chapter 16 metric spaces of curvature ^ i conrad plaut department of mathematics, university

CHAPTER 16

M e t r i c S p a c e s o f C u r v a t u r e ^ I

Conrad PlautDepartment of Mathematics, University of Tennessee, Knoxville, TN 37996, USA

E-mail: [email protected]

Contents1. Introduction 821

1.1. Overview 8211.2. Historical comments 822

2. Metric fundamentals 8242.1. Inner metric spaces 8242.2. Space forms and cosine laws 8262.3. Distance geometry, monotonicity 828

3. Curvature bounded below 8313.1. Wald-Berestovskii curvature 3= it 8313.2. The extended Hopf-Rinow theorem 8323.3. Alexandrov's comparisons 8343.4. The global comparison theorem 8373.5. The space of directions 8403.6. Differentiability of the distance function 844

4. Constructions 8454.1. Induced metrics, gluing, scaling 8464.2. Gromov—Hausdorff convergence 8474.3. Products 8494.4. Submetrics, quotients 8504.5. Cones, joins and simplicial complexes 851

5. Examples 8536. Nonnegatively curved spaces 8557. Analytical touls 856

7.1. The tangent cone 8567.2. Concave functions 8587.3. Development of curves and quasigeodesics 8607.4. Differential and gradient 8627.5. Gradient curves S66

R. Dimension R708.1. Spherical sets 8708.2. Regularity 870

HANDBOOK OF GEOMETRIC TOPOLOGYEdited by (U. Daverman and R.B. Sher© 2002 Elsevier Science B,V. All rights reserved

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8.3. Equivalence of Hausdorff and covering dimension 8719. Alexandrov spaces 874

9.1. Basic properties 8749.2. -Open maps 8769.3. Stratification and local cone structure 8779.4. DER functions 8789.5. Admissible functions 8809.6. Topological arguments 8819.7. Extremal sets 8829.8. Quasigeodesics and gradient curves 8849.9. Boundary 884

10. Differentiable structures 88510.1. Curvature bounded below and above 88610.2. Other differentiable structures 887

11. Convergence and stability 88811.1. Convergence 88811.2. Stability theorems 88811.3. Approximation 890

12. Metric invariants and recognition , 89012.1. Recognition problem 89012.2. Some metric invariants 89112.3. Sphere theorems 89212.4. Related results 893

13. Infinite dimensional spaces 894References 896

Metric spaces of curvature ^ к 821

1. Introduction

1.1. Overview

This article is intended to be an introduction to the theory of metric spaces of curvaturebounded below, and a survey of recent results. Time and space constraints have preventedit from being as comprehensive as we originally had planned, but we hope that this articlewill provide a good beginning point for a student or mathematician interested in this area,and a common reference for future papers on the subject. Through Section 9 we have triedto present good enough sketches of nearly all arguments, so that a dedicated reader can fillin the remaining details without too much difficulty. In Section 9 we provide fewer details,and in the last sections we mostly survey known results.

We do not assume in this article that the reader has any knowledge of Riemannian geom-etry, although such knowledge is helpful for understanding the motivation behind certaintopics. We also occasionally put results into perspective by stating what they mean in termsof Riemannian manifolds, but these parts of the article are not essential. A good workingknowledge of topology and basic real analysis should be sufficient background.

Most of the results described in this article were obtained in the current decade. Some ofthe papers have not been published yet (occasionally due to delays by publishers); some,apparently, will never be published. The field is in a state of rapid development, includingsimplification of earlier proofs. We have tried to be as inclusive as possible in at leastmentioning all closely related (correct) papers of which we were aware when this paperwas written in the summer of 1996, and apologize for any omissions.

We have limited ourselves to the subject of spaces of curvature bounded below in thesenses of Alexandrov or Berestovskii-Wald, which generalize the notion of bounded sec-tional curvature in Riemannian manifolds. Although they are included in this class ofspaces, we have not specifically discussed the theory of convex surfaces - except by way ofhistorical comments and examples. We have not included results about generalized Riccicurvature (cf. [22]). We include results on the metric geometry of Riemannian manifoldsinasmuch as they are also Alexandrov spaces, and we comment on methods from Alexan-drov space geometry for attacking problems in Riemannian geometry, but we go no furtherthan this. Good references for metric geometry of Riemannian manifolds are [33] and [35].

Many geometric topologists became interested in questions of convergence of Rie-mannian manifolds, and the accompanying singular spaces, as a result of the paper ofGrove, Petersen, and Wu [43]. One of the main results (Corollary 219 in the present paper)of [43] is a finiteness theorem for a class of Riemannian manifolds. Their proof uses manytools from geometric topology, such as the disjoint disk property and manifold resolutions.The reader may wonder what has happened to such techniques, which do not appear in thepresent article. Since [43] was written, great progress has been made in understanding thegeometric structure of the limit spaces that occur in L43]. The stronger geometric propertiesthat have been discovered have resulted in a smaller reliance on techniques from geomet-ric topology. The main tool for this kind of finiteness theorem is now Perelman's StabilityTheorem (Theorem 217), and the topological tools for its proof come entirely from Sieben-mann's paper [83]. Nonetheless, we hope that this paper shows that there are other, newerreasons why geometric topologists should remain interested in geometric singular spaces.

822 С. Plant

In particular, the relationship between geometric topology and metric geometry still seemsfar from fully understood. For example, the basic question of whether any nonsmooth topo-logical manifolds admit inner metrics of curvature bounded below remains unanswered.

We sincerely apologize for the lack of pictures in this article; time pressure is our onlyexcuse. Thanks are due Valera Berestovskii for providing historical information.

1.2. Historical comments

The foundations of the intrinsic geometry of metric spaces lie in Gauss' 1827 work, "Gen-eral investigations on curved surfaces". In mis paper he considers sets S in R3 having theproperty that there is a differentiable homeomorphism from an open set in the plane to aneighborhood of any point p e S, whose differential is one-to-one. Such a "regular sur-face" has an intrinsic geometry in which the distance between two points can be measuredas the shortest length of any curve joining those points. Here the length of a curve is mea-sured simply by considering it as a curve in the ambient Euclidean space. On the otherhand, the surface has a tangent plane, and the restriction of the ambient inner product tothis plane provides a "linear element" which is sufficient to measure the lengths of curves.Gauss' Theorema Egregium states that the Gaussian curvature depends only on this lin-ear element and not on the particular embedding of the surface. That is, curvature is anintrinsic property of the surface, not an extrinsic one. Riemann generalized the notion ofregular surface by freeing it entirely of its Euclidean confines: in modern terminology, aRiemannian manifold is a smooth manifold endowed with an inner product on each tangentspace (called the Riemannian metric), which varies smoothly in a natural sense. One canagain measure the lengths of curves by integrating the lengths (in terms of the Riemannianmetric) of their tangent vectors, and this measurement in turn gives rise to an intrinsic met-ric. It is possible to define a notion of sectional curvature which generalizes the notion ofGaussian curvature to higher dimensions.

