0,12(−1)e1+e2ψi(e1, e2) =

∂2

∂s1∂s2ψi(t1, t2).

10.7. BOUNDS FOR D1 PREVECTOR FIELDS 97

Therefore∣∣∆2

v1,v2X∣∣ =

∣∣∣∑

(e1,e2)∈0,12(−1)e1+e2X i(p+ e1v1 + e2v2)

∣∣∣

=∣∣∣

∑

(e1,e2)∈0,12(−1)e1+e2ψi(e1, e2)

∣∣∣

=∣∣∣ ∂2

∂s1∂s2ψi(t1, t2)

∣∣∣

≤ Ci‖v1‖‖v2‖where Ci is determined by a bound for all second partial derivativesof X i in U .

Since this is valid for each X i, i = 1, . . . , n, there is a K ∈ R suchthat∥∥∥

∑

(e1,e2)∈0,12(−1)e1+e2X(p+ e1v1 + e2v2)

∥∥∥ ≤ K‖v1‖‖v2‖ (10.6.1)

for every p ∈ U and v1, v2 ∈ Rn such that p + s1v1 + s2v2 ∈ U forall 0 ≤ s1, s2 ≤ 1, si ∈ R.

Applying the transfer principle to (10.6.1), we conclude that thesame formula holds, with the same K, for all p ∈ ∗U and v1, v2 ∈ ∗Rn

provided that p + s1v1 + s2v2 ∈ ∗U for all 0 ≤ s1, s2 ≤ 1, si ∈ ∗R. Inparticular this is true for p = a and all infinitesimal v1, v2 ∈ ∗Rn.

10.7. Bounds for D1 prevector fields

Recall that a D1 prevector field Φ satisfies the condition

Φ(a) − a− Φ(b) + b ≺ λ‖a− b‖whenever (a, b) ∈ Pa (see Section 10.3). We note that a D1 prevectorfield Φ satisfies the following bounds.

Proposition 10.7.1. If Φ is a D1 prevector field on M , and a, b ∈hM with a− b ≺ λ, then

‖a− b‖ ≺ ‖Φ(a) − Φ(b)‖ ≺ ‖a− b‖.Proof. By the triangle inequality we have∣∣‖Φ(a) − Φ(b)‖ − ‖a− b‖

∣∣ ≤ ‖Φ(a) − Φ(b) − a+ b‖= Kλ‖a− b‖ (10.7.1)

where the hyperreal constant K is defined by the last equality in(10.7.1), where K is finite by the D1 condition. In other words, wehave

−Kλ‖a− b‖ ≤ ‖Φ(a) − Φ(b)‖ − ‖a− b‖ ≤ Kλ‖a− b‖.


2

Rearranging terms, we obtain

(1 −Kλ)‖a− b‖ ≤ ‖Φ(a) − Φ(b)‖ ≤ (1 +Kλ)‖a− b‖,as required.

CHAPTER 11

Prevectorfield bounds via underspill, walks, flows

11.1. Operations on prevector fields

We will now show that addition of D1 prevector fields in coordinatesis equivalent to their composition (in the sense of the relation ≡). Moreprecisely, we show the following.

Proposition 11.1.1. Let Φ be a D1 prevector field and G anyprevector field. Then for every a ∈ hM , we have the following relationamong ivectors:

−−−−−−−→a Φ(G(a)) =

−−−−→a Φ(a) +

−−−−→a G(a).

In particular if both Φ, G are D1 then Φ G ≡ G Φ.

Proof. In a coordinate chart, let

x = a + (Φ(a) − a) + (G(a) − a) = Φ(a) +G(a) − a.

Passing to equivalence classes, we obtain the following relation amongivectors: −−−−→

a Φ(a) +−−−−→a G(a) = −→a x.

Now let b = G(a). Then

Φ(G(a)) − x = Φ(b) − Φ(a) − b+ a ≺ λ‖b− a‖ ≺≺ λ

by the D1 condition on Φ, proving the proposition.

We will now show that the composition of D1 prevector fields is D1.

Proposition 11.1.2. If prevector fields Φ, G are D1 then prevectorfield Φ G is D1.

Proof. Let c = G(a) and d = G(b). We have by the triangleinequality

‖Φ G(a) − Φ G(b) − a+ b‖≤ ‖Φ G(a) − Φ G(b) −G(a) +G(b)‖ + ‖G(a) −G(b) − a+ b‖= ‖Φ(c) − Φ(d) − c+ d‖ + ‖G(a) −G(b) − a+ b‖≺ λ‖c− d‖ + λ‖a− b‖ ≺ λ‖a− b‖

99

100 11. PREVECTORFIELD BOUNDS VIA UNDERSPILL, WALKS, FLOWS

where Proposition 10.7.1 is applied in order to bound c − d in termsof a− b.

A similar proposition can be proved for D2.1

11.2. Sharpening the bounds via underspill

Here we are interested in sharpening the ≺ bounds for prevector-fields on M in terms of a specific real constant.

Proposition 11.2.1. Let Φ be a prevector field on W , where W ⊆M is an open coordinate neighborhood with image U ⊆ Rn. Let B ⊆ Ube a closed ball. We will identify W with U to lighten the notation.Then we have the following.

(1) There is C ∈ R such that ‖Φ(a) − a‖ ≤ Cλ for all a ∈ ∗B.(2) If G is another prevector field, then there is a finite β ∈ ∗R

such that ‖Φ(a) −G(a)‖ ≤ βλ for all a ∈ ∗B.(3) If furthermore Φ ≡ G then the constant β ∈ ∗R in (2) can be

chosen to be infinitesimal.

Proof. Since the first statement is a special case of the second(take G(a) = a for all a), it suffices to prove the second statement. Let

A = n ∈ ∗N : ‖Φ(a) −G(a)‖ ≤ nλ for every a ∈ ∗B.Every infinite n ∈ ∗N is in A, so that

∗N \ N ⊆ A.

1Similarly, we have the following.

Proposition 11.1.3. If Φ, G are D2 then Φ G is D2.

Proof. In some coordinates let p = G(a), x = G(a+ v)−G(a) and y = G(a+w)−G(a), and so by Propositions 13.1.2 and 10.7.1 we have x ≺ ‖v‖ and y ≺ ‖w‖.Also by Propositions 13.1.2 and 10.7.1 we have

‖Φ(p + x + y) − Φ(G(a + v + w))‖ ≺ ‖p + x + y −G(a + v + w)‖= ‖ −G(a) + G(a + v) + G(a + w) −G(a + v + w)‖ ≺ λ‖v‖‖w‖.

Now

‖Φ G(a) − Φ G(a + v) − Φ G(a + w) + Φ G(a + v + w)‖= ‖Φ(p) − Φ(p + x) − Φ(p + y) + Φ G(a + v + w)‖

≤ ‖Φ(p)−Φ(p+x)−Φ(p+ y) + Φ(p+ x+ y)‖+ ‖Φ(p+ x+ y)−Φ(G(a+ v +w))‖≺ λ‖x‖‖y‖ + λ‖v‖‖w‖ ≺ λ‖v‖‖w‖.

11.3. FURTHER BOUNDS VIA UNDERSPILL 101

But A is an internal set, being defined in terms of internal entities Φand G.2 Therefore by underspill there is a finite integer C in A. Recallthat underspill is the fact that if A ⊆ ∗N is an internal set, and Aincludes all infinite n then it must also include a finite n. This isbased on the fact that infinite hypernaturals form an external set; seeSection 5.11.

To prove the third statement (for Φ ≡ G), let

A =

n ∈ ∗N : ‖Φ(a) −G(a)‖ ≤ λ

nfor every a ∈ ∗B

.

Every finite n ∈ ∗N is in A and so by overspill there is an infinite n ∈ ∗Nin A. We can therefore choose the infinitesimal value β = 1

n.

11.3. Further bounds via underspill

Proposition 11.3.1. Let Φ be a D1 prevector field onM . Let W ⊆M be an open coordinate neighborhood with image U ⊆ Rn. Let B ⊆ Ube a closed ball. Then there is K ∈ R such that

‖Φ(a) − a− Φ(b) + b‖ ≤ Kλ‖a− b‖ (11.3.1)

for all a, b ∈ ∗B.It follows that

(1 −Kλ)‖a− b‖ ≤ ‖Φ(a) − Φ(b)‖ ≤ (1 +Kλ)‖a− b‖. (11.3.2)

Proof. Let N = ⌊1/λ⌋. Let a, b ∈ ∗B. We construct a hyperfinitepartition of the segment between a and b. For each k = 0, . . . , N let

ak = a+k

N(b− a).

Then ak − ak+1 ≺ λ. Let Cab ∈ ∗R be the maximum of the ratios

‖Φ(ak) − ak − Φ(ak+1) + ak+1‖λ‖ak − ak+1‖

over all k = 0, . . . , N−1. Since Φ is D1, it follows that the constant Cabis finite. For every 0 ≤ k ≤ N − 1 we have

‖Φ(ak) − ak − Φ(ak+1) + ak+1‖ ≤ Cabλ‖ak − ak+1‖ = Cabλ‖a− b‖N

.

2Kanovei to comment on strong transitivity.


Therefore by the triangle inequality

‖Φ(a) − a−Φ(b) + b‖

≤N−1∑

k=0

‖Φ(ak) − ak − Φ(ak+1) + ak+1‖

≤ Cabλ‖a− b‖.

(11.3.3)

Now let

A = n ∈ ∗N : ‖Φ(a)− a−Φ(b) + b‖ ≤ nλ‖a− b‖ for each a, b ∈ ∗B.Since each Cab as in (11.3.3) is finite, every infinite n ∈ ∗N is in A.Hence by underspill, A also contains a finite integer K, proving (11.3.1).The estimate (11.3.2) assertion follows from the first as in the proof ofProposition 10.7.1.

A similar proposition can be proved for D2.3

11.4. Global prevector fields, walks, and flows

In this section we will define the hyperreal walk of a prevector fieldand relate it to the classical flow. Recall that a prevector field Φ is amap Φ : ∗M → ∗M from ∗M to itself. We wish to view a prevector fieldas the first step of its walk at time λ > 0, and will define the hyperreal

3Similarly we have the following.

Proposition 11.3.2. Let Φ be a D2 prevector field. If W is a coordinateneighborhood with image U ⊆ Rn and B ⊆ U is a closed ball, then there is K ∈ Rsuch that

‖Φ(a) − Φ(a + v) − Φ(a + w) + Φ(a + v + w)‖ ≤ Kλ‖v‖‖w‖for all a ∈ ∗B and v, w ∈ ∗ Rn such that a + v, a + w, a + v + w ∈ ∗B.

Proof. The proof is similar to that of Proposition 11.3.1. Let N = ⌊1/λ⌋.Given a ∈ ∗B and v, w ∈ ∗ Rn such that a + v, a + w, a + v + w ∈ ∗B, let ak,l =

a + kN v + l

Nw, 0 ≤ k, l ≤ N . Let Cavw be the maximum of

‖Φ(ak,l) − Φ(ak+1,l) − Φ(ak,l+1) + Φ(ak+1,l+1)‖λ‖v/N‖‖w/N‖

for 0 ≤ k, l ≤ N − 1 then Cavw is finite. For every 0 ≤ k, l ≤ N − 1 we have

‖Φ(ak,l) − Φ(ak+1,l) − Φ(ak,l+1) + Φ(ak+1,l+1‖ ≤ Cavwλ‖v/N‖‖w/N‖.Summing over 0 ≤ k, l ≤ N − 1 we get

‖Φ(a) − Φ(a + v) − Φ(a + w) + Φ(a + v + w)‖ ≤ Cavwλ‖v‖‖w‖.By underspill as in the proof of Proposition 11.3.1, there is one finite K whichworks for all a, v, w.

11.4. GLOBAL PREVECTOR FIELDS, WALKS, AND FLOWS 103

walk and the real flow accordingly (see below). The flow at arbitrarytime t is defined by iterating Φ the appropriate number of times.

Remark 11.4.1. The idea of a vector field being the infinitesimalgenerator of its flow receives a literal meaning in the TIDG setting.

We view Φ as an element Φ ∈ ∗Map(M). Consider the map∗Map(M) × ∗N → ∗Map(M)

which is the natural extension of the map Map(M) × N → Map(M)taking (φ, n) to φn = φ φ · · · φ (altogether n times). Thus Φn isdefined for all hypernatural n.

We will now define the hyperfinite flow Φt of a prevector field Φ.The flow will be defined at times t ∈ ∗R that are hyperinteger multiplesof the base infinitesimal λ.

Definition 11.4.2. Let Φ : ∗M → ∗M be a global prevector field.For each hypernatural n ∈ ∗N, let t = nλ and define the hyperrealwalk Φt of Φ at time t by setting

Φt(a) = Φn(a). (11.4.1)

Definition 11.4.3. We define the real flow φt on M by setting

φt(x) = st(Φnλ(x))

for all x ∈M , where nλ ≤ t < (n+ 1)λ, i.e.,

n =

⌊t

λ

⌋.

The real flow will be defined for all real t that are sufficiently smallso that the standard part is well-defined; see material around (11.8.1)for the details.

Theorem 11.4.4. Let Φ be a prevector field of the class D1 on M ,and let Φt be its hyperfinite walk as in (11.4.1). Then prevector field Φis invariant under Φt.

Proof. The differential dΦt of the map Φt = Φn acts on eachprevector (a,Φ(a)) by

dΦt(a,Φ(a)) = (Φn(a),Φn(Φ(a)))

= (Φn(a),Φ Φ · · · Φ(a))

= (Φn(a),Φ(Φn(a)))

= (b,Φ(b))

by the associativity of composition, where b = Φn(a), proving the in-variance of the prevector field Φ. Here the relation Φn Φ = Φ Φn isproved by internal induction (see Section 11.5).


Remark 11.4.5. This proof of invariance compares favorably tothe proof of invariance under the flow in the A-track framework (seeTheorem 7.8.2).

The hyperfinite walk for a local prevector field is defined similarly.A bit of care is required to specify its domain.

Given a local prevector field Φ : ∗U → ∗V , we extend Φ to Φ′ : ∗V →∗V by defining Φ′(a) = a for all a ∈ ∗V − ∗U , and let Yn = a ∈ ∗U :(Φ′)n(a) ∈ ∗U. The domain of Φt will be Yn(t) (here as before n(t) isthe integer part of t/λ), where it is defined by Φt(a) = Φ′

t(a).We would also like to consider Φt for t ≤ 0. This can be defined for a

global prevector field Φ which is bijective on ∗M , or for a local prevectorfield which is bijective in the sense of Remark 13.2.5, in particular a D1

local prevector field. Namely, we define Φt for t ≤ 0 to be (Φ−1)−t.Note that for any global prevector field Φ, the hyperreal walk Φt is

defined for all t ≥ 0. This is of course not the case for the classical flowof a classical vector field. Similarly Φt is defined for all t ≤ 0 if Φ isbijective.

11.5. Internal induction

Internal induction is treated in [Goldblatt 1998, p. 129].

Theorem 11.5.1. If X ⊆ ∗N be an internal subset containing 1and closed under the successor function4 n 7→ n+ 1. Then X = ∗N.

Proof. If the set difference Y = ∗N\X is nonempty, then since Y isinternal, it has a least element n ∈ Y (see Section 5.11). Hence n−1 ∈X . However, by closure under successor, the number n must also bein X , yielding a contradiction.

This will be used in the proof of Theorem 11.6.2.

11.6. Dependence on initial condition and prevector field

In this section we deal with initial conditions.5

Remark 11.6.1. We showed in Proposition 11.3.1 that each D1

prevector field Φ defined in an open neighborhood of a closed Euclideanball B satisfies the bound ‖Φ(a) − a − Φ(b) + b‖ ≤ Kλ‖a − b‖ forall a, b ∈ ∗B for an appropriate finite K. Such a K will be exploited inTheorem 11.6.2.

4“peulat ha’okev” according to hebrew wikipedia at peano axioms5tnai hatchalati

11.6. DEPENDENCE ON INITIAL CONDITION AND PREVECTOR FIELD 105

Theorem 11.6.2. Let Φ be a localD1 prevector field on ∗U . Given p ∈U and a coordinate neighborhood of p with image W ⊆ Rn, let B′ ⊆B ⊆ W be closed balls of radii r/2, r around the image of p. Sup-pose ‖Φ(a) − a − Φ(b) + b‖ ≤ Kλ‖a − b‖ for all a, b ∈ ∗B, with K afinite constant,6 then

(1) there is a positive T ∈ R such that Φt(a) ∈ ∗B for all a ∈ ∗B′

and −T ≤ t ≤ T . Furthermore, for all a, b ∈ ∗B′ and 0 ≤ t ≤T : ‖Φt(a) − Φt(b)‖ ≤ eKt‖a− b‖.

(2) If we take a slightly larger constant K ′ = K/(1 −Kλ)2, thenfor all a, b ∈ ∗B′ and −T ≤ t ≤ T :

e−K′|t|‖a− b‖ ≤ ‖Φt(a) − Φt(b)‖ ≤ eK

′|t|‖a− b‖.

Proof. We will give a proof for positive t.7 Let C ∈ R be as inProposition 11.2.1 (on sharpening bounds via underspill and overspill),so that ‖Φ(a) − a‖ ≤ Cλ for all a ∈ ∗B. We define T by setting

T =r

2C,

so that 0 < T ∈ R. Let a ∈ ∗B′ and 0 ≤ t ≤ T , and set

n = n(t) = ⌊t/λ⌋. (11.6.1)

By internal induction, we obtain a bound for Φn(a) − a by means of ahyperfinite sum as follows:

‖Φn(a) − a‖ ≤n∑

m=1

‖Φm(a) − Φm−1(a)‖ ≤ nCλ ≤ r

2.

By the triangle inequality we obtain

‖Φn(a)‖ ≤ ‖Φ(a) − a‖ + ‖a‖ ≤ r

2+r

2= r,

and therefore Φn(a) ∈ ∗B. By Proposition 11.3.1(2) we have

(1 −Kλ)‖a− b‖ ≤ ‖Φ(a) − Φ(b)‖ ≤ (1 +Kλ)‖a− b‖

6Such finite K exists by Proposition 11.3.1.7The case of negative t then follows from Remark 13.3.2.


for all a, b ∈ ∗B, and so by internal induction (see Theorem 11.5.1)8 weobtain

(1 −Kλ)n‖a− b‖ ≤ ‖Φn(a) − Φn(b)‖ ≤ (1 +Kλ)n‖a− b‖,proving the theorem in view of the fact that

(1 +Kλ)n =

(1 +

Kt

n

)n≈ eKt (11.6.2)

by formula (11.6.1) defining n. The choice of K ′ results by elementaryalgebra.

11.7. Dependence on initial conditions for local prevectorfield

In this section we will provide bounds on how fast the flows of apair of prevector fields can diverge from each other.

In particular, this will show that if prevector fields are equivalentthen the corresponding classical flows coincide.

We will deal with a more general situation (than that dealt within Section 11.6) where A′ is the set of initial conditions, whereas A isthe set where we assume that the flow is contained until time T . Theset A will typically be larger than A′.

Let Φ be a D1 local prevector field on ∗U and let G be any localprevector field on ∗U . Given a coordinate neighborhood included in Uwith image W ⊆ Rn, let A′ ⊆ A ⊆ ∗W be internal sets.

Assume that β is such that

‖Φ(a) −G(a)‖ ≤ βλ (11.7.1)

for all a ∈ A (in particular if β is infinitesimal then the prevector fieldsare equivalent). Assume also that K > 0 is a finite constant such that

‖Φ(a) − a− Φ(b) + b‖ ≤ Kλ‖a− b‖for all a, b ∈ A. Assume that 0 < T ∈ R is such that Φt(a) and Gt(a)are in A for all a ∈ A′ and 0 ≤ t ≤ T .

8This step requires internal induction and cannot be obtained by iterating theformula n times, writing down the resulting formula with integer parameter n, andapplying the transfer principle. The problem is that the entities occurring in theformula are internal rather than standard, so there is no real statement to applytransfer to. However, one can perhaps use the same trick as in the definition ofthe hyperfinite iterate in the first place, by introducing an additional parameter asin Section 11.4. Namely, the formula holds for all triples (Φ, G, n) ∈ map(M) ×map(M) × N, and then apply the star transform.

11.7. DEPENDENCE ON INITIAL CONDITIONS FOR LOCAL PREVECTOR FIELD107

Theorem 11.7.1. In the notation above, for all a ∈ A′ and 0 ≤t ≤ T , we have the following bound:

‖Φt(a) −Gt(a)‖ ≤ β

K(eKt − 1) ≤ βteKt.

If G−1 exists, e.g., if G is also D1, and if Φt(a) and Gt(a) are in A forall a ∈ A′ and −T ≤ t ≤ T , then

‖Φt(a) −Gt(a)‖ ≤ β

K(eK|t| − 1) ≤ β|t|eK|t|

for all −T ≤ t ≤ T .

Proof. It suffices to provide a proof for positive t. We will proveby internal induction that

‖Φn(a) −Gn(a)‖ ≤ β

K

((1 +Kλ)n − 1

). (11.7.2)

This implies the statement when applied to n = n(t); cf. (11.6.2). Thebasis of the induction is provided by the estimate (11.7.1). By Propo-sition 11.3.1(2) we have

‖Φn+1(a) −Gn+1(a)‖≤ ‖Φ(Φn(a)) − Φ(Gn(a))‖ + ‖Φ(Gn(a)) −G(Gn(a))‖≤ (1 +Kλ)‖Φn(a) −Gn(a)‖ + βλ.

Exploiting the inductive hypothesis (11.7.2), we obtain

‖Φn+1(a) −Gn+1(a)‖

≤ (1 +Kλ)β

K

((1 +Kλ)n − 1

)+ βλ

≤ β

K(1 +Kλ)n+1 − β

K− βKλ

K+ βλ

=β

K

((1 +Kλ)n+1 − 1

)

proving the inductive step from n to n + 1.

Corollary 11.7.2. Assume Φ and G are as in Theorem 11.7.1.If we have Φ ≡ G, then there is a β ≺≺ 1 for the statement of Theo-rem 11.7.1, and therefore Φt(a) ≈ Gt(a) for all 0 ≤ t ≤ T .

Proof. By Proposition 11.2.1.

9

9

Remark 11.7.3. If the internal sets A′ ⊆ A given in Theorem 11.7.1 areballs B′ ⊆ B of radii r′ < r ∈ ∗ R, and say we are not given that both Φt(a)


11.8. Inducing a real flow

The hyperreal walk Φt of a prevector field Φ induces a classical flowon M as follows.

Definition 11.8.1. Let Φ be a D1 local prevector field. Let B′ ⊆Rn and [−T, T ] be as in Theorem 11.6.2. We define the real flow

φt : B′ →M

induced by Φ by setting

φt(x) = st(Φt(x)). (11.8.1)

Recall that Φt is defined by hyperfinite iteration (of Φ) ⌊t/λ⌋ times.

The following are immediate consequences of Theorems 11.6.2 andCorollary 11.7.2.

Theorem 11.8.2. Given a D1 prevector field Φ the following hold:

(1) the flow φt is Lipschitz10 continuous with constant eK|t|.

(2) the real flow φt is injective.(3) If G is another D1 prevector field with real flow gt, and Φ ≡ G,

then φt = gt.

Remark 11.8.3. Suppose Φ is obtained from a classical vectorfield X by the procedure of Example 10.1.7. Then [Keisler 1976,Theorem 14.1] shows that φt is in fact the flow of the classical vectorfield X in the classical sense. By Theorem 11.8.2(3), each prevectorfield Φ that realizes X will yield a hyperfinite walk whose standardpart will produce the same classical flow.

The results of this subsection have the following application to thestandard setting.

Corollary 11.8.4. Consider an open set U ⊆ Rn. Let X, Y : U →Rn be classical vector fields, where X is Lipschitz with constant K,

and Gt(a) are in B for all a ∈ B′ and 0 ≤ t ≤ T , but rather we are given thatonly one of them remains in some ball B′ ⊆ B′′ ⊆ B with radius r′ < r′′ < r suchthat β ≺≺ r−r′′. Then it follows from the bound of Theorem 11.7.1 that the otherflow remains in B. To be precise, one needs to repeat the induction of the proof ofTheorem 11.7.1.

This needs to be rewritten.10 This follows from the estimates given if we interpret the flow as referring to

the map M → M at time t. However, if this refers to the full flow M × I → Mthen the claim apparently does not immediately follow and may not even be truefor a general prevectorfield.

11.9. GEODESIC FLOW 109

and ‖X(x) − Y (x)‖ ≤ b for all x ∈ U . If x(t), x(t) are integral curvesof X then

‖x(t) − x(t)‖ ≤ eKt‖x(0) − x(0)‖.If y(t) is an integral curve of Y with x(0) = y(0) then

‖x(t) − y(t)‖ ≤ b

K(eKt − 1) ≤ bteKt.

Proof. Define prevector fields on ∗U as usual by setting Φ(a)−a =λX(a) andG(a)−a = λY (a) as in Example 10.1.7, and apply Theorems11.6.2, 11.7.1, and Remark 11.8.3.

11.9. Geodesic flow

One advantage of the hyperreal approach to solving a differentialequation is that the hyperreal walk exists for all time, being definedcombinatorially by iteration of a selfmap of the manifold. The focustherefore shifts away from proving the existence of a solution, to estab-lishing the properties of a solution.

Thus, our estimates show that given a uniform Lipschitz bound onthe vector field, the hyperreal walk for all finite time stays in the finitepart of the ∗M . If M is complete then the finite part is nearstandard.Our estimates they imply that the hyperreal walk for all finite timedescends to a real flow on M .

In particular, the geodesic equation on an n-dimensional mani-fold M ,

αk′′

+ Γkijαi′αj

′= 0, (11.9.1)

can be interpreted as a first-order ODE on the (2n − 1)-dimensionalmanifold SM (union of unit spheres in all tangent spaces). The latteris known to satisfy a uniform Lipschitz bound. Therefore the geodesicflow on a complete Riemannian manifold M exists for all time t ∈ R.

In more detail, consider a Riemannian metric 〈 , 〉 with metric co-efficients written in coordinates as (gij) meaning that

⟨∂∂ui, ∂∂uj

⟩= gij.

The Γkij can be expressed as Γkij = gkℓ

2(giℓ;j − gij;ℓ + gjℓ;i). A smooth

curve in M with components (αi) is a geodesic if and only if (11.9.1)holds for each k.

CHAPTER 12

Universes, transfer

This chapter develops a detailed rigorous set-theoretic setting forthe TIDG framework.

12.1. Universes

The idea of nonstandard analysis is to enlarge the mathematicalworld we are studying, in a way that does not change any of its proper-ties, this leading to new insights on the original world. So, our startingpoint is some set X , which in the case of differential geometry maybe (the underlying set of) a smooth n-dimensional manifold M , or thedisjoint union of M with finitely many other sets one might want torefer to, say Rn, R, and N.

Let us review some set-theoretic background related to universes.Recall that given a set A, the symbol P(A) denotes the set of all subsetsof A. We define the universe V (X) as follows. Let

V0(X) = X.

At the next step we set V1 = V0(X) ∪ P(V0(X)). More generally, wedefine recursively for n ∈ N,

Vn+1(X) = Vn(X) ∪ P(Vn(X)).

Finally the universe V (X) is defined as follows.

Definition 12.1.1. The universe, or superstructure, over a set Xis the set

V (X) =⋃

n∈NVn(X).

Corollary 12.1.2. If a ∈ V (X) rX then a ⊆ V (X).

Exercise 12.1.3. By definition, Vm(X) ⊆ Vm+1(X). Prove that infact Vm(X) $ Vm+1(X), and even, by Cantor, Vm+1(X) has a strictlybigger cardinality than Vm(X).

For special needs of nonstandard analysis, one usually requires Xto have the following key technical property; see e.g., [Keisler 1976,15B] or [Chang & Keisler 1990, 4.4].

111

112 12. UNIVERSES, TRANSFER

Definition 12.1.4 (base sets). A set X is a base set if X 6= ∅ andif x ∈ X then x is an atom in V (X), so that x 6= ∅ while x∩V (X) = ∅.

The following are important consequences.

Corollary 12.1.5 ([Chang & Keisler 1990], Corollary 4.4.1).Assume that X is a base set. If a ∈ V (X) and a ∈ b ∈ Vm(X) thenm ≥ 1 and a ∈ Vm−1(X).

Corollary 12.1.6. Assume that X is a base set. If a, b ∈ V (X)and a ∩ V (X) = b ∩ V (X) then either a = b or a, b ∈ X ∪ ∅.

Proof. Suppose that a /∈ X . Then a ∈⋃m≥1 Vm(X) by construc-

tion, and hence a ⊆ V (X) and a ∩ V (X) = a. If also b /∈ X thensimilarly b ∩ V (X) = b, thus a ∩ V (X) = b ∩ V (X) implies a = b. Ifb ∈ X then b ∩ V (X) = ∅ (as X is a base set), and we conclude thata = a ∩ V (X) = ∅, as required.

For examples of base sets see Section 12.2.

12.2. The reals as the base set

We recall Dedekind’s construction of the real field.

Theorem 12.2.1. Each real x is defined following Dedekind as thepair x = Qx, Q

′x of two complementary sets of rationals Qx = q ∈

Q : q < x and Q′x = q ∈ Q : q ≥ x.

The key property of R which implies that R is a base set, is the factthat each real x consists of two elements Qx and Q′

x, which are sets ofvon Neumann rank (see below) equal exactly to ω.

For those not versed in set theory we provide a definition.

Definition 12.2.2. The von Neumann rank of a set x is an ordi-nal rank(x) defined so that rank(∅) = 0 and if x 6= ∅ then rank(x) isthe least ordinal strictly bigger than all ordinals rank(y), y ∈ x.

The existence and uniqueness of rank(x) follow from the axiomsof modern Zermelo–Fraenkel set theory ZFC, with the key role of theregularity, or foundation axiom in the argument. Informally, rank(x)can be viewed as the finite or transfinite length of the cumulative con-struction of the set x from the empty set.

Definition 12.2.3. Let γ be an ordinal. A set X is a γ-base set ifX 6= ∅, X consists of non-empty elements, and we have rank(a) = γwhenever a ∈ x ∈ X .

Lemma 12.2.4 ([Chang & Keisler 1990], Exercise 4.4.1). If γ isan infinite ordinal, and X is a γ-base set, then X is a base set.

12.3. MORE SETS IN UNIVERSES 113

Proof. If x ∈ X = V0(X) then rank(x) = γ + 1 by the choiceof X .

If x ∈ V1(X) then either x ∈ X with rank(x) = γ+1, or x = ∅ (theempty set) with rank(∅) = 0, or x ∈ V1(X) \X with rank(x) = γ+ 2.

Similarly if x ∈ V2(X) then either x ∈ X with rank(x) = γ + 1,or rank(x) ≤ 1 (for x = ∅ and x = ∅), or x ∈ V2(X) \ X withγ + 2 ≤ rank(x) ≤ γ + 3. Extending this argument, one easily showsby induction that if m ≥ 1 then every set y ∈ Vm(X)\X satisfies eitherγ+2 ≤ rank(y) ≤ γ+m+1 or rank(Y ) < m, hence never rank(y) = γ.This rules out the possibility of y ∈ x for any x ∈ X = V0(X).

Now consider the most important case X = R (the reals). Recallthat ω is the least infinite ordinal.

Lemma 12.2.5. The set X = R is an ω-base set, hence, a base set.

Proof. Set theory defines natural numbers so that 0 = ∅ and n =0, 1, 2, . . . , n − 1 for all n > 0. It follows that rank(n) = n for anynatural number n. Next, according to the set theoretic definition of anordered pair 〈x, y〉 as

〈x, y〉 = x, x, y,all pairs and triples of natural numbers have finite ranks. Thereforeeach rational number q, identified with a triple 〈s,m, n〉, where s ∈0, 1, 2, m = 1, 2, 3, . . . , n = 0, 1, 2, 3, . . . , and q = (s−1) · n

m, also has

a finite rank rank(q).We conclude that every infinite set S of rational numbers satis-

fies rank(S) = ω exactly. Therefore each Dedekind real x consists oftwo sets (infinite sets of rationals Qx and Q′

x) of rank rank(Qx) =rank(Q′

x) = ω. It remains to apply Lemma 12.2.4 with γ = ω.

From now on we concentrate on the case X = R.

12.3. More sets in universes

By definition the set V0(R) = R has cardinality card(R) = c (thecontinuum), and then by induction card(Vn(R)) = expn(c). This arrayof cardinalities easily exceeds all needs of conventional mathematics.For instance, the underlying set |M | of a smooth connected manifold Mof dimension n ≥ 1 is a set of cardinality c. Therefore we can iden-tify |M | with a set in the complement V1(R) \ V0(R). (This choiceexcludes the possibility of nonempty intersection of |M | and R.) Thenevery object of interest for studying the smooth manifold M will be anelement of V (R), e.g.,

(1) The topology of M is an element of V3(R);


(2) With the usual encoding of an ordered pair 〈a, b〉 as a, a, b,the set M ×M is an element of V4(R);

(3) The set Map(M) of all maps from M to itself is an elementof V5(R).

Similarly, the field operations of R, the vector space operations of Rn,and the natural embedding of N into R, are elements of V (R), as iseach function from Rn to R, etc.

See also Lemma 4.4.3 in [Chang & Keisler 1990].

12.4. Membership relation

Statements about V (X) are made using a first order language witha binary relation ∈, the membership relation, as the only primary re-lation. This language will be called ∈-language.

Definition 12.4.1. An ∈-formula φ is called bounded if and onlyif the quantifiers ∀x and ∃x always appear in φ in the form

∀x(x ∈ a⇒ · · · )and

∃x(x ∈ a ∧ · · · )where a is either a variable or a constant.

These two forms are usually abbreviated respectively as ∀x ∈ a (· · · )and ∃x ∈ a (· · · ).

12.5. Ultrapower construction of nonstandard universes

In this section we will present a generalisation of the ultrapowerconstruction of Section 4.7.

For the purpose of such a generalisation, recall that f, g, . . . denotefunctions in a standard universe V (X), while their extensions in anonstandard extension ∗V (X) will be ∗f, ∗g, . . .

Definition 12.5.1. A nonstandard extension of V (X) consists ofa base set ∗X with X $ ∗X , a subuniverse ∗V (X) ⊆ V (∗X), and a map

∗ : V (X) → ∗V (X) (12.5.1)

satisfying the following two conditions:

(I) one has ∗X ⊆ ∗V (X), and the map ∗ is an injective map fromV (X) into ∗V (X) such that if x ∈ X then ∗x ∈ ∗X;

(II) for every bounded formula φ(x1, . . . , xk), and every k-tupleof constants a1, . . . , ak ∈ V (X), the formula φ(a1, . . . , ak) istrue in V (X) if and only if the formula φ(∗a1, . . . ,

∗ak) is truein ∗V (X).

12.5. ULTRAPOWER CONSTRUCTION OF NONSTANDARD UNIVERSES 115

Example 12.5.2. An example is a formula expressing the statementof the extreme value theorem (EVT) for a function f ; see Section 5.8.Therefore by (II) the same formula holds in the extension for ∗f wherethe quantification is now over the extended domain.

Remark 12.5.3. If a 6= b then ∗a 6= ∗b by (II), so ∗ is a bijectionof V (X) onto a subset of ∗V (X). Yet we introduce the bijectivitycondition separately by (I) for the sake of convenience.

Also, still by (II), if a ∈ A ∈ V (X) then ∗a ∈ ∗A, so if one identifies awith its image ∗a, then one may think of A as a subset of ∗A, that is,one can think of ∗A as an extension of A. Since functions are also sets,for every function f ∈ V (X), f : A → B, we can consider the set ∗f ,which is also a function, ∗f : ∗A→ ∗B. If A and B are considered to besubsets of ∗A and ∗B respectively, then the function ∗f is an extensionof the function f .

Theorem 12.5.4. Nonstandard extensions exist.

This is a well-known result presented in many sources, including thefundamental monograph on model theory [Chang & Keisler 1990].We will provide a proof here so as to make the exposition self-contained.

Proof. The construction we employ is called reduced (or bounded)ultrapower. It consists of two parts:

(1) the ultrapower itself, along with the Los theorem, namelyLemma 12.6.2;

(2) the transformation of the bounded ultrapower into a universe.

We start with an arbitrary ultrafilter F ⊆ P(N) containing all cofi-nite sets X ⊆ N, namely a nonprincipal ultrafilter. The starting pointfor the construction of the full ultrapower is the product set V (X)N

which consists of all infinite sequences 〈xn〉 = 〈xn : n ∈ N〉 of ele-ments xn ∈ V (X), or equivalently, all functions f : N → V (X), viathe identification of each f with the sequence 〈f(n) : n ∈ N〉. Thesubsets Vn(X)N ⊆ V N are defined accordingly.

Definition 12.5.5. The set union

V (X)Nbd

=⋃

m

(Vm(X)N

)$ V (X)N

is the starting point for the construction of the bounded ultrapower.

Exercise 12.5.6. Find an element in V (X)N r V (X)Nbd

.

Let m be a natural number.

Definition 12.5.7. A sequence 〈xn〉 ∈ V (X)Nbd

is of type m if theset Dm = n : xn ∈ Vm(X) \ Vm−1(X) belongs to F .


Definition 12.5.8. In the notation as above, a sequence is totallyof type m if Dm = N.