Gauss is also responsible for the first observation that geometry can exert control overtopology. The Gauss-Bonnet Theorem, in its simplest global form, states that fg К da =2N x (S), where К is the Gaussian curvature, the first integral is over the entire surface andX (S) is the Euler characteristic of S. In particular, an orientable surface with everywherepositive Gaussian curvature must be homeomorphic to a sphere. One local form of thetheorem states fR К da = 2n — £ ft > where the first integral is over a region R boundedby a triangle whose sides are geodesies (shortest paths) and the sum on the right is of thethree exterior angles of the triangle. Note that in a triangle of fixed area in a space of high(positive) curvature, the interior angles of the triangle need to be fairly large: the triangleis "fat", and its perimeter is small relative to its area. With very negative curvature, atriangle of me same area must be very "thin", with larger relative perimeter. In other words,if we fix the side lengths of the triangle, then the above integral, in a positively curvedspace, integrates a higher curvature over a bigger area; the corresponding interior anglesshould be larger than in a triangle of same side lengths in a negatively curved space. Thisobservation is the basis for Alexandrov's definition of bounded curvature, which comparesthe angles of any small triangle in a space with intrinsic metric, to corresponding angles ina "representative" triangle in a "model" surface of constant curvature.


A natural extension of regular surfaces is convex surfaces - the boundaries of convexregions in Euclidean space. A (geometrically) nonsmooth example is the boundary of astandard simplex in R3. As in the case of a regular surface, the metric in a convex surfaceis defined by taking the length of the shortest curve (in the surface) joining two points.Convex surfaces generally have no smooth manifold structure naturally compatible withtheir geometry; clearly they are not always smooth submanifolds of Euclidean space. Thereare (possibly dense!) "singular points" with no well-defined tangent plane. Nonetheless,such spaces have a well-defined "tangent cone" at every point, consisting of all vectorstangent to shortest paths starting at a point. Following Alexandrov in the 40's [2], muchwork has been done to understand the structure of convex surfaces - mostly in the Russiangeometrical school. The connection to the present article is contained in the followingtheorem, due to Alexandrov:

THEOREM 1. A metric space with intrinsic metric is isometric to a convex surface if andonly if it is homeomorpkic to S2 und has curvature (in terms of triangle comparisons) > 0.

Here intrinsic (or inner) metric means that the distance between two points is the infi-mum of the lengths of curves joining them. Alexandrov's theorem shows that, in same theway that regular surfaces can, using Riemannian geometry, be freed from their embeddingsin Euclidean space, convex surfaces can be "abstracted" from their Euclidean confines byconsidering inner metric spaces having a lower curvature bound. This very large class ofspaces, which includes convex surfaces and Riemannian manifolds, as well as their quo-tients, cones, suspensions, and limits, is the main subject of this article.

It is interesting (given that convex surfaces have a lower curvature bound and notan upper one) that the Russian school initially worked mostly with metric spaces hav-ing an upper curvature bound. The primary exception to this trend was work on thequestion of synthetic differential geometry. This extension of Hilbert's Fifth Problemasks to what degree differential geometry can be developed from basic principles, with-out assuming the existence of coordinate charts. The problem of synthetic differentialgeometry was solved by Berestovskii and Nikolaev by 1980 [6,55]. They they showedthat a locally compact inner metric space having three metrically defined properties -curvature bounded both above and below, and "geodesic completeness" - are smoothmanifolds with C1 " Riemannian metrics (see Theorem 210). In the 80's the interestin convergence of Riemannian manifolds created by Gromov's Precompactness Theo-rem [32] turned attention towards the singular spaces which can occur as limits of Rie-mannian manifolds. Quickly it was recognized that if the manifolds have a uniform lowersectional curvature bound, then the limit spaces have a lower curvature bound in thesense of Alexandrov. There followed throughout the 90's an explosion of work on suchspaces.

One of the main themes of this article is the tremendous influence that a curvature boundcan have over topology, at the infinitesimal, local, and global levels, as well as on a globallevel across whole families of spaces with geometrically defined restrictions. In this regardmodern results in the field remain very close, in spirit, to their earliest ancestor, the Gauss-Bonnet Theorem.

824 С. Plant

2. Metric fundamentals

Throughout this article, (X, d) will denote a complete metric space. In some sections orsubsections we will add further restrictions to X; these will be stated in the first paragraphof the section or subsection in question. We will use the notation d to represent the metricin any metric space, as long as no confusion will result. The diameter of X is diam(X) =sup{d(x, y)\ x,у e X}. An isometry is a distance preserving map; a local isometry is amap which is an isometry when restricted to a small enough open set about any point.

DEFINITION 2. For any s > 0, an e-net A/" in X is a subset of X such that for all x e Xthere exists some >•ejV such that d(x, y) < s.

Since X is complete, it is compact if and only if for every s > 0 there exists a finite

If c : / —> X is a curve, we denote the length of с by L(c) = sup{£<i(*,-, *,--и)}, wherethe supremum is taken overall partitions {X\ < • • • < _*„). By the triangle inequality, L(c)is also the limit as the partitions become fine, of YJd(xt,x,+i)- If L{c) is finite we saythat с is rectifiable. A rectifiable curve can always be reparameterized in a monotone wayso that the parameter is proportional to arclength (cf. [78]). Unless otherwise stated, wewill assume that all rectifiable curves are parameterized proportional to arclength. A unitcurve is one that is parameterized by arclength. Note that any family of unit curves isautomatically equicontinuous. We will often use this fact without further comment whenapplying Ascoli's Theorem.

REMARK 3. Let yt '•[я,b]—• X be arclength-proportionalcurves in a metric space X, i =0,1 ,....We write yt —• ydto denote that y-, converges uniformly to yo- More generally,suppose that the y, are not defined on the same interval. Then we write y,- —> yo if, wheneach yi, i > 0, is reparameterized proportional to arclength on the same interval as yo, theresulting functions converge uniformly. Note that if L(y,) is bounded above then у, —* щif and only if y, converges to j ^ pointwise. This situation will occur frequently, and wewill use the contents of this remark without further reference.

REMARK 4. When we denote a constant in the form c(ct \ a ^ ) this means that theconstant с depends only on the values a \,..., o*.

2.1. Inner metric spaces

Let с be a curve in X joining points x and y. If d(x, y) = L(c) then с is called minimal.Henceforth the notation уаь will always denote a minimal curve joining a and b. A curveу: I —¥ X K called a geodesic if for each interior value t of / , у is minimal when restrictedto a small interval centered a t t . The metric of X is called inner (or intrinsic or length) iffor all xt у e X, d(x, y) = inf{L(c): с joins x and >'}. If each pair of points in X can bejoined by a minimal curve, X is sometimes called a geodesic space.


Inner metric spaces can be considered as geometric objects, as opposed to merely topo-logical or analytical objects. For example, one can consider the subspace metric on the unitsphere in Euclidean space. This metric is compatible with the subspace topology, and soprovides an adequate rough measure of "closeness" between points. However, this metricis inadequate for describing the internal geometry of the sphere. It is better to considerthe inner metric induced by the subspace metric - i.e., the distance between two points isthe length of the shortest curve joining them. One then obtains the metric of the standardgeometry of the sphere with Gaussian curvature 1, This geometry describes the sphere as itwould appear to a creature confined to live in the surface of the sphere, who, for example,would measure the distance between antipodal points as Л, not 2.

DEFINITION 5. For any x, y, z e X, we define the excess of the triple by s(y\ x, i) :=d(x,y) + d(y,z) — d(x,z)- If s(y;x,z) = 0 and x, y,z are distinct, we say that у is be-tween x and г; in this case, if d{x, y) = cf(y, z), then we say that у is a midpoint of x andy. We define the strong excess of the triple to be

s(y;x,z)ff(y;x,z) =

The next lemma is an immediate consequence of the definition and the triangle inequal-

ity.

LEMMA 6. Let w, x, y, z be points in X. Then(1) s(z\ y,w) + е(у;х,w) = e(y;z,x') + e(z; w,x), and(2) e(z;y',w) ^ e(y;z,x) + e(z; w,x).

PROPOSITION 7. X is an inner metric space if and only if for every x, z, e X and s > 0there exists a у e X such that d(x,y)=d(x,z)/2 and e{y;x,z) < e.

A point у as in the proposition will be called an "almost midpoint" between x and z.To prove the proposition, note that if X is an inner metric space, then by choosing a curvec joining of length sufficiently close to d(x, z), then the point у which bisects с is analmost midpoint. Conversely, we suppose, for simplicity, that d(x, z) = 1. We choose apoint0(1/2) such that

Continuing in this way, we can construct a mapping ф from the dyadic rationals in [0, 1 ]into X with the property that if, for dyadic rationals w, y, d(w, y) = 2~", then

Clearly ф can be extended (using the completeness of X) to all of [0,1j to produce a curvejoining X and г whose length is less than 1 — е.

826 С Plaut

In a metric space in general, the closure of an open metric ball B(p,r) = {x: d(x, p) <r) \s contained in, but not equal to the closed metric ball {q: d(p,q) ^ r } . However, equal-ity is easily seen to be true in an inner metric space. For this reason, we will use the notationB(p,r) to denote {q: d(p,q) < r] in an inner metric space.

The following is a theorem of Cohn-Vossen [26].

THEOREM 8. Аи inner metric space X is locally compact if and only if every closed,bounded set is compact. Every pair of points in a locally compact inner metric space canbe joined by a minimal curve.

The hardest part of the theorem is showing that local compactness implies that closedmetric balls are compact. To do this, one can show that, for any p, if W = {r: B(p, r)is compact} then sup W = oo. Here the main step is showing that if s € W for all s < гthen r ^ W. Given any sequence {at} in B(p,r) one can use the inner metric to find,for any n, points Щ such that Щ е Ъ(р,r - 2 ~ n ) and d(a,,b1) ^ 2 ~ n + l . Since we canextract a convergent subsequence of {b"), we can assume, by taking a subsequence, thatd(ai,aj) ^ 2~" + ? . The proof is completed using a standard diagonal argument. Minimalcurves can now be constructed using Ascoli's Theorem.

Any "reasonable" metric space (X, d) has an induced inner metric di (see Definition 65and subsequent comments).

One property of inner metrics that distinguishes them from ordinary metrics is that theyneed only be defined locally. Indeed, inner metrics can be defined by merely producing areasonable way to measure the lengths of sufficiently many curves in a neighborhood ofevery point - as is the case with Riemannian manifolds (see Example 94). We illustratethis point with the following proposition:

PROPOSITION 9. Let я :X —• Y be a covering map and suppose d is an inner metric onY. Then if X is urcwise connected, there exists a unique inner metric d on X such that Яis a local isometry.

The metric d , which is called the lift of d, is defined in the following way: for any pairof points x, у € X,d(x,>) — 'тЦИл(с))}, where the infimum is over all curves с joiningx andу in X. In essence we are measuring the length of a curve in X by projecting it andmeasuring the length of the resulting curve in Y. This proposition, as well as the followingcorollary, are false for metric spaces in general.

COROLLARY 10.IfX and Y are inner metric spaces and ф: X -v У is a one-to-one localisometry, then ф is an isometry.

2.2. Space forms and cosine laws

We denote by 5^ the n -dimensional simply connected space form (sphere, Euclidean space,or hyperbolic space) of curvature k. When n = 2 we will omit the n. The spaces S£ are


metrically characterized in the following way. First, S" is a locally compact inner metricspace. For any unit geodesies ^|, yj starting at a point, the quantity

.n) : 2 1 i m s i n

exists and satisfies the following cosine laws for all s, t:

cos*Md(y\(s),Y2(0) -cosVkscos~Jktcasa(Y\.n)= /=—.—7= f o r * > 0 , (1)

sin y/ks sin \/kt

cosa{y\, y2) = — for & = 0, (2)

coshsf\k\d{Y\ 0) , MO) ~ <=<»h VHfrcosh J\k\tfor*: < 0,

(3)

Note that, with the usual relationship between trigonometric and hyperbolic functions (e.g.,cosh* = cos be), the cosine laws for к <0andк > 0 are essentially the same. Furthermore,if we take the limit as к -*- 0, these formulas approach the cosine law for к = 0. The for-mula can be readily seen for к > 0 from l i m ^ o ~- — 1 and the following reformulationof the spherical cosine law [78]:

s,n (—,-2

sin i ~Jk(d{y\ {s), nU)) + s - t)sin ± Vk(d(yi (s), y2{t)) - s +t)— *t * id 1

sin *fks sin <JktIt is also evident from formula (4) (and its hyperbolic counterpart) that, if к is fixed and

.i or f gets small, the right-hand side of formula (4) tends to the Euclidean formula. In otherwords, the space forms are "infinitesimally" Euclidean.

The metric characterization is not the most useful way to understand the space forms;however, one of our goals is to minimize the knowledge of differential geometry that thereader must have. Therefore we confine our comments to only the most general ones,with no proofs given. For more details, consult one of the books on differential geom-etry in the bibliography. For the rest of this paragraph assume к > 0. The sphere ££ isthe sphere of radius 1/V& in R"+1, with the Riemannian metric induced by the inclu-sion. Equivalently, we can measure the distance between two points by the length of theshortest smooth (equivalently piecewise smooth) curve joining them, where the lengthis measured in the usual way for smooth curves in Euclidean space. With this distance,the sphere is isometric to the one given by the cosine law above. Being a smooth sub-manifold of R"+ l . S% has a tangent space at every point. Tangent to each unit vectoris a unique great circle, which is a geodesic in both the metric and differential geomet-ric senses. The angle between two tangent vectors (measured using the Euclidean inner

828 C. Plaut

product on the tangent space) is the same as the (metric) angle defined above, betweenthe corresponding geodesies. The great circles are the only geodesies, and are minimiz-ing up to length nj°Jk, which is the (intrinsic!) diameter of S£. We can therefore referto the space of directions at a point p, which will mean both the unit sphere in the tan-gent space at p, and equivalenfly the space of unit geodesies starting at p with the an-gle as metric. In either case, the space of directions is isometric to S"~'. Finally, notethat every S£ can be obtained from the unit sphere S" by scaling: multiplying the dis-tance function by the factor l/л/к. When scaling the metric, geodesies clearly remain thesame. In addition, it follows from the fact, mentioned above, that the space forms are in-finitesimally Euclidean, that the angle between geodesies is not changed by scaling. Inother words, scaling really just changes the size, not the essential geometry, of the spaceform.