In case m = 0, we naturally assume that V−1(X) = ∅, so thatD0 = n : xn ∈ V0(X), and 〈xn〉 is totally of type 0 if and only ifxn ∈ X = V0(X) for all n.

Lemma 12.5.9. Assume that 〈xn〉 ∈ V (X)Nbd. Then there is a

unique number m ∈ N such that 〈xn〉 is of type m.

Proof. By definition we have 〈xn〉 ∈ VM(X)N for some M andtherefore we have a pairwise disjoint union N =

⋃m≤M Dm. The lemma

now follows from the fact that F is an ultrafilter.

Following Section 5.1, we define an equivalence relation ∼ (or moreprecisely ∼F as the relation depends on F) on V (X)N

bdas follows:

〈xn〉 ∼ 〈yn〉 if and only if n ∈ N : xn = yn ∈ F .1

Lemma 12.5.10. Assume that 〈xn〉 ∈ V (X)Nbd

is of type m. Thereis a sequence 〈yn〉 ∈ V (X)N

bdtotally of type m, such that 〈xn〉 ∼F 〈yn〉.

Proof. If n ∈ Dm then simply let yn = xn. If n /∈ Dm thenput yn = a, where a is a fixed element in Vm(X) \ Vm−1(X). (Ora ∈ X = V0(X) in the case m = 0.)

We continue with the proof of Theorem 12.5.4. The F -quotient

V (X)Nbd/F = 〈xn〉F : 〈xn〉 ∈ V (X)N

bd

consists of all F -classes, or ∼F -classes

〈xn〉F = 〈yn〉 ∈ V (X)Nbd

: 〈xn〉 ∼ 〈yn〉of sequences 〈xn〉 ∈ V (X)N

bd. We define

Vm(X)N/F = 〈xn〉F : 〈xn〉 ∈ Vm(X)Nfor all m, so that V (X)N

bd/F =

⋃m Vm(X)N/F .

Corollary 12.5.11. Each element of V (X)Nbd/F has the form

〈xn〉F , where 〈xn〉 ∈ V (X)Nbd

is a sequence totally of type m for some(unique) m, so that xn ∈ Vm \ Vm−1(X) for all n.

Proof. Use lemmas 12.5.9, 12.5.10.

If 〈xn〉F , 〈yn〉F belong to V (X)Nbd

then we define a binary relation

〈xn〉F ∗∈ 〈yn〉F if and only if n ∈ N : xn ∈ yn ∈ F .1In this case there is a set X ∈ F such that we have xn = yn, for all n ∈ X .

12.6. LOS THEOREM 117

Lemma 12.5.12. If 〈xn〉F , 〈yn〉F belong to V (X)Nbd

and we have〈xn〉F ∗∈ 〈yn〉F then there is a set K ∈ F and an index m ≥ 1 suchthat xn ∈ Vm−1(X), yn ∈ Vm(X), and xn ∈ yn, for all n ∈ K.

Proof. By definition, the set L = n ∈ N : xn ∈ yn belongsto F , and, as 〈xn〉F , 〈yn〉F ∈ V (X)N

bd, there is an index M such that

xn, yn ∈ VM(X) for all n. Consider the sets

Km = n ∈ L : xn ∈ Vm−1(X) and yn ∈ Vm(X), 1 ≤ m ≤M.

It follows by Corollary 12.1.5 that L =⋃

1≤m≤M Km, therefore at leastone of Km belongs to F , and we can set K = Km.

If x ∈ V (X) then we can let 〈x〉 be the infinite constant sequence〈x, x, x, . . . 〉 = 〈xn〉, where xn = x forall n. We let 〈x〉F denote its F -class as above.

12.6. Los theorem

Our main goal in Chapter 12 is the proof of Transfer, an assertionto the effect that, roughly speaking, any sentence true or false in thestandard universe remains true, resp., false in the nonstandard uni-verse, Corollary 12.7.1 and ultimately Corollary 12.7.9. This is not areally easy task, in part because such a typical tool of logic as proofby induction on the logical complexity of the sentence considered, doesnot seem to work. However there is another, and stronger claim whichhappens to admit such an inductive proof. This is the result below, infact a cornerstone of the theory of ultraproducts. The theorem con-nects the truth of sentences related to V (X)N

bdwith their “truth sets”

in the ultrafilter F .

Definition 12.6.1. The truth set T ⊆ N of a formula φ is definedby setting

Tφ(〈a1n〉,...,〈akn〉) =n ∈ N : φ(a1n, . . . , a

kn) is true in V (X)

.

Lemma 12.6.2 (the Los theorem). If φ(x1, . . . , xk) is an arbitrarybounded ∈-formula and sequences 〈a1n〉, . . . , 〈akn〉 belong to V (X)N

bd, then

the sentence φ(〈a1n〉F , . . . , 〈akn〉F) is true in 〈V (X)Nbd/F ; ∗∈〉 if and only

if the truth set Tφ(〈a1n〉,...,〈akn〉) is a member of F .

Here we use an upper index in the enumeration of free variablesto distinguish it from the enumeration of terms of infinite sequencesin V (X)N

bd.

Example 12.6.3. Let φ be the formula x ∈ y. Assume that se-quences 〈an〉, 〈bn〉 belong to V (X)N

bd. Then the formula 〈an〉F ∈ 〈bn〉F


is true in 〈V (X)Nbd/F ; ∗∈〉 if and only if 〈an〉F ∗∈ 〈bn〉F if and only if (by

definition) the set T〈an〉∈〈bn〉 = n ∈ N : an ∈ bn belongs to F .

Proof. The proof of Lemma 12.6.2 proceeds by induction on theconstruction of formula φ out of elementary formulas x ∈ y by means oflogical connectives and quantifiers. Among them, it suffices to consideronly ¬ (the negation), ∧ (conjunction), and ∃ (the existence quantifier)since the remaining ones are known to be expressible via combinationsof those three. For instance, Φ ∨ Ψ is equivalent to ¬(Φ ∧ Ψ), and thequantifier ∀ is equivalent to ¬∃¬.

The base of induction is the case when φ is the formula x ∈ y,already checked in Example 12.6.3. As we go through the inductionsteps, we will reduce the arbitrarily long list of free variables x1, . . . , xk

to a minimally nontrivial amount for each step, for the sake of brevity.The step ¬. Let φ(x) be a bounded ∈-formula with a single free

variable x, and 〈an〉 ∈ V (X)Nbd

. Suppose that the lemma is establishedfor φ(〈an〉), and let us prove it for ¬ φ(〈an〉).

Indeed, formula ¬ φ(〈an〉F) is true in 〈V (X)Nbd/F ; ∗∈〉 if and only

if φ(〈an〉F) is false in 〈V (X)Nbd/F ; ∗∈〉. The latter holds if and only if

(by the inductive hypothesis) Tφ(〈an〉) /∈ F or equivalently T¬φ(〈an〉) ∈ Fsince obviously Tφ = N r T¬φ, and one and only one set in a pair ofcomplementary sets belongs to F (as F is an ultrafilter).

The step ∧. Here one proves the result for φ(a) ∧ ψ(a) assumingthat the lemma is established separately for φ(a) and ψ(a). We leaveit as an exercise.

The step ∃. Assume that the lemma is established for a boundedformula φ(x, y, z), and prove it for (∃z ∈ x)φ(x, y, z). Let 〈an〉, 〈bn〉 ∈V (X)N

bd. We have to prove that the statement

(A) (∃z ∈ 〈an〉F) φ(〈an〉F , 〈bn〉F , z) is true in 〈V (X)Nbd/F ; ∗∈〉

is equivalent to

(B) the set L = T∃z∈〈an〉φ(〈an〉,〈bn〉,z) belongs to F .

Note that L = n : ∃z ∈ an (φ(an, bn, z) is true in V (X)).By definition, (A) is equivalent to

(C) there is a sequence 〈cn〉 ∈ V (X)Nbd

such that both 〈cn〉F ∗∈ 〈an〉Fand φ(〈an〉F , 〈bn〉F , 〈cn〉F) are true in 〈V (X)N

bd/F ; ∗∈〉.

By the inductive hypothesis for φ and the base of induction result,the requirements of (C) hold if and only if each of the sets

U ′〈cn〉 = T〈cn〉∈〈an〉 = n : cn ∈ an, and

U ′′〈cn〉 = Tφ(〈an〉,〈bn〉,〈cn〉) = n : φ(an, bn, cn) is true in V (X)

12.7. ELEMENTARY EMBEDDING 119

belongs to F , which is equivalent (as F is an ultrafilter) to U〈cn〉 ∈ F ,where

U〈cn〉 = U ′〈cn〉 ∩ U ′′

〈cn〉 = T〈cn〉∈〈an〉∧φ(〈an〉,〈bn〉,〈cn〉).

On the other hand, the set L of (B) satisfies U〈cn〉 ⊆ L because if n ∈U〈cn〉 then taking z = cn satisfies φ(an, bn, z) in V (X). Therefore if T ∈F then L ∈ F as F is an ultrafilter. This ends the proof that (A)implies (B).

To prove the converse suppose that L ∈ F . Recall that V (X)Nbd

=⋃m Vm(X)N, hence the sequence 〈an〉 belongs to some Vm(X). If n ∈ L

then by definition there is some z = cn ∈ an such that φ(an, bn, z) is truein V (X). It follows that m > 0 and cn ∈ Vm−1(X), by Corollary 12.1.5If n /∈ L then let cn ∈ Vm−1(X) be arbitrary. The construction ofthe sequence 〈cn : n ∈ N〉 ∈ Vm−1(X)N uses the axiom of choice, ofcourse. Thus 〈cn〉 ∈ V (X)N

bdand if n ∈ L then n belongs to the

set U〈cn〉 = U ′〈cn〉 ∩ U ′′

〈cn〉 as above by construction. It follows that the

sets U ′〈cn〉, U

′′〈cn〉 belong to F , and hence we have (C) and (A).

This completes the proof of the Los theorem.

12.7. Elementary embedding

Corollary 12.7.1. The map x 7−→ 〈x〉F is an elementary embed-ding of the structure 〈V (X);∈〉 to 〈V (X)N

bd/F ; ∗∈〉.

In other words, if φ(x1, . . . , xk) is a bounded ∈-formula and we havea1, . . . , ak ∈ V (X), then the sentence φ(a1, . . . , ak) is true in V (X) ifand only if the sentence φ(〈a1〉F , . . . , 〈ak〉F) is true in 〈V (X)N

bd/F ; ∗∈〉.

Proof. Recall that 〈aj〉 is the infinite sequence 〈ajn : n ∈ N〉 suchthat ajn = aj for all n. Therefore the truth set Tφ(〈a1〉,...,〈ak〉) is equal

to N if φ(a1, . . . , ak) is true in V (X) and is equal to ∅ otherwise. Nowan application of Lemma 12.6.2 concludes the proof.

This result ends the first part of the proof of Theorem 12.5.4. Wedefined an embedding x 7−→ 〈x〉F of the universe 〈V (X);∈〉 to

〈V (X)Nbd/F ; ∗∈〉

which satisfies Transfer by Corollary 12.7.1. The final step of the proofof the theorem is embedding 〈V (X)N

bd/F ; ∗∈〉 into the universe V (∗X),

where ∗X = 〈an〉F : 〈an〉 ∈ XN, and, we recall, X = V0(X).

Lemma 12.7.2. Assume that γ is an infinite ordinal and X isa γ-base set. Then ∗X is a (γ + 4)-base set, hence a base set byLemma 12.2.4.


Proof. By definition, each 〈an〉F ∈ ∗X is a non-empty set ofpairwise ∼F -equivalent sequences 〈an〉 = 〈an : n ∈ N〉 of elementsan ∈ X . It remains to prove that rank(〈an〉) = γ + 4. Note that asequence of this form is formally equal to the set of all ordered pairs〈n, an〉 = n, n, an : n ∈ N, where

rank(n) = n,

rank(an) = γ + 1 (as X is a γ-base set),

rank(n) = n + 1,

rank(n, an) = γ + 2,

rank(n, n, an) = γ + 3,

rank(〈an : n ∈ N〉) = γ + 4,

as required.

The lemma shows that V (∗X) is a legitimate superstructure whichsatisfies all results in sections 12.1 and 12.2.

Definition 12.7.3. Define a map h : V (X)Nbd/F → V (∗X) as

follows. Suppose that 〈an〉F ∈ V (X)Nbd/F . It can be assumed, by

Corollary 12.5.11, that 〈an〉 is totally of type m for some m, so thatan ∈ Vm(X) \ Vm−1(X) (or just an ∈ X = V0(X) provided m = 0) forall n. The definition of h(〈an〉F) goes on by induction on m.

(1) If m = 0 then we put h(〈an〉F) = 〈an〉F .In other words, if 〈an〉 ∈ XN then h(〈an〉F) = 〈an〉F ∈ ∗X .

(2) If m ≥ 1 then we put

h(〈an〉F) =h(〈bn〉F) : 〈bn〉 ∈ Vm−1(X)N and 〈bn〉F ∗∈ 〈an〉F

.

This type of definitions is called the Mostowski collapse, see Theorem4.4.9 in [Chang & Keisler 1990].

Exercise 12.7.4. Prove by induction on m that if 〈an〉 ∈ Vm(X)N

then h(〈an〉F) ∈ Vm(∗X).

Lemma 12.7.5. The map h is a bijection, and h is an isomorphismin the sense that if 〈an〉, 〈bn〉 ∈ V (X)N

bdthen

〈an〉F ∗∈ 〈bn〉F if and only if h(〈an〉F) ∈ h(〈bn〉F) .

Proof. To prove that h is a bijection, suppose that 〈an〉, 〈bn〉 ∈V (X)N

bdand 〈an〉F 6= 〈bn〉F . Then the truth set T = n ∈ N : an 6= bn

belongs to F by Lemma 12.6.2. It can be assumed, by Corollary 12.5.11,that 〈an〉, 〈bn〉 are totally of type m, resp., m′, for some m,m′.

Case 1 : m = m′ = 0, so that an, bn ∈ X = V0(X) for all n. Thenh(〈an〉F) = 〈an〉F and h(〈bn〉F) = 〈bn〉F by construction, and on theother hand, we have 〈an〉F 6= 〈bn〉F since T ∈ F .

12.7. ELEMENTARY EMBEDDING 121

Case 2 : m′ = 0 and m ≥ 1, so that bn ∈ X = V0(X) andan ∈ Vm(X) \ Vm−1(X) for all n. Then h(〈bn〉F) = 〈bn〉F ∈ ∗X whileh(〈an〉F) ⊆ V (∗X) by construction, so that h(〈bn〉F) 6= h(〈an〉F) byLemma 12.7.2. (Recall Definition 12.1.4.)

Case 3 : m = 0 and m′ ≥ 1, similar.

Case 4 : m = m′ ≥ 1. Then, for any n, an and bn belong toVm(X) \ Vm−1(X), therefore an ⊆ Vm−1(X) and bn ⊆ Vm−1(X). Inaddition, if n ∈ T then an 6= bn, hence there exists some cn ∈ Vm−1(X)satisfying cn ∈ an \ bn or cn ∈ bn \ an. It follows that one of the sets

T ′ = n ∈ T : cn ∈ an \ bn, T ′′ = n ∈ T : cn ∈ bn \ anbelongs to F ; suppose that T ′ ∈ F . If n /∈ T then let cn ∈ Vm−1(X)be arbitrary. Then 〈cn〉 ∈ Vm−1(X). Now we have 〈cn〉 ∗∈ 〈an〉 but¬ 〈cn〉 ∗∈ 〈bn〉, hence, by Definition 12.7.3, h(〈cn〉F) ∈ h(〈an〉F) buth(〈cn〉F) /∈ h(〈bn〉F). Therefore h(〈an〉F) 6= h(〈bn〉F).

Case 5 : m,m′ ≥ 1 and m 6= m′, say m′ < m. For any n wehave an ∈ Vm(X) \ Vm−1(X), hence an ⊆ Vm−1(X). Note that an ⊆Vk(X) is impossible for k < m − 1 as otherwise an ∈ Vk+1(X) ⊆Vm−1(X), a contradiction. Thus there is an element cn ∈ an, cn ∈Vm−1(X) \ Vm−2(X), in particular, cn ∈ Vm−1(X) \ Vm′−1(X). Then〈cn〉 ∈ Vm−1(X) and 〈cn〉 ∗∈ 〈an〉, hence h(〈cn〉F) ∈ h(〈an〉F). On theother hand, we have cn /∈ bn by Corollary 12.1.5, hence ¬ 〈cn〉 ∗∈ 〈bn〉and h(〈cn〉F) /∈ h(〈bn〉F). Thus still h(〈an〉F) 6= h(〈bn〉F).

The claim that h is a bijection is established.Now prove the isomorphism claim. Let 〈an〉, 〈bn〉 be sequences in

V (X)Nbd

. As above, it can be assumed that 〈an〉, 〈bn〉 are totally oftype m, resp., m′, for some m,m′ ∈ N. Suppose that 〈an〉F ∗∈ 〈bn〉F .By definition the set N = n ∈ N : an ∈ bn belongs to F . Thenan ∈ Vm′−1(X) for all n ∈ N by Corollary 12.1.5, and hence we haveh(〈an〉F) ∈ h(〈bn〉F) by Definition 12.7.3.

If conversely h(〈an〉F) ∈ h(〈bn〉F) then immediately 〈an〉F ∗∈ 〈bn〉Fstill by Definition 12.7.3.

Definition 12.7.6. Let ∗V (X) = h(〈an〉F) : 〈an〉 ∈ V (X)Nbd/F,

the range of h. If m ∈ N then let∗Vm(X) = h(〈an〉F) : 〈an〉 ∈ Vm(X)N/F.

Exercise 12.7.7. Prove, using the results of Lemma 12.7.5, thatthe sets ∗Vm(X) satisfy ∗Vm(X) = ∗V (X) ∩ Vm(∗X), ∗Vm+1(X) ⊆P(∗Vm(X)), and finally ∗V (X) =

⋃m

∗Vm(X) ⊆ V (∗X).

Thus ∗V (X) is a subuniverse of the full universe V (∗X) over ∗X =∗V0(X). Let’s use letters ξ, η to denote arbitrary elements of V (X)N

bd/F ,


so that if ξ ∈ V (X)Nbd/F then h(ξ) ∈ ∗V (X). As h is an isomorphism

by Lemma 12.7.5, we immediately obtain the following.

Corollary 12.7.8. If φ(x1, . . . , xk) is a bounded ∈-formula andξ1, . . . , ξk ∈ V (X)N

bd/F , then φ(ξ1, . . . , ξk) is true in 〈V (X)N

bd/F ; ∗∈〉 if

and only if the sentence φ(h(ξ1), . . . , h(ξk)) is true in 〈∗V (X);∈〉.

Now we are ready to accomplish the proof of Theorem 12.5.4.We claim that ∗X , ∗V (X), and the map x 7−→ ∗x is a nonstandard exten-sion of V (X). The proof is maintained by combining Corollaries 12.7.1and 12.7.8. Namely, if x ∈ V (X) then let ∗x = h(〈x〉F), so that x 7−→ ∗xmaps V (X) onto ∗V (X) ⊆ V (∗X). Further, Lemma 12.7.5 ensures thatcondition (I) of Section 12.5 holds. Now let’s validate condition (II) of12.5. Consider a bounded formula φ(x1, . . . , xk), and arbitrary elementsa1, . . . , ak ∈ V (X). Then the formula φ(a1, . . . , ak) is true in V (X) ifand only if φ(〈a1〉F , . . . , 〈ak〉F) is true in 〈V (X)N

bd/F ; ∗∈〉 (by Corollary

12.7.1). if and only if the formula φ(∗a1, . . . , ∗ak) is true in ∗V (X) (byCorollary 12.7.8).

Note that both V (X) and ∗V (X) are considered as ∈-structureswith the true membership relation ∈.

It remains to prove that X $ ∗X properly, or, to be more precise,there is an element y ∈ ∗X not equal to any ∗x, x ∈ X .

Let a0, a1, a2, . . . be an infinite sequence of pairwise distinct ele-ments an ∈ X . Then 〈an〉 = 〈an : n ∈ N〉 belongs to XN = V0(X)N. Itfollows that ξ = h(〈an〉F) ∈ ∗X . Prove that if x ∈ X then ξ 6= ∗x . Bydefinition one has to show that 〈an〉 6∼ 〈x〉, that is, the set N = n :an 6= x belongs to F . However N contains all natural numbers with apossible exception of a single index n satisfying an = x. Hence N ∈ F .

This completes the proof of Theorem 12.5.4.

Rephrasing condition (II) of Section 12.5, we obtain:

Corollary 12.7.9 (Transfer). The map x 7−→ ∗x is an elemen-tary ∈-embedding of the structure 〈V (X);∈〉 to 〈∗V (X);∈〉. In otherwords, if φ(x1, . . . , xk) is a bounded ∈-formula and a1, . . . , ak ∈ V (X),then φ(a1, . . . , ak) is true in V (X) if and only if φ(∗a1, . . . , ∗ak) is truein ∗V (X).

12.8. Definability

In Chapter 4 we presented a construction of the hyperreal line viafilters and ultrapowers. For a broader picture, it may be useful tocomment on the issue of definability.

12.9. CONSERVATIVITY 123

Many people felt that the hyperreal line is not definable sinceits definition involves nonconstructive foundational material. There-fore it came as a surprise to many people when Kanovei and Shelahproved in 2004 that, in fact, the hyperreal line is indeed definable.See http://www.ams.org/mathscinet-getitem?mr=2039354 and thearticle itself [Kanovei & Shelah 2004].

What this means is that there exists a specific set-theoretic formulathat defines the hyperreal line. This might seem surprising. To clarifythe situation, one would compare it with that for Lebesgue measure.It is widely known and accepted as fact that the Lebesgue measureis σ-additive. Two items need to be pointed out here:

(1) to define the Lebesgue measure, one does not need the axiomof choice.

(2) to prove that the Lebesgue measure is σ-additive, one needsthe axiom of choice.

Here item (2) follows from the existence of a model of ZF where thereal line is a countable union of countable sets. This model is called theFeferman-Levy model [Feferman & Levy 1963]. See also [Jech 1973,chapter 10].

We see that, while the Lebesgue measure can be defined in ZF, aproof of its σ-additivity necessarily requires Choice.

With regard to its foundational status, the situation is similar fora hyperreal line. One can define a hyperreal line without Choice (asKanovei and Shelah did), but actually to prove something about it onewould need some version of Choice.

12.9. Conservativity

The various frameworks developed by Robinson and since, includingthose of Nelson and Hrbacek, are conservative extensions of ZFC, andin this sense are part of classical mathematics.

As far as conservativity is concerned, the following needs to be keptin mind. Robinson showed that, given a proof exploiting infinitesimals,there always exists an infinitesimal-free paraphrase.

However, this is merely a theoretical result. In practice, such aparaphrase may involve an exponential increase in the complexity ofthe proof, and correspondingly an exponential decrease in its compre-hensibility, at least in principle.

From the point of computer scientist one should certainly be ableto appreciate the difference.

Similarly, one can note that any result using the real numbers canbe restated in terms of the rationals using sequences, and even in terms

http://www.ams.org/mathscinet-getitem?mr=2039354


of the integers, etc. The ancient Greeks used formulations that wouldbe very difficult for us to follow if we did not use modern translations,in terms of extended number systems that have since become widelyaccepted and are considered intuitive. The related controversy overUnguru’s proposal is well known.

The following two additional points shouuld be kept in mind.First, the conservativity property of the pair ZFC, IST does not hold

for pairs like An, A∗n, where An is the n-th order arithmetic and A∗

n itsnonstandard extension, in fact A∗

n is rather comparable to the stan-dard An+1. This profound result of Henson–Keisler near 1994 is well-known to NSA people (provide references and exact formulations).

Second, IST yields some principally new mathematics with respectto ZFC, in the sense that there are IST sentences not equivalent toany ZFC sentence. This is established in [Kanovei & Reeken 2004].This can be compared to the obvious fact that any sentence about Cadmits an equivalent sentence about R.

The following comments by Henson and Keisler usefully illustratethe point:

It is often asserted in the literature that any theoremwhich can be proved using nonstandard analysis canalso be proved without it. The purpose of this paper isto show that this assertion is wrong, and in fact thereare theorems which can be proved with nonstandardanalysis but cannot be proved without it. There iscurrently a great deal of confusion among mathemati-cians because the above assertion can be interpretedin two different ways. First, there is the followingcorrect statement: any theorem which can be provedusing nonstandard analysis can be proved in Zermelo-Fraenkel set theory with choice, ZFC, and thus isacceptable by contemporary standards as a theoremin mathematics. Second, there is the erroneous con-clusion drawn by skeptics: any theorem which canbe proved using nonstandard analysis can be provedwithout it, and thus there is no need for nonstandardanalysis. The reason for this confusion is that theset of principles which are accepted by current math-ematics, namely ZFC, is much stronger than the setof principles which are actually used in mathematicalpractice. It has been observed (see [F] and [S]) that al-most all results in classical mathematics use methods

12.9. CONSERVATIVITY 125

available in second order arithmetic with appropriatecomprehension and choice axiom schemes. This sug-gests that mathematical practice usually takes placein a conservative extension of some system of secondorder arithmetic, and that it is difficult to use thehigher levels of sets. In this paper we shall considersystems of nonstandard analysis consisting of secondorder nonstandard arithmetic with saturation princi-ples (which are frequently used in practice in nonstan-dard arguments). We shall prove that nonstandardanalysis (i.e. second order nonstandard arithmetic)with the ω1-saturation axiom scheme has the samestrength as third order arithmetic. This shows that inprinciple there are theorems which can be proved withnonstandard analysis but cannot be proved by theusual standard methods. [Henson & Keisler 1986,p. 377]

In short, A∗n is comparable to the standard An+1 rather than to An

itself.

CHAPTER 13

More regularity

13.1. D2 implies D1

Lemma 13.1.1. Let B ⊆ Rn be an open ball around the origin 0,let a ∈ hal(0), and let 0 6= v ∈ ∗Rn with v ≺≺ 1. If G : ∗B → ∗Rn isan internal function satisfying

G(x) − 2G(x+ v) +G(x + 2v) ≺ ‖v‖‖G(a) −G(a+ v)‖for all x ∈ hB, then there is m ∈ ∗N such that a+mv ∈ hB and

G(a) −G(a + v) ≺ ‖v‖‖G(a) −G(a+mv)‖.Proof. Let N = ⌊r/‖v‖⌋ where 0 < r ∈ R is slightly smaller

than the radius of B, and for 0 ≤ x ∈ ∗ R, ⌊x⌋ ∈ ∗N is the integerpart of x. So any m ≤ N satisfies that a + mv ∈ hB. Let A =G(a) − G(a + v). For 0 ≤ j ≤ N let xj = a + jv, then by ourassumption on G we have G(xj) − 2G(xj+1) + G(xj+2) = Cj‖v‖‖A‖with Cj ≺ 1. Let C be the maximum of C0, · · · , CN , then C ≺ 1and G(xj) − 2G(xj+1) + G(xj+2) ≤ C‖v‖‖A‖ for all 0 ≤ j ≤ N .Given k ≤ N we have∥∥∥A−

(G(xk)−G(xk+1)

)∥∥∥ =∥∥∥(G(x0)−G(x1)

)−(G(xk)−G(xk+1)

)∥∥∥

≤k−1∑

j=0

∥∥∥(G(xj) −G(xj+1)

)−(G(xj+1) −G(xj+2)

)∥∥∥ ≤ Ck‖v‖‖A‖.

So for any m ≤ N ,

∥∥∥mA−(G(x0)−G(xm)

)∥∥∥ =∥∥∥m−1∑

k=0

(A−(G(xk)−G(xk+1)

))∥∥∥ ≤ Cm2‖v‖‖A‖,

so∥∥∥mA −

(G(x0) − G(xm)

)∥∥∥ = Km2‖v‖‖A‖ with K ≺ 1. It follows

that

m‖A‖ −Km2‖v‖‖A‖ ≤ ‖G(x0) −G(xm)‖,and so, multiplying by ‖v‖, we have m‖v‖‖A‖(1−Km‖v‖) ≤ ‖v‖‖G(x0)−G(xm)‖ .

127

128 13. MORE REGULARITY

Now let m = min N , ⌊ 12K‖v‖⌋ , then

m‖v‖‖A‖/2 ≤ ‖v‖‖G(x0) −G(xm)‖.By definition of N and since K ≺ 1 we have that m‖v‖ is appreciable,i.e. not infinitesimal, and so finally A ≺ ‖v‖‖G(x0)−G(xm)‖ , that is,G(a) −G(a + v) ≺ ‖v‖‖G(a) −G(a+mv)‖.

Proposition 13.1.2. If F is D2 for some choice of coordinatesin W , then F is D1.

Proof. Given a ∈ hW , in the given coordinates take some ball Baround st(a). Define G(x) = F (x)−x, then we must show for any v ≺λ that G(a) − G(a + v) ≺ λ‖v‖ (here v = b − a in Definition 10.3.3).If G(a)−G(a+v) ≺≺ λ‖v‖ then we are certainly done. Otherwise λv ≺‖G(a) −G(a+ v)‖, and on the other hand G(x) − 2G(x+ v) +G(x+2v) = F (x) − 2F (x + v) + F (x + 2v) ≺ λ‖v‖2, by taking v = w inDefinition 10.4.1. Together we have G(x) − 2G(x + v) + G(x + 2v) ≺‖v‖‖G(a) − G(a + v)‖, so by Lemma 13.1.1 there is m ∈ ∗N suchthat a+mv ∈ hB and

G(a)−G(a+v) ≺ ‖v‖‖G(a)−G(a+mv)‖ ≤ ‖v‖(‖G(a)‖+‖G(a+mv)‖

)≺ ‖v‖λ

since F is a prevector field and so G(x) = F (x) − x ≺ λ for all x.

13.2. Injectivity

When speaking about local D1 or D2 prevector fields, wheneverneeded we will assume, perhaps by passing to a smaller domain, thata constant K as in Propositions 11.3.1, 11.3.2 exists.

Corollary 13.2.1. If F is a D1 prevector field then F is injectiveon hM .

Proof. Let a 6= b ∈ hM . If st(a) 6= st(b) then clearly F (a) 6= F (b).Otherwise there exists a B including a, b as in Proposition 11.3.1, and(2) of that proposition implies F (a) 6= F (b).

We will now show that a D1 prevector field is in fact bijective onhM . We first prove local surjectivity, as follows.

Proposition 13.2.2. Let B1 ⊆ B2 ⊆ B3 ⊆ Rn be closed ballscentered at the origin, of radii r1 < r2 < r3. If F : ∗B2 → ∗B3 is alocal D1 prevector field then F (∗B2) ⊇ ∗B1.

Proof. Fix 0 < s ∈ R smaller than r2 − r1 and r3 − r2. We willapply transfer to the following fact: For every function f : B2 → B3, if‖f(x)−f(y)‖ ≤ 2‖x−y‖ for all x, y ∈ B2 and ‖f(x)−x‖ < s for all x ∈

13.3. TIME REVERSAL 129

B2 then f(B2) ⊇ B1. This fact is indeed true since our assumptionson f imply that it is continuous, and that for every x ∈ ∂B2 the straightinterval between x and f(x) is included in B3−B1, so f |∂B2

is homotopicin B3−B1 to the inclusion of ∂B2. Now if some p ∈ B1 is not in f(B2)then f |∂B2

is null-homotopic in B3 − p, and so the same is true forthe inclusion of ∂B2, a contradiction. Applying transfer we get thatfor every internal function f : ∗B2 → ∗B3, if ‖f(x) − f(y)‖ ≤ 2‖x− y‖for all x, y ∈ ∗B2 and ‖f(x)− x‖ < s for all x ∈ ∗B2 then f(∗B2) ⊇ ∗B1.In particular this is true for a D1 prevector field F : ∗B2 → ∗B3, byProposition 11.3.1(2).

The following is immediate from Corollary 13.2.1 and Propostion 13.2.2.

Theorem 13.2.3. If F : ∗M → ∗M is a D1 prevector field then F |hM :hM → hM is bijective.

Remark 13.2.4. On all ∗M , a D1 prevector field may be noninjec-tive and nonsurjective, e.g. take M = (0, 1) and F : ∗M → ∗M givenby F (x) = λ for x ≤ λ and F (x) = x otherwise. (Recall that thedefinition of D1 prevector field imposes no restrictions in ∗M − hM).

Remark 13.2.5. For D1 prevector field F , the map F |hM : hM →hM and its inverse (F |hM)−1 are not internal if M is noncompact, sincetheir domain is not internal. On the other hand, for any A ⊆M , F |∗Ais internal. Furthermore, on ∗B1 of Proposition 13.2.2, F has an in-verse F−1 : ∗B1 → ∗B2 in the sense that F F−1(a) = a for all a ∈ ∗B1,and F−1 is internal. So, for a local D1 prevector field F : ∗U → ∗V wemay always assume (perhaps for slightly smaller domain) that F−1 :∗U → ∗V also exists, in the above sense. As mentioned, we will usuallynot mention the range ∗V but rather speak of a local prevector fieldon ∗U .

13.3. Time reversal

Proposition 13.3.1. If F is D1 then F−1 is D1, where F−1 is asin Remark 13.2.5.

Proof. Let x = F−1(a), y = F−1(b), then

‖F−1(a)−F−1(b)−a+b‖ = ‖x−y−F (x)+F (y)‖ ≺ λ‖x−y‖ ≺ λ‖F (x)−F (y)‖ = λ‖a−b‖by Lemma 10.7.1.

Remark 13.3.2. If one follows the proofs of Proposition 13.3.1 andLemma 10.7.1 one sees that if ‖F (a) − F (b) − a + b‖ ≤ Kλ‖a− b‖ insome domain ∗U ⊆ ∗Rn, then

‖F−1(a) − F−1(b) − a+ b‖ ≤ K ′λ‖a− b‖

130 13. MORE REGULARITY

in a corresponding domain for F−1, with K ′ only slightly larger than K,namely K ′ = K/(1 −Kλ).

We conclude this section with the following observations.

Lemma 13.3.3. Let F,G be D1 prevector fields. If F (a) ≡ G(a) forall standard a, then F (b) ≡ G(b) for all b, i.e. F ≡ G.

Proof. Let K be as in Proposition 11.3.1 for both F and G.Let a = st(b), then

‖F (b)−G(b)‖ = ‖F (b)−F (a)−b+a+F (a)−G(a)+G(a)−G(b)−a+b‖≤ ‖F (b) − F (a) − b+ a‖ + ‖F (a) −G(a)‖ + ‖G(a) −G(b) − a+ b‖

≤ Kλ‖a− b‖ + ‖F (a) −G(a)‖ +Kλ‖a− b‖ ≺≺ λ.

Recall that Definition 10.2.1, which defines when a prevector field Frealizes a classical vector field X , involves only standard points. Itfollows from Lemma 13.3.3 that if F is D1 then this determines F upto equivalence. Namely, we have the following.

Corollary 13.3.4. Let U ⊆ Rn be open, and X : U → Rn avector field. If F,G are two D1 prevector fields that realize X then F ≡G. In particular, if X is Lipschitz and G is a D1 prevector field thatrealizes X, then F ≡ G, where F is the prevector field obtained from Xas in Example 10.1.7.

Part 2

500 yrs of infinitesimal math:Stevin to Nelson

CHAPTER 14

16th century

14.1. Simon Stevin

This section contains a discussion of Simon Stevin, unending dec-imals, and the real numbers. The material in this section is based in[B laszczyk, Katz & Sherry 2013].

Nearly a century before Newton and Leibniz, Simon Stevin (Stev-inus) sought to break with an ancient Greek heritage of working ex-clusively with relations among natural numbers,1 and developed anapproach capable of representing both “discrete” number ( ’αριθµoς)2

composed of units (µoναδων) and continuous magnitude (µεγεθoς) ofgeometric origin.3 According to van der Waerden, “[Stevin’s] generalnotion of a real number was accepted, tacitly or explicitly, by all laterscientists” [Van der Waerden 1985, p. 69].

D. Fearnley-Sander wrote that “the modern concept of real num-ber . . . was essentially achieved by Simon Stevin, around 1600, andwas thoroughly assimilated into mathematics in the following two cen-turies” [Fearney-Sander 1979, p. 809]. D. Fowler points out that“Stevin . . . was a thorough-going arithmetizer: he published, in 1585,the first popularization of decimal fractions in the West . . . in 1594, hedescribed an algorithm for finding the decimal expansion of the rootof any polynomial, the same algorithm we find later in Cauchy’s proofof the Intermediate Value Theorem” [Fowler 1992, p. 733]. Fowleralso emphasizes that important foundational work was yet to be done

1The Greeks counted as numbers only 2, 3, 4, . . .; thus, 1 was not a number, norare the fractions: ratios were relations, not numbers. Consequently, Stevin had tospend time arguing that the unit (µoνας) was a number.

2Euclid [Euclid 2007, Book VII, def. 2].3Euclid [Euclid 2007, Book V]. See also Aristotle’s Categories, 6.4b, 20-23:

“Quantity is either discrete or continuous. [. . . ] Instances of discrete quantities arenumber and speech; of continuous, lines, surfaces, solids, and besides these, timeand place”.

133

134 14. 16TH CENTURY

by Dedekind, who proved that the field operations and other arith-metic operations extend from Q to R.4 Meanwhile, Stevin’s decimalsstimulated the emergence of power series (see below) and other de-velopments. We will discuss Stevin’s contribution to the IntermediateValue Theorem in Subsection 14.3 below.

In 1585, Stevin defined decimals in La Disme as follows:

Decimal numbers are a kind of arithmetic based onthe idea of the progression of tens, making use of theArabic numerals in which any number may be writtenand by which all computations that are met in busi-ness may be performed by integers alone without theaid of fraction. (La Disme, On Decimal Fractions, tr.by V. Sanford, in [Smith 1920, p. 23]).

By numbers “met in business” Stevin meant finite decimals,5 andby “computations” he meant addition, subtraction, multiplications, di-vision and extraction of square roots on finite decimals.

Stevin argued that numbers, like the more familiar continuous mag-nitudes, can be divided indefinitely, and used a water metaphor toillustrate such an analogy:

As to a continuous body of water corresponds a con-tinuous wetness, so to a continuous magnitude corre-sponds a continuous number. Likewise, as the con-tinuous body of water is subject to the same divisionand separation as the water, so the continuous num-ber is subject to the same division and separation asits magnitude, in such a way that these two quantitiescannot be distinguished by continuity and discontinu-ity [Stevin 1585, p. 3]; quoted in [Malet 2006].

Stevin argued for equal rights in his system for rational and irrationalnumbers. He was critical of the complications in Euclid [Euclid 2007,Book X], and was able to show that adopting the arithmetic as a wayof dealing with those theorems made many of them easy to understandand easy to prove. In his Arithmetique [Stevin 1585], Stevin proposedto represent all numbers systematically in decimal notation. P. Ehrlichnotes that Stevin’s

4Namely, there is no easy way from representation of reals by decimals, tothe field of reals, just as there is no easy way from continuous fractions, anotherwell-known representation of reals, to operations on such fractions.

5But see Stevin’s comments on extending the process ad infinitum in main textat footnote 11.

14.2. DECIMALS FROM STEVIN TO NEWTON AND EULER 135

viewpoint soon led to, and was implicit in, the an-alytic geometry of Rene Descartes (1596-1650), andwas made explicit by John Wallis (1616-1703) andIsaac Newton (1643-1727) in their arithmetizationsthereof [Ehrlich 2005, p. 494].

14.2. Decimals from Stevin to Newton and Euler

Stevin’s text La Disme on decimal notation was translated intoEnglish in 1608 by Robert Norton (cf. Cajori [Cajori 1928, p. 314]).The translation contains the first occurrence of the word “decimal” inEnglish; the word will be employed by Newton in a crucial passage63 years later.6 Wallis recognized the importance of unending decimalexpressions in the following terms:

Now though the Proportion cannot be accurately ex-pressed in absolute Numbers: Yet by continued Ap-proximation it may; so as to approach nearer to it,than any difference assignable (Wallis’s Algebra, p. 317,cited in [Crossley 1987]).

Similarly, Newton exploits a power series expansion to calculate de-tailed decimal approximations to log(1 + x) for x = ±0.1,±0.2, . . .(Newton [1]).7 By the time of Newton’s annus mirabilis, the idea of un-ending decimal representation was well established. Historian V. Katzcalls attention to “Newton’s analogy of power series to infinite decimalexpansions of numbers” [Katz, V. 1993, p. 245]. Newton expressedsuch an analogy in the following passage:

Since the operations of computing in numbers andwith variables are closely similar . . . I am amazed thatit has occurred to no one (if you except N. Mer-cator with his quadrature of the hyperbola) to fitthe doctrine recently established for decimal numbersin similar fashion to variables, especially since theway is then open to more striking consequences. For

6See main text at footnote 9.7Newton scholar N. Guicciardini kindly provided a jpg of a page from New-

ton’s manuscript containing such calculations, and commented as follows: “It wasprobably written in Autumn 1665 (see Mathematical papers 1, p. 134). White-side’s dating is sometimes too precise, but in any case it is a manuscript that wascertainly written in the mid 1660s when Newton began annotating Wallis’s Arith-metica Infinitorum” [Guicciardini 2011]. The page from Newton’s manuscriptcan be viewed at http://u.math.biu.ac.il/~katzmik/newton.html

http://u.math.biu.ac.il/~katzmik/newton.html


since this doctrine in species has the same relation-ship to Algebra that the doctrine in decimal numbershas to common Arithmetic, its operations of Addi-tion, Subtraction, Multiplication, Division and Root-extraction may easily be learnt from the latter’s pro-vided the reader be skilled in each, both Arithmeticand Algebra, and appreciate the correspondence be-tween decimal numbers and algebraic terms continuedto infinity . . . And just as the advantage of decimalsconsists in this, that when all fractions and roots havebeen reduced to them they take on in a certain mea-sure the nature of integers, so it is the advantage ofinfinite variable-sequences that classes of more com-plicated terms (such as fractions whose denominatorsare complex quantities, the roots of complex quanti-ties and the roots of affected equations) may be re-duced to the class of simple ones: that is, to infiniteseries of fractions having simple numerators and de-nominators and without the all but insuperable en-cumbrances which beset the others [Newton 1671].8

In this remarkable passage dating from 1671, Newton explicitly namesinfinite decimals as the source of inspiration for the new idea of infi-nite series.9 The passage shows that Newton had an adequate numbersystem for doing calculus and real analysis two centuries before thetriumvirate.10

In 1742, John Marsh first used an abbreviated notation for repeat-ing decimals [Marsh 1742, p. 5]; cf. [Cajori 1928, p. 335]. Euler ex-ploits unending decimals in his Elements of Algebra in 1765, as when hesums an infinite series and concludes that 9.999 . . . = 10 [Euler 1770,p. 170].

14.3. Stevin’s cubic and the IVT

In the context of his decimal representation, Stevin developed nu-merical methods for finding roots of polynomials equations. He de-scribed an algorithm equivalent to finding zeros of polynomials (seeCrossley [Crossley 1987, p. 96]). This occurs in a corollary to prob-lem 77 (more precisely, LXXVII) in (Stevin [Stevin 1625, p. 353]).

8We are grateful to V. Katz for signaling this passage.9See footnote 6.10The expression “the great triumvirate” is used by Boyer in [Boyer 1949,

p. 298] to describe Cantor, Dedekind, and Weierstrass.

14.3. STEVIN’S CUBIC AND THE IVT 137

Here Stevin describes such an argument in the context of finding aroot of the cubic equation (which he expresses as a proportion to con-form to the contemporary custom)

x3 = 300x+ 33915024.

Here the whimsical coefficient seems to have been chosen to emphasizethe fact that the method is completely general; Stevin notes further-more that numerous addional examples can be given. A textual dis-cussion of the method may be found in Struik [Stevin 1958, p. 476].

Centuries later, Cauchy would prove the Intermediate Value The-orem (IVT) for a continuous function f on an interval I by a divide-and-conquer algorithm. Cauchy subdivided I into m equal subinter-vals, and recursively picked a subinterval where the values of f haveopposite signs at the endpoints [Cauchy 1821, Note III, p. 462]. Toelaborate on Stevin’s argument following [Stevin 1958, §10, p. 475-476], note that what Stevin similarly described a divide-and-conqueralgorithm. Stevin subdivides the interval into ten equal parts, result-ing in a gain of a new decimal digit of the solution at every iteration ofhis procedure. Stevin explicitly speaks of continuing the iteration adinfinitum:11

Et procedant ainsi infiniment, l’on approche infini-ment plus pres au requis [Stevin 1625, p. 353, last 3lines].

Arguably, the abstract existence proofs for the real numbers are notneeded here since Stevin gives an algorithm for producing an explicitdecimal representation of the solution. The IVT for polynomials wouldresurface in Lagrange before being generalized by Cauchy to the newlyintroduced class of continuous functions.

One frequently hears sentiments to the effect that mathematiciansbefore 1870 did not and could not have provided rigorous proofs, sincethe real number system was not even built yet, but such an attitudemay be anachronistic. It overemphasizes the significance of the tri-umvirate project in an inappropriate context. Stevin is concerned withconstructing an algorithm, whereas Cantor is concerned with develop-ing a foundational framework based upon the existence of the totalityof the real numbers, as well as their power sets, etc. The latter projectinvolves a number of non-constructive ingredients, including the axiomof infinity and the law of excluded middle. But none of it is neededfor Stevin’s procedure, because he is not seeking to re-define “number”

11Cf. footnote 5.


in terms of alternative (supposedly less troublesome) mathematical ob-jects. Many historians emphasize the approaches developed by Cantor,Dedekind, and Weierstrass to proofs of the existence of real numbers.Sometimes this is done at the expense, and almost to the exclusion, ofStevin’s approach. Stevin’s contribution to the construction of the realnumbers deserves more explicit acknowledgment.

CHAPTER 15

17th century

15.1. Pierre de Fermat

15.2. James Gregory

15.3. Gottfried Wilhelm von Leibniz

139

CHAPTER 16

18th century

16.1. Leonhard Euler

141

CHAPTER 17

19th century

17.1. Augustin-Louis Cauchy

143

CHAPTER 18

20th century

18.1. Thoralf Skolem

18.2. Edwin Hewitt

18.3. Jerzy Los

18.4. Abraham Robinson

18.5. Edward Nelson

145

Bibliography

[Albeverio et al. 2009] Albeverio, S.; Fenstad, J.; Hoegh-Krohn, R.; Lindstrom, T.Nonstandard Methods in Stochastic Analysis and Mathematical Physics.Academic Press 1986, Dover 2009.

[Almeida, Neves & Stroyan 2014] Almeida, R.; Neves, V.; Stroyan, K. “Infinites-imal differential geometry: cusps and envelopes.” Differ. Geom. Dyn.Syst. 16 (2014), 1–13.

[Bair et al. 2013] Bair, J.; B laszczyk, P.; Ely, R.; Henry, V.; Kanovei, V.; Katz,K.; Katz, M.; Kutateladze, S.; McGaffey, T.; Schaps, D.; Sherry, D.;Shnider, S. “Is mathematical history written by the victors?” Notices ofthe American Mathematical Society 60 (2013) no. 7, 886-904.See http://www.ams.org/notices/201307/rnoti-p886.pdfand http://arxiv.org/abs/1306.5973

[Benoit 1997] Benoit, E. “Applications of Nonstandard Analysis in Ordinary Dif-ferential Equations.” Nonstandard Analysis, Theory and Applications.Editors: Leif O. Arkeryd, Nigel J. Cutland, C. Ward Henson NATO ASISeries Volume 493, 1997, pp 153-182.

[B laszczyk, Katz & Sherry 2013] B laszczyk, P.; Katz, M.; Sherry, D. “Ten miscon-ceptions from the history of analysis and their debunking.” Foundationsof Science, 18 (2013), no. 1, 43-74.See http://dx.doi.org/10.1007/s10699-012-9285-8and http://arxiv.org/abs/1202.4153

[Boothby 1986] Boothby, W. An introduction to differentiable manifolds and Rie-mannian geometry. Second edition. Pure and Applied Mathematics, 120.Academic Press, Inc., Orlando, FL, 1986.

[Borovik & Katz 2012] Borovik, A., Katz, M. “Who gave you the Cauchy–Weierstrass tale? The dual history of rigorous calculus.” Foundationsof Science 17, no. 3, 245-276.See http://dx.doi.org/10.1007/s10699-011-9235-x

[Boyer 1949] Boyer, C. The concepts of the calculus. Hafner Publishing Company.[Bradley & Sandifer 2009] Bradley, R., Sandifer, C. Cauchy’s Cours d’analyse. An

annotated translation. Sources and Studies in the History of Mathematicsand Physical Sciences. Springer, New York.

[Cajori 1928] Cajori. A history of mathematical notations, Volumes 1-2. The OpenCourt Publishing Company, La Salle, Illinois, 1928/1929. Reprinted byDover, 1993.

[Cauchy 1821] Cauchy, A. L. Cours d’Analyse de L’Ecole Royale Polytechnique.Premiere Partie. Analyse algebrique (Paris: Imprimerie Royale, 1821).

147

http://dx.doi.org/10.1007/s10699-012-9285-8

http://arxiv.org/abs/1202.4153

http://dx.doi.org/10.1007/s10699-011-9235-x

148 BIBLIOGRAPHY

[Chang & Keisler 1990] Chang, C.; Keisler, H.J. Model Theory. Studies in Logicand the Foundations of Mathematics (3rd ed.), Elsevier (1990) [Origi-nally published in 1973].

[Crossley 1987] Crossley, J. The emergence of number. Second edition. World Sci-entific Publishing Co., Singapore.

[Davis 1977] Davis, M.: Applied nonstandard analysis. Pure and Applied Math-ematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. Reprinted by Dover, NY, 2005, seehttp://store.doverpublications.com/0486442292.html

[Do79] Dombrowski, P.: 150 years after Gauss’ “Disquisitiones generales circasuperficies curvas”. With the original text of Gauss. Asterisque, 62.Societe Mathematique de France, Paris, 1979.

[Ehrlich 2005] Ehrlich, P. “Continuity.” In Encyclopedia of philosophy, 2nd edition(Donald M. Borchert, editor), Macmillan Reference USA, FarmingtonHills, MI, 2005 (The online version contains some slight improvements),pp. 489–517.

[Euclid 2007] Euclid. Euclid’s Elements of Geometry, edited, and provided with amodern English translation, by Richard Fitzpatrick, 2007.See http://farside.ph.utexas.edu/euclid.html

[Euler 1770] Euler, L. Elements of Algebra (3rd English edition). John Hewlett andFrancis Horner, English translators. Orme Longman, 1822.

[Feferman & Levy 1963] Feferman, S.; Levy, A. “Independence results in set theoryby Cohen’s method II.” Notices Amer. Math. Soc. 10 (1963).

[Fowler 1992] Fowler, D. “Dedekind’s theorem:√

2 ×√

3 =√

6.” American Math-ematical Monthly 99, no. 8, 725–733.

[Goldblatt 1998] Goldblatt, R. Lectures on the hyperreals. An introduction to non-standard analysis. Graduate Texts in Mathematics, 188. Springer-Verlag,New York.

[Gordon et al. 2002] Gordon, E.; Kusraev, A.; Kutateladze, S. Infinitesimal analy-sis. Updated and revised translation of the 2001 Russian original. Trans-lated by Kutateladze. Mathematics and its Applications, 544. KluwerAcademic Publishers, Dordrecht, 2002.

[Grobman 1959] Grobman, D. “Homeomorphisms of systems of differential equa-tions.” Doklady Akademii Nauk SSSR 128, 880–881.

[Guicciardini 2011] Guicciardini, N. Private communication, 27 november 2011.[Hartman 1960] Hartman, P. “A lemma in the theory of structural stability of

differential equations.” Proc. A.M.S. 11 (4), 610–620.[Henson & Keisler 1986] Henson, C. W.; Keisler, H. J. “On the strength of non-

standard analysis.” J. Symbolic Logic 51 (1986), no. 2, 377–386.[Herzberg 2013] Herzberg, F. Stochastic Calculus with Infinitesimals. Lecture Notes

in Mathematics Volume 2067, Springer, 2013[Hewitt 1948] Hewitt, E. “Rings of real-valued continuous functions. I.” Trans.

Amer. Math. Soc. 64 (1948), 45–99.[Jech 1973] Jech, T. The axiom of choice. Studies in Logic and the Foundations

of Mathematics, Vol. 75. North-Holland Publishing Co., Amsterdam-London; Amercan Elsevier Publishing Co., Inc., New York.

http://farside.ph.utexas.edu/euclid.html

BIBLIOGRAPHY 149

[Kanovei, Katz & Nowik 2016] Kanovei, V.; Katz, K.; Katz, M.; Nowik, T. “Smalloscillations of the pendulum, Euler’s method, and adequality.” QuantumStudies: Mathematics and Foundations, to appear.See http://dx.doi.org/10.1007/s40509-016-0074-x

and http://arxiv.org/abs/1604.06663

[Kanovei & Reeken 2004] Kanovei, V.; Reeken, M. Nonstandard analysis, axiomat-ically. Springer Monographs in Mathematics, Berlin: Springer.

[Kanovei & Shelah 2004] Kanovei, V.; Shelah, S. A definable nonstandard modelof the reals.” J. Symbolic Logic 69 (2004), no. 1, 159–164.

[Katz & Sherry 2012] Katz, M.; Sherry, D. “Leibniz’s laws of continuity and ho-mogeneity.” Notices of the American Mathematical Society 59 (2012),no. 11, 1550-1558.See http://www.ams.org/notices/201211/rtx121101550p.pdfand http://arxiv.org/abs/1211.7188

[Katz & Sherry 2013] Katz, M., Sherry, D. “Leibniz’s infinitesimals: Their fiction-ality, their modern implementations, and their foes from Berkeley toRussell and beyond.” Erkenntnis 78, no. 3, 571–625.See http://dx.doi.org/10.1007/s10670-012-9370-yand http://arxiv.org/abs/1205.0174

[Katz, V. 1993] Katz, V. J. “Using the history of calculus to teach calculus.” Sci-ence & Education. Contributions from History, Philosophy and Sociologyof Science and Education 2, no. 3, 243–249.

[Keisler 1986] Keisler, H. J.: Elementary Calculus: An Infinitesimal Approach.Second Edition. Prindle, Weber & Schimidt, Boston, 1986. See onlineversion at http://www.math.wisc.edu/∼keisler/calc.html

[Keisler 1976] H. J. Keisler. Foundations of Infinitesimal Calculus. On-line Edition,2007, at the author’s web page:http://www.math.wisc.edu/∼keisler/foundations.html(Originally published by Prindle, Weber & Schmidt 1976.)

[Klein 1908] Klein, F. Elementary Mathematics from an Advanced Standpoint.Vol. I. Arithmetic, Algebra, Analysis. Translation by E. R. Hedrick andC. A. Noble [Macmillan, New York, 1932] from the third German edition[Springer, Berlin, 1924]. Originally published as Elementarmathematikvom hoheren Standpunkte aus (Leipzig, 1908).

[Leibniz 1989] Leibniz, G. W.: La naissance du calcul differentiel. (French) [Thebirth of differential calculus] 26 articles des Acta Eruditorum. [26 ar-ticles of the Acta Eruditorum] Translated from the Latin and with anintroduction and notes by Marc Parmentier. With a preface by MichelSerres. Mathesis. Librairie Philosophique J. Vrin, Paris, 1989.

[ Los 1955] Los, J. “Quelques remarques, theoremes et problemes sur les classes defi-nissables d’algebres.” In Mathematical interpretation of formal systems,98–113, North-Holland Publishing Co., Amsterdam, 1955.

[Lutz & Goze 1981] Lutz, R.; Goze, M. Nonstandard Analysis, a Practical Guidewith Applications. Lecture Notes in Mathematics 881, Springer-Verlag,1981.

[Malet 2006] Malet, A. “Renaissance notions of number and magnitude.” HistoriaMathematica 33, no. 1, 63–81.

[Marsh 1742] Marsh, J. Decimal arithmetic made perfect. London.

http://dx.doi.org/10.1007/s40509-016-0074-x


http://www.ams.org/notices/201211/rtx121101550p.pdf


http://dx.doi.org/10.1007/s10670-012-9370-y


http://www.math.wisc.edu/~keisler/calc.html

http://www.math.wisc.edu/~keisler/foundations.html

150 BIBLIOGRAPHY

[Newton 1671] Newton, I.: A Treatise on the Methods of Series and Fluxions, 1671.In Whiteside [4] (vol. III), pp. 33–35.

[1] Newton, I.: Cambridge UL Add. 3958.4, f. 78v[2] Newton, I. 1946. Sir Isaac Newton’s mathematical principles of natural

philosophy and his system of the world, a revision by F. Cajori of A.Motte’s 1729 translation. Berkeley: Univ. of California Press.

[3] Newton, I. 1999. The Principia: Mathematical principles of natural phi-losophy, translated by I. B. Cohen & A. Whitman, preceded by A guideto Newton’s Principia by I. B. Cohen. Berkeley: Univ. of CaliforniaPress.

[Nowik & Katz 2015] Nowik, T., Katz, M. “Differential geometry via infinitesimaldisplacements.” Journal of Logic and Analysis 7:5 (2015), 1–44. Seehttp://www.logicandanalysis.org/index.php/jla/article/view/237/106

and http://arxiv.org/abs/1405.0984

[Robinson 1966] Robinson, A. Non-standard analysis. North-Holland PublishingCo., Amsterdam 1966.

[Rudin 1976] Rudin, W. Principles of mathematical analysis. Third edition. Inter-national Series in Pure and Applied Mathematics. McGraw-Hill BookCo., New York-Auckland-Dusseldorf, 1976.

[Fearney-Sander 1979] Fearnley-Sander, D. “Hermann Grassmann and the creationof linear algebra.” Amer. Math. Monthly 86, no. 10, 809–817.

[Sherry & Katz 2014] Sherry, D., Katz, M. “Infinitesimals, imaginaries, ideals, andfictions.” Studia Leibnitiana 44 (2012), no. 2, 166–192 (the article ap-peared in 2014 even though the year given by the journal is 2012). Seehttp://arxiv.org/abs/1304.2137

[Smith 1920] Smith, D. Source Book in Mathematics. Mc Grow-Hill, New York.[Spivak 1979] Spivak, M. A comprehensive introduction to differential geometry.

Vol. 4. Second edition. Publish or Perish, Inc., Wilmington, Del., 1979.[Stevin 1585] Stevin, S. (1585) L’Arithmetique. In Albert Girard (Ed.), Les Oeu-

vres Mathematiques de Simon Stevin (Leyde, 1634), part I, p. 1–101.[Stevin 1625] Stevin, S. L’Arithmetique. In Albert Girard (Ed.), 1625, part II.

Online athttp://www.archive.org/stream/larithmetiqvedes00stev#page/353/mode/1up

[Stevin 1958] Stevin, S. The principal works of Simon Stevin. Vols. IIA, IIB: Math-ematics. Edited by D. J. Struik. C. V. Swets & Zeitlinger, Amsterdam1958. Vol. IIA: v+pp. 1–455 (1 plate). Vol. IIB: 1958 iv+pp. 459–976.

[Stroyan 1977] Stroyan, K. “Infinitesimal analysis of curves and surfaces.” In Hand-book of Mathematical Logic, J. Barwise, ed. North-Holland PublishingCompany, 1977.

[Stroyan 2015] Stroyan, K. Advanced Calculus using Mathematica: NoteBook Edi-tion.

[Stroyan & Luxemburg 1976] Stroyan, K.; Luxemburg, W. Introduction to the The-ory of Infinitesimals. Academic Press, 1976.

[Sun 2000] Sun, Y. “Nonstandard Analysis in Mathematical Economics.” In: Non-standard Analysis for the Working Mathematician, Peter A. Loeb andManfred Wolff, eds, Mathematics and Its Applications, Volume 510,2000, pp 261-305.

http://www.logicandanalysis.org/index.php/jla/article/view/237/106



http://www.archive.org/stream/larithmetiqvedes00stev#page/353/mode/1up

BIBLIOGRAPHY 151

[Tao 2014] Tao, T. Hilbert’s fifth problem and related topics. Graduate Studiesin Mathematics, 153. American Mathematical Society, Providence, RI,2014.

[Tao & Vu 2016] Tao, T.; Van Vu, V. “Sum-avoiding sets in groups.”See http://arxiv.org/abs/1603.03068

[Tarski 1930] Tarski, A. “Une contribution a la theorie de la mesure.” FundamentaMathematicae 15, 42–50.

[Van den Berg 1987] Van den Berg, I. Nonstandard asymptotic analysis. LectureNotes in Mathematics 1249, Springer 1987.

[Van der Waerden 1985] Van der Waerden, B. L. A history of algebra. From al-Khwarizmi to Emmy Noether. Springer-Verlag, Berlin.

[We55] Weatherburn, C. E.: Differential geometry of three dimensions. Vol. 1.Cambridge University Press, 1955.

[4] Whiteside, D. T. (Ed.): The mathematical papers of Isaac Newton. Vol.III: 1670–1673. Edited by D. T. Whiteside, with the assistance in publica-tion of M. A. Hoskin and A. Prag, Cambridge University Press, London-New York, 1969.

[Zygmund 1945] Zygmund, A. Smooth functions. Duke Math. J. 12. 47–76.


CHAPTER 19

Theorema Egregium

19.1. Preliminaries to the Theorema Egregium of Gauss

In this section and the two following we will give a proof of Gauss’stheorem to the effect that the Gaussian curvature function of a surfaceis an intrinsic invariant of a surface.

We denote the coordinates in R2 by (u1, u2). In R3, let 〈 , 〉 be theEuclidean inner product. Let x = x(u1, u2) : R2 → R3 be a regularparametrized surface. Here “regular” means that the vector valuedfunction x has Jacobian of rank 2 at every point. Consider partialderivatives xi = ∂

∂ui(x), where i = 1, 2. Thus, vectors x1 and x2 form a

basis for the tangent plane to the surface at every point. Similarly, let

xij =∂2x

∂ui∂uj

be the second partial derivatives. Let n = n(u1, u2) be a unit normalto the surface at the point x(u1, u2), so that 〈n, xi〉 = 0.

Definition 19.1.1 (symbols gij, Lij , Γkij). The first fundamentalform (gij) at a point of the surface is given in coordinates by gij =〈xi, xj〉. The second fundamental form (Lij) is given in coordinatesby Lij = 〈n, xij〉. The symbols Γkij are uniquely defined by the decom-position

xij = Γkijxk + Lijn

= Γ1ijx1 + Γ2

ijx2 + Lijn,(19.1.1)

where the repeated (upper and lower) index k implies summation, inaccordance with the Einstein summation convention.

Definition 19.1.2 (symbols Lij). The Weingarten map (Lij) is anendomorphism of the tangent plane, namely Rx1⊕R x2. It is uniquelydefined by the decomposition of nj = ∂

∂uj(n) as follows:

nj = Lijxi

= L1jx1 + L2

jx2.

Note the relationLij = −Lkjgki.

153

154 19. THEOREMA EGREGIUM

Definition 19.1.3. We will denote by square brackets [ ] the anti-symmetrisation over the pair of indices found in between the brackets,e.g.

a[ij] = 12(aij − aji).

Note that g[ij] = 0, L[ij] = 0, and Γk[ij] = 0. We will use the

notation Γkij;ℓ for the ℓ-th partial derivative of the symbol Γkij.

Proposition 19.1.4. We have the following relation

Γqi[j;k] + Γmi[j Γqk]m = −Li[jLqk] (19.1.2)

for each set of indices i, j, k, q (with, as usual, an implied summationover the index m).

Proof. The formula will result by antisymmetrizing a formula forthe third partial.

Let us calculate the tangential component, with respect to the ba-sis x1, x2, n, of the third partial derivative xijk. Recall that nk =Lpkxp and xjk = Γℓjkxℓ + Likn. Thus, we have

(xij)k =(Γmijxm + Lijn

)k

= Γmij;kxm + Γmijxmk + Lijnk + Lij;kn

= Γmij;kxm + Γmij

(Γpmkxp + Lmkn

)+ Lij (Lpkxp) + Lij;kn.

Grouping together the tangential terms, we obtain

(xij)k = Γmij;kxm + Γmij

(Γpmkxp

)+ Lij (Lpkxp) + (. . .)n

=(

Γqij;k + ΓmijΓqmk + LijL

qk

)xq + (. . .)n

=(

Γqij;k + ΓmijΓqkm + LijL

qk

)xq + (. . .)n,

since the symbols Γqkm are symmetric in the two subscripts. Now thesymmetry in j, k (equality of mixed partials) implies the following iden-tity:

(xi[j)k] = 0.

Therefore

0 = (xi[j)k]

=(

Γqi[j;k] + Γmi[jΓqk]m + Li[jL

qk]

)xq + (. . .)n,

and therefore Γqi[j;k] + Γmi[jΓqk]m + Li[jL

qk] = 0 for each q = 1, 2.

19.2. THE THEOREMA EGREGIUM OF GAUSS 155

19.2. The theorema egregium of Gauss

Definition 19.2.1. The Gaussian curvature functionK = K(u1, u2)is the determinant of the Weingarten map:

K = det(Lij) = 2L1[1L

22]. (19.2.1)

Theorem 19.2.2. We have the following three equivalent formulasfor Gaussian curvature K:

(a) K = det(Lij) = 2L1[1L

22];

(b) K =det(Lij)

det(gij);

(c) K = − 2g11L1[1L

22].

Proof. The first formula is our definition of K. To prove the sec-ond formula, we use the relation Lij = −Lkjgki. By the multiplicativityof determinant with respect to matrix multiplication, we have

det(Lij) = (−1)2det(Lij)

det(gij).

Let us now prove formula (c). The proof is a calculation. Note that bydefinition, 2L1[1L

22] = L11L

22 − L12L

21. Hence

−2L1[1L22] =

(L1

1g11 − L21g21

)L2

2 −(L1

2g11 − L22g21

)L2

1

= g11L11L

22 − g21L

21L

22 − g11L

12L

21 + g21L

22L

21

= g11(L1

1L22 − L1

2L21

)

= g11det(Li j) = g11K.(19.2.2)

Note that we can either calculate using the formula Lij = −Lkjgki,or the formula Lij = −Lkigkj. Only the former one leads to the appro-priate cancellations as in (19.2.2).

Theorem 19.2.3 (Theorema egregium). The Gaussian curvaturefunction K = K(u1, u2) can be expressed in terms of the coefficients ofthe first fundamental form alone (and their first and second derivatives)as follows:

K =2

g11

(Γ21[1;2] + Γj1[1Γ

22]j

), (19.2.3)

where the symbols Γkij can be expressed in terms of the derivatives of gijbe the formula Γkij = 1

2(giℓ;j − gij;ℓ + gjℓ;i)g

ℓk, where (gij) is the inversematrix of (gij).

This theorem will be proved in Section 19.3.


Weingartenmap (Lij)

Gaussian cur-vature K

mean cur-vature H

plane

(0 00 0

)0 0

cylinder

(1 00 0

)0 1

2

invarianceof curvature

yes no

Table 19.3.1. Plane and cylinder have the same intrinsicgeometry (K), but different extrinsic geometries (H)

19.3. Intrinsic versus extrinsic

Theorem 19.3.1. Unlike Gaussian curvature K, the mean curva-ture

H = 12Lii = tr(W )

cannot be expressed in terms of the gij and their derivatives.

Indeed, the plane and the cylinder have parametrisations with iden-tical gij, but with different mean curvature, cf. Table 19.3.1. To sum-marize, Gaussian curvature K is an intrinsic invariant, while meancurvature H = 1

2Lii is an extrinsic invariant, of the surface.

Proof of theorema egregium. We present a streamlined version ofdo Carmo’s proof [?, p. 233]. The proof is in 3 steps.

(1) We express the third partial derivative xijk in terms of the Γ’s(intrinsic information) and the L’s (extrinsic information).

(2) The equality of mixed partials yields an identification of anappropriate expression in terms of the Γ’s, with a certain com-bination of the L’s.

(3) The combination of the L’s is expressed in terms of Gaussiancurvature.

The first two steps were carried out in Proposition 19.1.4. Choosingthe valus i = j = 1 and k = q = 2 for the indices in equation (19.1.2),we obtain

Γ21[1;2] + Γm1[1Γ

22]m = −L1[1L

22]

= g1iLi[1L

22]

= g11L1[1L

22]

19.4. GAUSSIAN CURVATURE AND LAPLACE-BELTRAMI OPERATOR 157

Γ1ij j = 1 j = 2

i = 1 µ1 µ2

i = 2 µ2 −µ1

Γ2ij j = 1 j = 2

i = 1 −µ2 µ1

i = 2 µ1 µ2

Table 19.4.1. Symbols Γkij of a metric e2µ(u1,u2)δij

since the term L2[1L

22] = 0 vanishes. Applying Theorem 19.2.2(c),

we obtain the desired formula for K and complete the proof of thetheorema egregium.

19.4. Gaussian curvature and Laplace-Beltrami operator

In this section we will complement the material of section 19.2.A (u1, u2) chart in which the metric becomes conformal (see Defi-

nition 20.8.1) to the standard flat metric, is referred to as isothermalcoordinates. The existence of the latter is proved in [?].

Recall that a metric λ(x, y)(dx2 + dy2), where u1 = x and u2 = y,has metric coefficients g11 = g22 = λ whereas g12 = 0.

Definition 19.4.1. The Laplace-Beltrami operator for a metric g =λ(x, y)(dx2 +dy2), where u1 = x and u2 = y, in isothermal coordinatesis

∆LB =1

λ

(∂2

∂(u1)2+

∂2

∂(u2)2

).

The notation means that when we apply the operator ∆LB to afunction f = f(u1, u2), we obtain

∆LB(f) =1

λ

(∂2f

∂(u1)2+

∂2f

∂(u2)2

).

In more readable form for f = f(x, y), we have

∆LB(f) =1

λ

(∂2f

∂x2+∂2f

∂y2

).

Theorem 19.4.2. Given a metric in isothermal coordinates withmetric coefficients gij = λ(u1, u2)δij, its Gaussian curvature is minushalf the Laplace-Beltrami operator applied to the log of the conformalfactor λ:

K = −1

2∆LB log λ. (19.4.1)

Here log x is the natural logarithm so that (log x)′ = 1x.


Proof. Let λ = e2µ. We have from Table 19.4.1:

2Γ21[1;2] = Γ2

11;2 − Γ212;1 = −µ22 − µ11.

It remains to verify that the ΓΓ term in the expression (19.2.3) for theGaussian curvature vanishes:

2Γj1[1Γ22]j = 2Γ1

1[1Γ22]1 + 2Γ2

1[1Γ22]2

= Γ111Γ

221 − Γ1

12Γ211 + Γ2

11Γ222 − Γ2

12Γ212

= µ1µ1 − µ2(−µ2) + (−µ2)µ2 − µ1µ1

= 0.

Then from formula (19.2.3) we have

K =2

λΓ21[1;2] = −1

λ(µ11 + µ22) = −∆LBµ.

Meanwhile, ∆LB log λ = ∆LB(2µ) = 2∆LB(µ), proving the result.

Setting λ = f 2, we restate the theorem as follows.

Theorem 19.4.3. Given a metric in isothermal coordinates withmetric coefficients gij = f 2(u1, u2)δij, its Gaussian curvature is minusthe Laplace-Beltrami operator of the log of the conformal factor f :

K = −∆LB log f. (19.4.2)

Either one of the formulas (19.2.3), (19.4.1), or (19.4.2) can serveas the intrinsic definition of Gaussian curvature, replacing the extrinsicdefinition (19.2.1), cf. Remark 34.7.1.

19.5. Duality in linear algebra

Let V be a real vector space. We will assume all vector spaces tobe finite dimensional unless stated otherwise.

Example 19.5.1. Euclidean space Rn is a real vector space of di-mension n.

Example 19.5.2. The tangent plane TpM of a regular surface Mat a point p ∈ M is a real vector space of dimension 2.

Definition 19.5.3. A linear form, also called 1-form, φ on V is alinear functional

φ : V → R .

Definition 19.5.4. The dual space of V , denoted V ∗, is the spaceof all linear forms on V :

V ∗ = φ : V → R | φ is a 1-form on V .

19.6. POLAR, CYLINDRICAL, AND SPHERICAL COORDINATES 159

Evaluating φ at an element x ∈ V produces a scalar φ(x) ∈ R.

Definition 19.5.5. The natural pairing between V and V ∗ is alinear map

〈 , 〉 : V × V ∗ → R,defined by setting 〈x, y〉 = y(x), for all x ∈ V and y ∈ V ∗.

If V admits a basis of vectors (xi), then V ∗ admits a unique basis,called the dual basis (yj), satisfying

〈xi, yj〉 = δij, (19.5.1)

for all i, j = 1, . . . , n, where δij is the Kronecker delta function.

Example 19.5.6. The vectors ∂∂x

and ∂∂y

form a basis for the tan-

gent plane TpE of the Euclidean plane E at each point ∈ E. The dualspace is denoted T ∗

p and called the cotangent plane. The basis dual to(∂∂x, ∂∂y

)is denoted (dx, dy).

Polar coordinates will be defined in detail in Subsection 19.6. Theyprovide helpful examples of vectors and 1-forms, as follows.

Example 19.5.7. In polar coordinates, we have a basis ∂∂r, ∂∂θ

for thetangent plane TpE of the Euclidean plane E at each point p ∈ E \ 0.The dual space T ∗

p has a basis dual to(∂∂r, ∂∂θ

)and denoted (dr, dθ).

Example 19.5.8. In polar coordinates, the 1-form

r dr

occurs frequently in calculus. This 1-form vanishes at the origin r = 0,and gets “bigger and bigger” as we get further away from the origin(see next section).

19.6. Polar, cylindrical, and spherical coordinates

Definition 19.6.1. Polar coordinates (koordinatot koteviot) (r, θ)satisfy r2 = x2 + y2 and x = r cos θ, y = r sin θ.

In R2 \ 0, one way of defining the ranges for the variables is torequire

r > 0 and

θ ∈ [0, 2π).

It is shown in elementary calculus that the area of a region D inthe plane in polar coordinates is calculated using the area element

dA = r dr dθ.


Thus, an integral is of the form∫

D

dA =

∫∫rdrdθ.

Cylindrical coordinates in Euclidean 3-space are studied in VectorCalculus.