As in the case of the sphere, if к < О, S^ can be obtained from the hyperbolic space5" ] by scaling by the factor 1/V~&• Hyperbolic space is homeomorphic to Euclideanspace. Unit geodesies starting at a point with the angle metric form a space of directionshomeomorphic to S"~[. The geodesies starting from a point spread out much more rapidlythan geodesies of the same angle in Euclidean space; consequently geodesies in S^. p, likethose in Euclidean space but unlike those in the sphere, never come back together again.Also, every segment of a geodesic in 5", is minimizing (as in Euclidean space, but notthe sphere). In the space forms it should now be clear that, the lower the curvature, themore rapidly geodesies diverge. Later we will see that measuring the rate of divergence ofgeodesies is an excellent way to understand curvature.

DEFINITION 11. X is said lo be (metrically) n-pmnt homogeneous if for every [x\,,..,xn],iyi У п } С X such that d(xj,xj)~d(yi,yj) for all /, j , there is an isometry ф: Х -> Xsuch that 0(JCJ) = >7 for all i. A 1-point homogeneous space is simply called homogeneous.

Using Cartan's Theorem (cf. [27]), the following can be shown by induction on m.

PROPOSITION 12. S% ism-point homogeneous for all m.

2.3. Distance geometry, monotonicity

Let M be a finite metric space, A basic question in distance geometry is when M canbe embedded in a space form. If M — [x,y,£} has three points, it is not hard to seegeometrically that M can be (isometrically) embedded in any space form S so long asd(x, y) + d{x, z) + d(y, z) < 2я/*/к. When £ ^ 0 we always take k~^2 — oo (e.g., whenconsidering radii of metric balls). The image of such an embedding (which is uniqueup to isometric motion of S$, by Proposition 12) gives rise to a representative triangleT = T(x,y,z)inijfc,

DEFINITION 13. A triple (>-; x, z) is subset of three points in a metric space. The distin-guished point у is the vertex of the triple. Wedenotebyotkiy,x,z) e [0, N~\ the angle of arepresentative triangle in S* at the point corresponding to y.


As can be seen from the cosine laws and the results of the previous section, the functiona/ciy, x,z) is a continuous function of k, d\ := d(x,z), dj '.—d(y,x) and d-j, = d{y,г),and is monotone increasing in either of the first two variables (leaving the other variablesfixed). Also, ak{y\ x,z) = n if and only if у is between* andг, andat(у; х,z) = 0 if andonly if one of x, z is between у and the other. In general, а^(у; х, z) is defined for all кsuch that d(x, y) + d(x, z) + d(y, z) <2nj*J

REMARK 14. We will refer to the monotonicity of ak{x\> ' , г ) in к and d\ as the firstmonotonicity and second monotonicity, respectively. If d(x, y) + d(x, z) + d(y, z) >Injyfk then we take Щ{у; x, z) = ос.

There is a close relationship between u(h; a,c) and ак(Ь; o, c).The following lemma isevident from the Cosine Laws or elementary Riemannian geometry;

LEMMA 15. Let к be fixed. Then u{b; a,c) is small if and only if a^ib: a, c) is close ton.

More precisely, for every s > 0 there exist constants S\(k),S2(k) > 0 such that ifa(.h;a,c) <S\ then ctk(b;a,c) > n — s and if a^(b; a,c) > л — Si then a(b;a,c) < в.Often we are interested in showing that a&(b; a, c) —> n as a, b, с converge to some point.We can use the following criterion:

LEMMA 16. Let M ^ 1 and к be fixed. Then for every e > 0 there exists a S(kt M) > 0such that ifd{a,b) ^ M and ^ < ^ ^ < M1 and e(b; a,c)<S• (d(a,b) + d(b,c) +d{a,c)) then а^(Ь\а,с) ^ n — g.

To prove the lemma, observe that by the first monotonicity it is sufficient to prove itfor к < 0. If d(a,b) = M, then d(a,c) ^ 1/M2 and the statement follows easily from thecosine laws or a compactness argument. The proof is now complete by observing that,if Г is a triangle in Sk (k < 0) with side lengths S\, £2, Si, and T is another triangle in5* with side lengths mSi,mS2,mST, for some m > 1, then the angles in T are all largerthan the corresponding angles in T . (This fact follows from the first monotonicity andelementary Euclidean geometry.) Thus by "enlarging" a given triangle until d(a, b) = M,we are finished.

Later we will be using representative angles to approximate angles (which will bedefined later). The following facts are convenient. First, we say that two sequences{(xt2",Xn,Xi})] and {(yal ytuy u i ) } are (metrically) equivalent if

\d(Xij,xjk)d(yij,yik)\hm ' — — - — = 0 forall 7,*.l d ( ) }

PROPOSITION 17. Let {(br,6i>c{j] be a sequence of triples such that ak(i) :=ff k (ty;a, , Ci) is defined far all i. Then

l ( S : ) V , ) i(2) If {{b\\«[,cj)} is a sequence of triples equivalent to {(bi\ai,Ci)} such that

afki.i) = щф'^а ';,^) is defined for all i, then in O f - K » & k ( i) exists if and only if

li ^ ( ) exists, andifboth exist they are equal.

830 C. Plata

The first part of the proposition can be proved using the fact that the space forms areinnnUesimal ly Euclidean (cf. the comments after formula (4)). To prove the second part,note first mat the argument can be reduced to the case when the sets of triples have two"side lengths" in common, by replacing (bi\( t { , Ci) with a triple having one side length incommon with (b'; ;a't,cj), then replacing it with one having two sides in common. But ifthe triples have two sides in common, then the second part follows from the first part andthe triangle inequality.