Definition 19.6.2. Cylindrical coordinates (koordinatot gliliot)

(r, θ, z)

are a natural extension of the polar coordinates (r, θ) in the plane.

The volume of an open region D is calculated with respect to cylin-drical coordinates using the volume element

dV = r dr dθ dz.

Namely, an integral is of the form∫

D

dV =

∫∫∫rdr dθ dz.

Example 19.6.3. Find the volume of a right circular cone withheight h and base a circle of radius b.

Spherical coordinates1

(ρ, θ, ϕ)

in Euclidean 3-space are studied in Vector Calculus.

Definition 19.6.4. Spherical coordinates (ρ, θ, ϕ) are defined asfollows. The coordinate ρ is the distance from the point to the origin,satisfying

ρ2 = x2 + y2 + z2,

or ρ2 = r2+z2, where r2 = x2+y2. If we project the point orthogonallyto the (x, y)-plane, the polar coordinates of its image, (r, θ), satisfy x =r cos θ and y = r sin θ. The last coordiate ϕ of the spherical coordinatesis the angle between the position vector of the point and the third basisvector e3 in 3-space (pointing upward along the z-axis). Thus

z = ρ cosϕ

while

r = ρ sinϕ .

1koordinatot kaduriot

19.7. DUALITY IN CALCULUS; DERIVATIONS 161

Here the ranges of the coordinates are often chosen as follows:

0 ≤ ρ

0 ≤ θ ≤ 2π

and

0 ≤ ϕ ≤ π

(note the different upper bounds for θ and ϕ). Recall that the area ofa spherical region D is calculated using a volume element of the form

dV = ρ2 sinϕ dρ dθ d ϕ,

so that the volume of a region D is∫

D

dV =

∫∫∫

D

ρ2 sinϕ dρ dθ d ϕ .

Example 19.6.5. Calculate the volume of the spherical shell be-tween spheres of radius α > 0 and β ≥ α.

The area of a spherical region on a sphere of radius ρ = β is calcu-lated using the area element

dA = β2 sinϕ dθ d ϕ .

Thus the area of a spherical region D on a sphere of radius β is givenby the integral ∫

D

dA =

∫∫β2 sinϕ dθ d ϕ .

Example 19.6.6. Calculate the area of the spherical region on asphere of radius β included in the first octant, (so that all three Carte-sian coordinates are positive).

19.7. Duality in calculus; derivations

Let E be a Euclidean space of dimension n, and let p ∈ E be afixed point.

Definition 19.7.1. Let

Dp = f : f ∈ C∞

be the space of real C∞-functions f defined in a neighborhood of p ∈ E.

Note that Dp is infinite-dimensional as it includes all polynomials.


Theorem 19.7.2. A partial derivative ∂∂ui

at p a 1-form

∂

∂ui: λp → R

on the space λp, satisfying Leibniz’s rule

∂(fg)

∂ui

∣∣∣∣p

=∂f

∂ui

∣∣∣∣p

g(p) + f(p)∂g

∂ui

∣∣∣∣p

for all f, g ∈ λp.

The formula can be written briefly as∂∂ui

(fg) = ∂∂ui

(f)g + f ∂∂ui

(g),

at the point p. This was proved in calculus. The formula above moti-vates the following more general definition of a derivation.

Definition 19.7.3. A derivation X at p is an R-linear form

X : Dp → R

on the space Dp satisfying Leibniz’s rule:

X(fg) = X(f)g(p) + f(p)X(g) (19.7.1)

for all f, g ∈ Dp.

Remark 19.7.4. Linearity of a derivation is required only withregard to multiplication by scalars from R, not with regard to multi-plication by functions.

CHAPTER 20

Derivations, tori

20.1. Space of derivations

Let M be an n-dimensional manifold, and let p ∈ M . Recall thata derivation X is a 1-form on the space Dp of functions defined near psuch that X satisfies the Leibniz rule (see Section 19.7). The followingtwo results are familiar from advanced calculus.

Proposition 20.1.1. The space of all derivations at p is a vectorspace of dimension n, and is called the tangent space Tp at p.

Proof in case 1. Let us prove the result in the case n = 1 of asingle coordinate u = u1 at the point p = 0. Let X be a derivation.Then

X(1) = X(1 · 1) = 2X(1)

by Leibniz’s rule. Therefore X(1) = 0, and similarly for any constantby linearity of X . Now consider the monic polynomial u = u1 ofdegree 1, i.e., the linear function

u ∈ Dp=0.

We evaluate the derivation X at u and set c = X(u) ∈ R. By theTaylor formula, any function f ∈ Dp=0, f = f(u), can be written as

f(u) = a+ bu+ g(u)u

where g is smooth and g(0) = 0. Now we have by Linearity and Leibnizrule

X(f) = X(a+ bu+ g(u)u)

= bX(u) +X(g)u(0) + g(0) · 1

= bc

= c∂

∂u(f).

Thus X coincides with c ∂∂u

for all f ∈ Dp. Hence the tangent space is1-dimensional, proving the theorem in this case.

163

164 20. DERIVATIONS, TORI

Proof in case 2. Let us prove the result in the case of two vari-ables u, v at the point p = 0 (the general case is similar). Let X be aderivation. Then

X(1) = X(1 · 1) = 2X(1)

by Leibniz’s rule. Therefore X(1) = 0, and similarly for any constantby linearity of X . Now consider the monic polynomial u = u1 ofdegree 1, i.e., the linear function

u ∈ Dp=0.

We evaluate the derivation X at u and set c = X(u). Similarly, considerthe monic polynomial v = v1 of degree 1, i.e., the linear function

v ∈ Dp=0.

We evaluate the derivation X at v and set c′ = X(v).By the Taylor formula, any function f ∈ Dp=0, f = f(u, v), can be

written as

f(u, v) = a + bu+ b′v + g(u, v)u2 + h(u, v)uv + k(u, v)v2,

where the functions g(u, v), h(u, v), and k(u, v) are smooth. Note thatthe coefficients b and b′ are the first partial derivatives of f . Now wehave

X(f) = X(a+ bu+ b′v + g(u, v)u2 + h(u, v)uv + k(u, v)v2)

= bX(u) + b′X(v) +(X(g)u2 + g(u, v)2uX(u) + · · ·

)|u=0,v=0

= bX(u) + b′X(v)

= bc + b′c′

= c∂

∂u(f) + c′

∂

∂v(f).

Thus X coincides with the derivation

c∂

∂u+ c′

∂

∂v

for all test functions f ∈ Dp. Hence the tangent space sis spanned bythe tangent vectors ∂

∂uand ∂

∂vis therefore 2-dimensional, proving the

theorem in this case.

We have the following immediate corollary.

Corollary 20.1.2. Let (u1, . . . , un) be coordinates in a neighbor-hood of p ∈ M . Then a basis for the space Tp is given by the n partialderivatives (

∂

∂ui

), i = 1, . . . , n.

20.3. FIRST FUNDAMENTAL FORM 165

Definition 20.1.3. The vector space dual to the tangent space Tpis called the cotangent space, and denoted T ∗

p .

Thus an element of a tangent space is a vector, while an element ofa cotangent space is called a 1-form, or a covector.

Definition 20.1.4. The basis dual to the basis ( ∂∂ui

) is denoted

(dui), i = 1, . . . , n.

Thus each dui is by definition a linear form on Tp, or a cotan-gent vector (covector for short). We are therefore working with dualbases

(∂∂ui

)for vectors, and (dui) for covectors. The pairing as in

(19.5.1) gives ⟨∂

∂ui, duj

⟩= duj

(∂

∂ui

)= δji , (20.1.1)

where δji is the Kronecker delta: δji = 1 if i = j and δji = 0 if i 6= j.

Example 20.1.5. Examples of 1-forms in the plane are dx, dy, dr, rdr,dθ.

20.2. Constructing bilinear forms out of 1-forms

Recall that the polarisation formula allows one to reconstruct asymmetric bilinear form B, from the quadratic form Q(v) = B(v, v),at least if the characteristic is not 2:

B(v, w) =1

4(Q(v + w) −Q(v − w)). (20.2.1)

Similarly, one can construct bilinear forms out of the 1-forms dui,as follows. Consider a quadratic form defined by a linear combinationof the rank-1 quadratic forms (dui)2. Polarizing the quadratic form,one obtains a bilinear form on the tangent space Tp.

20.3. First fundamental form

Let M be a surface, and p ∈M a point on the surface.

Definition 20.3.1. The first fundamental form g on M is a sym-metric positive definite bilinear form on the tangent space Tp = TpMat p:

g : Tp × Tp → R,

defined for all p ∈M and varying continuously and smoothly in p.


Example 20.3.2. Assume M is imbedded in R3. Euclidean 3-space R3 is equipped with a standard inner product which we de-note 〈 , 〉R3 . Its restriction to the tangent plane of M gives a firstfundamental form on M :

g : Tp × Tp → R, g(v, w) := 〈v, w〉R3 .

The first fundamental form is traditionally expressed by a matrixof coefficients called metric coefficients gij, with respect to coordi-nates (u1, u2) on the surface.

Thus, if a surface is imbedded in Euclidean space R3, the metricis obtained by restricting the Euclidean inner product to the tangentplane of the surface, as follows.

Definition 20.3.3. The metric coefficients gij are given by theinner product of the i-th and the j-th vector in the basis:

gij =

⟨∂x

∂ui,∂x

∂uj

⟩

R3

.

In particular, the diagonal coefficient satisfies

gii =

∥∥∥∥∂x

∂ui

∥∥∥∥2

,

namely it is the square length of the i-th basis vector ∂x∂ui

.

20.4. Expressing the metric in terms of 1-forms

In the case of surfaces, at every point p = (u1, u2), we have themetric coefficients gij = gij(u

1, u2). Each metric coefficient is thus afunction of two variables.

Consider the case when the matrix is diagonal. This can always beachieved at a point by a change of coordinates. If this is true everywherein a coordinate chart, such coordinates are called isothermal. In thenotation developed in Section 20.2, we can write the first fundamentalform in isothermal coordinats as follows:

g = g11(u1, u2)(du1)2 + g22(u

1, u2)(du2)2. (20.4.1)

For example, if the metric coefficients form an identity matrix, weobtain the standard flat metric

g =(du1)2

+(du2)2, (20.4.2)

where the interior superscript denotes an index, while exterior super-script denotes the squaring operation.

20.6. SURFACE OF REVOLUTION 167

Sometimes it is convenient to simplify notation by replacing thecoordinates (u1, u2) simply by (x, y). Then the flat metric (20.4.2) willappear simply as

dx2 + dy2.

20.5. Hyperbolic metric

The hyperbolic metric can expressed by the bilinear form

1

y2(dx2 + dy2).

Note that this expression is undefined for y = 0. Traditionally oneconsiders the upper half plane defined by the condition y > 0. Thehyperbolic metric in the upper half plane is a complete metric. ItsGaussian curvature K = K(x, y) satisfies K = −1 at every pointfollows from the Laplace-Beltrami operator form of the formula forthe Gaussian curvature (see Section 19.4), and results from the factthat the second derivative of log y is − 1

y2.

20.6. Surface of revolution

In the previous sections, we have outlined a formalism for describinga metric on an arbitrary surface, in coordinates denoted (u1, u2). Inthe special case of a surface of revolution, it is customary to use thenotation u1 = θ and u2 = ϕ. Then the metric (20.4.1) can be writtenas

g = g11(θ, ϕ)(dθ)2 + g22(θ, ϕ)(dϕ)2.

The starting point in the construction of a surface of revolution in R3

is a curve C in the xz-plane, parametrized by a pair of functions x =f(ϕ), z = g(ϕ). We will assume that f(ϕ) > 0. The surface of revolu-tion (around the z-axis) defined by C is parametrized as follows:

x(θ, ϕ) = (f(ϕ) cos θ, f(ϕ) sin θ, g(ϕ)).

The condition f(ϕ) > 0 ensures that the resulting surface is imbeddedin R3. It will be an imbedded torus if the original curve C itself isa Jordan curve. The pair of functions (f(ϕ), g(ϕ)) gives an arclengthparametrisation of the curve if

(df

dϕ

)2

+

(dg

dϕ

)2

= 1.


Proposition 20.6.1. Assume (f(ϕ), g(ϕ)) gives an arclength para-metrisation of the curve. Then in the notation of Section 20.4, themetric g of the surface of revolution is

g = f 2(ϕ)dθ2 + dϕ2.

Example 20.6.2. Setting f(ϕ) = sinϕ and g(ϕ) = cosϕ, we obtaina parametrisation of the sphere S2 in spherical coordinates:

gS2 = sin2 ϕdθ2 + dϕ2.

Proof of Proposition 20.6.1. To calculate the first fundamen-tal form of a surface of revolution, note that

x1 =∂x

∂θ= (−f sin θ, f cos θ, 0),

while

x2 =∂x

∂ϕ=

(df

dϕcos θ,

df

dϕsin θ,

dg

dϕ

),

so that we have g11 = f 2 sin2 θ + f 2 cos2 θ = f 2, while

g22 =

(df

dϕ

)2

(cos2 θ + sin2 θ) +

(dg

dϕ

)2

=

(df

dϕ

)2

+

(dg

dϕ

)2

and

g12 = −f dfdϕ

sin θ cos θ + fdf

dϕcos θ sin θ = 0.

Thus we obtain the first fundamental form

(gij) =

(f 2 0

0(dfdϕ

)2+(dgdϕ

)2), (20.6.1)

completing the proof.

In fact, equation (20.6.1) has the following stronger consequence.

Proposition 20.6.3. Assume (f, g) is a regular parametrisation ofthe curve. In the notation of Section 20.4, the metric g of the surfaceof revolution is

g = f 2dθ2 +

((df

dϕ

)2

+

(dg

dϕ

)2)dϕ2.

In Section 20.10, we will express the metric of a surface of revolutionby a first fundamental form expressed by a scalar matrix, after anappropriate change of coordinates.

20.7. COORDINATE CHANGE 169

20.7. Coordinate change

In this section we will work with coordinate charts in the Euclideanplane. Given a metric in one coordinate chart, we would like to under-stand how the metric coefficients tranform under change of coordinates.Consider a change from a coordinate chart

(ui), i = 1, 2,

to another coordinate chart, denoted

(vα), α = 1, 2.

In the overlap of the two domains, the coordinates can be expressed interms of each other, e.g. v = v(u). Denote by gij the metric coefficientswith respect to the chart (ui), and by gαβ the metric coefficients withrespect to the chart (vα).

Definition 20.7.1 (Einstein summation convention). The rule isthat whenever a product contains a symbol with a lower index andanother symbol with the same upper index, take summation over thisrepeated index.

Proposition 20.7.2. Under a coordinate change, the metric coef-ficients transform as follows:

gαβ = gij∂ui

∂vα∂uj

∂vβ.

Proof. In the case of a metric induced by a Euclidean imbedding

x : R2 → R3,

defined by x = x(u) = x(u1, u2), we obtain a new parametrisation

y(v) = x(u(v)).

Applying chain rule, we obtain

gαβ =

⟨∂y

∂vα,∂y

∂vβ

⟩

=

⟨∂x

∂ui∂ui

∂vα,∂x

∂uj∂uj

∂vβ

⟩

=∂ui

∂vα∂uj

∂vβ

⟨∂x

∂ui,∂x

∂uj

⟩

= gij∂ui

∂vα∂uj

∂vβ,

completing the proof.


20.8. Conformal equivalence

Definition 20.8.1. Two metrics, g = gijduiduj and h = hijdu

iduj,on a surface M are called conformally equivalent , or conformal forshort, if there exists a function f = f(u1, u2) > 0 such that

g = f 2h,

in other words,

gij = f 2hij ∀i, j. (20.8.1)

Definition 20.8.2. The function f above is called the conformalfactor (note that sometimes it is more convenient to refer, instead, tothe function λ = f 2 as the conformal factor).

Remark 20.8.3. Note that the length of every vector at a pointwith coordinates (u1, u2) is multiplied precisely by f(u1, u2). Morespeficially, a vector v = vi ∂

∂uiwhich is a unit vector for the metric h, is

“stretched” by a factor of f , i.e. its length with respect to g equals f .Indeed, the new length of v is

√g(v, v) = g

(vi

∂

∂ui, vj

∂

∂uj

) 1

2

=

(g

(∂

∂ui,∂

∂uj

)vivj

) 1

2

=√gijvivj

=√f 2hijvivj

= f√hijvivj

= f.

20.9. Uniformisation theorem for tori

Definition 20.9.1. An equivalence class of metrics on M confor-mal to each other is called a conformal structure (mivneh) on M .

Theorem 20.9.2. Locally, every metric on a surface can be writtenas f(x, y)2(dx2 + dy2) with respect to suitable coordinates (x, y).

There is a global version of this theorem for tori. The uniformisationtheorem in the genus 1 case can be formulated as follows.

Recall that by Pythagoras’ theorem, the square-length of a vectorin the plane is the sum of the squares of its components, or

ds2 = dx2 + dy2,

20.10. ISOTHERMAL COORDINATES 171

where dx and dy measure the components and ds is the arclength inthe plane.

Definition 20.9.3. A lattice L ⊆ R2 a subgroup isomorphic to Z2

which is not included in any line in R2.

Definition 20.9.4. A function f(x, y) on R2 is called L-periodic iff(x+ ℓ) = f(x) for all ℓ ∈ L.

Definition 20.9.5. A flat torus T2 is a quotient R2 /L, where L isa lattice in the plane.

Theorem 20.9.6 (Uniformisation theorem). For every metric g onthe 2-torus T2, there exists a lattice L ⊆ R2 and a positive L-periodicfunction f(x, y) on R2 such that the torus (T2, g) is isometric to

(R2 /L, f 2ds2

), (20.9.1)

where ds2 = dx2 + dy2 is the standard flat metric of R2.

More generally, there is also a global form of the theorem in arbi-trary genus, which can be stated in several ways. One of such waysinvolves Gaussian curvature.

Theorem 20.9.7 (uniformisation). Every metric on a connected(kashir) surface is conformally equivalent to a metric of constant Gauss-ian curvature.

From the complex analytic viewpoint, the uniformisation theoremstates that every Riemann surface is covered by either the sphere, theplane, or the upper halfplane. Thus no notion of curvature is neededfor the statement of the uniformisation theorem. However, from thedifferential geometric point of view, what is relevant is that every con-formal class of metrics contains a metric of constant Gaussian curva-ture. See [?] for a lively account of the history of the uniformisationtheorem.

20.10. Isothermal coordinates

Definition 20.10.1. Coordinates (u1, u2) on a surface M are calledisothermal if the matrix (gij) of the first fundamental form of M , withrespect to (u1, u2), is a scalar matrix at every point of M .

The following lemma expresses the metric of a surface of revolutionin isothermal coordinates.


Theorem 20.10.2. Suppose (f(ϕ), g(ϕ)), where f(ϕ) > 0, is anarclength parametrisation of the generating curve1 of a surface of rev-olution. Then the change of variable

ψ =

∫1

f(ϕ)dϕ,

produces a new parametrisation (in terms of variables θ, ψ), with respectto which the first fundamental form is given by a scalar matrix (gij) =(f 2δij), i.e. the metric is

f 2(ϕ(ψ))(dθ2 + dψ2).

In other words, we obtain an explicit conformal equivalence be-tween the metric on the surface of revolution and the standard flatmetric on the quotient of the (θ, ψ) plane. Such coordinates are re-ferred to as “isothermal coordinates” in the literature. The existenceof such a parametrisation is predicted by the uniformisation theorem(see Theorem 20.9.6) in the case of a general surface.

Proof. Let ϕ = ϕ(ψ). By chain rule, dfdψ

= dfdϕ

dϕdψ

. Now consider

again the first fundamental form of Proposition 20.6.3. To impose the

condition g11 = g22, we need to solve the equation f 2 =(dfdψ

)2+(dgdψ

)2,

or

f 2 =

((df

dϕ

)2

+

(dg

dϕ

)2)(

dϕ

dψ

)2

.

In the case when the generating curve is parametrized by arclength, weare therefore reduced to the equation f = dϕ

dψ, or

ψ =

∫dϕ

f(ϕ).

Thus we obtain a parametrisation of the surface of revolution in co-ordinates (θ, ψ), such that the matrix of metric coefficients is a scalarmatrix.

20.11. Tori of revolution

We will compute the conformal parameter of the specific torus ofrevolution (x − a)2 + y2 = b2, where b < a. Note that the latticesobtained in the previous section are all rectangular, with θ varying

from 0 to 2π and ψ varying from 0 to∫ L0

dϕf(ϕ)

. The lemma of the

previous section has the following corollary.

1This curve does not have to be closed. We will specialize to Jordan curves inSection 20.11.

20.11. TORI OF REVOLUTION 173

Corollary 20.11.1. Consider a torus of revolution in R3 formedby rotating a Jordan curve of length L > 0, with unit speed param-etisation (f(ϕ), g(ϕ)) where ϕ ∈ [0, L]. Then the torus is conformallyequivalent to a flat torus R2 /Lc,d, defined by a rectangular lattice

Lc,d = cZ⊕dZ,where c = 2π and d =

∫ L0

dϕf(ϕ)

, while the metric expressed in coordi-

nates (θ, ψ) isf 2(ϕ(ψ))(dθ2 + dψ2),

for the change of coordinate ψ =∫

dϕf(ϕ)

.

CHAPTER 21

Conformal parameter

21.1. Conformal parameter τ of standard tori of revolution

Figure 21.1.1. Torus: lattice (left) and imbedding (right)

Definition 21.1.1. Every flat torus C/L is similar to the torusspanned by τ ∈ C and 1 ∈ C, where τ is in the standard fundamentaldomain D = z = x + iy ∈ C : |x| ≤ 1

2, y > 0, |z| ≥ 1. The

parameter τ is called the conformal parameter of the torus.

In the previous section we studied the torus of revolution gener-ated by a Jordan curve of length L with arclength parametrisation(f(ϕ), g(ϕ)). We showed that the conformal class of the torus containsthe flat rectangular torus, whose lattice Lc,d is spanned by perpendic-

ular vectors of length c = 2π and d =∫ L0

dϕf(ϕ)

. We therefore obtain the

following corollary.

Corollary 21.1.2. The conformal parameter τ of a torus of rev-olution is pure imaginary:

τ = iσ2

of absolute value

σ2 = max

c

d,d

c

≥ 1.

175

176 21. CONFORMAL PARAMETER

Proof. The proof is immediate from the fact that the lattice isrectangular.

21.2. Comformal parameter of standard torus of rev

We would like to compute the conformal parameter τ of the stan-dard tori imbedded in R3. We first modify the parametrisation so asto obtain a generating curve parametrized by arclength:

f(ϕ) = a+ b cosϕ

b, g(ϕ) = b sin

ϕ

b, (21.2.1)

where ϕ ∈ [0, L] with L = 2πb, and b < a.

Theorem 21.2.1. The corresponding flat torus is given by the lat-tice in the (θ, ψ) plane of the form Lc,d = cZ+dZ where c = 2π and

d =2π√

(a/b)2 − 1.

Thus the conformal parameter τ of the flat torus satisfies

τ = imax

(1√

(a/b)2 − 1,√

(a/b)2 − 1

).

Proof. By Corollary 21.1.2, replacing ϕ by ϕ(ψ) produces isother-mal coordinates (θ, ψ) for the torus generated by (21.2.1), where

ψ =

∫dϕ

f(ϕ)=

∫dϕ

a + b cos ϕb

,

and therefore the flat metric is defined by a lattice in the (θ, ψ) planewith c = 2π and

d =

∫ L=2πb

0

dϕ

a+ b cos ϕb

.

Changing the variable to to

t =ϕ

b

we obtain

d =

∫ 2π

0

dt

(a/b) + cos t.

Thus we are evaluating the definite integral

d =

∫ 2π

0

dt

(a/b) + cos t,

21.2. COMFORMAL PARAMETER OF STANDARD TORUS OF REV 177

where a/b > 1. Let α = a/b. We will use the residue theorem. Firstnote that

d =

∫ 2π

0

dt

α + cos t

=

∫dt

α+ Re(eit)

=

∫2dt

2α+ eit + e−it.

The change of variables z = eit yields

dt =−idzz

and along the unit circle we have

d =

∮ −2idz

z(2α + z + z−1)

=

∮ −2idz

z2 + 2αz + 1

=

∮ −2idz

(z − λ1)(z − λ2),

(21.2.2)

where

λ1 = −α +√α2 − 1, λ2 = −α −

√α2 − 1. (21.2.3)

From (21.2.3) we obtain

λ1 − λ2 = 2√α2 − 1. (21.2.4)

The root λ2 is outside the unit circle. Thus z = λ1 is the only singu-larity of the meromorphic function under the integral sign in (21.2.2)inside the unit circle. Hence we need the residue at λ1 to apply theresidue theorem. The residue at λ1 is obtained by multiplying throughby (z − λ1) and then evaluating at z = λ1. We therefore obtainfrom (21.2.4)

Resλ1 =−2i

λ1 − λ2=

−2i

2√α2 − 1

=−i√α2 − 1

.

The integral is determined by the residue theorem in terms of theresidue at the pole z = λ1. Therefore the lattice parameter d fromCorollary 21.1.2 equals

d =(2πiResλ1

)=

2π√(α)2 − 1

,

where α = a/b, proving the theorem.


21.3. Curvature expressed in terms of θ(s)

In Section 19.4 we dealt with the Gaussian curvature of surfaces.In the present section we treat a different notion of curvature, that ofa differentiable curve in the plane.

Let s be an arclength parameter of a plane curve α(s). Using theidentification R2 = C, the tangent vector v(s) = α′(s) can be expressedas

v(s) = eiθ(s),

where θ(s) is the angle measured counterclockwise between the posi-tive x-axis and the vector v(s).

The (geodesic) curvature kα(s) is by definition the function kα(s) =|α′′(s)|. Note that one has kα(s) ≥ 0 by definition.

Theorem 21.3.1. We have the following expression for the curva-ture kα of the curve α:

kα(s) =

∣∣∣∣dθ

ds

∣∣∣∣ . (21.3.1)

Proof. Recall that v(s) = eiθ(s). By chain rule, we have

dv

ds= ieiθ(s)

dθ

ds,

and therefore the curvature kα(s) satisfies

kα(s) = |α′′(s)| =

∣∣∣∣dv

ds

∣∣∣∣ =dθ

ds,

since |i| = 1.

Corollary 21.3.2. Let α(t) = (x(t), y(t)) be an arbitrary parametri-sation (not necessarily arclength) of a curve. We have the followingexpression for the curvature kα:

kα =|x′y′′ − y′x′′|

|α′|3 (21.3.2)

Proof. With respect to an arbitrary parameter t, the componentsof the tangent vector v(t) = α′(t) are x′(t) and y′(t). Hence we have

tan θ =y′

x′. (21.3.3)

Differentiating (21.3.3) with respect to t, we obtain

1

cos2 θ

dθ

dt=x′y′′ − x′′y′

x′2. (21.3.4)

21.4. TOTAL CURVATURE OF A CONVEX JORDAN CURVE 179

Meanwhile

x′ =dx

dt=dx

ds

ds

dt= cos θ

ds

dtand

dθ

dt=dθ

ds

ds

dt. (21.3.5)

Furthermore, dtds

= |v(t)| = |α′(t)| = |dαdt|. Solving (21.3.5) for dθ

ds, we

obtain from (21.3.4) that

dθ

ds=

|x′y′′ − y′x′′||α′|3

and by (21.3.1) we obtain (21.3.2). This is equivalent to the calcula-tion (??).

21.4. Total curvature of a convex Jordan curve

The result on the total curvature1 of a Jordan curve is of interest inits own right. Furthermore, the result serves to motivate an analogousstatement of the Gauss–Bonnet theorem for surfaces in Section 21.6.

Definition 21.4.1. The total curvature Tot(C) of a curve C withcounterclockwise arclenth parametrisation α(s) : [a, b] → R2 is theintegral

Tot(C) =

∫ b

a

kα(s)ds. (21.4.1)

When a curve C is closed, i.e, α(a) = α(b), it is more convenient towrite the integral (21.4.1) using the notation of a line integral

Tot(C) =

∮

C

kα(s)ds. (21.4.2)

Theorem 21.4.2. The total curvature of each strictly convex2 Jor-dan curve C with arclength parametrisation α(s) equals 2π, i.e.,

Tot(C) =

∮

C

kα(s)ds = 2π.

Proof. Let α(s) be a unit speed parametrisation so that α(0) bethe lowest point of the curve, and assume the curve is parametrizedcounterclockwise. Let v(s) = α′(s). Then v(0) = ei0 = 1 and θ(0) = 0.

1Akmumiyut kolelet2Kmura with a “kuf”.


The function θ = θ(s) is monotone increasing from 0 to 2π. Applyingthe change of variable formula for integration, we obtain

Tot(C) =

∮

C

kα(s)ds =

∮

C

∣∣∣∣dv

ds

∣∣∣∣ ds =

∮

C

dθ

dsds =

∫ 2π

0

dθ = 2π,


Remark 21.4.3. We can also consider the normal vector n(s) tothe curve. The normal vector satisfies n(s) = ei(θ+

π2) = iei(θ) and |dn

ds| =

|dvds| = dθ

ds. Therefore we can also calculate the total curvature as follows:

∮

C

kα(s)ds =

∮

C

∣∣∣∣dn

ds

∣∣∣∣ ds =

∮

C

dθ

dsds =

∫ 2π

0

dθ = 2π,

with n(s) in place of v(s).

Remark 21.4.4. The theorem in fact holds for an arbitrary regularJordan curve, provided we adjust the definition of curvature to allow fornegative sign (in such case θ(s) will not be a monotone function). Wehave dealt only with the convex case in order to simplify the topologicalconsiderations. See further in Section 21.5.

Remark 21.4.5. A similar calculation will yield the Gauss–Bonnettheorem for convex surfaces in Section 21.6.

21.5. Signed curvature kα of Jordan curves in the plane

Let α(s) be an arclength parametrisation of a Jordan curve in theplane. We assume that the curve is parametrized counterclockwise. Asin Section 21.3, a continuous branch of θ(s) can be chosen where θ(s)is the angle measured counterclockwise from the positive x-axis to thetangent vector v(s) = α′(s).

Note that if the Jordan curve is not convex, the function θ(s) willnot be monotone and at certain points its derivative may take negativevalues: θ′(s) < 0. Once we have such a continuous branch, we can

define the signed curvature kα as follows.

Definition 21.5.1. The signed curvature kα(s) of a plane Jordancurve α(s) oriented counterclockwise is defined by setting

kα(s) =dθ

ds.

We have the following generalisation of Theorem 21.4.2 on the totalcurvature of a convex curve.

21.5. SIGNED CURVATURE kα OF JORDAN CURVES IN THE PLANE 181

Theorem 21.5.2. The total signed curvature kα of each Jordancurve C with arclength parametrisation α(s) equals 2π, i.e.,

Tot(C) =

∫ b

a

kα(s)ds = 2π.

Proof. We exploit the continuous branch θ(s) as in Theorem 21.4.2,but without the absolute value signs on the derivative of θ(s). Applyingthe change of variable formula for integration, we obtain

Tot(C) =

∮

C

kα(s)ds =

∮

C

dθ

dsds =

∫ 2π

0

dθ = 2π,


Remark 21.5.3. For nonconvex Jordan curves, the Gauss map tothe circle will not be one-to-one, but will still have an algebraic de-gree one. This phenomenon has a far-reaching analogue for imbeddedsurfaces where the algebraic degree is proportional to the Euler char-acteristic.

Definition 21.5.4. The algebraic degree of a map θ : C → S1 at apoint z ∈ S1 is defined to be the sum

∑

θ−1(z)

sign

(dθ

ds

),

where the sum runs over all points in the inverse image of z.

For Jordan curves (i.e., imbedded curves), the algebraic degree isalways 1. This is the topological ingredient behind the theorem on thetotal curvature of a Jordan curve.

21.5.1. Tangent vector defined by a curve. This subsectioncan be skipped.

Let M be a manifold. A tangent vector X at a point p ∈ M wasdefined as a linear functional X : Dp → R on the space Dp of smoothfunctions defined in a neighborhood of p, where the 1-form is requiredto satisfy Leibniz’s rule (19.7.1). The tangent vectors at p form thetangent space Tp = TpM at p, which is an n-dimensional vector space.If (u1, . . . , un) are coordinates centered at p, then we can write down thefollowing basis for Tp:

(∂∂ui

), i = 1, . . . , n. If α(t) is a smooth curve

through p satisfying α(0) = p, the formula Xf = ddt

∣∣t=0

f α(t) =ddt

∣∣t=0

f(α(t)) for all f ∈ Dp, defines an initial vector3 X = α′(0). Thedirectional derivative only depends on the values of f along a curve

3hat’chalati


Figure 21.5.1. Three points on a curve with thesame tangent direction, but the middle one correspondsto θ′(s) < 0.

with initial vector X , by elementary calculus. Thus we can think of atangent vector as being defined by choosing a suitable smooth curve.

Example 21.5.5. Let p be the point at the center of a coordi-nate system with coordinates (u1, . . . , un). Consider the curve α1(t)defined in coordinates by the n-tuple (t, 0, . . . , 0). Similarly considerthe curves αi, where i = 1, . . . , n, so that αn(t) = (0, . . . , 0, t). Thenwe have ∂

∂ui= α′

i(0), i = 1, . . . , n.

The advantage of the more abstract definition of a tangent vectoris its independence of any choices. This will allow us to define thetangent bundle by considering all tangent vectors simultaneously.

21.5.2. Rotation index of a closed curve in the plane. Thissubsection is to be skipped.

Given a regular closed plane curve α(s) : [0, L] → R2, of length L,a continuous branch of

θ(s) =1

ilogα′(s)

can be chosen even if the curve is not simple, obtaining a map

θ : [0, L] → R .

21.6. COORDINATE PATCH DEFINED BY FAMILY OF NORMAL VECTORS183

As in Section 21.5, we can assume that θ(0) = 0. Then θ(L) is neces-sarily an integer multiple of 2π. See Millman & Parker [?, p. 55].

Definition 21.5.6. The rotation index ια of a closed unit speedplane curve α(s) is the integer

ια =θ(L) − θ(0)

2π.

Example 21.5.7. The rotation index of a figure-8 curve is 0.

Theorem 21.5.8. Let α(s) be an arclength parametrisation of ageometric curve C. Then the rotation index is related to the total signed

curvature Tot(C) as follows:

ια =Tot(C)

2π. (21.5.1)

The proof (by change of variable in integration) is the same as inSection 21.5. An analogous relation between the Euler characteristicof a surface and its total Gaussian curvature appears in Section 21.7.

21.6. Coordinate patch defined by family of normal vectors

The Gauss–Bonnet theorem for surfaces is to a certain extent anal-ogous to the theorem on the total curvature of a plane curve (Theo-rem 21.5.2 and Theorem 21.5.8). In both cases, an integral of curvatureturns out to have topological significance.

In the notation of Section 19.1, we have the metric coefficients (gij)of the parametrisation x(u1, u2) of the surface Σ in a neighborhood ofa point p ∈ Σ.

The Gauss mapG : M → S2

is given by the unit normal vector n(u1, u2).

Remark 21.6.1. The Gauss map defines a coordinate patch on thesphere S2 near the point G(p). In other words, we obtain a parametri-sation of a neighborhood of the point G(p) ∈ S2.

Definition 21.6.2. In the new coordinate patch, we obtain themetric coefficients (gα,β) of the sphere S2 relative to the parametrisa-tion n(u1, u2).

Here each coefficient gαβ = gαβ(u1, u2) is a function of u1 and u2,as well. These metric coefficients are defined as usual by taking thescalar products of hte first partial derivatives of the parametrisation,this time given by n(u1, u2):

gαβ(u1, u2) = 〈nα, nβ〉R3 , α, β = 1, 2.


Lemma 21.6.3. We have the following identity relating the metriccoefficients of the metric (gαβ) on the sphere S2, on the one hand, andthe metric coefficients (gij) of the metric on M :

det(gαβ) =(K(u1, u2)

)2det(gij)

where K = K(u1, u2) is the Gaussian curvature of the surface M .

Proof. Let L = (Lij) be the matrix of the Weingarten map withrespect to the basis (x1, x2). By definition of curvature we have K =det(L). Recall that the coefficients Lij of the Weingarten map aredefined by

nα = Liαxi = xiLiα. (21.6.1)

Consider the 3 × 2-matrices A = [x1 x2] and B = [n1 n2]. Then(21.6.1) implies by definition that

B = AL.

Therefore the Gram matrices satisfy

Gram(n1, n2) = BtB = (AL)tAL = LtAtAL

= Lt Gram(x1, x2) L.(21.6.2)

Now we apply the determinant to both sides. We have

det(Gram(x1, x2)) = det(gij).

Similarly,

det(Gram(n1, n2)) = det(gαβ).

Therefore from (21.6.2) we obtain

det(gαβ) = det(gij)det(L)2,

and the theorem follows from the identity K = det(L). 4

4Note that by the Cauchy–Binet formula ??, the desired identity is equivalentto the formula

|n1 × n2| = |det(Lij)| |x1 × x2|. (21.6.3)

An alternative proof of (21.6.3) is to observe that a linear map multiplies the areaof parallelograms by (the absolute value of) its determinant. The Weingarten mapsends each vector xi to ni and therefore it sends the parallelogram spanned by x1

and x2 to the parallelogram spanned by n1 and n2. Therefore it modifies the areaof the parallegram by |det(Li

j)|.

21.7. TOTAL GAUSSIAN CURVATURE AND GAUSS–BONNET THEOREM 185

21.7. Total Gaussian curvature and Gauss–Bonnet theorem

Let Σ ⊆ R3 be an imbedded surface. Recall that the area ele-ment dA of a surface M is given locally in coordinates (u1, u2) by

dAM =√

det(gij)du1du2.

Here the factor√

det(gij) should be thought of as the area of the par-

allelogram spanned by ∂∂u1

and ∂∂u2

. For example, in polar coordinatesin the plane, the determinant is r2 and therefore we get dA = rdrdθ.On the unit sphere we have

dA = sinϕdθ dϕ.

Let M be a convex closed surface in R3, i.e., an imbedded closedsurface of positive Gaussian curvature (a typical example is an ellip-soid).

Theorem 21.7.1 (Special case of Gauss-Bonnet). The curvatureintegral of M satisfies∫

M

K(p) dAM = 4π = 2πχ(S2),

where K is the Gaussian curvature function defined at every point p ofthe surface.

Proof. The convexity of the surface guarantees that the map n isone-to-one.

We examine the integrand K dAM in a coordinate chart (u1, u2)where it can be written as K(u1, u2) dAM . By Lemma 21.6.3, we have

K(u1, u2) dAM = K(u1, u2)√

det(gij)du1du2 =

√det(gαβ)du1du2 = dAS2.

Thus, the expression K dAM coincides with the area element dAS2 ofthe unit sphere S2 in every coordinate chart. Hence we can write∫

M

K dAM =

∫

S2

dAS2 = 4π,


CHAPTER 22

Addendum on a Bernoullian framework

22.1. Ultrapower construction

We briefly describe a construction that yields a nonstandard ex-tension satisfying countable saturation. For every n ∈ N let Sn(X)be the set of all sequences 〈xi〉i∈N such that xi ∈ Vn(X) for all i, andlet S(X) =

⋃n∈N Sn(X). Let U ⊆ P(N) be an ultrafilter on N such

that every cofinite subset of N is in U . Define an equivalence relation ∼on S(X) as follows: 〈xi〉i∈N ∼ 〈yi〉i∈N if the set i ∈ N : xi = yi is in U .We denote the equivalence class of 〈xi〉i∈N by [〈xi〉i∈N]. We define a rela-tion ε on S(X) as follows: [〈xi〉i∈N] ε [〈yi〉i∈N] if the set i ∈ N : xi ∈ yiis in U .

A16. Let S ′(X) denote the set of equivalence classes of elementsof S(X), and let ∗X denote the set of equivalence classes of elementsof S0(X). We recursively define a function T : S ′(X) → V (∗X), asfollows: For a = 〈xi〉i∈N ∈ S0(X) simply T ([a]) = [a]. For n > 0and 〈Ai〉i∈N ∈ Sn(X) − Sn−1(X), let T ([〈Ai〉i∈N]) = T ([〈xi〉i∈N]) :[〈xi〉i∈N] ε [〈Ai〉i∈N] , which is an element of Vn(∗X). For A ∈ S(X)let kA denote the sequence with constant value A, then finally, let ∗ :V (X) → V (∗X) be defined by ∗A = T ([kA]). In this construction, theinternal objects are precisely those appearing in the image of T .

22.2. Saturation and topology

In Theorem we further assume that our nonstandard extensionsatisfies countable saturation, which is the following property: When-ever Ann∈N is a countable collection of internal sets such that

⋂0≤k≤nAk 6=

∅ for every n ∈ N, then⋂n∈NAn 6= ∅. Any nonstandard extension

constructed via the ultrapower construction described below, satisfiescountable saturation.

187

CHAPTER 23

More on Gauss-Bonnet

23.1. Second proof of Gauss–Bonnet theorem

We will use the existence of global coordinates (with singularitiesonly at the north and south poles) on the sphere to write down a moreconcrete version of the calculation.

Choose global coordinates u1 = θ, u2 = ϕ on M by precomposingthe spherical coordinates by the inverse G−1 of the Gauss map G :M → S2. Here we can dispense with local coordinate charts andcompute the Gauss–Bonnet integral in these global coordinates. Usingthe relation K(u1, u2)

√det(gij) =

√det(gαβ) we obtain as before

∫

M

K(u1, u2) dAM =

∫

M

K(θ, ϕ) dAM

=

∫∫

M

K(θ, ϕ)√

det(gij)dθdϕ

=

∫∫

S2

√det(gαβ)dθdϕ =

∫∫

S2

dAS2.

Here each of the double integrals∫∫

can be replaced by the iteratedintegral ∫ π

0

dϕ

(∫ 2π

0

dθ(. . .)

),

where the appropriate integrand is placed where the dots (. . .) are.

23.2. Euler characteristic

The Euler characteristic of a closed imbedded surface in Euclidean3-space can be defined via the total Gaussian curvature.

Definition 23.2.1. The Euler characteristic χ(M) of a surface Mis defined by the relation

2πχ(M) =

∫

M

K(p)dAM , (23.2.1)

where K(p) is the Gaussian curvature at the point p ∈M .

189

190 23. MORE ON GAUSS-BONNET

The relation (23.2.1) is similar to the line intergal expression forthe rotation index in formula ??. To show that this definition of theEuler characteristic agrees with the usual one, it is necessary to usethe notion of algebraic degree for maps between surfaces, similar tothe algebraic degree of a self-map of a circle.

23.3. Exercise on index notation

Theorem 23.3.1. Every 2×2 matrix A = (aij) satisfies the identity

aikakj + q δij = akka

ij , (23.3.1)

where q = det(A).

Proof. We will use the Cayley-Hamilton theorem which assertsthat pA(A) = 0 where pA(λ) is the characteristic polynomial of A. Inthe rank 2 case, we have pA(λ) = λ2 − (trA)λ + det(A), and thereforewe obtain

A2 − (trA)A+ det(A)I = 0,

which in index notation gives aikakj−akkaij+q δij = 0. This is equivalent

to (23.3.1).

For a matrix A of size 3×3, the characteristic polynomial pA(λ) hasthe form pA(λ) = λ3−Tr(A)λ2 + s(A)λ− q(A)λ0. Here q(A) = det(A)and s(A) = λ1λ2 + λ1λ3 + λ2λ3, where λi are the eigenvalues of A. ByCayley-Hamilton theorem, we have

pA(A) = 0. (23.3.2)

Exercise 23.3.2. Express the equation (23.3.2) in index notation.

Exercise 23.3.3. A matrix A is called idempotent if A2 = A. Writethe idempotency condition in indices with Einstein summation conven-tion (without Σ’s).

Exercise 23.3.4. Matrices A and B are similar if there exists aninvertible matrix P such that AP − PB = 0. Write the similaritycondition in indices, as the vanishing of the (i, j)th coefficient of thedifference AP − PB.

CHAPTER 24

Bundles

24.1. Trivial bundle (E,B, π), function as a section (mikta)

Before introducing the notion of a tangent bundle, let us make someelementary remarks about graphs. Let M be a surface (more generally,a manifold). Given a function

f : M → R

we can define its graph in M × R as the collection of pairs

(p, f(p)) ∈ M × R | p ∈M.

We view M × R as a trivial bundle over M , denoted π:

π : M × R →M.

We can also consider the section s : M → M×R defined by the obviousformula

s(p) = (p, f(p)).

The section thus defined clearly satisfies the identity

π s = IdM ,

in other words π(s(p)) = p for all p ∈ M . We will now develop thelanguage of bundles and sections in the context of the tangent andcotangent bundles of M .

The total space of a bundle is typically denoted E, the base isdenoted B, and the surjective bundle map is denoted π:

E

πB

The whole package is referred to as the triple

(E,B, π).

191

192 24. BUNDLES

24.2. Mobius band as a first example of a nontrivial bundle

The tangent bundle of a circle is a cylinder S1 × R, which can bethought of as a trivial bundle over the circle.

A first example of a non-trivial bundle is provided by the Mobiusband (tabaat mobius, rtzuat mobius). The latter can be thought of asa non-trivial bundle over the circle.

Consider the parametrized circle c(θ) = 10(cos θ, sin θ, 0) with nor-mal vector

n(θ) = (cos θ, sin θ, 0).

The circle can be imbedded in a parametrized Mobius band in R3. Thisis done by adding the endpoints of a “rotating” interval at every point:

µ(θ) = c(θ) ±(cos θ

2n(θ) + sin θ

2e3).

Now “filling in” the pair of points ±(cos θ

2n(θ) + sin θ

2e3)

by the in-terval joining them, we obtain a Mobius band

MB(θ, s) = c(θ) + s(cos θ

2n(θ) + sin θ

2e3),

where −1 ≤ s ≤ 1. The projection back to the circle c defines anon-trivial interval bundle over the circle.

24.3. Tangent bundle

The tangent bundle TM of a surface (or more generally mani-fold) M is by definition the collection of pairs

TM := (p,X)|p ∈M,X ∈ Tp .We have a natural projection

π : TM → M,

defined by (p,X) 7→ p, “forgetting” the second component. Clearly,the inverse image of a point is precisely the tangent plane Tp at thatpoint:

π−1(p) = Tp.

A vector field is a “section” of the tangent bundle, analogous to thesplitting map for a homomorphism. Thus a section s : M → TM is amap satisfying

π s = IdM .

In other words, if s is a section, we have π(s(p)) = p for all p ∈M .We first complete the preliminary material on bundles.

24.5. COTANGENT BUNDLE 193

24.4. Hairy ball theorem

In Section 24.2, we discussed the simplest example of a nontrivialbundle, given by the Mobius band. Another example of a nontrivialbundle is provided by the following famous result.

Theorem 24.4.1 (The hairy ball theorem). There is no nonvan-ishing continuous tangent vector field on a sphere.

Thus, if f is a continuous function that assigns a vector in R3 toevery point p on a sphere, such that f(p) is always tangent to the sphereat p, then there is at least one p such that f(p) = 0. The theorem wasfirst stated by Henri Poincare in the late 19th century.

The theorem is famously stated as “you can’t comb a hairy ballflat”, or sometimes, “you can’t comb the hair of a coconut”. It wasfirst proved in 1912 by Brouwer.

24.5. Cotangent bundle

In Section 24.3, we defined the tangent bundle of M . We now definethe dual object, the cotangent bundle.

Definition 24.5.1. The cotangent bundle of M is the collection ofpairs

T ∗M :=

(p, η)|p ∈M, η ∈ T ∗p

.

We have a natural projection (to the “first factor”)

π : T ∗M →M,

so that we obtain the usual triple (E,B, π) = T ∗M,M, π).Given a coordinate chart (u1, . . . , un) inM , we obtain a basis du1, . . . , dun)

at every point in the chart. Every cotangent vector ω at a point withcoordinates (u1, . . . , un) is a linear combination

ω = ωidui = ω1du

1 + ω2du2 + · · · + ωndu

n,

and therefore we can coordinatize T ∗M by the 2n-tuple

(u1, . . . , un, ω1, . . . , ωn).

With respect to this coordinate chart in T ∗M , the projection π :T ∗M → M “forgets” the last n coordinates.

Definition 24.5.2. A differential 1-form ω on M is by definitiona section of the cotangent bundle of M .

Thus at every point p ∈M , a differential form ω gives a linear form

ωp : TpM → R .

194 24. BUNDLES

In coordinates, a differential 1-form ω is given by

ω = ωi(u1, . . . , un)dui,

where the ωi are functions of n variables.

24.6. Exterior derivative of a function

Given a smooth function f ∈ C∞(M), we define a differential 1-form, denoted df , on M , in terms of the directional derivative by setting

df(X) = Xf,

where Xf is the directional derivative of the function f in the di-rection of the vector X . This formula applies at every point of M .Denoting ω = df , we see that

ωp(Xp) = Xpf ∈ R

where Xp is the value of the vector field X at the point p.

Theorem 24.6.1. In coordinates (u1, . . . , un), we have

df =∂f

∂uidui, (24.6.1)

with Einstein summation convention.

The proof is a restatement of chain rule in several variables.

Definition 24.6.2. The collection of all differential 1-forms on Mis denoted

Ω1(M).

Note that Ω1(M) is an infinite-dimensional vector space. The space Ω1(M)is by definition the space of sections of the cotangent bundle of M . Inthis notation, the exterior derivative is an R-linear map

d : C∞(M) → Ω1(M)

from the space C∞(M) of functions on M to the space Ω1(M) of 1-forms on M .

24.7. Extending gradient and exterior derivative

Given a function f on an inner product space V , one defines thegradient ∇f of f by setting

〈∇f,X〉 = Xf,

for every vector X . Note that we have a canonical isomorphism

: V → V ∗, X 7→ X.

24.8. PLAN FOR BUILDING DE RHAM COHOMOLOGY 195

The isomorphism is defined by setting

X(Y ) = 〈X, Y 〉,for every vector Y ∈ V . The inverse is denoted

♯ : V ∗ → V, ω 7→ ω♯,

for every 1-form ω. If the coordinates are chosen in such a way thatthe basis ( ∂

∂u1, . . . , ∂

∂un) is an orthonormal basis of TpM , then

(∂

∂ui

)= dui,

and

(dui)♯ =∂

∂ui.

Theorem 24.7.1. The exterior derivative of a function f at p ∈Mis related to the gradient of f at p ∈M as follows:

∇f = ♯(df),

at every point p of the surface M .

The proof is immediate in an orthonormal basis.

24.8. Plan for building de Rham cohomology

The construction of de Rham cohomology of a manifold involvesseveral stages. It is helpful to keep these stages in mind as we buildup the relevant mathematical machinery step-by-step. We start withlinear algebra and end with linear algebra.

(1) Linear-algebraic stage: from a vector space V to its exterioralgebra

ΛV = ⊕kΛkV.

(2) Topological stage: from a smooth manifold M to its tangentbundle TM and its cotangent bundle T ∗M .

(3) A vector field as a section of TM .(4) A section of T ∗M is a differential 1-form on M .(5) The exterior bundle ΛM = ⊕kΛ

k(T ∗M) is the bundle of ex-terior algebras parametrized by points of M . This is a finite-dimensional manifold.

(6) A section of Λk(T ∗M) is a differential k-form on M .(7) The space Ωk(M) of sections of Λk(T ∗M) is an infinite-dimensional

vector space: we are back to linear algebra.(8) The exterior derivative d turns Ω∗(M) into a differential graded

algebra.

196 24. BUNDLES

24.9. Exterior algebra, area, determinant, rank

The exterior product or wedge product of vectors is an algebraicconstruction generalizing certain features of the cross product to higherdimensions.

Like the cross product, and the scalar triple product, the exteriorproduct of vectors is used in Euclidean geometry to study areas, vol-umes, and their higher-dimensional analogs.

In linear algebra, the exterior product provides a basis-independentmanner for describing the determinant and the minors of a linear trans-formation, and is fundamentally related to ideas of rank and linearindependence.

The exterior algebra over a vector space V is generated by an oper-ation called exterior product. It is a key building block in the definitionof the algebra of differential forms.

24.10. Formal definition, alternating property

Formally, the exterior algebra is a certain unital associative (chokkibutz) algebra over the field K, including V as a subspace. It isdenoted by

Λ(V )

and its multiplication is also known as the wedge product or the exteriorproduct and is written as

∧

Definition 24.10.1. The wedge product is an associative and bi-linear operation:

∧ : Λ(V ) × Λ(V ) → Λ(V ),

sending

(α, β) 7→ α ∧ β,with the essential feature that it is alternating on V :

v ∧ v = 0 for all v ∈ V. (24.10.1)

This implies in particular

u ∧ v = −v ∧ u (24.10.2)

for all u, v ∈ V , and

v1 ∧ v2 ∧ · · · ∧ vk = 0 (24.10.3)

whenever v1, . . . , vk ∈ V are linearly dependent.

24.12. AREAS IN THE PLANE 197

Remark 24.10.2. Unlike associativity and bilinearity which are re-quired for all elements of the algebra Λ(V ), the last three propertiesare imposed only on the elements of the subspace V of the algebra.The defining property (24.10.1) and property (24.10.3) are equivalent;properties (24.10.1) and (24.10.2) are equivalent unless the character-istic of K is two.

24.11. Example: rank 1

The exterior algebra Λ(R) over the vector space V = R1 is 2-dimensional, and isomorphic to the subalgebra of the algebra of 2 by 2matrices consisting of uppertriangular matrices with a pair of identicaleigenvalues, i.e., matrices (

x y0 x

)

where x, y ∈ R. As a vector space, the algebra is the sum Λ(R) =R ·1 ⊕ R ·e1, where e1 is the matrix

e1 =

(0 10 0

).

The matrix e1 is nilpotent: e21 = 0, as required by (24.10.1). Here V =R e1.

24.12. Areas in the plane

The purpose of this section is to motivate the skewness of the ex-terior product on vectors in V .

The area of a parallelogram can be expressed in terms of the deter-minant of the matrix of coordinates of two of its vertices.

The Cartesian plane R2 is a vector space equipped with a basis.The basis consists of a pair of unit vectors

e1 = (1, 0), e2 = (0, 1).

Suppose that

v = v1e1 + v2e2, w = w1e1 + w2e2

are a pair of given vectors in R2, written in components. There is aunique parallelogram having v and w as two of its sides. The area ofthis parallelogram is given by the standard determinant formula:

A =∣∣det

[v w

]∣∣= |v1w2 − v2w1|.

198 24. BUNDLES

Consider now the exterior product of v and w and exploit the propertiesstipulated above:

v ∧ w = (v1e1 + v2e2) ∧ (w1e1 + w2e2)

= v1w1e1 ∧ e1 + v1w2e1 ∧ e2 + v2w1e2 ∧ e1 + v2w2e2 ∧ e2,

so that

v ∧ w = (v1w2 − v2w1)e1 ∧ e2,

where the first step uses the distributive law for the wedge product, andthe last uses the fact that the wedge product is alternating. Note thatthe coefficient in this last expression is precisely the determinant of thematrix [v w]. The fact that this may be positive or negative has theintuitive meaning that v and w may be oriented in a counterclockwise orclockwise sense as the vertices of the parallelogram they define. Suchan area is called the signed area of the parallelogram: the absolutevalue of the signed area is the ordinary area, and the sign determinesits orientation.

The fact that this coefficient is the signed area is not an accident.In fact, it is relatively easy to see that the exterior product should berelated to the signed area if one tries to axiomatize this area as analgebraic construct.

In detail, if A(v, w) denotes the signed area of the parallelogramdetermined by the pair of vectors v and w, then A must satisfy thefollowing properties:

(1) A(av, bw) = abA(v, w) for any real numbers a and b, sincerescaling either of the sides rescales the area by the sameamount (and reversing the direction of one of the sides reversesthe orientation of the parallelogram).

(2) A(v, v) = 0, since the area of the degenerate (mathemat-ics)—degenerate parallelogram determined by v (i.e., a linesegment) is zero.

(3) A(w, v) = −A(v, w), since interchanging the roles of v and wreverses the orientation of the parallelogram.

(4) A(v + aw,w) = A(v, w), since adding a multiple of w to vaffects neither the base nor the height of the parallelogramand consequently preserves its area.

(5) A(e1, e2) = 1, since the area of the unit square is one.

With the exception of the last property, the wedge product satis-fies the same formal properties as the area. In a certain sense, thewedge product generalizes the final property by allowing the area of aparallelogram to be compared to that of any “standard” chosen paral-lelogram. In other words, the exterior product in two-dimensions is a

24.13. VECTOR AND TRIPLE PRODUCTS 199

basis-independent formulation of area. This axiomatization of areas isdue to Leopold Kronecker and Karl Weierstrass.

24.13. Vector and triple products

For vectors in R3, the exterior algebra is closely related to the vectorproduct and triple product. Using the standard basis e1, e2, e3, thewedge product of a pair of vectors

u = u1e1 + u2e2 + u3e3

and

v = v1e1 + v2e2 + v3e3

is

u∧v = (u1v2−u2v1)(e1∧e2)+(u1v3−u3v1)(e1∧e3)+(u2v3−u3v2)(e2∧e3),

where e1 ∧ e2, e1 ∧ e3, e2 ∧ e3 is the basis for the three-dimensionalspace Λ2(R3). This imitates the usual definition of the vector productof vectors in three dimensions. Bringing in a third vector

w = w1e1 + w2e2 + w3e3,

the wedge product of three vectors is

u∧v∧w = (u1v2w3+u2v3w1+u3v1w2−u1v3w2−u2v1w3−u3v2w1)(e1∧e2∧e3),

where e1∧e2∧e3 is the basis vector for the one-dimensional space Λ3(R3).This imitates the usual definition of the triple product.

The vector product and triple product in three dimensions eachadmit both

(1) geometric and(2) algebraic

interpretations. The vector product u×v can be interpreted as a vectorwhich is perpendicular to both u and v and whose magnitude is equalto the area of the parallelogram determined by the two vectors. Itcan also be interpreted as the vector consisting of the minors of thematrix with columns u and v. The triple product of u, v, and w isgeometrically a (signed) volume. Algebraically, it is the determinant ofthe matrix with columns u, v, and w. The exterior product in three-dimensions allows for similar interpretations. In fact, in the presence ofa positively oriented orthonormal basis, the exterior product generalizesthese notions to higher dimensions.

200 24. BUNDLES

24.14. Anticommutativity of the wedge product

Theorem 24.14.1. The wedge product is anticommutative on ele-ments of V .

Proof. Let x, y ∈ V . Then

0 = (x + y) ∧ (x + y) = x ∧ x + x ∧ y + y ∧ x + y ∧ y = x ∧ y + y ∧ x.

Hence

x ∧ y = −y ∧ x.

More generally, if x1, x2, ..., xk are elements of V , and σ is a permu-tation of the integers [1, ..., k], then

xσ(1) ∧ xσ(2) ∧ · · · ∧ xσ(k) = sgn(σ)x1 ∧ x2 ∧ · · · ∧ xk,

where sgn(σ) is the signature of a permutation—signature of the per-mutation σ. A proof of this can be found in greater generality inBourbaki (1989).

24.15. The k-exterior power; simple (decomposable)multivectors

The k-th exterior power of V , denoted Λk(V ), is the vector subspaceof Λ(V ) spanned by elements of the form

x1 ∧ x2 ∧ · · · ∧ xk, xi ∈ V, i = 1, 2, . . . , k.

Definition 24.15.1. If α ∈ Λk(V ), then α is said to be a k-multivector. If, furthermore, α can be expressed as a wedge productof k elements of V , then α is said to be decomposable, or simple.

Although decomposable multivectors span Λk(V ), not every ele-ment of Λk(V ) is decomposable.

Example 24.15.2. In R4, the following 2-multivector is not decom-posable:

α = e1 ∧ e2 + e3 ∧ e4. (24.15.1)

(This is in fact a symplectic form, since α ∧ α 6= 0. See S. Sternberg[?, III.6].

24.17. RANK OF A MULTIVECTOR 201

24.16. Basis and dimension of exterior algebra

Theorem 24.16.1. If the dimension of V is n and e1, ..., en is abasis of V , then the set

ei1 ∧ ei2 ∧ · · · ∧ eik | 1 ≤ i1 < i2 < · · · < ik ≤ n (24.16.1)

is a basis for Λk(V ).

Proof. Given any wedge product of the form

v1 ∧ · · · ∧ vkthen every vector vj can be written as a linear combination of thebasis vectors ei; using the bilinearity of the wedge product, this canbe expanded to a linear combination of wedge products of those basisvectors. Any wedge product in which the same basis vector appearsmore than once is zero; any wedge product in which the basis vectorsdo not appear in the proper order can be reordered, changing the signwhenever two basis vectors change places.

In general, the resulting coefficients of the basis k-vectors can becomputed as the minors of the matrix that describes the vectors vj interms of the basis ei.

By counting the basis elements, the dimension of Λk(V ) is the bi-nomial coefficient (

n

k

)=

n!

k!(n− 1)!.

In particular, Λk(V ) = 0 for k > n.

Theorem 24.16.2. The dimension of Λ(V ) equals 2n.

Proof. Any element of the exterior algebra can be written as asum of multivectors. Hence, as a vector space the exterior algebra is adirect sum

Λ(V ) = Λ0(V ) ⊕ Λ1(V ) ⊕ Λ2(V ) ⊕ · · · ⊕ Λn(V )

(where by convention Λ0(V ) = K and Λ1(V ) = V ), and thereforeits dimension is equal to the sum of the binomial coefficients, whichis 2n.

24.17. Rank of a multivector

If α ∈ Λk(V ), then it is possible to express α as a linear combinationof decomposable multivectors:

α = α(1) + α(2) + · · · + α(s)

202 24. BUNDLES

where each α(i) is decomposable, say

α(i) = α(i)1 ∧ · · · ∧ α(i)

k , i = 1, 2, . . . , s.

Definition 24.17.1. The rank of the multivector α is the minimalnumber of decomposable multivectors in such an expansion of α.

Rank is particularly important in the study of 2-multivectors.

Theorem 24.17.2. The rank of a 2-multivector α equals half therank of the matrix of coefficients of α in a basis.

Thus if ei is a basis for V, then α can be expressed uniquely as

α =∑

i,j

aijei ∧ ej

where aij = −aji (the matrix of coefficients is skew-symmetric). Therank of α agrees with the rank of the matrix (aij).

For example, the symplectic from (24.15.1) has rank 2, whereas therank of its matrix of coefficients is 4.

Theorem 24.17.3. In characteristic 0, the 2-multivector α hasrank p if and only if the p-fold product satisfies

α ∧ · · · ∧ α︸︷︷︸p

6= 0

andα ∧ · · · ∧ α︸︷︷︸

p+1

= 0.

CHAPTER 25

Multilinear forms; exterior differential complex

25.1. Formal construction of the exterior algebra

The tensor product V ⊗W of two vector spaces V and W over afield K is defined as follows.

Consider the set of ordered pairs in the Cartesian product V ×W .For the purposes of this construction, regard this Cartesian product asa set rather than a vector space.

Definition 25.1.1. The free vector space F on V ×W is the vectorspace in which the elements of V ×W are a basis.

Thus, F (V ×W ) is the collection of all finite linear combinationsof the form

F (V ×W ) =αie(vi,wi)

∣∣ αi ∈ K, (vi, wi) ∈ V ×W,

where we have used the symbol e(v,w) to emphasize that these are takento be linearly independent by definition for distinct pairs

(v, w) ∈ V ×W.

The tensor product arises by defining the following equivalence re-lations in F (V ×W ):

e(v1+v2,w) ∼ e(v1,w) + e(v2,w)

e(v,w1+w2) ∼ e(v,w1) + e(v,w2)

ce(v,w) ∼ e(cv,w)

ce(v,w) ∼ e(v,cw)

where v1, v2 ∈ V , w1, w2 ∈ W , and c ∈ K. Denote by R ⊆ F (V ×W )the space generated by these four equivalence relations, in other words,by the differences

e(v1+v2,w) − (e(v1,w) + e(v2,w))e(v,w1+w2) − (e(v,w1) + e(v,w2))ce(v,w) − e(cv,w)ce(v,w) − e(v,cw)

Definition 25.1.2. The tensor product of the two vector spaces Vand W is then the quotient space V ⊗W = F (V ×W )/R.

203

204 25. MULTILINEAR FORMS; EXTERIOR DIFFERENTIAL COMPLEX

One readily checks with respect to a basis that the dimensions of Vand W multiply:

dim(V ⊗W ) = dimV dimW.

Definition 25.1.3. The exterior product V ∧ W of V and W isthe the vector space obtained as the quotient of V ⊗W by the ideagenerated by all the sums v ⊗ w + w ⊗ v.

The image of v ⊗ w under the projection V ⊗ W → V ∧ W isdenoted v ∧ w. Denote by (ei) a standard basis for V . One readilychecks that ei ∧ ej , where i < j, form a basis for

V ∧ V = Λ2(V ).

Similar remarks apply to the higher exterior powers Λk(V ).

25.2. Exterior differential complex

Associating to every cotangent space T ∗p its exterior algebra Λ(T ∗

p ),we obtain the exterior bundle

ΛM := Λ(T ∗M).

Similarly putting together the k-exterior powers ΛkT ∗p , we obtain

the subbundle ΛkM := Λk(T ∗M).

Definition 25.2.1. A differential k-form is a section of the ex-terior bundle ΛkM of k-multivectors. The space of such sections isdenoted Ωk(X).

Recall that we have an exterior derivative d defined on functions fby

df =∂f

∂uidui, (25.2.1)

with Einstein summation convention, see (24.6.1). A similar formuladefines the exterior derivative for an arbitrary k-form.

Theorem 25.2.2. The exterior derivative d gives rise to an exteriordifferential complex

0 → C∞(M) → Ω1(M) → Ω2(M) → . . .→ Ωn(M) → 0,

satisfying d2 = d d = 0.

For instance, the differential d : Ω1(M) → Ω2(M) is defined on a1-form fdu by

d(fdu) = df ∧ du.Remark 25.2.3. The exterior differential complex leads to the def-

inition of de Rham cohomology of M , see Section 28.9.

25.3. ALTERNATING MULTILINEAR FORMS 205

25.3. Alternating multilinear forms

Let V be a vector space over a field K. Let k ∈ N.

Definition 25.3.1. An alternating multilinear function

f : V k → K (25.3.1)

is called an alternating multilinear form on V .

The set of all alternating k-multilinear forms is is a vector space,as the sum of two such maps, or the multiplication of such a map witha scalar, is again alternating.

Note that for k = 1, we obtain the space V ∗ = Λ1(V ∗). In general,we have the following duality.

Theorem 25.3.2. If V has finite dimension n, then the space ofalternating multilinear forms can be identified with Λk(V ∗), where V ∗

denotes the dual space of V .

In particular, the dimension of the space of anti-symmetric mapsfrom V k to K is the binomial coefficient

(nk

).

Proof. Let us make such an identification explicit. Consider a k-tuple

x = (x1, . . . , xk) ∈ V k.

We will denote by yj, elements of the dual space V ∗.Given a decomposable element

y = y1 ∧ . . . ∧ yk ∈ ΛkV ∗,

we would like to define a multilinear map f = fy as in (25.3.1), associ-ated with y. We define it by setting

fy(x1, . . . , xk) =∑

σ∈Sk

sgn(σ) y1(xσ(1)) . . . yk(xσ(k)). (25.3.2)

In other words,

fy(x) = det(yi(xj))

is a k× k-determinant. Note that we do not multiply by 1k!

. In this wefollow S. Sternberg [?, p. 20].

Theorem 25.3.3. The space Λk(V ∗) is naturally dual to Λk(V ).

The duality is given on simple elements of Λk(V ) by the pairing

〈φ , v1 ∧ . . . ∧ vk〉 = φ(v1, . . . , vk).

The case k = 2 will be examined in detail in the next section.


25.4. Case of 2-forms

Let y1, y2 ∈ V ∗ be 1-forms. Consider the decomposable (simple)exterior 2-form

ω = y1 ∧ y2 ∈ Λ2(V ∗).

Then ω defines an alternating bilinear function

fω : V 2 → K

defined as follows. Let u, v ∈ V . Following (25.3.2), we set

fω(u, v) = y1(u)y2(v) − y1(v)y2(u) = det

(y1(u) y1(v)y2(u) y2(v)

)

where yi(u) is the evaluation of the covector yi ∈ V ∗ on the vector u ∈V .

Example 25.4.1. In the (x, y)-plane, the form ω = dx∧ dy definesa bilinear function fω whose geometric meaning is the signed area ofthe parallelogram spanned by the pair of vectors u, v.

Proposition 25.4.2. The value of fω on the pair u, v in fact de-pends only on its image ξ = u ∧ v in Λ2(V ).

Proof. The proof is immediate from the interpretation of fω(u, v)as a generalized signed area of the parallelogram spanned by u and v.Namely, fω is proportional to the standard area form dx∧dy since Λ2

((R2)∗

)

is 1-dimensional, and dx∧dy calculates the area A if the parallelogramspanned by u and v, where u ∧ v = Ae1 ∧ e2 ∈ Λ2(R2).

Remark 25.4.3. We will sometimes write ω(ξ) in place of fω(u, v),where ξ = u ∧ v. Thus by definition,

ω(ξ) = fω(u, v).

25.5. Norm on 1-forms

Given a norm ‖ ‖ on a vector space V , there is a natural norm onthe dual space V ∗, defined as follows. We will denote the new norm ‖ ‖∗for the purposes of this section.

Definition 25.5.1. The norm ‖ ‖∗ on V ∗ is defined for y ∈ V ∗ bysetting

‖y‖∗ = sup y(x) | x ∈ V, ‖x‖ ≤ 1 .In other words, we calculate the dual norm of the covector y by

maximizing its value over vectors x ∈ V of norm at most 1.

25.6. POLAR COORDINATES AND DUAL NORMS 207

It is clear that the inequality ‖x‖ ≤ 1 can be replaced by equalityin this definition:

‖y‖∗ = sup y(x) | x ∈ V, ‖x‖ = 1 ,i.e. the norm of y ∈ V ∗ can be calculated over the unit vectors x ∈ V .

Example 25.5.2. If (∂

∂x,∂

∂y

)

is an orthonormal basis for the Euclidean plane V = R2, then the dualspace V ∗ admits an orthonormal basis

(dx, dy)

which is dual to the basis(∂∂x, ∂∂y

). Indeed, dx( ∂

∂x) = 1 and ‖dx‖∗ = 1.

25.6. Polar coordinates and dual norms

The Euclidean metric in the plane E defines a norm on the tangentplane by the usual identification of a space and its tangent space.

Lemma 25.6.1. The 1-form xdy − ydx has norm r.

Proof. This is immediate from the fact that dx and dy are or-thonormal.

Consider the polar coordinates

(r, θ)

in the plane. Note that θ is undefined at the origin. Then (dr, dθ) isa basis for the cotangent plane at every point of E \0. However, thebasis is not orthonormal.

Since tan θ = yx, we have

x tan θ = y,

hence

tan θ + xdθ/dx

cos2 θ= dy/dx,

so that

tan θ dx+ xdθ

cos2 θ= dy.

Multiplying by r cos θ, we obtain

r sin θ dx +x

cos θrdθ = r cos θdy,

orydx+ r2dθ = xdy.


We then obtain r2dθ = xdy − ydx. By lemma 25.6.1, we obtain

‖dθ‖∗ =1

r,

so that it is the covector rdθ that has unit norm. It follows that inthe (r, θ) coordinates, we obtain an orthonormal basis for the cotangentspace consisting of the pair of covectors

(dr, rdθ).

25.7. Dual bases and dual lattices in a Euclidean space

The discussion is similar to the situation with a pair of dual vectorspaces treated in Section 19.5. Thus, let V be a Euclidean vector spaceequipped with a scalar product (inner product), denoted 〈 , 〉. Givena basis (xi) for V , there exists a unique basis (yj) satisfying

〈xi, yj〉 = δij,

where δij is the Kronecker delta function. We will view Rn as equippedwith its standard dot product, denoted 〈x, y〉.

Definition 25.7.1. Given a lattice L ⊆ Rn, the dual lattice L∗ ⊆Rn is the lattice

L∗ = y ∈ Rn | ∀x ∈ L, 〈x, y〉 ∈ Z .Thus the pairing between a point in a lattice and a point in its dual

is by definition always an integer.

Theorem 25.7.2. Consider a Z-basis (xi) for L. Consider the ba-sis (yj) dual to the basis (xi). Then (yj) is a Z-basis for the duallattice L∗.

Definition 25.7.3. Given a lattice L ⊆ Rn with its Euclideannorm | |, denote by λ1(L) the least length of a nonzero vector in L:

λ1(L) = min|v|∣∣ v ∈ L \ 0

.

We now consider the rank 1 case.

Proposition 25.7.4. Let L ⊆ R be a lattice and L∗ ⊆ R the duallattice of L. Then

λ1(L∗) =

1

λ1(L). (25.7.1)

Proof. Given a lattice L ⊆ R, denote by α > 0 its generator, sothat L = Zα. For an element y ∈ R to pair integrally with α, it mustbe a multiple of β = 1

α. Thus

L∗ = Z β

25.7. DUAL BASES AND DUAL LATTICES IN A EUCLIDEAN SPACE 209

and λ1(L∗) = β = 1

α.

Remark 25.7.5. Note that the reciprocal relation (25.7.1) does nothold in general for rank greater than 1. Already in rank 2, the productλ1(L

∗)λ1(L) can be greater than 1. For the Eisenstein lattice LE ⊆ Cspanned by the cube roots of unity, we obtain the value λ1(L

∗E)λ1(LE) =

2/√

3.

CHAPTER 26

Comass norm, Hermitian product, symplectic form

There are two distinct natural norms on k-multivectors: the Eu-cldean norm and the comass norm. First we complete the material onexterior algebra.

26.1. Λ2(V ) is isomorphic to the standard model Λ20(R

n)

We letΛ2

0(Rn) = Span ei ∧ ej : 1 ≤ i < j ≤ n ,

so that dim Λ20(R

n) =(n2

). Let V be an n-dimensional vector space. As

in Section 25.1, we set

F (V ) = Spane(v,w) : v, w ∈ V

,

and define the equivalence relation ∼ as in Section 25.1, with the ad-ditional relation stipulating that

ev,w ∼ −ew,v, (26.1.1)

and defineΛ2(V ) := F (V )/ ∼ .