It is not possible, generally, to embed a given four point metric space in any space form.For example, the quadruple M = {a,b, c,d} such that d(a,b) = d(b,c) = d(b.d) — 1and d(a, c) = d(a,d) = 2 is branching; that is, there is a point (b) that is simultaneouslybetween two other pairs of points ({a, c) and (a, d)), but M cannot be embedded in thereal line. If this quadruple were embedded as a quadruple {A, B, C, D] in a space form,the minimal geodesies joining Л to С and A to D would coincide as a minimal geodesicfrom A to B; that is, the geodesic from A to В would "branch" at B. Since geodesiesin a space form (or any Riemannian manifold!) never branch, M cannot be embedded inany S£. A quadruple M will be said to be linear if it can be embedded in the real line,and nondegenerate if no point is between two omer points. If a quadruple M = {a,b,c,d}can be embedded in £|, then we say that к is an embedding curvature for M. If M canbe embedded in Sjt, we say к is я planar embedding curvature for M. Note that M can beembedded in S\ if and only if M ean be embedded in S£ for all n > 3. 1n [10] the followingtheorem is proved:

THEOREM 18. Let M be a (metric) quadruple. Then exactly one of the following is true:(1) M is nondegenerate and the embedding curvatures are an interval of the form

[K[, K2] or (—00, K], and the endpoints of these intervals are precisely the pla-nar embedding curvatures for M.

(2) M is branching and so cannot embed in any space form.(3) M is linear.(4) M has a unique embedding curvature k, which is planar.

We will give a sketch of the proof of the theorem, for which we need the followingnotation:

DEFINITION 19. Let M = [x], ^2,^5,^4} be a (metric) quadruple. For any £, and i,j,m,ndistinct, we let

Vk(xi) = ak(xi; XJ,xm)+ ak(XJ; xj, xn) + ak (x,; xm, xn),

From the properties of a/,, we see that V& (XJ ) is also continuous and monotone increasingin k. Now suppose that M = [a, b, c, d] is isometrically embedded in 5 | and the image ofan embedding is the points A, B, C, D. Then the geodesies from A to the other three pointsare directions yB, yc, YD at A, that is, they form a triple of points embedded in the space ofdirections at Л, which is S\. The converse is also true: if ak (a; b, с),а^(а\ с,d),ак(а; b,d)


are the distances for some triple in Sj then M embeds in £|. From what we have alreadyseen about embedding triples of points in space forms, we immediately get the followingproposition. (The nondegeneracy is needed to ensure none of the angles is 0.)

PROPOSITION 20. A nondegenerate quadruple M = {x\, xj, хг, *4} embeds in £ | if andonly if Ak(M) < 2Л and the triangle inequality holds for any three values ak(x;;xj,xm),uk(xi\Xj,xn), ctk(Xi\xm,xn), Furthermore, if ak(xr,xj,xm) = ak(xj\Xj,xn) +<** (xi ;XmiXn) for some i then the embedding is planar.

Now, if for some a e M , V/,(a) = 2л then each angle is ^ n but the sum of any two is^ 7Г, so the triangle inequality is always true. M can be embedded in S^, but more exactly,the three corresponding directions span a 2-plane in the tangent space, so we have in factembedded M in 5*. It is shown in [10, Lemma 3], that if K is the largest number for whichAK(M) is defined then AK{M) > 2 Л . On the other hand, the properties of a* indicate thatfor some k < K, Ak{M) < 2n. By continuity there must be an m such that Am(M) =2x.Thus we have a planar embedding of M in Sm, and m is the maximal possible value forwhich M embeds in S^• It is also shown in [10, Lemma 4], that if there is another planarembedding, i.e., an embedding of M in S/,, withh < m then there are no embeddings of Uin Si for any k < h. We see that there are at least one and at most two planar embeddings,and they are extremal. In light of Proposition 20 and the above comments, we see that Mhas a nonplanar embedding in S^ if and only if Ak(M) < 2it and the triangle inequalityis satisfied and strict for all triples of angles. Therefore the set of all such k is open, andits boundary consists of values m such that M embeds in Sm. The remainder of the proofof Theorem 18 is a case-by-case analysis. A similar case-by-case analysis can be used toprove the following, which is not stated in [10].

COROLLARY 21, Let M be a quadruple. Then M has a planar embedding number m ^kif and only if Ak(M) ^ 2 Л .

3. Curvature bounded below

3.1. Wald-Berestovskii curvature ^ k

The following definition is due to Berestovskii [10], and is a modification of one due toWald[88].

DEFINITION 22. X is said to have curvature ^ k if for all x € X there is an open setU С X containing x such that every quadruple of points {a, b, c, d] in U isometricallyembeds (as a metric space with the induced metric) in Sm for some m > k. Such a set Uwill be called a region of curvature ^ k.

A natural question to ask is: Why four points? Certainly the results of the previous sec-tion show that three points is too few. On the other hand, Berestovskii has shown [10] thatif one replaces quadruples by quintuples in the above definition, then one gets essentially

832 С. Plaut

only spaces of constant curvature. So only quadruples allow the possibility of an interestingtheory beyond the geometry of space forms. As an immediate consequence of Corollary 21we get the following formulation of bounded curvature, which has the advantage of involv-ing only the single model space Si.

PROPOSITION 23. An open set U in X is a region of curvature ^ к if and only if everyquadruple [a,b,c,d] in U satisfies Vk(a) ^ 2n.

It is well-known that every separable compact metrizable space embeds topologicallyin the Hilbert Cube H = /( x h x • • •, where /„ = [0,2~"1- Since H itself is naturallya convex subset of the (geometrically Euclidean) separable Hilbert space, it has curvature^ 0 with the induced metric (in fact it is "flat" in some sense). It follows that every compactmetrizable space admits ametricofnonnegative curvature. We see that simply having sucha metric imposes no topological restrictions on the space. As we have mentioned earlier, itis also true that simply having an inner metric imposes few restrictions on a metric space.It is therefore somewhat remarkable that the combination of these two conditions imposes,as we will see, a very strong geometric and topological structure.

3.2. The extended Hopf-Rinow theorem

DEFINITION 24. Let уаь be a minimal curve in X. The curve у is called extendable be-yond b if у is the restriction of a minimal curve having b in its interior, and almost extend-able beyond b if for every £ > 0 there exists a c e X\{a,b] such that a(b;a,c) < E.

Note that Definition 24 is stronger than the definition of "almost extendable" given in[76]. It easy to see that an extendable minimal curve is almost extendable.

DEFINITION 25. An inner metric space with the property that every minimal curve Yahhas a restriction that is extendable beyond b is called geodesically complete.

Geodesic completeness is equivalent to the maximal domain of definition of every geo-desic being all of R. The Hopf-Rinow Theorem in Riemannian geometry states, for a Rie-mannian manifold M, the equivalence of metric completeness and geodesic completeness,and, if M is complete in this way, gives the existence of minimal curves between all pairsof points. In a Riemannian manifold, locally, joins by minimal curves are unique. We willsee that geodesic completeness generally fails for spaces of curvature bounded below (andcan fail badly even for 2-dimensional spaces). Almost extendability of geodesies serves asits replacement.

DEFINITION 26. For any p e X let J,, be the set of all q e X such that there exists aunique minimal curve ypq which is almost extendable beyond q.

THEOREM 27. If X is an inner metric space of curvature ^ к then Jp contains a denseG$ subset.