Lemma 26.1.1. We have dim Λ2(V ) ≤(n2

).

Proof. Choose a basis a1, . . . , an for V . Each v ∈ V decomposesas a sum v = viai with vi ∈ R. Then

e(v,w) = e(viai,wjaj) ∼ viwje(ai,aj) ∼∑

i<j

viwje(ai,aj) +∑

i>j

viwje(ai,aj).

By 26.1.1, we have

e(v,w) ∼∑

i<j

viwje(ai,aj) −∑

j<i

viwje(aj ,ai).

Hencee(v,w) ∼

∑

i<j

(viwj − vjwi)e(ai,aj). (26.1.2)

This proves the lemma.

Theorem 26.1.2. Λ2(V ) is isomorphic to Λ20(R

n).

211

212 26. COMASS NORM, HERMITIAN PRODUCT, SYMPLECTIC FORM

Proof. We define a map

ζ : Λ2(V ) → Λ20(R

n)

by setting

ζ(e(v,w)) =∑

i<j

(viwj − vjwi)ei ∧ ej ∈ Λ20(V ).

In particular,

ζ(e(ai,aj)

)= ei ∧ ej ∈ Λ2

0(Rn),

showing that ζ is onto. Combined with Lemma 26.1.1, this proves thetheorem.

26.2. Euclidean norm on k-multivectors and k-forms

A natural Euclidean norm can be defined on the space Λk(Rn) of k-multivectors by declaring the basis

ei1 ∧ ei2 ∧ · · · ∧ eik | 1 ≤ i1 < i2 < · · · < ik ≤ n (26.2.1)

of (24.16.1) to be orthonormal. Then each simple k-form

ei1 ∧ . . . ∧ eikhas unit Euclidean norm:

|ei1 ∧ . . . ∧ eik | = 1,

and they are all mutually perpendicular. Since we have identified thespace of k-multivectors with the space of k-linear forms, this gives aEuclidean norm on the k-linear forms, as well.

Example 26.2.1. For the symplectic form α = e1 ∧ e2 + e3 ∧ e4on V = R4, we have |α| =

√2.

In the next section, we will define a different norm on the space offorms.

26.3. Comass norm of a form

Let V be an inner product space, for instance Rn. Recall that anexterior form is called simple (or decomposable) if it can be expressedas a wedge product of 1-forms.

Lemma 26.3.1. The comass norm for a simple k-form coincideswith the natural Euclidean norm on k-forms.

26.4. SYMPLECTIC FORM FROM A COMPLEX VIEWPOINT 213

Proof. A k-tuple of 1-forms span a k-blade, or a k-dimensionalsubspace P ⊆ V . The k-tuple can be replaced by an orthonormal k-tuple forming a basis for P . This can be done by means of a volume-preserving transformation, for instance by applying the Gram-Schmidtprocess.

In general, the comass is defined as follows. Recall that a k-multivector of V ∗ can be viewed as a k-linear form on V via (25.3.2).

Definition 26.3.2. The comass of a k-linear form is its maximalvalue on a k-tuple of unit vectors.

In formulas, the comass norm of a k-linear form ω on Rn is

‖ω‖ = max ω(e1, . . . , ek) | ei ∈ Rn, |ei| = 1 for 1 ≤ i ≤ k . (26.3.1)

Here |e| denotes the Euclidean norm on Rn.

Example 26.3.3. For the symplectic form α = e1 ∧ e2 + e3 ∧ e4on V = R4, we have |α| =

√2, while ‖α‖ = 1. In other words, its

comass norm is smaller than its Euclidean norm:

‖α‖ < |α|.This will be proved in the context of Wirtinger’s inequality.

Lemma 26.3.4. We have ‖ω‖ ≤ |ω| with equality if and only if ω issimple.

Indeed, the Euclidean norm |ω| of ω is defined by formula similarto (26.3.1), where the maximum is taken over all k-forms in Λk(V ) andnot merely the simple (decomposable) ones (by the finite-dimensionalversion of the Riesz representation). This proves the inequality. Acharacterisation of the boundary case of equality is postponed.

26.4. Symplectic form from a complex viewpoint

Example 26.4.1. The 2-form dx ∧ dy viewed as a bilinear formon R2 is represented by the antisymmetric matrix

(0 1−1 0

).

Consider the ν-dimensional complex vector space Cν . Denote by Zjthe j-th coordinate

Zj : Cν → C,with the usual decomposition

Zj = Xj + iYj


into the real and imaginary parts. The complex conjugate Zj is bydefinition

Zj = Xj − iYj ,

for all j = 1, . . . , ν. Then the exterior product Zj∧Zj can be expressedas follows:

Zj ∧ Zj = (Xj + iYj) ∧ (Xj − iYj) = −2iXj ∧ Yj =2

iXj ∧ Yj.

Thusi

2Zj ∧ Zj = Xj ∧ Yj.

Example 26.4.2 (Symplectic form). Let Z1, . . . , Zν be the coordi-nate functions in Cν . Then the standard symplectic 2-form (24.15.1),

denoted A ∈∧2((Cν)∗), is given by

A = i2

ν∑

j=1

Zj ∧ Zj. (26.4.1)

Remark 26.4.3. A more traditional way of writing the form is interms of the real basis, by the expression

A =∑

j

Xj ∧ Yj = X1 ∧ Y1 + . . .+Xν ∧ Yν.

We have preferred the form (26.4.1) emphasizing the complex structurebecause of future applications.

Example 26.4.4. The corresponding skew-symmetric matrix is ablock diagonal matrix with ν diagonal blocks of the form

(0 1−1 0

)

corresponding to each of the complex coordinates.

26.5. Hermitian product

Let V be a ν-dimensional vector space over C. Let H = H(v, w)be a Hermitian product on V .

Example 26.5.1. The standard Hermitian product on V = Cν isgiven by

ν∑

j=1

vjwj (26.5.1)

where v = (vj) and w = (wj).

26.6. ORTHOGONAL DIAGONALISATION 215

Remark 26.5.2. Here we adopt the convention that a Hermitianproduct H is complex linear in the second variable. There are varyingconventions in textbooks regarding this issue. Federer’s book [?] usesthe convention (26.5.1).

Recall that we have

H(v, w) = H(w, v). (26.5.2)

Consider its real part v · w, and imaginary part A = A(v, w):

H(v, w) = v · w + iA(v, w). (26.5.3)

Then v · w is a scalar product on V viewed as a 2ν-dimensional realvector space.

Lemma 26.5.3. The imaginary part A is skew-symmetric and there-fore can be viewed as an element in A ∈

∧2 V , the second exterior powerof V .

Proof. We have

A(w, v) = ℑ(H(w, v)) = ℑ(H(v, w)

)

by (26.5.2). Therefore A(w, v) = −ℑ(H(v, w)) = −A(v, w).

Lemma 26.5.4. The comass of the standard symplectic form A sat-isfies ‖A‖ = 1.

Proof. It suffices to evaluate A on a pair ξ = v ∧ w (see Re-mark 25.4.3), where v and w are orthonormal. Thus v · w = 0.We therefore have H(v, w) = iA(v, w). Since A(ξ) is real, the pair-ing 〈ξ, A〉 = A(ξ) can be evaluated as follows:

A(ξ) = A(v, w) = H(iv, w) = ℜ(H(iv, w)) = (iv) · w ≤ 1 (26.5.4)

by the Cauchy-Schwarz inequality; equality holds if and only if onehas iv = w.

26.6. Orthogonal diagonalisation

In this section, we recall some standard material from linear algebra.

Theorem 26.6.1. Every skew-symmetric real matrix can be orthog-onally diagonalized into 2 by 2 blocks.

Alternatively, the theorem can be formulated as follows (see Theo-rem 26.6.3).

Definition 26.6.2. An endomorphism f of Rn is anti-selfadjoint if

〈f(x), y〉 = −〈x, f(y)〉.


Theorem 26.6.3. Given an anti-selfadjoint endomorphism f of Rn,there exists an orthogonal decomposition of Rn into subspaces Vj invari-ant under f , where dimVj ≤ 2 for all j.

Recall that the rank of a 2-form α is the least possible number ofsimple (decomposable) 2-forms α(j) occurring in a presentation

α =∑

j

α(j).

Corollary 26.6.4. The rank of a 2-form on V = Rn is at most n2.

Proof. A 2-form is given by a skew-symmetric matrix A, whichcan be thought of as an anti-selfadjoint endomorphism fA. Applyingthe diagonalisation theorem, we obtain an orthonormal basis e1, . . . , enwith respect to which the endomorphism decomposes into µ blocks ofsize 2 by 2, with possibly an additional 1-dimensional block on whichthe endomorphism vanishes. With respect to this basis, the 2-form willhave the following form. We have an orthonormal set ω1, . . . , ω2µ ∈ V ∗

and nonnegative numbers λ1, . . . , λµ, so that

fA =

µ∑

j=1

λj (ω2j−1 ∧ ω2j) . (26.6.1)

Therefore the rank of A is at most µ.

With respect to the new basis, the matrix will consist of 2 by 2blocks of the form (

0 λj−λj 0

)

(or half of it, depending on normalisation).

Remark 26.6.5. Some comments on the diagonalisation theorem:suppose that A ∈ Rn×n is anti-symmetric. Its eigenvalues are all pureimaginary. Suppose that v = v1 + iv2 is an eigenvector (separated intoreal and imaginary parts) corresponding to eigenvalue iλ. Then Av =iλv is equivalent to Bv0 = 0, where B ∈ R2n×2n is the symmetric block

matrix

[λI −AA λI

]and v0 =

v1v2

is a column vector of length 2n.

This formulation is the standard eigenvalue problem for the real

symmetric matrix0 A

−A 0, and each real solution v0 =

v1v2

gives a 2-

dimensional real A-invariant subspace spanv1, v2. Of course, the

eigenvector w0 =−v2v1

gives the same subspace. There may be an issue

in case of multiple eigenvalues, but this can be resolved.

CHAPTER 27

Wirtinger inequality, projective spaces

27.1. Wirtinger inequality

We exploit the material presented earlier to prove Wirtinger’s in-equality for 2-forms. Then we proceed to the complex projective space.

Recall that the standard symplectic form on V (where V is isomor-phic to Cν) is denoted

A ∈ Λk(V ∗)

where Λk(V ∗) is the space of all k-linear alternating forms on V (seeTheorem 25.3.3). The form A can be thought of as the imaginary partof the standard Hermitian product as in (27.1.1) below.

Following H. Federer [?, p. 40], we prove an optimal upper boundfor the comass norm ‖ ‖, cf. Definition 26.3.2, of the exterior powersof a 2-form.

We will use the decomposition

H(v, w) = v · w + A(v, w) (27.1.1)

of a Hermitian inner product into the sum of a scalar product and thesymplectic form, see (26.5.3), similar to the decomposition

zz′ = (x− iy)(x′ + iy′) = (xx′ + yy′) + i(xy′ − x′y)

(the convention of “barring” the first variable was discussed in Re-mark 26.5.2).

Theorem 27.1.1 (Wirtinger inequality). Let V be a ν-dimensional

complex vector space. Let µ ≥ 1. If ξ ∈∧2µ V and ξ is simple, then

〈ξ, Aµ〉 ≤ µ! |ξ|;equality holds if and only if there exist elements v1, . . . , vµ ∈ V suchthat

ξ = v1 ∧ (iv1) ∧ · · · ∧ vµ ∧ (ivµ).

Consequently, the comass satisfies ‖Aµ‖ = µ!

Proof. The main idea is that in real dimension 2µ, every 2-formsplits into a sum of at most µ orthogonal simple (decomposable) pieces.

217

218 27. WIRTINGER INEQUALITY, PROJECTIVE SPACES

We assume that |ξ| = 1, where | | is the natural Euclidean normin∧2µ V . We therefore need to show that 〈ξ, Aµ〉 ≤ µ! for all such ξ.The case µ = 1 was treated in Lemma 26.5.4.In the general case µ ≥ 1, we consider the 2µ dimensional sub-

space T associated with the simple 2µ-vector ξ. Let f : T → V be theinclusion map (hachala, not hachlala). Recall that

A ∈∧

2 (V ∗)

is a 2-linear alternating form on V .

Definition 27.1.2. The restriction of A to the subspace T ⊆ V isdenoted

(∧2f)A ∈∧

2 (T ∗).

Lemma 27.1.3. The restiction of Aµ to T is the µ-th power of therestriction of A to T .

The proof is trivial. We orthogonally diagonalize the skew-symmetric 2-form (∧2f)A. Thus, we decompose the form (∧2f)A into 2×2 diagonalblocks. Thus, we can choose dual orthonormal bases e1, . . . , e2µ of T

and ω1, . . . , ω2µ of∧1(T ∗), and nonnegative numbers λ1, . . . , λµ, so that

(∧2f)A =

µ∑

j=1

λj (ω2j−1 ∧ ω2j) . (27.1.2)

By Lemma 26.5.4 we have ‖A‖ = 1 and therefore

λj = A(e2j−1, e2j) ≤ ‖A‖ = 1 (27.1.3)

for each j = 1, . . . , 2µ.Noting that ξ = ǫe1 ∧ · · · ∧ e2µ with ǫ = ±1, we use Lemma 27.1.3

to compute

(∧2µf

)Aµ = µ! λ1 . . . λµ ω1 ∧ · · · ∧ ω2µ

by Lemma 28.1.1 below. Therefore

〈ξ, Aµ〉 = ǫµ! λ1 . . . λµ ≤ µ! (27.1.4)

since λj ≤ 1 from (27.1.3). Note that equality occurs in (27.1.4) if andonly if ǫ = 1 and λj = 1 for each j. Applying the proof of Lemma 26.5.4,we conclude that e2j = ie2j−1, for each j = 1, . . . , 2µ.

27.2. WIRTINGER INEQUALITY FOR AN ARBITRARY 2-FORM 219

27.2. Wirtinger inequality for an arbitrary 2-form

Recall from Section 20.2 that the polarisation formula allows one toreconstruct a symmetric bilinear formB, from the quadratic formQ(v) =B(v, v), at least if the characteristic is not 2:

B(v, w) =1

4(Q(v + w) −Q(v − w)). (27.2.1)

This will be exploited in the proof of Proposition 27.2.1 below.There is a useful generalisation of Wirtinger’s inequality for arbi-

trary 2-forms. Let T = Cµ.

Proposition 27.2.1. Given an orthonormal basis (ω1, . . . , ω2µ) of∧1 T ∗, and nonzero real numbers λ1, . . . , λµ, the form

α =

µ∑

j=1

λj (ω2j−1 ∧ ω2j) (27.2.2)

has comass ‖α‖ = maxj |λj|.Proof. We can assume without loss of generality that each λj is

positive. This can be attained in one of two ways. One can permutethe coordinates, by applying the transposition flipping ω2j−1 and ω2j,so as to change the sign of λj . Alternatively, one can replace, say, ω2j

by −ω2j , which similarly changes the sign of λj.

Setting ω′2j−1 = λ

1/2j ω2j−1 and ω′

2j = λ1/2j ω2j , we can write

α =

µ∑

j=1

ω′2j−1 ∧ ω′

2j

We now exploit the polarisation formula (27.2.1).

Remark 27.2.2. The polarisation works on a real slice; one thencomplexifies to go from a real dot product to a Hermitian product.

Consider the hermitian inner product Hα on T obtained by polar-izing the quadratic form

∑

j

(ω′2j−1

)2+(ω′2j

)2=∑

j

(λ1/2j ω2j−1

)2+(λ1/2j ω2j

)2,

where the squares are understood in the sense of the symmetric product(see (20.2.1)).

Lemma 27.2.3. TheHα-norm |v|α of eachH-unit vector v is boundedabove by (maxj λj)

1/2:

|v|α ≤ maxj

(λj)1/2.

220 27. WIRTINGER INEQUALITY, PROJECTIVE SPACES

The Hermitian inner product Hα is set up in such a way as to satisfy

Hα(v, w) = iα(v, w),

orα(v, w) = −iHα(v, w)

for each Hα-orthogonal pair v, w.Now let ζ be a unit 2-vector such that

||α|| = α(ζ), (27.2.3)

Consider its 2-plane T ⊆ V . In the 2-plane T , the unit disc of the scalarproduct defined by Hα is an ellipse with respect to the scalar productof H . We will exploit its (perpendicular) principal axes. Let v, w bean orthonormal pair proportional to the principal axes. We can thenwrite ζ = v ∧ w. The H-orthonormal pair v, w is also Hα-orthogonal(though in general not Hα-orthonormal), as well. Then, as in (26.5.4),we have

α(ζ) = −iHα(ζ) = Hα(iv, w) ≤ |iv|α |w|α ≤ maxjλj

by Lemma 27.2.3. This proves the lemma.

The lemma generalizes the the µ-th powers of a 2-form, giving ageneralisation of Wirtinger’s inequality.

Corollary 27.2.4. Every real alternating 2-form A satisfies thecomass bound

‖Aµ‖ ≤ µ!‖A‖µ. (27.2.4)

Proof. We may assume that ‖A‖ = 1. By Lemma 27.2.1, wehave λj ≤ 1 for each j = 1, . . . , ν.

An inspection of the proof Proposition 27.1.1 reveals that the or-thogonal diagonalisation argument, cf. (27.1.3), applies to an arbi-trary 2-form A with comass ‖A‖ = 1.

27.3. Real projective space

The 0-dimensional sphere S0 ⊆ R1 consists of two points, ±1. Wecan therefore represent it as a disjoint union

S0 = e0 ∪ a(e)0,

where a(v) = −v is the antipodal map. The 1-dimensional sphereS1 ⊆ R2 can similarly be partitioned into a union

S1 = e0 ∪ a(e0) ∪ e1 ∪ a(e1),

and similarly

Sn = e0 ∪ a(e0) ∪ e1 ∪ a(e1) ∪ . . . ∪ en ∪ a(en).

27.4. COMPLEX PROJECTIVE LINE CP1 221

The quotient of Sn by the antipodal map gives a decomposition

RPn = e0 ∪ e1 ∪ . . . ∪ en. (27.3.1)

Thus, the real projective space admits a cell decomposition with asingle cell in each dimension.

In the context of the de Rham cohomology, we will be interested ina similar decomposition of complex projective space.

27.4. Complex projective line CP1

In projective geometry, one starts with an arbitrary field K. Thefamiliar construction of “adding a point ∞ at infinity” then producesthe projective line

KP1 = K ∪ ∞over K. When the field K = C is that of the complex numbers, wethereby obtain the compactification C∪∞ of C, namely the Riemannsphere S2:

CP1 = C ∪ ∞ = S2.

The familiar stereographic projection provides an identification of thecomplement of a point in S2 and the complex numbers C.

CHAPTER 28

Complex projective spaces, de Rham cohomology

28.1. A clarification

A havhara. We used a combinatorial result in the proof of Wirtinger’sinequality.

Lemma 28.1.1. In(∑µ

j=1 λjxj

)µwith commuting variables xj, the

coefficient of x1x2 · · ·xµ is

λ1 · · ·λµ µ!

Proof. The product of µ parentheses

(λ1x1 + λ2x2 + . . .+ λµxµ) (λ1x1 + λ2x2 + . . .+ λµxµ) · · · (λ1x1 + λ2x2 + . . .+ λµxµ)

can be analyzed combinatorially as follows: we have µ possibilities ofchoosing the variable x1 out of the µ parenthetical expressions. Outof the remaining µ − 1 parenthetical expressions, we have as manypossibilities for choosing x2, etc., resulting in µ! possibilities.

In the calculation of the comass of the powers of the symplecticform, we applied this combinatorial fact to the variables xj given bythe 2-forms

xj = ω2j−1 ∧ ω2j .

28.2. Complex projective space CPn

Example 28.2.1. More generally, we have

CPn = Cn ∪ CPn−1,

where CPn−1 is thought of as the hyperplane at infinity.

Complex projective n-space CPn is a complex manifold that maybe described by n + 1 complex coordinates as

(z0, z1, . . . , zn) ∈ Cn+1, (z1, z2, . . . , zn) 6= (0, 0, . . . , 0)

where the tuples differing by an overall rescaling are identified:

(z0, z1, . . . , zn) ≡ (λz0, λz1, . . . , λzn); λ ∈ C, λ 6= 0. (28.2.1)

223

224 28. COMPLEX PROJECTIVE SPACES, DE RHAM COHOMOLOGY

These are homogeneous coordinates in the traditional sense of projec-tive geometry. A set of homogeneous coordinates of a point in CPn isdenoted

[z0, z1, . . . , zn]

(using the traditional square brackets).

28.3. Real projective space as a quotient

Recall that the real projective space can be thought of as an an-tipodal quotient of the sphere:

RPn = Sn/G,

where G = S0 = ±1 is the cyclic group of order 2. There is a similarpresentation of complex projective space, where in place of the cyclicgroup G, we have the unitary group U(1).

28.4. Unitary group

The unitary group U(n) is the group of complex n× n matrices Msatisfying

MM t = In.

Thus, the group U(1) is the circle of complex numbers of unit absolutevalue: U(1) = S1 ⊆ C.

In the next section, we will exploit the action of U(1) on Cn+1 byscalar multiplication as in (28.2.1).

28.5. Hopf bundle

Theorem 28.5.1. The space CPn is a quotient space of the unit (2n+1)-sphere S2n+1 in Cn+1 under the action of U(1):

CPn = S2n+1/U(1).

Thus, the mapS2n+1 → CPn

is a bundle (eged) with fiber (siv) S1.

Proof. Every complex line Cv in Cn+1 intersects the unit spherein a circle. We restrict the vectors in (28.2.1) to the unit sphere, andthe scalars λ in (28.2.1) to the unit circle in C. We then identify underthe natural action of U(1) to obtain CPn.

Example 28.5.2. For n = 1, this construction yields the classicalHopf bundle:

S2 = S3/U(1),

where S2 = CP1.

28.6. FLAGS AND CELL DECOMPOSITION OF CPn 225

28.6. Flags and cell decomposition of CPn

Recall that the real projective space has a cell decomposition witha cell in each dimension, see formula (27.3.1). An analogous decompo-sition, but with cells

Definition 28.6.1. A complete flag (Vj) (degel) in V = Cn+1 is achoice of a nested (mekanenet) sequence of subspaces of dimensions

0, 1, 2, . . . , n+ 1,

namely:0 = V0 ⊆ V1 ⊆ V2 ⊆ · · · ⊆ Vn+1 = V.

Theorem 28.6.2. A choice of a complete flag (Vj) determines aunique decomposition of CPn into cells in each even dimension:

CPn = e0 ∪ e2 ∪ · · · ∪ e2n. (28.6.1)

Proof. The 2j-dimensional cell e2j ⊆ CPn consists of points withhomogeneous coordinates [z0, z1, . . . , zn] contained in the j-th set-theoreticdifference:

(z0, z1, . . . , zn) ∈ Vj \ Vj−1.

Definition 28.6.3. The standard flag is the nested sequence

0 = V0 ⊆ C1 ⊆ C2 ⊆ · · · ⊆ Cn+1,

where Cj is the subspace of points whose first j coordinates are arbi-trary and the remaining n+ 1 − j coordinates vanish.

With respect to the standard flag, we obtain

e2j = (z0, z1, . . . , zj , 0, . . . , 0) ∈ Cn+1 | zj 6= 0.Proposition 28.6.4. The collection e2j of points with coordinates

in Vj \ Vj−1 is diffeomorphic to Cj.

Proof. A diffeomorphism

e2j → Cj

can be specified as follows:

(z0, . . . , zj−1, zj, 0, . . . , 0) 7→(z0zj, . . . ,

zj−1

zj

)∈ Cj,

proving the proposition.

Theorem 28.6.5. The closure e2j ⊆ CPn of each cell e2j is thecomplex projective subspace CPj ⊆ CPn:

e2j = CPj.


Proof. Consider a nonzero point

(z0, . . . , zj−1, 0, . . . , 0)

whose j-th complex coordinate vanishes. Such a point can be approx-imated by points with non-vanishing j-th coordinate, by consideringthe sequence

(z0, . . . , zj−1,1

k. . . , 0), k = 1, 2, . . . .

Therefore every point of CPj−1 is in the closure of the 2j-cell, andtherefore

e2j = Cj ∪ CPj−1 = CPj,completing the proof.

28.7. g, A, and J in a complex vector space

Consider the relation

sin(θ) = cos(π2− θ)

in elementary trigonometry. The corresponding relation in terms ofscalar products and determinants is the following. Let v, w ∈ R2.Then the 2 by 2 determinant satisfies

det[v, w] = (iv) · w,where i denotes the rotation by π

2resulting from the usual identifica-

tion R2 = C. In other words, we have

A(v, w) = g(iv, w), (28.7.1)

where A is the standard symplectic form dx∧dy, and g is the standardmetric dx2 + dy2 in C.

This illustrates the passage from a symmetric form to an antisym-metric form. More generally, let V be a complex vector space, forinstance Cµ. Thus, V is equipped with an endomorphism

J : V → V

satisfying J2 = −Id. The endomorphism is called an almost complexstructure. In Cµ, we have

J(v) = (iv1, iv2, . . . , ivµ).

A Hermitian product H(v, w) on V decomposes as

H(v, w) = g(v, w) + iA(v, w),

where g is a scalar product in Cµ = R2µ and A is the symplectic form.Note that g is symmetric in the two variables while A is antisymmetric.Moreover, we have the following relation among J , g, and A.

28.8. FUBINI-STUDY 2-FORM ON CP1 227

Proposition 28.7.1. One has the relation

A(v, w) = g(Jv, w).

Proof. We have

g(iv, w) = ℜ(H(iv, w)) = ℜ(−iH(v, w)).

Hence

g(iv, w) = ℜ(−i(iA(v, w))) = −i2A(v, w) = A(v, w),


Remark 28.7.2. In this way, a metric on a complex manifold canbe used to construct a symplectic form on the manifold, as we illustratein Section 28.8.

28.8. Fubini-Study 2-form on CP1

We now examine in detail the case of the complex projective lineCP1, i.e. the 2-sphere.

Returning to CP1, we similarly obtain the differential 2-form fromthe metric by formula (28.7.1).

The round metric of the sphere can be expressed in stereographiccoordinates as

g =dx2 + dy2

(1 + r2)2

where r =√x2 + y2. This normalisation results in a round sphere of

radius 12.

The Fubini-Study form is merely the area form of the metric.

Theorem 28.8.1. In an affine neighborhood, we have the followingformulas for the metric g and the corresponding Fubini-Study 2-form A(the area form of the metric) on the complex projective line:

g =dx2 + dy2

(1 + r2)2

and

A =dx ∧ dy(1 + r2)2

The proof is immediate from (32.4.2) and (28.7.1).


28.9. Exterior differential complex

Recall that given an n-dimensional manifold M , we have an exteriordifferential complex

0 → C∞(M) → Ω1(M) → Ω2(M) → . . .→ Ωn(M) → 0. (28.9.1)

Here each Ωi(M) is the space of sections of the exterior bundle Λi(T ∗M),while the differential

di : Ωi → Ωi+1

raising the degree by 1, is defined as follows. For f ∈ C∞(M), wedefine the 1-form df ∈ Ω1(M) by setting

df = d0f =∂f

∂ujduj.

Similarly, the differential

d = d1 : Ω1(M) → Ω2(M)

is defined on a 1-form fdu by

d(fdu) = df ∧ du, (28.9.2)

and similarly for the higher differentials.

Definition 28.9.1. A differential form ω is called closed if dω = 0.

Proposition 28.9.2. The sequence of maps as in (28.9.1) is a dif-ferential complex, in the sense that d d = 0, or more explicitly

dk dk−1 = 0 ∀ k.Proof. The condition d d = 0 is sometimes written as d2 = 0

where d2 is understood as the composition of two consecutive differen-tials. We check the condition d2 = 0 at the level of Ω1(M). Given afunction f ∈ C∞(M), we first apply the differential d = d0 to obtain adifferential 1-form df = ∂f

∂uidui. Next, we apply the differential

d = d1 : Ω1(M) → Ω2(M)

to the 1-form ∂f∂uidui. Using (28.9.2), we obtain

d

(∂f

∂uidui)

= d

(∂f

∂ui

)dui =

∂2f

∂ui∂ujduj ∧ dui,

with a double summation with indices i and j running from 1 to nindependently of each other. Since dui∧dui = 0, we exploit the equalityof mixed partials to write

d2f = d

(∂f

∂uidui)

=∑

i<j

∂2f

∂ui∂uj(duj ∧ dui + dui ∧ duj

)= 0,

28.11. DE RHAM COHOMOLOGY RING OF M , BETTI NUMBERS 229

since exterior (alternating) 1-forms anti-commute by definition.Similarly, for a typical 1-form fdu1, we have

d(fdu1 ∧ du2) =∂f

∂uidui ∧ du1.

Therefore

d2(fdu1 ∧ du2) = d

(∂f

∂uidui ∧ du1

).

Thus,

d2(fdu1 ∧ du2) =∑

i<j

∂2f

∂ui∂uj(duj ∧ dui + dui ∧ duj

)∧ u1 = 0

as before.

We can restate the proposition as follows.

Corollary 28.9.3. We have

Im(dk−1) ⊆ Ker(dk) ⊆ Ωk(M).

This leads to the definition of de Rham cohomology ofM , as follows.

28.10. De Rham cocycles and coboundaries

Definition 28.10.1. The group

ZkdR(M) = Ker(dk) ⊆ Ωk(M)

is called the group of de Rham k-cocycles.

Thus, a de Rham k-cocycle is by definition a closed differentialk-form.

Definition 28.10.2. The group

BkdR(M) = Im(dk−1) ⊆ Ωk(M)

is called the group of de Rham k-coboundaries.

An alternative term for a de Rham coboundary is an exact differ-ential form.

28.11. De Rham cohomology ring of M , Betti numbers

Definition 28.11.1. The k-th de Rham cohomology group of M ,denoted Hk

dR(M), is by definition the group

HkdR(M) = Zk

dR(M)/BkdR(M) = Ker(dk)/Im(dk−1)

(cocycles modulo the coboundaries).


Even though the object HkdR(M) is traditionally referred to as a

group, it is in fact a real vector space.

Definition 28.11.2. The k-th Betti number bk(M) of M is bydefinition the dimension of the vector space Hk

dR(M):

bk(M) = dimHkdR(M).

CHAPTER 29

de Rham cohomology continued

29.1. Comparison of topology and de Rham cohomology

Proposition 29.1.1. Every connected manifold M has unit 0-thBetti number b0(M) = 1, i.e. H0

dR(M) = R.

Proof. Consider the segment

0 → C∞(M) → Ω1(M)

of the exterior differential complex. The space of the 0-coboundariesis trivial. The space of 0-cocycles consists of all functions f ∈ C∞(M)satisfing df = 0. Thus at every point, all the partial derivatives of fmust vanish. Such a function must be locally constant. Since M isconnected, the function f must also be globally constant. We thereforeobtain an identification

H0dR(M) ≃ R ⊆ C∞(M)

of 0-th de Rham cohomology group with the space of constant func-tions.

Theorem 29.1.2. Every closed connected orientable n-dimensionalmanifold M satisfies bn(M) = 1, i.e. Hn

dR(M) = R.

Proof. The proof of this result is more difficult than the previousone, and is postponed.

29.2. de Rham cohomology of a circle

Theorem 29.2.1. We have H1dR(S1) = R, i.e., b1(S1) = 1.

Proof. Let M = S1. Consider the space of 1-cocycles Z1dR(M).

We define a homomorphism

Φ : Z1dR(M) → R

as follows. Let ω ∈ Z1dR(M) be a de Rham 1-cocycle, i.e., a 1-form. We

set

Φ(ω) =

∮

M

ω.

231

232 29. DE RHAM COHOMOLOGY CONTINUED

Using the coordinate θ on the circle, we can express ω in terms of thestandard 1-form dθ ∈ Ω1(M) by writing ω = f(θ)dθ. Thus,

Φ(ω) =

∫ 2π

0

f(θ)dθ = F (2π) − F (0),

where F : [0, 2π] → R is an antiderivative for f , so that dF = f(θ)dθ.Note that F does not descend to a single-valued function on the circle(only to a multiple-valued function). We have to show that the 1-form

ω − Φ(ω)dθ

is a 1-coboundary. Consider the function

g(θ) = F (θ) − Φ(ω)θ.

The function g satisfies g(0) = g(2π) and hence descends to a smoothsingle-valued function on the circle. Therefore by definition,

dg ∈ B1dR(M).

Then

dg = fdθ − Φ(ω)dθ,

and therefore the kernel of Φ consists precisely of the de Rham 1-coboundaries, i.e., the exact forms. By the isomorphism theorem fromelementary group theory, we conclude that the first cohomology groupof the circle is isomorphic to R.

29.3. 2-dimensional cohomology of CP1

We will show later that H2dR(CP1) is isomorphic to R. A spe-

cific representative of the non-trivial 2-dimensional cohomology classinH2

dR(CP1) called the Fubini-Study formA was defined in Section 28.8.A more general formula for such a form on CPn will be defined later,and generates H2

dR(CPn). The form A is a globally defined closed 2-form on CP1. Its cohomology class

[A] ∈ H2dR(CP1)

spans H2dR(CP1) ≃ R. The 2-form has unit comass at every point.

Therefore we have

vol(CP1) =

∫

CP1

A =

∫∫

C

1

(1 + r2)2dx ∧ dy.

Hence

vol(CP1) =

∫∫rdrdθ

(1 + r2)2= 2π

∫rdr

(1 + r2)2= π

∫ ∞

1

du

u2= π,

29.4. 2-DIMENSIONAL COHOMOLOGY OF THE TORUS 233

where we used the u-substitution u = 1 + r2. The area is π sincewe are dealing with a 2-sphere of Gaussian curvature 4, radius 1

2, and

Riemannian diameter π2.

Remark 29.3.1. Later we will define an integer lattice in coho-mology. The element 1

πA ∈ H2

dR(CP1) is a generator of the integer

lattice L2dR(CP1) ≃ Z.

29.4. 2-dimensional cohomology of the torus

Theorem 29.4.1. We have H2dR(T2) ≃ R.

Proof. Consider a two-torus whose dimensions are 1 by 1. Let xand y be the standard coordinates on the torus, so they both go fromzero to one.

Let f(x, y) be a function with total integral 0. We would like toshow that the 2-form fdA is exact. It suffices to find functions g and hon the torus such that

fdxdy = d(gdx+ hdy). (29.4.1)

Let

a(x) =

∫ 1

0

f(x, t)dt,

and set

g(x, y) = −∫ y

0

f(x, t)dt+ a(x)y,

where a(x) was chosen so that g(x, 1) = g(x, 0). Here

a(0) =

∫ 1

0

f(0, t)dt =

∫ 1

0

f(1, t)dt = a(1)

since f is periodic in x. Thus, g is periodic in the x-direction, as well.Then (29.4.1) holds iff

− ∂

∂yg +

∂

∂xh = f.

Since ∂∂yg = −f + a, we need ∂

∂xh = a(x). Thus we define h to be

h(x) =

∫ x

0

a(s)ds

so h depends only on x. Note that

h(1) =

∫ 1

0

a(s)ds =

∫ 1

0

∫ 1

0

f(s, t)dtds = 0

since f has total integral zero. Of course h(0) = 0 also, so h is smoothon the torus.

234 29. DE RHAM COHOMOLOGY CONTINUED

To do the sphere case, use spherical coordinates φ and θ so that dA =cos2(φ)dφdθ. Then solve

fdA = d(gdφ+ h dθ)

making the necessary changes to allow for the cos2(φ).

CHAPTER 30

Elements of homology theory

30.1. Relation between Fubini-Study 2-form and volumeform on CPn

The Fubini-Study metric g and 2-form A on CP1 was defined inSection 28.8 as follows. The metric g is

g =dx2 + dy2

(1 + r2)2

and the corresponding Fubini-Study 2-form A (the area form of themetric) on the complex projective line is

A =dx ∧ dy(1 + r2)2

.

We will define the Fubini-Study 2-form A on CPn later. We areprimarily interested in the the following theorem.

Theorem 30.1.1. The wedge power

An

of the Fubini-Study form is a nondegenerate volume form at every pointof CPn.

Proof. It suffices to check such a relationship at a point. TheFubini-Study form at a point is the symplectic form

∑nj=1 dxj ∧ dyj.

Its top exterior power is

n! dx1 ∧ dy1 ∧ · · · ∧ dxn ∧ dyn,

as a special case of the calculation we saw in the context of Wirtinger’sinequality.

The following section should be viewed as “general culture” placingthis fact in a more general setting of the cup product.

235

236 30. ELEMENTS OF HOMOLOGY THEORY

30.2. Cup product

Definition 30.2.1. The cup-product ∪ of cohomology classes isthe operation

∪ : HadR(M) ×Hb

dR(M) → Ha+bdR (M),

defined by the alternating product in the exterior algebra Λ(T ∗p ) at

every point p ∈M and in the complex of differential forms Ω(M).

Theorem 30.2.2. The product in cohomology is independent of thechoice of representatives at the level of the closed differential forms.

Thus, we have

α ∪ β ∈ Ha+bdR (M)

if α ∈ HadR(M) and β ∈ Hb

dR(M).

Definition 30.2.3. The de Rham cohomology ring is the gradedring

⊕kHkdR(M), k = 0, . . . , n

whether the additive structure comes from the real vector space struc-ture on the individual groups, whereas the multiplicative structure isgiven by the cup product ∪.

30.3. Cohomology of complex projective space

Theorem 30.3.1. The Betti numbers of complex projective space CPn

satisfy bk(CPn) = 1 for all even k = 0, 2, . . . , 2n, and bk(CPn) = 0 forall other k.

Remark 30.3.2. The geometric counterpart of this result is thedecomposition of complex projective space with a single cell in eacheven dimension, see (28.6.1).

The above theorem can be refined as follows, taking into accountthe ring structure.

Theorem 30.3.3. The cohomology ring of CPn is the truncatedpolynomial ring in a single 2-dimensional variable. Namely, the ringis generated by a single class ω ∈ H2

dR(CPn), with the unique relation

ωn+1 = 0.

Corollary 30.3.4. The 2n-dimensional class

ωn ∈ H2ndR(CPn)

spans the 1-dimensional vector space H2ndR(CPn) over R.

30.5. THE FUNDAMENTAL GROUP π1(M) 237

30.4. Abelianisation in group theory

Definition 30.4.1. Given a group π defined by generators gi andrelations rk, one usually writes

π = 〈gi | rk〉.The commutator of two elements g, h of a group is the element

[g, h] = ghg−1h−1.

By adding the commutation relations between each pair of generators,we obtain an abelian group called the abelianisation of π.

Example 30.4.2. The group Z can be presented as follows:

Z = 〈g1 | ∅〉.Example 30.4.3. The group Zn = Z /nZ can be presented as fol-

lows:

Z /nZ = 〈g1 | gn1 〉,where the right-hand side is written multiplicatively. Thus, we canthink of it as the subgroup of n-th roots of unity in S1 ⊆ C.

Example 30.4.4. The group Z2 can be presented as follows:

Z2 = 〈g1, g2 | [g1, g2]〉.Definition 30.4.5. Given a group π = 〈gi | rk〉, with neutral

element 1, its abelianisation πab is the group

πab = 〈gi | rk, [gi, gj] = 1〉obtained by adding commutation relations for each pair of generators.

Example 30.4.6. The abelianisation of the fundamental group ofa surface of genus g is Z2g.

30.5. The fundamental group π1(M)

This is the group defined as follows. We consider based loops in(M, ∗) where ∗ is a fixed basepoint, and define equivalence using ho-motopy as in Nowik’s class.

Remark 30.5.1. A based loop defines a trivial class in the fun-damental group if and only if the map S1 → M can be “filled” by adisk D, i.e., extended to a map D → M where ∂D = S1.

Example 30.5.2. The fundamental group of RP2 is the finite cyclicgroup Z2, while the fundamental group of T2 is the infinite group Z2.


30.6. The second homotopy group π2(M)

Similarly to the fundamental group π1(M), one defines the secondhomotopy group π2(M) as the group generated by based maps

S2 → M

modulo based homotopy connecting a pair of based maps. Similarly tothe case of π1, a map belongs to a trivial class in π2(M) if it can beextended to a map of the 3-ball: B3 →M .

Remark 30.6.1. Unlike the fundamental group, the second homo-topy group π2(M) is always abelian.

Example 30.6.2. The second homotopy group of the 2-sphere S2 =CP1 is infinite cyclic: π2(S

2) = Z.

More generally, we have the following example.

Example 30.6.3. The second homotopy group of the CPn is infinitecyclic for all n: π2(CPn) = Z.

Discussion of long exact sequence of fibration and application inthe case of the Hopf fibration.

30.7. First singular homology group

The singular homology groups with integer coefficients, Hk(M ;Z)for k = 0, 1, . . . of M are abelian groups which are homotopy invariantsof M . Developing the singular homology theory is time-consuming.The case that we will be primarily interested in as far as these notesare concerned, is that of the 1-dimensional homology group:

H1(M ;Z)

(as well as the 2-dimensional homology group H2(M ;Z), see below).In this case, the homology groups can be characterized easily withoutthe general machinery of singular simplices.

Let S1 ⊆ C be the unit circle, which we think of as a 1-dimensionalmanifold with an orientation given by the counterclockwise direction.

Remark 30.7.1. The existence of orientations on parametrizedloops in C is familiar from the course in complex functions. See Sec-tion 32.1 for a more general treatment of orientations.

Definition 30.7.2. A 1-cycle α on a manifold M is an integerlinear combination

α =∑

i

nifi

30.7. FIRST SINGULAR HOMOLOGY GROUP 239

where ni ∈ Z is called the multiplicity (ribui), while each

fi : S1 →M

is a loop given by a smooth map from the circle to M , and each loopis endowed with the orientation coming from S1.

Definition 30.7.3. The space of 1-cycles onM is denoted Z1(M ;Z).

Let (Σg, ∂Σg) be a surface with boundary ∂Σg , where the genus gis irrelevant for the moment and is only added so as to avoid confusionwith the summation symbol

∑.

The boundary ∂Σg is a disjoint union of circles.

Theorem 30.7.4. Assume the surface Σg is oriented. Then theorientation induces an orientation on each boundary circle.

See Section 32.1 for a more detailed treatment of orientations.Thus we obtain an orientation-preserving identification of each bound-

ary component with the standard unit circle S1 ⊆ C (with its counter-clockwise orientation).

Given a map Σg → M , its restriction to the boundary thereforeproduces a 1-cycle

∂Σg ∈ Z1(M ;Z),

and 1-boundaries are precisely the 1-cycles that can be obtained in thisway.

Definition 30.7.5. A 1-boundary in M is a 1-cycle∑

i

nifi ∈

Z1(M ;Z) such that there exists a map of a suitable oriented sur-face Σg →M (for some g which may depend on the 1-cycle) satisfying

∂Σg =∑

i

nifi.

Definition 30.7.6. The space of all 1-boundaries is denoted

B1(M ;Z) ⊆ Z1(M ;Z).

Definition 30.7.7. The 1-dimensional homology group of M withinteger coefficients is the quotient group

H1(M ;Z) = Z1(M ;Z)/B1(M ;Z).

Definition 30.7.8. Given a cycle C ∈ Z1(M ;Z), its homologyclass will be denoted [C] ∈ H1(M ;Z).

We see that the definition of the homology group is somewhat moreinvolved than that of the fundamental group. The relation betweenthem is given by the following theorem.


Theorem 30.7.9. The 1-dimensional homology group H1(M ;Z) isthe abelianisation of the fundamental group π1(M):

H1(M ;Z) = (π1M)ab .

Remark 30.7.10. A significant difference between the fundamen-tal group and the first homology group is the following. While onlybased loops participate in the definition of the fundamental group, thedefinition of H1(M ;Z) involves free (not based) loops.

Example 30.7.11. The fundamental groups of the real projectiveplane RP2 and the 2-torus T2 are already abelian. Therefore one ob-tains

H1(RP2;Z) = Z /2Z,and

H1(T2;Z) = Z2 .

Example 30.7.12. The fundamental group of an orientable closedsurface Σg of genus g is known to be a group on 2g generators with asingle relation which is a product of g commutators. Therefore one has

H1(Σg;Z) = Z2g .

CHAPTER 31

Stable norm in homology

31.1. Length, area, volume of cycles

Assume the manifold M has a Riemannian metric. Given a smoothloop f : S1 → M , we can measure its volume (length) with respect tothe metric of M . We will denote this length by

vol(f)

with a view to higher-dimensional generalisation. Namely,

vol(f) =

∫ √gij αi

′(t)αj ′(t)dt

where αi are the components of the loop with respect to a suitablecoordinate chart. As usual, if the loop travels through several charts,we work with partitions.

Definition 31.1.1. The volume (length) of a 1-cycle C =∑

i nifiis the combined length of all the individual loops, with multiplicity:

vol(C) =∑

i

|ni| vol(fi).

Definition 31.1.2. Let α ∈ H1(M ;Z) be a 1-dimensional homol-ogy class. We define the volume of α as the infimum of volumes ofrepresentative 1-cycles:

vol(α) = inf vol(C) | C ∈ α ,where the infimum is over all cycles C =

∑i nifi representing the

class α ∈ H1(M ;Z).

Example 31.1.3. In the case of the real projective plane, there isonly one nontrivial class. The volume of this class with respect to agiven metric g on RP2 is then the least length of a noncontractible loopfor this metric.

31.2. Length in 1-dimensional homology of surfaces

The following phenomenon occurs for orientable surfaces.

241

242 31. STABLE NORM IN HOMOLOGY

Theorem 31.2.1. LetM be an orientable surface, i.e. 2-dimensionalmanifold. Let α ∈ H1(M ;Z). For all j ∈ N, we have

vol(jα) = j vol(α), (31.2.1)

where jα denotes the class α+ α + . . .+ α, with j summands.

Proof. To fix ideas, let j = 2. By Lemma 31.2.2 below, a mini-mizing loop C representing a multiple class 2α will necessarily intersectitself in a suitable point p. Then the 1-cycle represented by C can bedecomposed into the sum of two 1-cycles (where each can be thoughtof as a loop based at p). The shorter of the two will then give a min-imizing loop in the class α which proves the identity (31.2.1) in thiscase. The general case follows similarly.

We have the following elementary fact about the topology of a cylin-der .

Lemma 31.2.2. A loop going around a cylinder twice necessarilyhas a point of self-intersection.

Proof. We think of the loop as the graph of a 4π-periodic func-tion f(t) (or alternatively a function on [0, 4π] with equal values at theendpoints). Consider the difference g(t) = f(t) − f(t + 2π). Then gtakes both positive and negative values. Thus, if g(t) is positive,then g(t+ π) is negative.

Therefore by the intermediate function theorem, the function gmust have a zero t0. Then f(t0) = f(t0 + 2π) hence t0 is a pointof self-intersection of the loop.

Another way of writing this identity is as follows:

vol(α) =1

jvol(jα). (31.2.2)

31.3. Stable norm for higher-dimensional manifolds

The phenomenon captured by (31.2.2) is no longer true for higher-dimensional manifolds. Namely, the volume of a homology class is nolonger multiplicative. However, the limit as j → ∞ exists and is calledthe stable norm.

Definition 31.3.1. Let M be a compact manifold of arbitrarydimension. The stable norm ‖ ‖ of a class α ∈ H1(M ;Z) is the limit

‖α‖ = limj→∞

1

jvol(jα). (31.3.1)

31.4. STABLE 1-SYSTOLE 243

Remark 31.3.2. It is obvious from the definition that one has

‖α‖ ≤ vol(α).

However, the inequality may be strict in general. As noted above, for2-dimensional manifolds we have ‖α‖ = vol(α).

Proposition 31.3.3. The stable norm vanishes for a class of finiteorder.

Proof. If α ∈ H1(M ;Z) is a class of finite order, one has finitelymany possibilities for vol(jα) as j varies. The factor of 1

jin (31.3.1)

shows that ‖α‖ = 0.

Similarly, if two classes differ by a class of finite order, their stablenorms coincide. Thus the stable norm passes to the quotient latticedefined below.

Definition 31.3.4. The torsion subgroup of H1(M ;Z) will be de-noted T1(M) ⊆ H1(M ;Z). The quotient lattice L1(M) is the lattice

L1(M) = H1(M ;Z)/T1(M).

Proposition 31.3.5. The lattice L1(M) is isomorphic to Zb1(M),where b1 is called the first Betti number of M .

Proof. This is a general result in the theory of finitely generatedabelian groups.

31.4. Stable 1-systole

Definition 31.4.1. Let M be a manifold endowed with a Riemann-ian metric, and consider the associated stable norm ‖ ‖. The stable 1-systole of M , denoted stsys1(M), is the least norm of a 1-homologyclass of infinite order:

stsys1(M) = inf‖α‖ | α ∈ H1 (M ;Z) \ T1(M)

= λ1(L1(M), ‖ ‖

).

The 1-systole sys1(M, g) of a metric g on a surface M is the leastlength of a noncontractible loop on M .

Example 31.4.2. For an arbitrary metric on the 2-torus T2, the1-systole and the stable 1-systole coincide by Theorem 31.2.1:

sys1(T2) = stsys1(T

2),

for every metric on T2.


31.5. 2-dimensional homology and systole

Let M be a manifold. An example of particular interest to us is thecomplex projective space M = CPn.

Definition 31.5.1. A 2-cycle in M is a linear combination∑nifi,

where ni ∈ Z, of maps fi : Σgi → M of closed surfaces of arbitrarygenus into M .

A significant difference from the 1-dimensional case is that we allowarbitrary genus, whereas in the 1-dimensional case we used a uniquemanifold (the circle). Thus, each fi is a map fi : Σgi → M where Σgi

is a closed surface of genus gi.A singular 1-boundary was defined in an earlier section as the

boundary of a map from a surface to M . We define 2-boundaries in asimilar way.

Definition 31.5.2. A 2-boundary is a sum∑nifi realizable as the

boundary of a map of a 3-manifold into M .

For the orientation induced on the boundary, see Section 32.1 con-taining a more detailed treatment of orientations.

Let Z2(M ;Z) andB2(M ;Z) be the spaces of 2-cycles and 2-boundaries,respectively. Then 2-dimensional homology is defined as before as thequotient group

H2(M ;Z) = Z2(M ;Z)/B2(M ;Z).

Example 31.5.3. We have H2(Sn;Z) = Z if n = 2 and 0 otherwise.

Example 31.5.4. We have H2(CPn;Z) = Z for all n = 1, 2, . . ..

The stable norm and the stable systole are defined identically tothe 1-dimensional case. Thus, we have

stsys2(M) = inf‖α‖

∣∣ α ∈ H2 (M ;Z) \ T2(M)

= λ1(L2(M), ‖ ‖

),

where L2(M) is the lattice given by the quotient by the torsion sub-group T2 ⊆ H2(M ;Z). The stable norm is defined in the next section.

31.6. Stable norm in two-dimensional homology

If M is a manifold with a metric g, the stable norm ‖ ‖ in H2(M ;Z)is defined in a way similar to the case k = 1, using the stabilisationformula (31.2.2). To define the area of a map into M , we proceedas follows. Given a domain U ⊆ R2 with coordinates (u1, u2), and amap x : U → M , we form the tangent vectors x1 = ∂f

∂u1and x2 = ∂f

∂u2,

31.7. PU’S INEQUALITY 245

in TpM , calculate the area ‖x1 ∧ x2‖ of the parallelogram in TpMspanned by x1 and x2 with respect to the metric on M . The area isthen the integral ∫

U

‖x1 ∧ x2‖du1du2.

The area of the parallelogram can be calculated as usual as

‖x1 ∧ x2‖ = |x1||x2| sinαwhere α is the angle between them. Here |xi| = g(xi, xi)

1/2, and

cosα =g(x1, x2)

|x1| |x2|where g is the metric on M .

Gromov’s stable systolic inequality (see chapter 32.5) for complexprojective space is a higher dimensional analogue of Pu’s systolic in-equality for the real projective space, which we review below.

31.7. Pu’s inequality

Recall that the real projective plane RP2 is a smooth non-orientablesurface. Among its elementary properties are the following.

Theorem 31.7.1. The real projective plane has the following prop-erties:

(1) π1(RP2) = H1(RP2;Z) = Z2;(2) The natural orientable double cover of RP2 is the sphere S2.(3) Every metric g on RP2 naturally lifts to a Z2-invariant met-

ric g on the double cover S2.

Thus we have a smooth map

S2 → RP2

which is 2-to-1. In other words, RP2 is the quotient of S2 by the actionof Z /2Z where the nontrivial element

τ : S2 → S2 (31.7.1)

is the antipodal map of S2, namely τ(p) = −p when S2 is thought ofas the unit sphere in R3.

Definition 31.7.2. The systole sys1(RP2) can be thought of in two

equivalent ways:

(1) the least g-length of a loop representing the nontrivial homol-ogy class in H1(RP2;Z);

(2) the least g-length of a path joining a pair of antipodal points, pand τ(p), of the double cover.


Theorem 31.7.3 (P. M. Pu). Every metric g on the real projectiveplane RP2 satisfies the optimal inequality

sys1(g)2 ≤ π

2area(g),

where sys1 is the systole. The boundary case of equality is attainedprecisely when the metric is of constant Gaussian curvature.

Remark 31.7.4. There is an intriguing similarity between Pu’s in-equality and the isoperimetric inequality for Jordan curves in the plane.Both inequalities relate a length and an area, and both are sharp.

Our main interest in Pu’s inequality in this chapter is as motiva-tion for the complex case. In the complex case, we have an analogousinequality

stsys2(g)2 ≤ 2 vol(g),

for every metric g on CP2, with equality for the Fubini-Study metric.Before stating and proving Gromov’s stable systolic inequality for

complex projective space, we review some material from calculus onmanifolds.

31.8. Integration of a 1-form over a circle

In previous chapters, we have developed a cohomology theory (deRham cohomology), and a homology theory (singular homology). Inthis section we will begin to relate the two theories.

Consider a circle C(r) of radius r > 0 in C. The counterclockwiseorientation turns the circle into a 1-cycle. Thus we can think of C(r) asan element of the space of cycles with integer coefficients, Z1(C(r);Z).

Consider the polar coordinates (r, θ) in C. The differential 1-form dθis defined everywhere in C \ 0. We will denote its restriction to thecircle C(r) ⊆ C by the same symbol dθ.

The function θ itself is a multi-valued function on C \ 0 and alsoon the circle C(r). We can parametrize the circle by the interval [0, 2π],for instance by means of the parametrisation

reiθ, θ ∈ [0, 2π]. (31.8.1)

Thus we can think of θ as a single-valued function on the circle with apoint removed:

θ : C(r) \ rei0 → R .

Here the point rei0 ∈ R ⊆ C corresponds to the values 0 and 2π of theparameter θ.

31.9. VOLUME FORM OF A SURFACE 247

Remark 31.8.1. The closed 1-form dθ, inspite of appearances, isnot exact, i.e. it is not a coboundary. This is because the function θis not a true function on the circle but only a multi-valued one. Infact, rdθ is the volume form of Cr.

Proposition 31.8.2. We have the relation∫

Cr

dθ = 2π,

independent of r.

Proof. We use the parametrisation (31.8.1) to perform the inte-gration using the fundamental theorem of calculus as follows:

∫

Cr

dθ = θ]2π0 = 2π ,


Remark 31.8.3. The residue calculation of 1z

at the origin gives∮

|z|=r

dz

z= i

∫ 2π

0

dθ = 2πi,

showing that the residue is 1 by Cauchy’s theorem.

31.9. Volume form of a surface

The area of a surface is calculated by means of a familar formulafrom classical differential geometry. The area of a surface in a coordi-nate chart U ⊆ R2 with coordinates x = u1, y = u2, with respect tothe first fundamental form gij is expressed by a double integral

area =

∫

U

√g11g22 − g12g21 dxdy.

From the differential geometric viewpoint, a more correct form of theexpression for the area would be in terms of the absolute value of theintegral of a 2-form:

area =

∣∣∣∣∫

U

√det(gij) dx ∧ dy

∣∣∣∣ ,

where the symbol “∧” stands for the wedge product. In other words,the area 2-form dA of the surface is the form dA :=

√det(gij) dx∧ dy.

Definition 31.9.1. The expression√

det(gij) dx ∧ dy, or alterna-tively f 2dx ∧ dy, is called the area form or the volume form.


Suppose the metric is expressed in isothermal coordinates in termsof a conformal factor f(x, y) > 0, so that

g = f 2(dx2 + dy2)

Remark 31.9.2. Here the squares of dx and dy in the expressionfor the metric g are the symmetric, not alternating, powers.

Then the area becomes

area =

∫

U

f 2(x, y) dx ∧ dy.

In Section 32.2 we will define the notion of the volume form foran arbitrary Riemannian manifold. The key property of the volumeform η of a Riemannian manifold M is the expression for the totalvolume vol(M) of M in terms of the volume form:

vol(M) =

∫

M

η.

In line with the traditional notation of calculus, the volume form issometimes denoted

dvolM ,

so that we have an identity

vol(M) =

∫

M

dvolM .

For any function h ∈ C∞(M) on the manifold, we have the obviousbound ∫

M

h dvolM ≤ (max h) vol(M),

where maxh is the maximal value of h on M .

31.10. Duality of homology and de Rham cohomology

We have defined k-cycles only in the case k = 1 and k = 2. Thesetwo values are meant throughout this section.

Recall that Ωk(M) on M is the space of sections of the exteriorbundle of the manifold M . As illustrated in the previous sections,a differential k-form η ∈ Ωk(M) on M can be integrated over a k-cycle C ∈ Zk(M) to obtain a real number

∫

C

η ∈ R .

Integration thus defines a pairing, which we will denote by the in-tegration symbol

∫.

31.10. DUALITY OF HOMOLOGY AND DE RHAM COHOMOLOGY 249

Definition 31.10.1. The pairing is between the space Ωk(M) ofdifferential forms on M , and the space Zk(M ;Z) of k-cycles in M :∫

: Ωk(M) × Zk(M ;Z) → R .

Theorem 31.10.2 (Stokes theorem). Let M be a closed manifold.Let η ∈ Ωk(M) be a differential form. Assume η is closed, i.e. dη = 0.Let C ∈ Zk(M ;Z) be a k-cycle. Then the value of the integral

∫Cη only

depends on the cohomology class ω = [η] ∈ HkdR(M) and the homology

class α = [C] ∈ Hk(M ;Z).

Moreover, the integral vanishes over any torsion class. Thereforewe obtain a pairing ∫

: HkdR(M) × Lk(M) → R,

where Lk(M) = Hk(M ;Z)/Tk(M).

Definition 31.10.3. We define an integral lattice LkdR(M) in deRham cohomology Hk

dR(M) to be the lattice dual to the homology lat-tice Lk(M) (here duality is understood in the sense of Definition 25.7.1).In more detail, we have

LkdR(M) =

ω ∈ Hk

dR(M)

∣∣∣∣∫

C

η ∈ Z (∀C ∈ α ∈ Lk(M)) (∀η ∈ ω)

.

Alternatively, choosing a basis (y1, . . . , yb) for Lk(M), we can definethe dual lattice as follows:

LkdR(M) =

ω ∈ Hk

dR(M)

∣∣∣∣∫

yi

ω ∈ Z ∀i = 1, . . . , b

.

Theorem 31.10.4. The pairing∫

between Lk(M) and LkdR(M) isnon-degenerate for every closed orientable n-manifold M .

CHAPTER 32

Orientation and fundamental class

32.1. Orientation and induced orientation

Let M be an n-dimensional manifold, so that for every p ∈ M , wehave dim(T ∗

pm) = n. The n-th exterior algebra at p is 1-dimensional,i.e.,

dim Λn(T ∗pM) ≃ R .

Thus, any n-form η ∈ Ωn(M) locally can be written as

η = f(u1, . . . , un)du1 ∧ · · · ∧ dun.In this section, we will only be concerned with n-forms which are

nonvanishing at every point, i.e., f(u1, . . . , un) 6= 0. Such an n-form iscalled a volume form. We will say that two volume forms η and η′ areequivalent if they differ by a positive function g = g(p) > 0 on M :

η ∼ η′ ⇐⇒ η(p) = g(p)η′(p).

Definition 32.1.1. An orientation on M is an equivalence class ofvolume forms on M .

Example 32.1.2. On the circle S1, we have the orientation givenby dθ (or any positive multiple of it), and the “opposite” orientationgiven by −dθ. This can be thought of as an arrow indicating theclockwise or counterclockwise direction.

Recall that an n-form η can be thought of as an antisymmetricmultilinear form. Thus, for a 2-form we have η = η(v, w).

Definition 32.1.3. Let v be a vector and η = η a 2-form. Theinterior product

vy η ∈ Ω1

is a 1-form, defined by setting

(vy η)(·) = η(v, ·),where the dot denotes the variable.

In other words, (vy η)(w) = η(v, w), for every vector w. One sim-ilarly defines the interior product for n-forms, by substituting v intothe first variable. Then vy η ∈ Ωn−1.

251

252 32. ORIENTATION AND FUNDAMENTAL CLASS

Example 32.1.4. Consider the area form η = dx∧ dy in the plane.If v = ∂

∂xthen vy η = dy. Similarly, the orthonormal basis

∂∂ry η = rdθ

since dr, rdθ is an orthonormal basis for the cotangent plane.

Let M be an n-manifold with boundary. Thus, we have a bound-ary S ⊆ M where S is (n − 1)-dimensional, and a neighborhood ofthe boundary has the form I × N where I = [0, 1] is an interval.Let u1, . . . , u

n−1 be coordinates in S, and let t ∈ I parametrize theinterval, so that ∂

∂tis a vector field not tangent to the boundary.

Definition 32.1.5. Let η ∈ Ωn(M) generate an orientation on M .Then the induced orientation on the boundary S ⊆M is generated bythe interior product ∂

∂ty η ∈ Ωn−1(S).

This is the formal way of defining the induced orientation on theboundary.

32.2. Fundamental class in homology

Let M is a closed (i.e. compact without boundary) orientable man-ifold of dimension n. As in the case of the circle, a choice of an orien-tation on M gives an isomorphism Hn(M ;Z) ≃ Z.

Definition 32.2.1. The element which is a positive generator iscalled the fundamental class of M and denoted [M ] ∈ Hn(M ;Z).

Thus, [M ] corresponds to the element 1 ∈ Z under the isomorphismHn(M ;Z) ≃ Z. Similarly, we have an isomorphism

LndR(M) ≃ Z .

By the nondegeneracy of the pairing, we obtain the following.

Corollary 32.2.2. We can choose a generator ω = [η] ∈ LndR(M)in such a way as to satisfy ∫

M

η = 1. (32.2.1)

This is immediate from Theorem 31.10.4. In an earlier chapter, wedefined the notion of comass on alternating (exterior) forms.

Definition 32.2.3. A volume form is a nonzero element ζ ∈ Ωn(M)such that the pointwise comass norm ‖ζp‖ equals 1 for every p ∈ M .

The volume form satisfies∫

M

ζ = vol(M).

32.3. DUALITY OF COMASS AND STABLE NORM 253

Example 32.2.4. In the case of the unit circle S1, the 1-form ζ = dθhas unit pointwise norm, therefore∫

S1

dθ = 2π = vol(S1).

Note that the class [ζ ] is not in the lattice L1dR(S1). Meanwhile, the

form1

2πζ =

1

2πdθ

represents an element of H1dR(S1) which is a generator of the integer

lattice L1dR(S1), since it integrates to 1 over S1.

32.3. Duality of comass and stable norm

Definition 32.3.1. The comass norm ‖ ‖∞ of a differential k-formis the supremum of the pointwise comass norms, cf. Definition 26.3.2:

‖η‖∞ = sup ‖ηp‖ | p ∈M .Next, we define a notion of comass norm in de Rham cohomology,

as follows.

Definition 32.3.2. The comass norm ‖ω‖∗ of a class ω ∈ HkdR(M)

in de Rham cohomology is the infimum of comass norms ‖η‖∞ of therepresentative differential forms η ∈ ω.

Thus we have

‖ω‖∗ = infη

supp‖ηp‖ | p ∈M, η ∈ ω,

where ‖ ‖ denotes the comass norm in the exterior algebra Λ(T ∗pM).

Theorem 32.3.3 (Federer; Gromov). The stable norm ‖ ‖ on Lk(M)and the comass norm ‖ ‖∗ on LkdR(M) ⊆ Hk

dR(M) are dual to eachother.

Thus, the pair of normed lattices (Lk(M), ‖ ‖) and (LkdR(M), ‖ ‖∗)are dual to each other.

For the purpose of this introductory discussion, assume n ≥ 2.There are at least three equivalent ways of defining the Fubini-Studymetric on CPn:

(1) It is the unique metric on CPn whose sectional curvature Ksatisfies 1 ≤ K ≤ 4;

(2) it is defined by a distance function d(v, w) expressed in termsof the Hermitian inner product in homogeneous coordinates;

(3) it is given by a metric ds2 written by a specific length elementds appearing below.


The case of the complex projective line CP1 is special in that the Gauss-ian curvature K is constant.

The Fubini-Study metric on the complex projective line is dis-cussed in Section 32.4, and the Fubini-Study 2-form is discussed inSection 28.8.

32.4. Fubini-Study metric on complex projective line

The complex projective line

CP1 = C ∪∞ = S2

carries a natural metric called the Fubini-Study metric ds2.The distance function d(v, w), associated with the Fubini-Study

metric, takes its simplest form in homogeneous coordinates [Z0, Z1].Namely, given a pair of unit vectors v, w ∈ C2, we have the corre-sponding complex 1-dimensional subspaces Cv ⊆ C2 and Cw ⊆ C2,thought of as points in the projective line. The distance between themwill be denoted, briefly, d(v, w). The distance is defined in terms of thestandard Hermitian product H(v, w) as follows.

Definition 32.4.1. The distance between points v, w in CP1 isdefined by

d(v, w) = arccos |H(v, w)|, (32.4.1)

i.e.

cos d(v, w) = |H(v, w)| = |v1w1 + v2w2|.Remark 32.4.2. The distance vanishes (i.e. cos(d(v, w)) = 1) if v

and w differ by a unit complex scalar: v = λw. This is consistentwith the fact that the vectors are supposed to define the same point incomplex projective space.

In an affine neighborhood Z1 6= 0 in CP1, we introduce a coordinate

z = Z0/Z1 = x + iy.

Theorem 32.4.3. With respect to coordinate z, the Fubini-Studymetric ds2 takes the form

ds2 =dzdz

(1 + zz)2=dx2 + dy2

(1 + r2)2=

dr2

(1 + r2)2. (32.4.2)

Here r2 = zz = x2 + y2, where r is the first of the usual polar coordi-nates (r, θ).

We present some elementary global-geometric properties of the met-ric.

32.5. FUBINI-STUDY METRIC ON COMPLEX PROJECTIVE SPACE 255

Theorem 32.4.4. The maximal distance between a pair of pointsof CP1 with respect to the distance (32.4.1) is π

2, i.e. its Riemannian

diameter is π2.

Indeed, the arccosine of a positive value does not exceed π2.

Note that the unit sphere metric has Riemannian diameter π, whichis twice the value that appears in the theorem.

Theorem 32.4.5. The complex projective line CP1 with the Fubini-Study metric gFS is isometric to the sphere of radius 1

2in R3 and total

area π.

Proof. The Gaussian curvature K of the Fubini-Study metric gFSis K = 4 at every point. The curvature can be calculated directly usingthe formula K = −∆LB log f where f is the conformal factor f = 1

1+r2

as in (32.4.2). Thus we obtain a metric on the 2-sphere CP1 of constantGaussian curvature 4.

By a general result in Riemannian geometry, such a metric is iso-metric to a concentric sphere in R3. The normalisation K = 4 forcesthe radius of the sphere to be 1

2. Hence the total area is π. Alterna-

tively, this results from the Gauss-Bonnet theorem.

32.5. Fubini-Study metric on complex projective space

The Fubini-Study metric on the complex projective line was dis-cussed in Section 32.4.

The complex projective space CPn carries a natural metric calledthe Fubini-Study metric ds2. The distance function d(v, w), associatedwith the Fubini-Study metric, takes its simplest form in homogeneouscoordinates [Z0, . . . , Zn]. Namely, given a pair of unit vectors v, w ∈Cn+1, we have

cos d(v, w) = |H(v, w)| = |∑

j

vjwj|.

In an affine neighborhood Zn 6= 0, we introduce affine coordinates

zj = Zj/Zn = xj + iyj .

With respect to these coordinates, the Fubini-Study metric takes thefollowing form. The Fubini-Study metric gFS(X,X) at a point u ∈ Cn

is given by the formula

gFS(X,X) =H(X,X) (1 + |u|2) − |H(X, u)|2

1 + |u|2 . (32.5.1)

In the case n = 1, we have |H(X, u)| = |X| |u| and the formula reducesto (32.4.2).


Unlike the 1-dimensional case, sectional curvature is not constant.

Theorem 32.5.1. The sectional curvature of the Fubini-Study met-ric on CPn satisfies 1 ≤ K ≤ 4 at every point and in every 2-dimensionaldirection.

The Fubini-Study 2-form A is defined as before by the formulaA = g(Jv, w) where J is the complex structure, i.e., multiplicationby i. One can write down an explicit formula, as well.1

Proposition 32.5.2. A is a globally defined closed 2-form on CPn.

Its cohomology class [A] spans H2dR(CPn) and has unit comass at

every point.The proof of Gromov’s inequality exploits the duality between the

stable norm and the comass, defined in earlier chapters. It also relieson our generalized version of the Wirtinger inequality to obtain anoptimal constant in the inequality.

32.6. Gromov’s inequality for complex projective space

Recall that if ω is a generator of L2dR(CPn) then ωn is a fundamental

cohomology class for CPn, i.e. a generator of L2ndR(CPn).

Theorem 32.6.1 (M. Gromov). Every Riemannian metric g oncomplex projective space CPn satisfies the inequality

stsys2(g)n ≤ n! vol2n(g);

equality holds for the Fubini-Study metric on CPn.

So as to clarify the nature of the proof, we will state the followingslightly more general inequality. A similar proof works for the moregeneral theorem.

Theorem 32.6.2. Let M be a 2n-dimensional Riemannian mani-fold, satisfying the following conditions:

(1) b2(M) = 1, in other words H2dR(M) = R;

(2) If ω 6= 0 in H2dR(M) then ω∪n 6= 0 in H2n

dR(M).

Then every Riemannian metric g on M satisfies the inequality

stsys2(g)n ≤ n! vol2n(g).

1More explicitly, one can write down A in an affine neighborhood as follows:

A = i

(∑j dzj ∧ dzj

1 + r2− (∑

zjdzj) ∧(∑

zjdzj)

(1 + r2)2

)

related to (32.5.1).

CHAPTER 33

Proof of Gromov’s theorem

33.1. Homology class α and cohomology class ω

Following Gromov’s notation in [?, Theorem 4.36], we let

α ∈ H2(CPn;Z)) ≃ Z (33.1.1)

be the positive generator in homology. Recall that α is represented bythe inclusion of the 2-sphere CP1 ⊆ CPn. Let

L2dR(CPn) ≃ Z

be the lattice in de Rham cohomology H2dR(CPn) dual to H2(CPn;Z).

Let

ω ∈ L2dR(CPn)

be the dual generator in cohomology, so that∫αω = 1, where integra-

tion is understood in the sense of a representative cycle and 2-form.Here ω is represented by the form 1

πA where A is the Fubini-Study

form:

ω =[1πA]∈ H2.

Then the cup power ωn is a generator of L2ndR(CPn) ≃ Z, i.e. a fun-

damental cohomology class for CPn. Let η ∈ ω be a closed differen-tial 2-form. Since wedge product ∧ in Ω∗(X) descends to cup productin H∗

dR(X), we have from (32.2.1)

1 =

∫

CPn

η∧n. (33.1.2)

33.2. Comass and application of Wirtinger inequality

Let g be a metric on CPn. The metric defines a norm on eachtangent and cotangent space, as well as all the exterior powers. Wecan then calculate the pointwise comass ‖ηp‖ at a point p, and thecomass ‖η‖∞ of a differential form η.

The comass norm of a differential k-form is, by definition, thesupremum of the pointwise comass norms, cf. Definition 26.3.2. Let

257

258 33. PROOF OF GROMOV’S THEOREM

M = CPn. Choose a point p ∈ M . By the Wirtinger inequality andCorollary 27.2.4, we obtain the following bound for the norm at p:

‖η∧np ‖ ≤ n! (‖ηp‖)n ≤ n! (‖η‖∞)n , (33.2.1)

where ‖ ‖∞ is the comass norm on differential forms. Since this is truefor every p ∈ M , we obtain the following inequality for the comass ofthe form ηn:

‖η∧n‖∞ ≤ n! (‖η‖∞)n .

33.3. Volume calculation

Now let dvol be the volume form of the metric (M, g). At everypoint, we have

ηnp = ‖ηnp‖ dvolM .Then

1 =

∫

CPn

ηn

=

∫

CPn

‖ηn‖ dvol

≤ n! (‖η‖∞)n vol2n(CPn, g)

We therefore obtain

1 ≤ n! (‖η‖∞)n vol2n(CPn, g). (33.3.1)

This is true for every form η representing the class ω ∈ H2dR(M). The

infimum of (33.3.1) over all η ∈ ω gives the following bound involvingthe comass ‖ω‖∗ of the class ω:

1 ≤ n! (‖ω‖∗)n vol2n (CPn, g) , (33.3.2)

where ‖ ‖∗ is the comass norm in cohomology. Denote by ‖ ‖ the stablenorm in homology. Let L2(M) = H2(M ;Z). Recall that the normedlattices

(L2(M), ‖ ‖) and (L2dR(M), ‖ ‖∗)

are dual to each other [?]. Since b1(CPn) = 1, it follows that the class αof (33.1.1) satisfies

‖α‖ =1

‖ω‖∗ ,

and hencestsys2(g)n = ‖α‖n ≤ n! vol2n(g). (33.3.3)

Equality is attained by the two-point homogeneous Fubini-Study met-ric, since the standard CP1 ⊆ CPn is calibrated by the Fubini-StudyKahler 2-form, which satisfies equality in the Wirtinger inequality atevery point.