Metric .spaces of curvature ^ k 833

We call the above theorem an "extended" Hopf-Rinow Theorem, even though the ter-minology is somewhat misleading. Even for Riemannian manifolds, this theorem does notfollow from the Hopf-Rinow Theorem: being geodesically complete does not mean thatevery minimal curve is extendable in the sense of Definition 24. (A Riemannian manifoldwith the latter property must be diffeomorphie to its tangent space via the exponential map.Theorem 27 also includes information about the cut locus of the manifold.)

The proof of the theorem needs some preliminaries.

PROPOSITION 28. Let X be an innermetric space andsuppose p,q e X have the propertythat В := B(q, r) is a region of curvature > к for some r > 0, and there exist qt with q, ->qandaiq;qi,p)-*• 0. For any positive t < min{r, d(p, q)} and S > 0 there exists an E > 0such that if a J € B, j = 1,2, satisfy d(q, a j) = t and£(я,; p,q) < s , thend{a\,a2) < S,

PROOF. Suppose d(q,aj) = t. The proposition will follow from Lemma 15 if we canshow that for any ц > 0 there exists an t > 0 and i large enough that if s{a y , p,q) < rthen a(q\aj,q;) < r\. For then we know that a* (g;#,,#,) is close to Ж, and by Proposi-tion 23 щ(а;а],а2} is small, and by Proposition 17 d{a\,a2) is small. Choose i so thato(q;qi, p) К f)/2 and.d(q,qt) < t. By Lemma 6, if

s(ay, p,q) < f, e(q; ah q.) < f

so we need only choose f < r)d(q,qi)/2. D

We return to the proof of Theorem 27. Let L, be the set of all у е X such that for somel, a(y- p, z.),d(y, z) < 2" ; , and let J'r - f ] ^ l Lt. Obviously J'p is a G&. The proof thatJ'r is dense requires only properties of the inner metric (this fact allows us to get the globalconclusion even though the curvature assumption is only local). Basically, we start with aconstant po > 0, begin at an arbitrary point x, and choose a point x\ such that d(x\,x) =A)/2 and A(X\ ; p, x) < 1/2. Continuity of the distance function implies that there is aP\ < A)/4 such that if some point у satisfies d(y\x\) < p\ then a(y; p,x) <. 1/2. Nowwe choose X2 such that d(x2,xi) = p\ and<r(x2\ p, x\) < 1/4. Continuing this process wecan construct a Cauchy sequence {_*,} such that г := Umx, is in J' and d(*, z) < po-

To complete the proof we need to construct a minimal curve from p to any q e Jp.As in the usual proof of the Hopf-Rinow Theorem in Riemannian geometry, we start atq and construct our curve "backwards" towards p. We suppose that B(q, r) is a regionof curvature > k. Let t < r be positive and choose points x; such that d(xj ,q) = f ande{xf, p,q) -*• 0. (The points Xi can be found by choosing almost minimal curves from pto q and using the intermediate value theorem to get d(X{,q) = t . ) Then Proposition 28implies that the sequence {л, } is Cauchy, and converges to a point y (t) between p and qsuch that d(y(t), q) = t. Doing this for every ( e (0, r) and letting y(0) = q we obtain ageodesic у:[0,r)->•X starting at q such that the restriction of у to any [0,t) with ( < ris minimal. Then lim,^r y(t) exists; we call it y(r). By continuity, у :[0, r] -» X is aminimal curve and y(r) is between p and q. For any;,- -* r. f,- < r , s(y(r); p, y{ti)) = 0

- y(r), so y(r) satisfies the hypothesis of Proposition 28. Now we can continue

834 С. Plaut

extending у beyond у (r) as above. By a standard open-and-closed argument we eventuallyreach p.

The above argument also proves the following corollary, which will be true globallyonce we have obtained the Global Comparison Theorem (Theorem 43),

COROLLARY 29. Let X be an inner metric space and suppose p € X has the propertythat В := B(p, r) is a region of curvature ^ k. Then q e JPH B if and only if there existpoints q\ with q\ ->• q ando(q; q\, p) -* 0.

By the Baire Category Theorem, the intersection of countably many dense G; sets isagain a dense G$, We immediately have the following:

COROLLARY 30. IfX has curvature ^kandx\,X2,... e X, there exist points *,- arbitrar-ily close to Xj such that every pair x\,x'• is joined by a unique minimal curve. Furthermore,we can take x\ =xj.

For example, if we are given three points x,y, z in X, we can find points >•',z! arbitrarilyclose to y, z, respectively, such that x, y', z1, form the corners of a triangle of minimalcurves in X.

Note that if у is a minimal curve from у to *, any point arbitrarily close to x in theinterior of у is in J},.

3.3. Alexandrov's comparisons

In this subsection X will always be assumed to be an inner metric space of curvature > k.and U will denote a region of curvature ^ к. A triangle in X consists of three minimalcurves T = (yab. Vac, Ybc)- We allow "degenerate" triangles, in which the geodesies maycoincide in part. A hinge is a pair of minimal curves (yah,Уас)- Wfe will always assume thatYah and yac are parameterized by arc length on R + so that уаьЩ = }V(0) — a.lid lieson yac and yac, yabС f /, then we can embed a,b,c,d in Sm, for some m > k; the imageof this quadruple consists of the corners of a representative triangle T(a,b,c) (having thesame side lengths) in Sn,, with a point d on y^. If we consider the representative triangleT(d\ b', c') in Sk and the point d' on y&c> s u c h that d(a', d') = d(a, d), thenby the firstmonotonkity (see Section 2.2), d(b,d) ^ d(b!,d!). We refer to the points d and d! asthe point corresponding to d in Sm, S k :, respectively. They are uniquely determined by therepresentative triangle. Choosing a point e on уаь and applying the above argument tothe hinge (уаь, yiHi) and the point e, together with the second monotonicity, we prove thefollowing statement, which is one of Alexandrov's curvature conditions.

AO Given any triangle T — {y(,ft, yw, Уса) in U, d e yac, and e e уаь, there exists arepresentative triangle T in S . If d, e are the points on T corresponding to d, e,respectively, then did, e) ^ d(d, ё).

REMARK 31. One can consider an a priori weaker condition than АО by fixing e = b inthe statement of АО and only allowing d to vary. It is a nice geometric exercise using the


second monotonicity to show that this "one-point" version of АО is in fact equivalent to theabove version. We will also refer to the one-point version as АО; often it is easier to verify.

An immediate consequence of АО and the second monotonicity is that for any hinge(Yah, KreK the function al, f) = а *(а; Yub($), Kac(O)' s monotone decreasing in each vari-able, into [0. тг]. Thus the following definition makes sense.

DEFINITION 32. Given a hinge (уа/„yac), with я е U, the angle of the hinge in X is thenumber

= hm am(a; yab(s). K,t (')) 6 [0, n\

for any fixed m.