33.3. VOLUME CALCULATION 259

Example 33.3.1. Every metric g on the complex projective planesatisfies the optimal inequality

stsys2(CP2, g)2 ≤ 2 vol4(CP2, g). (33.3.4)

CHAPTER 34

Eiseinstein, Hermite, Loewner, Pu, and Guth

Here we prove the two classical results of systolic geometry, namelyLoewner’s torus inequality as well as Pu’s inequality for the real pro-jective plane, as well as a more recent result by L. Guth.

34.1. Eisenstein integers and Hermite constant

We now review some arithmetic preliminaries.

Definition 34.1.1. The lattice LE ⊆ R2 = C of the Eisensteinintegers is the lattice in C spanned by the elements 1 and the sixthroot of unity.

To visualize the lattice, start with an equilateral triangle in C with

vertices 0, 1, and 12

+ i√32

, and construct a tiling of the plane by repeat-edly reflecting in all sides. The Eisenstein integers are by definition theset of vertices of the resulting tiling.

Definition 34.1.2. Let b ∈ N. The Hermite constant γb is definedin one of the following two equivalent ways:

(1) γb is the square of the biggest first successive minimum λ1(L)among all lattices L of unit covolume;

(2) γb is defined by the formula

√γb = sup

λ1(L)

vol(Rb /L)1/b

∣∣∣∣L ⊆ (Rb, | |), (34.1.1)

where the supremum is extended over all lattices L in Rb witha Euclidean norm | |.

A lattice realizing the supremum is called a critical lattice. A crit-ical lattice may be thought of as the one realizing the densest packingin Rb when we place balls of radius 1

2λ1(L) at the points of L.

The lattice of Eisenstein integers is critical.

261

262 34. EISEINSTEIN, HERMITE, LOEWNER, PU, AND GUTH

34.2. Standard fundamental domain and Eisenstein integers

Definition 34.2.1. The standard fundamental domain D ⊆ C isthe domain

D =z ∈ C

∣∣ |z| > 1, |ℜ(z)| < 12, ℑ(z) > 0

(34.2.1)

It is a fundamental domain for the action of the group PSL(2,Z)in the upperhalf plane of C.

Lemma 34.2.2. Given a lattice L ⊆ C, one can choose a basis τ, 1for a homothetic (similar) lattice in such a way that τ ∈ D.

Proof. Consider a lattice L ⊆ C = R2. Choose a “shortest” vec-tor z ∈ L, i.e. we have |z| = λ1(L). By replacing L by the lattice z−1L,we may assume that the complex number +1 ∈ C is a shortest elementin the lattice. We will denote the new lattice by the same letter L, sothat now λ1(L) = 1. Now complete the element +1 ∈ L to a Z-basis

τ,+1 (34.2.2)

for L. Consider the real part ℜ(τ). Clearly, we can adjust the basisby adding a suitable integer to τ , so as to satisfy the condition −1

2≤

ℜ(τ) ≤ 12. Since τ ∈ L, we have |τ | ≥ λ1(L) = 1. Then the basis

vector τ lies in the closure of the domain D of Definition 34.2.1.

The following result is an essential part of the proof of Loewner’storus inequality with isosystolic defect.

Lemma 34.2.3. When b = 2, we have the following value for theHermite constant: γ2 = 2√

3= 1.1547 . . .. The corresponding critical

lattice is homothetic to the Z-span of the cube roots of unity in C, i.e.the Eisenstein integers.

Proof. Clearly, multiplying L by nonzero complex numbers doesnot change the value of the quotient

λ1(L)2

area(C/L).

The imaginary part of τ chosen as in Lemma 34.2.2 satisfies ℑ(τ) ≥√32

, with equality possible in the following two cases: τ = eiπ3 ∈ D,

or τ = ei2π3 ∈ D. Finally, we calculate the area of the parallelogram

in C spanned by τ and +1, and write

area(C/L)

λ1(L)2= ℑ(τ) ≥

√3

2

to conclude the proof.

34.4. LOEWNER’S INEQUALITY WITH DEFECT 263

34.3. Loewner’s torus inequality

Recall that the systole of a Riemannian manifold M is defined tobe least length of a noncontractible loop on M . In the case of thetorus T2, the fundamental group is abelian. Hence the systole can beexpressed in this case as follows:

sys1(T2) = λ1

(H1 (T2;Z), ‖ ‖

),

where ‖ ‖ is the stable norm. In this sense, Gromov’s inequality (33.3.4)for the complex projective plane is a higher-dimensional analogue ofLoewner’s inequality.

Loewner’s torus inequality relates the total area, to the systole, i.e.least length of a noncontractible loop on the torus (T2, g).

Theorem 34.3.1 (Loewner’s torus inequality). Every metric on thetorus satisfies

area(g) −√32

sys1(g)2 ≥ 0. (34.3.1)

Theorem 34.3.2. The boundary case of equality in (34.3.1) is at-tained if and only if the metric is homothetic to the flat metric obtainedas the quotient of R2 by the lattice formed by the Eisenstein integers.

34.4. Loewner’s inequality with defect

Loewner’s torus inequality can be strengthened by introducing a“defect” (shegiya? she’erit?) term a la Bonnesen. To write it down, weneed to review the conformal representation theorem (uniformisationtheorem).

Theorem 34.4.1 (Conformal representation theorem). Every met-ric g on the torus T2 is isometric to a metric of the form

f 2(dx2 + dy2),

with respect to a unit area flat metric dx2 + dy2 on the torus T2 viewedas a quotient R2 /L of the (x, y) plane by a lattice.

See (20.9.1) for more details.The defect term in question is simply the variance (shonut) of the

conformal factor f above. The inequality with the defect term looksas follows.

Theorem 34.4.2. Every metric on the torus satisfies

area(g) −√32

sys(g)2 ≥ Var(f). (34.4.1)


Here the error term, or isosystolic defect, is given by the variance(shonut)

Var(f) =

∫

T2

(f −m)2 (34.4.2)

of the conformal factor f of the metric g = f 2(dx2 +dy2) on the torus,relative to the unit area flat metric g0 = dx2+dy2 in the same conformalclass.

Here

m =

∫

T2

fdxdy (34.4.3)

is the mean (tochelet) of f . More concretely, if (T2, g0) = R2 /Lwhere L is a lattice of unit coarea, and D is a fundamental domainfor the action of L on R2 by translations, then the integral (34.4.3) canbe written as

m =

∫

D

f(x, y)dxdy

where dxdy is the standard Lebesgue measure of R2.

34.5. Computational formula for the variance (shonut)

The proof of inequalities with isosystolic defect relies upon the fa-miliar computational formula for the variance of a random variable Xin terms of expected values.

Namely, we have the formula

Eµ(X2) − (Eµ(X))2 = Var(X), (34.5.1)

where µ is a probability measure. Here the variance is

Var(X) = Eµ((X −m)2

),

where m = Eµ(X) is the expected value (i.e. the mean) (tochelet).

34.6. An application

Keeping our differential geometric application in mind, we will de-note the random variable f .

Now consider a flat metric g0 of unit area on the 2-torus T2. Denotethe associated measure by µ. In other words, µ is given by the usualLebesgue measure dxdy. Since µ is a probability measure, we can applyformula (34.5.1) to it. Consider a metric g = f 2g0 conformal to the flatone, with conformal factor f > 0. Then we have

Eµ(f 2) =

∫

T2

f 2dxdy = area(g),

34.7. FUNDAMENTAL DOMAIN AND LOEWNER’S TORUS INEQUALITY 265

since f 2dx ∧ dy is precisely the area 2-form of the metric g.Equation (34.5.1) therefore becomes

area(g) − (Eµ(f))2 = Var(f). (34.6.1)

Next, we will relate the expected value Eµ(f) to the systole ofthe metric g. Then we will then relate (34.5.1) to Loewner’s torusinequality.

34.7. Fundamental domain and Loewner’s torus inequality

Let us prove Loewner’s torus inequality for the metric g = f 2g0using the computational formula for the variance. We first analyze theexpected value term Eµ(f) =

∫T2 fdxdy in (34.6.1).

By the proof of Lemma 34.2.3, the lattice of deck transformationsof the flat torus g0 admits a Z-basis similar to τ, 1 ⊆ C, where τbelongs to the standard fundamental domain (34.2.1). In other words,the lattice is similar to

Z τ + Z 1 ⊆ C.

Consider the imaginary part ℑ(τ) and set

σ2 := ℑ(τ) > 0.

Lemma 34.7.1. We have σ2 ≥√32, with equality if and only if τ is

the primitive cube or sixth root of unity.

Proof. This is immediate from the geometry of the fundamentaldomain.

Since g0 is assumed to be of unit area, the basis for its group ofdeck tranformations can therefore be taken to be

σ−1τ, σ−1,

where ℑ(σ−1τ) = σ. With these normalisations, we see that the flattorus is ruled by a pencil of horizontal closed geodesics, denoted

γy = γy(x),

each of length σ−1, where the “width” of the pencil equals σ, i.e. theparameter y ranges through the interval [0, σ], with γσ = γ0. Note thateach of these closed geodesics is noncontractible in T2, and thereforeby definition

length(γy) ≥ sys1(g).


By Fubini’s theorem, we pass to the iterated integral (nishneh) to ob-tain the following lower bound for the expected value:

Eµ(f) =

∫ σ

0

(∫

γy

f(x)dx

)dy

=

∫ σ

0

length(γy)dy

≥ σ sys(g),

see [?, p. 41, 44] for details. The theorem now follows from the followingmore general inequality.

Theorem 34.7.2. We have

area(g) − σ2 sys(g)2 ≥ Var(f). (34.7.1)

Proof. Substituting the inequality

Eµ(f) ≥ σ sys

into (34.6.1), we obtain the inequality

area(g) − σ2 sys(g)2 ≥ Var(f),

where f is the conformal factor of the metric g with respect to the unit

area flat metric g0. Since σ2 ≥√32

, we obtain in particular Loewner’storus inequality with isosystolic defect,

area(g) −√32

sys(g)2 ≥ Var(f). (34.7.2)

cf. [?].

34.8. Boundary case of equality

Corollary 34.8.1. A metric satisfying the boundary case of equal-ity in Loewner’s torus inequality (35.8.1) is necessarily flat and homo-thetic to the quotient of R2 by the lattice of Eisenstein integers.

Proof. If a metric f 2ds2 satisfies the boundary case of equalityin (35.8.1), then the variance of the conformal factor f must vanishby (34.7.2). Hence f is a constant function. The proof is completed byapplying Lemma 34.2.3 on the Hermite constant in dimension 2.

Now suppose τ is pure imaginary, i.e. the lattice L is a rectangularlattice of coarea 1.

Corollary 34.8.2. If τ is pure imaginary, then the metric g =f 2g0 satisfies the inequality

area(g) − sys(g)2 ≥ Var(f). (34.8.1)

34.10. TANGENT MAP 267

Proof. If τ is pure imaginary then σ =√

ℑ(τ) ≥ 1, and theinequality follows from (34.7.1).

In particular, every surface of revolution satisfies (34.8.1), since itslattice is rectangular, cf. Corollary 20.11.1.

34.9. Hopf fibration h

To prove Pu’s inequality, we will need to study the Hopf fibrationmore closely.

The circle action in C2 restricts to the unit sphere S3 ⊆ C2, whichtherefore admits a fixed point free circle action. Namely, the circle

S1 = eiθ : θ ∈ R ⊆ C

acts on a point (z1, z2) ∈ C2 by

eiθ.(z1, z2) = (eiθz1, eiθz2). (34.9.1)

The quotient manifold (space of orbits) S3/S1 is the 2-sphere S2 (fromthe projective viewpoint, the quotient space is the complex projectiveline CP1).

Definition 34.9.1. The quotient map

h : S3 → S2, (34.9.2)

called the Hopf fibration.

34.10. Tangent map

Proposition 34.10.1. Consider a smooth map f : M → N betweendifferentiable manifolds. Then f defines a natural map

df : TM → TN

called the tangent map.

Proof. We represent a tangent vector v ∈ TpM by a curve

c(t) : I →M,

such that c′(o) = v. The composite map σ = f c : I → N is a curvein N . Then the vector σ′(0) is the image of v under df :

df(v) = σ′(0),



34.11. Riemannian submersions

Consider a fiber bundle map f : M → N between closed manifolds.Let F ⊆M be the fiber over a point p ∈ N . The differential df : TM →TN vanishes on the subspace TxF ⊆ TM at every point x ∈ F . Moreprecisely, we have the vertical space (anchi)

ker(dfx) = TxF.

Given a Riemannian metric on M , we can consider the orthogonalcomplement of TxF in TxM , denoted Hx ⊆ TxM . Thus

TxM = TxF ⊕Hx

is an orthogonal decomposition.

Definition 34.11.1. A fiber bundle map f : M → N betweenRiemannian manifolds is called a Riemannian submersion if at everypoint x ∈ M , the restriction of df : Hx → Tf(x)N is an isometry.

Proposition 34.11.2. The Hopf fibration h : S3 → S2 is a Rie-mannian submersion, for which the natural metric on the base is ametric of constant Gaussian curvature +4.

The latter is explained by the fact that the maximal distance be-tween a pair of S1 orbits is π

2rather than π, in view of the fact that

a pair of antipodal points of S3 lies in a common orbit and thereforedescends to the same point of the quotient space.

A pair of points at maximal distance is defined by a pair v, w ∈ S3

such that w is orthogonal to Cv.

34.12. Hamilton quaternions

The quaternionic viewpoint will be applied in Section 36.7. Thealgebra of the Hamilton quaternions is the real 4-dimensional vectorspace with real basis 1, i, j, k, so that

H = R 1 + R i+ R j + R k

equipped with an associative (chok kibutz), non-commutative productoperation. This operation has the following properties:

(1) the center of H is R 1;(2) the operation satisfies the relations i2 = j2 = k2 = −1 and ij =

−ji = k, jk = −kj = i, ki = −ik = j.

Lemma 34.12.1. The algebra H is naturally isomorphic to the com-plex vector space C2 via the identification R 1 + R i ≃ C.

34.13. COMPLEX STRUCTURES ON THE ALGEBRA H 269

Proof. The subspace R j + R k can be thought of as

R j + R ij = (R 1 + R i)j,

and we therefore obtain a natural isomorphism

H = C1 + Cj,

showing that 1, j is a complex basis for H.

Theorem 34.12.2. Nonzero quaternions form a group under quater-nionic multiplication.

Proof. Given q = a+bi+cj+dk ∈ H, let N(q) = a2+b2+c2+d2,and q = a − bi − cj − dk. Then qq = N(q). Therefore we have amultiplicative inverse

q−1 =1

N(q)q,


34.13. Complex structures on the algebra H

Definition 34.13.1. A purely imaginary Hamilton quaternion q ∈H0 in H = R 1 + R i + R j + R k, is a real linear combination of i, j,and k.

Thus H0 ⊆ H is a real 3-dimensional subspace.

Proposition 34.13.2. A choice of a purely imaginary quaternion q ∈H0 specifies a natural complex structure on H.

Proof. We can assume without loss of generality that |q| = 1.We then have q2 = −1 and hence R 1 + R q is naturally isomorphicto C. Choosing a unit-norm quaternion r ∈ H orthogonal to the real2-dimensional subspace R 1 + R q ⊆ H, we obtain a natural isomor-phism H ≃ C + Cr.

CHAPTER 35

From surface tension to Loewner’s inequality

This chapter contains some introductory remarks.

35.1. Surface tension and shape of a water drop

Perhaps the most familiar physical manifestation of the 3-dimensionalisoperimetric inequality is the shape of a drop of water. Namely, a dropwill typically assume a symmetric round shape. Since the amount ofwater in a drop is fixed, surface tension forces the drop into a shapewhich minimizes the surface area of the drop, namely a round sphere.Thus the round shape of the drop is a consequence of the phenomenonof surface tension (metach panim). Mathematically, this phenomenonis expressed by the isoperimetric inequality.

35.2. Isoperimetric inequality in the plane

The solution to the isoperimetric problem in the plane is usuallyexpressed in the form of an inequality that relates the length L of aclosed curve and the area A of the planar region that it encloses. Theisoperimetric inequality states that

4πA ≤ L2,

and that the equality holds if and only if the curve is a round circle.The inequality is an upper bound for area in terms of length. It canbe rewritten as follows:

L2 − 4πA ≥ 0.

35.3. Central symmetry

Recall the notion of central symmetry: a Euclidean polyhedron iscalled centrally symmetric if it is invariant under the “antipodal” map

x 7→ −x.Thus, in the plane central symmetry is the rotation by 180 degrees.For example, an ellipse is centrally symmetric, as is any ellipsoid in3-space.

271

272 35. FROM SURFACE TENSION TO LOEWNER’S INEQUALITY

35.4. Property of a centrally symmetric polyhedron in3-space

There is a geometric inequality that is in a sense dual to the isoperi-metric inequality in the following sense. Both involve a length and anarea. The isoperimetric inequality is an upper bound for area in termsof length. There is a geometric inequality which provides an upperbound for a certain length in terms of area. More precisely it can bedescribed as follows.

Any centrally symmetric convex body of surface area A can besqueezed through a noose of length

√πA, with the tightest fit achieved

by a sphere. This property is equivalent to a special case of Pu’sinequality (see below), one of the earliest systolic inequalities.

Example 35.4.1. An ellipsoid is an example of a convex centrallysymmetric body in 3-space. It may be helpful to the reader to developan intuition for the property mentioned above in the context of thinkingabout ellipsoidal examples.

An alternative formulation is as follows. Every convex centrallysymmetric body P in R3 admits a pair of opposite (antipodal) pointsand a path of length L joining them and lying on the boundary ∂Pof P , satisfying

L2 ≤ π

4area(∂P ).

35.5. Notion of systole

The systole of a compact metric space X is a metric invariant of X ,defined to be the least length of a noncontractible loop in X . We willdenote it as follows:

sys(X).

When X is a graph, the invariant is usually referred to as the girth,ever since the 1947 article by W. Tutte [?]. Possibly inspired by Tutte’sarticle, Loewner started thinking about systolic questions on surfacesin the late 1940s, resulting in a 1950 thesis by his student P.M. Pu [?].The actual term “systole” itself was not coined until a quarter centurylater, by Marcel Berger.

This line of research was, apparently, given further impetus by aremark of Rene Thom, in a conversation with Berger in the library ofStrasbourg University during the 1961-62 academic year, shortly afterthe publication of the papers of R. Accola and C. Blatter. Referringto these systolic inequalities, Thom reportedly exclaimed: “Mais c’estfondamental!” [These results are of fundamental importance!]

35.7. PU’S INEQUALITY 273

Subsequently, Berger popularized the subject in a series of articlesand books, most recently in the march ’08 issue of the Notices of theAmerican Mathematical Society [?]. A bibliography at the Websitefor systolic geometry and topology currently contains over 160 articles.Systolic geometry is a rapidly developing field, featuring a number ofrecent publications in leading journals. Recently, an intriguing link hasemerged with the Lusternik-Schnirelmann category. The existence ofsuch a link can be thought of as a theorem in “systolic topology”.

35.6. The real projective plane RP2

In projective geometry, the real projective plane RP2 is defined asthe collection of lines through the origin in R3. The distance functionon RP2 is most readily understood from this point of view. Namely,the distance between two lines through the origin is by definition theangle between them, or more precisely the lesser of the two angles.This distance function corresponds to the metric of constant Gaussiancurvature +1.

Alternatively, RP2 can be defined as the surface obtained by iden-tifying each pair of antipodal points on the 2-sphere.

Other metrics on RP2 can be obtained by quotienting metrics on S2

imbedded in 3-space in a centrally symmetric way, see the discussionin Section 35.4.

Topologically, RP2 can be obtained from the Mobius strip by at-taching a disk along the boundary.

Among closed surfaces, the real projective plane is the simplestnon-orientable such surface.

35.7. Pu’s inequality

Pu’s inequality applies to general Riemannian metrics on RP2.A student of Charles Loewner’s, P.M. Pu proved in a 1950 thesis

(published in 1952) that every metric g on the real projective plane RP2

satisfies the optimal inequality

sys(g)2 ≤ π

2area(g),

where sys is the systole. The boundary case of equality is attained pre-cisely when the metric is of constant Gaussian curvature. Alternatively,the inequality can be presented as follows:

area(g) − 2

πsys(g)2 ≥ 0.

There is a vast generalisation of Pu’s inequality, due to M. Gromov,called Gromov’s systolic inequality for essential manifolds. To state his

274 35. FROM SURFACE TENSION TO LOEWNER’S INEQUALITY

result, we will need to review a topological notion of being essential inChapter 37.

35.8. Loewner’s torus inequality

Similarly to Pu’s inequality, Loewner’s torus inequality relates thetotal area, to the systole, i.e. least length of a noncontractible loop onthe torus (T2, g):

area(g) −√32

sys(g)2 ≥ 0. (35.8.1)

The boundary case of equality is attained if and only if the metric ishomothetic to the flat metric obtained as the quotient of R2 by thelattice formed by the Eisenstein integers.

35.9. Bonnesen’s inequality

The classical Bonnesen inequality [?] is the strengthened isoperi-metric inequality

L2 − 4πA ≥ π2(R− r)2, (35.9.1)

see [?, p. 3]. Here A is the area of the region bounded by a closed Jordancurve of length (perimeter) L in the plane, R is the circumradius of thebounded region, and r is its inradius. The error term π2(R − r)2 onthe right hand side of (35.9.1) is traditionally called the isoperimetricdefect (shegiya? she’erit?)

A similar strengthening of Loewner’s inequality exists and will bediscussed in Section

CHAPTER 36

Pu’s inequality

Pu’s inequality for the real projective plane is the inequality

sys1(g)2 ≤ π

2area(g)

for every metric g on RP2. The case of equality is attained preciselyby the metrics of constant Gaussian curvature on RP2.

The proof of the inequality exploits certain properties of circle fi-brations of the 3-sphere. A combination of a double fibration of the 3-sphere and a suitable integral-geometric identity will yield a proof ofPu’s inequality.

36.1. From complex structure to fibration

Recall that we have H = R+H0 where R = R 1 is the centerwhile H0 is the space of pure imaginary quaternions. A complex struc-ture on H ≃ R4 defined by q ∈ H0 leads to a fibration

fq : S3 → S2

of the unit 3-sphere, as in (34.9.2). Each fiber of fq is a circle S1 ⊆ S3

which is an orbit of the action (34.9.1). It will be convenient in thesequel to denote this data by a two-arrow diagram of the followingtype:

S1 → S3 → S2, (36.1.1)

where the first arrow denotes the inclusion of a fiber, while the secondarrow denotes the Hopf fibration.

Lemma 36.1.1. Choosing a pair of pure imaginary quaternions q, r ∈H0 which are orthogonal, results in a pair of circle fibrations fq, fr of S

3

such that a fiber of one fibration is perpendicular to a fiber of the otherfibration whenever two such fibers meet.

Such a pair of fibrations, illustrated in Figure 36.1.1 following thenotation of (36.1.1), will play a crucial role in the proof of Pu’s in-equality.

275

276 36. PU’S INEQUALITY

S1

AAA

AAAA

A S2

S3

fr>>

fq

AAA

AAAA

A

S1

>>S2

Figure 36.1.1. A pair of fibrations

36.2. Lie groups

A Lie group is simultaneously a group and a smooth manifold, insuch a way that the two structures are compatible. More precisely, wehave the following definition.

Definition 36.2.1. A Lie group is a manifold G with an associative(chok kibutz) operation µ : G × G → G and inverse ν : G → G withthe following properties:

(1) the operation defines the structure of a group on G, e.g. thereexists an element e ∈ G such that µ(e, x) = x for all x ∈ G,and furthermore µ(x, ν(x)) = e;

(2) both µ and ν are smooth maps.

Corollary 36.2.2. The manifold S3 admits the structure of a Liegroup.

Proof. Nonzero quaternions form a group by Theorem 34.12.2.The multiplication restricts to the unit sphere S3 ⊆ R4 ≃ H by therule q−1 = q for q ∈ S3.

36.3. A special isomorphism

Definition 36.3.1. Let M be a Riemannian manifold. The unitsphere tangent bundle T uM , or the unit tangent bundle for short, isthe subset of TM consisting of elements of unit norm:

T uM = v ∈ TM | |v| = 1.The sphere S3 can be thought of as the Lie group formed of the

unit (Hamilton) quaternions in H ≃ C2.In the sequel, an important role will be played by the Lie group SO(3).

It turns out SO(3) can be identified with S3/±1. Therefore the fi-bration (34.9.2) descends to a fibration

SO(3) → S2, (36.3.1)

36.4. SPHERE AS HOMOGENEOUS SPACE 277

exploited in the next section.

Lemma 36.3.2. The construction of two “perpendicular” fibrationsof Section 34.13 descends to the quotient by the natural Z2 action on S3,as well.

Proposition 36.3.3. The group SO(3) can be identified with theunit tangent bundle of S2.

Proof. Let G = SO(3). We will construct a natural identification

F : G→ T u(S2).

The group G acts naturally on the unit sphere S2 ⊆ R3 by matrixmultiplication. We choose a fixed unit tangent vector v ∈ T u(S2).

To construct the required identification f , we send an element g ∈G, acting on S2, to the image of v under the tangent map dg : TS2 →TS2. Namely,

F (g) = dg(v) ∈ T u(S2).

To summarize, we have the following three distinct ways of viewingthe same Lie group:

(1) The group SO(3) of orthogonal matrices;(2) the quotient S3/±1 of the 3-sphere;(3) the unit tangent bundle of the 2-sphere.

The underlying smooth manifold can in fact be identified with thereal projective space RP3, as well, as is obvious from item (2).

36.4. Sphere as homogeneous space

The sphere S2 is the homogeneous space of the Lie group SO(3),cf. (36.3.1), so that

S2 = SO(3)/SO(2),

cf. [?, p. 82]. The stabilizer of the north pole (0, 0, 1) ∈ S3 is clearlythe subgroup of matrices of the form

cos θ − sin θ 0sin θ cos θ 0

0 0 1

which is clearly isomorphic to S0(2). We will denote by SO(2)σ thefiber over (i.e. stabilizer of) a typical point σ ∈ S2. The projection

p : SO(3) → S2 (36.4.1)

is a Riemannian submersion for the standard metric of constant curva-ture. If we choose a metric of constant Gaussian curvature +1 on S2,


then the metric on SO(3) is of constant sectional curvature 14. We will

view this fibration as a fibration of the unit circle tangent bundle ofthe sphere:

p : T uS2 → S2 (36.4.2)

using the identification of T uS2 and SO(3) discussed in Section 36.3.

36.5. Dual real projective plane

Denote by RP2∗ the dual real projective plane, namely the collectionof lines in the real projective plane RP2 itself. The symbol

RP2∗

denotes the orientable double cover, diffeomorphic to the 2-sphere. Wewill avoid using the notation S2 for this space so as not to confuse twodistinct fibrations.

We think of RP2∗ as the configuration space of oriented great cir-cles ν on the sphere, with measure dν.

Theorem 36.5.1. Let M = S2. The unit tangent bundle T uM canbe represented as follows:

T uM =

(x, ν) ∈M × RP2∗∣∣∣ x ∈ ν

Proof. An oriented (directed) line through x ∈M defines a uniqueunit tangent vector at x, and vice versa.

36.6. A pair of fibrations

By definition the fibration p : T uS2 → S2 sends (x, ν) to x:

p(x, ν) = x.

By Theorem 36.5.1, there is a second fibration q of T uS2, defined by

q(x, ν) = ν.

Thus, the total space T uS2 ≃ SO(3) admits another Riemannian sub-mersion, which we denote

q : SO(3) → RP2∗, (36.6.1)

whose typical fiber ν is an orbit of the geodesic flow on SO(3) when thelatter is viewed as the unit tangent bundle of S2, cf. Remark 36.3.3.

Lemma 36.6.1. Each orbit ν ⊆ SO(3) projects under p to a greatcircle on the sphere.

Proof. A fiber of fibration q is the collection of unit vectors tan-gent to a given directed closed geodesic (great circle) on S2.

36.7. DOUBLE FIBRATION OF SO(3) AND INTEGRAL GEOMETRY ON S2279

SO(2)

##GGG

GGGG

GGG

(RP2, dν

)

SO(3)

q99ttttttttt

p

&&LLLL

LLLL

LL

ν

::uuuuuuuuuu

great circle// (S2, dσ)

Figure 36.6.1. Integral geometry on S2, cf. (36.4.1) and (36.6.1)

While fibration q may seem more “mysterious” than fibration p,the two are actually equivalent from the quaternionic point of view,cf. Sections 34.12 and 36.1.

The diagram of Figure 36.6.1 illustrates the maps defined so far.

36.7. Double fibration of SO(3) and integral geometry on S2

In this section, we present a self-contained account of an integral-geometric identity used in the proof of Pu’s inequality in Section 36.9.The identity has its origin in results of P. Funk [?] determining a sym-metric function on the two-sphere from its great circle integrals; see [?,Proposition 2.2, p. 59], as well as Preface therein.

Denote by Eg(ν) the energy, and by Lg(ν) the length, of a curve νwith respect to a metric g = f 2g0, where f > 0 is a function onthe 2-sphere, while g0 is the standard metric of constant Gaussian cur-vature +1.

Theorem 36.7.1. We have the following integral-geometric iden-tity:

area(g) =1

2π

∫∫

RP2

Eg(ν)dν.

Proof. Let ν(t) be a parametrisation of the great circle ν viewedas an orbit in T uS2 ≃ SO(3). Then p ν(t) is its projection to the 2-sphere. Note that by definition of energy, we have

Eg(ν) =

∫

ν

f 2 p ν(t)dt.


We apply Fubini’s theorem [?, p. 164] to the Riemannian submersion q,to obtain

∫∫

RP2

Eg(ν)dν =

∫∫

RP2

dν

(∫ 2π

0

f 2 p ν(t)dt

)

=

∫∫∫

SO(3)

f 2 p.(36.7.1)

Next, we apply Fubini’s theorem to the Riemannian submersion p, toobtain ∫∫

RP2

Eg(ν)dν =

∫∫∫

SO(3)

f 2 p

=

∫∫

S2

f 2 dσ

(∫

SO(2)σ

1

)

= 2π area(g),

(36.7.2)

completing the proof of the theorem.

36.8. An inequality

Theorem 36.8.1. Let f > 0 be a continuous function on S2. Thenthere is a great circle ν such that the following inequality is satisfied:

(∫

ν

f(ν(t))dt

)2

≤ π

∫

S2

f 2(σ)dσ, (36.8.1)

where t is the arclength parameter, and dσ is the Riemannian measureof the metric g0 of the standard unit sphere.

Corollary 36.8.2. There is a great circle of the sphere whoselength is L in the metric f 2g0, so that L2 ≤ πA, where A is the Rie-mannian surface area of the metric f 2g0. In the boundary case of equal-ity in (36.8.1), the function f must be constant.

Proof of Theorem 36.8.1. The proof is an averaging argumentand shows that the average length of great circles is short. Comparinglength and energy yields the inequality

(∫RP2

Lg(ν)2)

2π≤∫

RP2

Eg(ν)dν (36.8.2)

by the Cauchy-Schwarz inequality. Denoting the least length of a greatcircle ν by Lmin, we obtain

4π(Lmin)2

2π≤ 2π area g

36.9. PROOF OF PU’S INEQUALITY 281

by Theorem 36.7.1, since area(RP 2, g0) = 4π.In the boundary case of equality in (36.8.1), one must have equality

also in inequality (36.8.2) relating length and energy. It follows that theconformal factor f must be constant along every great circle. Hence fis constant everywhere on S2.

36.9. Proof of Pu’s inequality

A metric g on RP2 lifts to a centrally symmetric metric g on S2.Applying Proposition 36.8.1 to g, we obtain a great circle of g-length Lsatisfying L2 ≤ πA, where A is the Riemannian surface area of g. Thus

we have(L2

)2 ≤ π2

(A2

), where sysπ1(g) ≤ L

2while area(g) = A

2, proving

Pu’s inequality. See [?] for an alternative proof.

CHAPTER 37

Gromov’s essential inequality

This chapter deals with Gromov’s systolic inequality for essentialmanifolds.

37.1. Essential manifolds

To give some examples, all surfaces are essential with the excep-tion of the sphere S2. Real projective spaces RPn are essential. If amanifold M admits a metric of negative curvature, then M is essential.

More generally, to define the property of being essential for an n-dimensional closed manifold M , we need a minimum of homology the-ory. This will be done in section

37.2. Gromov’s inequality for essential manifolds

The deepest result in the field is Gromov’s inequality for the homo-topy 1-systole of an essential n-manifold M :

sys(M)n ≤ Cn vol(M),

where Cn is a universal constant only depending on the dimensionof M . Here the systole sys(M) is by definition the least length of anoncontractible loop in M .

Example 37.2.1. The inequality holds for all nonsimply connectedsurfaces, for real projective spaces, for all manifolds admitting a hy-perbolic metric.

The proof involves a new invariant called the filling radius, intro-duced by Gromov, defined as follows.

37.3. Filling radius of a loop in the plane

The filling radius of a simple loop C in the plane is defined as thelargest radius, R > 0, of a circle that fits inside C, see Figure 37.2.1:

Fillrad(C ⊆ R2) = R.

There is a kind of a dual point of view that allows one to generalizethis notion in an extremely fruitful way, as shown in [?]. Namely, we

283

284 37. GROMOV’S ESSENTIAL INEQUALITY

R

C

Figure 37.2.1. Largest inscribed circle has radius R

consider the ǫ-neighborhoods of the loop C, denoted UǫC ⊆ R2, seeFigure 37.3.1.

C

Figure 37.3.1. Neighborhoods of loop C

As ǫ > 0 increases, the ǫ-neighborhood UǫC swallows up more andmore of the interior of the loop. The “last” point to be swallowed upis precisely the center of a largest inscribed circle. Therefore we canreformulate the above definition by setting

Fillrad(C ⊆ R2) = inf ǫ > 0 | loop C contracts to a point in UǫC .

To define an absolute filling radius in a situation where M is equippedwith a Riemannian metric g, Gromov exploits an imbedding due toC. Kuratowski.

37.7. HOMOLOGY THEORY FOR GROUPS 285

37.4. The sup-norm

On the space L∞(X) of bounded Borel functions f on a set X , onecan consider the norm

‖f‖ = supx∈X

|f(x)|.

This norm is called the sup-norm.

37.5. Riemannian manifolds as spaces with a distancefunction

We consider a manifoldM equipped with a distance function d(x, y),where x, y ∈M . In particular, a Riemannian metric on M defines sucha distance function, by minimizing the length of all paths joining xto y.

37.6. Kuratowski imbedding

One imbeds M in the Banach space L∞(X) of bounded Borelfunctions on M , equipped with the sup norm ‖ ‖. Namely, we mapa point x ∈ M to the function fx ∈ L∞(M) defined by the for-mula fx(y) = d(x, y) for all y ∈ M , where d is the distance functiondefined by the metric. By the triangle inequality we have d(x, y) =‖fx−fy‖, and therefore the imbedding is strongly isometric, in the pre-cise sense that internal distance and ambient distance coincide. Sucha strongly isometric imbedding is impossible if the ambient space isa Hilbert space, even when M is the Riemannian circle (the distancebetween opposite points must be π, not 2!). We then set E = L∞(M)in the formula above.

37.7. Homology theory for groups

Recall that the n-dimensional homology group is nontrivial. If M isorientable, thenHn(M ;Z) = Z. IfM is non-orientable, thenHn(N,Z2) =Z2. In either case, a generator of the homology group is denoted [M ]and called the fundamental class.

There is a parallel homology theory for groups. In particular, onecan consider the homology of the fundamental group π = π1(M),namely Hn(π). One additional ingredient we need to know is the exis-tence of a natural homomorphism

φ : Hn(M) → Hn(π).

286 37. GROMOV’S ESSENTIAL INEQUALITY

Then M is called essential if its fundamental class [M ] maps to anonzero class in the homology of its fundamental group:

φ([M ]) 6= 0 ∈ Hn(π).

37.8. Relative filling radius

We first define a notion of a filling radius of M relative to an imbed-ding. Given an imbedding of M in Euclidean space E, let ǫ > 0, anddenote by UǫM the neighborhood in E consisting of all points at dis-tance at most ǫ from M . Let

φǫ : Hn(M) → Hn(E)

be the inclusion homomorphism induced by the inclusion of M in itsǫ-neighborhood UǫM in E. We set

Fillrad(M ⊆ E) = inf ǫ > 0 | φǫ([M ]) = 0 ∈ Hn(UǫM) ,

37.9. Absolute filling radius

We define the filling radius of M to be its relative filling radiusrelative to the canonical imbedding in L∞(M), by setting

FillRad(M) = FillRad (M ⊆ L∞(M)) .

Namely, Gromov proved a sharp inequality relating the systole andthe filling radius, :sysπ1 ≤ 6 FillRad(M), valid for all essential mani-folds M ; as well as an inequality

FillRad ≤ Cnvoln1n (M),

valid for all closed manifolds M .A summary of a proof, based on recent results in geometric measure

theory by S. Wenger, building upon earlier work by L. Ambrosio andB. Kirchheim, appears in Section 12.2 of the book ”Systolic geometryand topology” referenced below. A completely different approach tothe proof of Gromov’s inequality was recently proposed by L. Guth.

37.10. further

Proof of optimal inequality relating systole and filling radius.

mikhailg.katz vladimirkanovei tahlnowiku.cs.biu.ac.il/~katzmik/egreg826old3.pdf · 28.10. de rham...

Documents