The fact that the angle is independent of к follows from formula (4); in fact, we couldeven let m vary, as long as it stays bounded. Thus, we can measure angles, when conve-nient, using representatives in any space form, including the Euclidean plane. We summa-rize this property in the following convenient form:

LEMMA 33. Let y, ft be unit geodesies starting at peU. Then

d(y(t),

The following basic result is the triangle inequality for angles, and is a consequence ofthe triangle inequality for the metric of X, and Definition 32:

PROPOSITION 34. For any three minimal curves уаь, уас, yad in X,

b, Yad) 4t <a(yat>, yac)+Ot(yai:, Yad)-

The next two statements, which are also curvature conditions of Alexandrov, followfrom АО and are equivalent by the second monotonicity. Note that our meaning of anglecoincides with the usual one if X is a Riemannian manifold. Given a hinge H = (уаь, Уас)in X, a representative of H in Sk consists of a hinge H — (ya^, у^ъ) in 5* such thatd(a, b) = d{a, i>), d{a,c) m d{a, c) and a(yab, j v )

Al For any triangle Г в (yab t yac,}%,) in U, if f - (y^, y^, y^) is a representativeof T in Sk, then a(yah, yac) > а { ? ф Y 5 C ) -

A2 For any hinge H = (yah, Yac) in V, there exists a representative H = (y~^, y$$) ofH in S i ,andd(b,c)^d(b,c) .

It is important to consider when equality occurs in conditions Al and A2, which we willdenote by EAl and EA2 (cf. Appendix to [36]). In the locally compact case, suppose we

836 С Plant

have a triangle T = { уиь, Vac, уы) which satisfies EA1 and is contained in a much largerregion of curvature ^ k. Then applying АО and A2, we see that the distance between anypair of points on уаь and }v, respectively, must be equal to the corresponding distance ona representative triangle. For any d on уьс , and minimal curve yad, applying both Propo-sition 34 and Al, we see that both (j>a/>, yad) and (yad, >w) are EA2. Applying the sameargument to all points along уьс, we can "fill in" the triangle T:

PROPOSITION 35. IfX is locally compact and T is a triangle which satisfies EA1 and liesB(p, r), where B{p, 3r) is a region of curvature bounded below, then T spans a surfacein X which is isometric to the surface spanned by a representative TofT in S;,.

The proofs of the next two simple but important lemmas are immediate from the waythat angles are measured, together with A1, and Proposition 23, respectively.

LEMMA 36. Let у г,щ be minimal curves starting at pi —• p e X. Ify;—* у andctj —> afor some minimal curves y, a starting at p, then a(y, «) ^ liminf ct(y/, а,-).

LEMMA 37. If У\,У2,Уз are minimal curves starting at p, then ct(y\, yz) + а(у2, Уз) +

COROLLARY 38. Ify\,y%,ут, are minimal curves starting at p, andл — a(y\, уг) ^ £ forsome e ^ 0, then a(yi, уг) + а(уг, уз) <ТГ + S.

Although it is a trivial consequence of Lemma 37, we state the above corollary be-cause we will use it in this form frequently; it allows us to bound an angle from above bybounding its (almost) complementary angle from below using A1. For example, this canbe carried out at any point p which is in the interior of a geodesic, or is the endpoint ofan almost extendable geodesic. By Theorem 27, this means almost everywhere. Note thatthe case e = 0 is the "Theorem of Complementary Angles". A branching geodesic consistsof two geodesies with common endpoint which coincide on an interval, but one is not anextension of the other. If such occurred in a space of curvature ^ k, then at the "branchpoint" p, the sum of angles between three geodesies leaving p would be > 2JT . We obtain:

COROLLARY 39. In X, geodesies do not branch (bifurcate). In particular, the restrictionof any (minimal) уаь to a proper segment is the unique minimal curve joining its endpoints.

COROLLARY 40. Let Y bean inner metric space such that for every p eУ, Jp contains adense G&. If for some k, condition АО holds for all triangles in any ball B(x,r), then everyquadruple in B(x, r/2) isometrically embeds in Sm for some m^k.

The last corollary, which implies the essential equivalence of Alexandrov's definition(together with suitably many minimal curves) and Berestovskii's definition, can be provedas follows: We can equivalently show that for any quadruple [a,b, c, d\ in B{p,r/2),Vk(a) К 2 Л . By continuity and our assumption on Jp we can assume that all points can bejoined by minimal curves, and now the conclusion follows from Lemma 37 and Al.


REMARK 41. We can now completely understand "one-dimensional" spaces X of curva-ture bounded below. By "one-dimensional", we mean that there is some point p e X thathas at most two distinct unit geodesies leaving it. Since geodesies do not branch, there areexactly three possibilities: X is isometric to a closed interval, X is isometric to the reals,or X is isometric to a circle. Some of me arguments in the sequel require the existence ofat least three minimal curves starting at every point. From now on, we wilt assume, withoutfurther mention, that X is not one-dimensional.

REMARK 42. It follows from the Rauch Comparison Theorem [23,27] that in a Rie-mannian manifold, sectional curvature ^ к and Al are equivalent; hence curvature ^ кin the metric sense and sectional curvature ^ к are equivalent.

3.4. The global comparison theorem

THEOREM 43. IfX is an inner metric space of curvature ^ k , then all of X is a region ofcurvature ^ k.

The above theorem is known in Riemannian geometry as the Alexandrov-ToponogovTheorem, or simply Toponogov's Theorem. It was proved by Alexandrov for convex sur-faces (using approximation by polyhedra) [2] and by Toponogov for Riemannian mani-folds of any dimension [87]. The first completely metric proof of the theorem (even forRiemannian manifolds) may be found in [72], but the proof there requires geodesic com-pleteness. In the above generality, proofs may be found in [76] and [ 19]. (Note that theproof in [18] uses local compactness; in the published version, the existence of minimalcurves is assumed in a way that is not essential.) We outline the argument in [76], which inturn is a modified version of the proof in [72].

DEFINITION 44. For any p e X, the comparison radius at p is defined to be ct(p) :=sup[r: B(p,r) is a region of curvature > k}.

If p does not lie in a region of curvature ^ к then we can take c ip) = 0. The func-tion ck is obviously continuous, and proving the Global Comparison Theorem amountsto proving that c^ip) = °o for some point p. We begin by describing a constructionthat goes back to Alexandrov, and is a standard step in proofs of Toponogov's Theo-rem in Riemannian geometry. In describing it, and in the proofs which follow, we adoptan American colloquialism and say that a triangle or hinge which satisfies the relevantcurvature condition is AOK. Let p,q,r,s be points in X joined by minimal curves,where s is on the minimal curve joining q and r. We claim that if all the smaller hingesH(yqp, >%,), H(yUj, Ysp), H(ysp,ysr) are AOK, then the large hinge H{yqp, yqr) is AOK.For the proof, choose a representative triangle T(Q, P, S) of T(q, p, s) in S^ and ex-tend the shortest curve YQ$ beyond S to a geodesic FQR of length d(q, r). Then by Al,а(у*(Пfsrp) ?etkU;q,P) =ct(ysQ,ysp), so a(ysp, ySF) ^a(ySP,ySR) by Lemma 38, andit follows from A2 and the second monotonicity that dip,r) ^d(P, R). On the other hand.

838 C. Plant

Al also implies that a(y4P, yqr) = a(yqp,yqs) c T(YGP, YQR), so it follows from the sec-ond monotonicity that H(yqp, yqr) is AOK, This argument can be easily extended to aninductive argument in which the geodesic yqr has been subdivided finitely many times andeach of the smallest resulting wedges except the "last" is AOK. In the proof, the process ofalways going back to the beginning angle after working one's way forward is more exag-gerated. For that reason, we will refer to an argument of this type as a "b&f" (for backwardsand forwards) argument. For example, a b&f argument can be used to prove the followingproposition, by subdividing the nonminimal geodesic into minimal segments. Although itis of intrinsic interest, we will not use the proposition in this paper.

PROPOSITION 45. Let H(yt>4, y[lr) be a hinge in a sufficiently large region of curvature^ к such that ypq is minimal and ypr is a (possibly nonminimal) geodesic of length ^тт/s/k. Then H{ypq, ypr) is AOK.

Since all points may not be joined by minimal curves, we need to use a modifiedform of the b&f argument. Suppose, in the above case, the point s is not joined top by a minimal curve. Then by Corollary 30 we can find points л-, —>• s such thatSJ e Jp П Jq П Jlr. Since ak($i\q,r) —* я, we can apply the same argument (with afew E 's introduced) to reach the same conclusion, as long as we know that the hingesW(yqp, Yqst), H(ySiq, ySiP), HiYstp, y.sir) are all AOK. We will still refer to this type of anargument as a b&f argument.

Theorem 43 follows from the next proposition. In fact suppose that Q(X) — r < со forsome r > 0. Then Proposition 46 implies that there must exist a point x\ e B(x, 3r) suchthat Ck(x\) < 1/6. But then there must exist a point xj € B{x\, 1/2) such that Q(XT) <1/12. Continuing in this way we can construct a Cauchy sequence of points {*,} such thatck(xj) -> 0. But since X has curvature ^ k, x := limjQ has ci;(x) > 0, a contradiction tothe continuity of c*.

PROPOSITION 46. If there exists some К > 0 such that on the ball B(p, r), then

The basic idea of the proof is to fix a base point q € B(p, г/У). Choose positive x <such that if (yah, rac) is a hinge in Sk such that a(y, г) ^ Л/2 and L(r) < 4x then

for all t € [0, Z,(T)1, d(a,t(t)) ^ L(y) + 2#. The last condition can be accomplishedbecause, as was pointed out after formula 4, small and thin triangles in Sk are almostEuclidean. Of course the choice of x depends on, and is an increasing function of, &. Thenfor some suitably chosen small number x > 0 w e want to prove the following statementby induction:

S(«) If (a;qt b) is a triple such that q, b € Ja, d{q,b),d(q,a) < nx andd(a,b) < / ,the hinge (yai/, Yab) is AOK.

If we can prove this statement, then we can remove the requirement that d(a, b) < x bya b&f argument. One only has to be careful in the case of k > 0, as the induction takesone close to nx = nf-Jk, because then the existence of representatives in 5^ becomes a

Metric spaces of curvature ^ k 839

problem. By making x small, we can come arbitrarily close to nj°Jk.. Now suppose thatd{a, p) = n/\fkdxi&a and p are joined by a minimal curve y. If q e Ja r\JpnJr-, wherePi -• p along y, then applying S(«) for small x and passing to the limit shows that theminimal curves yiUj

a n d ypq together form a minimal curve from a to p. Since our spaceis not one-dimensional (see Remark 41), we could choose q not on y, so there are twodistinct minimal curves joining a and p. Since geodesies do not bifurcate and a and pwere arbitrary of distance n/Vk, we see that the diameter of X is at most njsfk. It is nowpossible to argue that any two geodesies starting at p must come back together at a, andcomplete the proof.

S(«) is proved as follows. The case n — I is trivial, since x is small. Suppose S(«)holds, and we have a triple (a;q,b) such that q, be Ja, d(q,h),d(q,a) < (n + l)x andd(a,b) < x- We go back along yq(S to a point с such that d{a,c) = 3x- Now x was chosensmall enough that the new triple (a; c, b) lies in a ball of radius 2e, so we will be done byab&f argument if we can show that any triple (d;q,b) such that

d(q,d)<(n-2)x and d(a?b) < 4x

(5)

is AOK.Let z = yqij and R] = y^h and suppose (Г, N) represents the hinge (r, RJ) in St. We

denote points in & by the capitals of die letters denoting points in X. Let в = УВД inSt. Note that if d{Q, B)'^ nx > d(q,b) then the hinge is A2 and we are finished, so wesuppose that d{Q, B) < nx- Now we can choose x small enough that the hinge (Г, N) issufficiently thin (recall that thin triangles in Sk are approximately Euclidean, or see [76]for more details) that the point F on в such that d(B, F) =2x has the following property:IfQ is minimal from D to G then for all t, d(Q, Q(t)) < n x -

To motivate the rest of the proof, we first show how, in a Riemannian manifold, theproof is now easily finished. Choose a geodesic со starting at d such that the direction of wis between the directions of ц and r, and a{&, r) =a{£2, T). Let / =o)(d(D, F)), Thena b&f argument shows inductively that со stays inside B(p, nx), that the hinge (GO, r) isAOK, The hinge (a>, 17) is small enough to be AOK, and now we get that (Т, fj) is AOKfrom the fact that а(т],т) = a(co, r) +ct(w, ф an& d(,b,q) ^ d(b, f) +d(f,q).

In the general case, we will see later (Proposition 53) that there is a direction GO havingthe above properties, but it may not be a geodesic direction - and even if it were, it mightnot extend to sufficient length. Instead, we show the existence of points which play the roleof points on со in the above argument. This kind of idea is carried further by the constructionof gradient curves and quasi-geodesics later in this paper. The actual argument is somewhattechnical; for our sketch here we assume that all points can be joined by minimal curves,and limit ourselves to showing the following: There exist points d' e X and D' e Sk suchthat

(1) cc(d';b,q)Kc((D';B,Q\(2) d(q,d')=d{Q,D')=d(q,d),(3) d(d',b)^d{D\B),(4) s(D';B,Q)<e(D;B,Q).

840 С. Plaut

This is the first, and essentially the "open" step in an open-closed argument that showswe can find points d" and D" satisfying conditions (2), (3), and s{D'\ B, Q) = 0. We thenhave

d{b,q) < d(b,d") + did",q) ^ d{B, D") + d ( D ' \ Q) = d{B, Q).

(The condition (1) is needed to ensure we can iterate the construction.) To construct D'and d\ choose a point E on YQD such that d(E, D) = % and let D' be the point on YEBsuch that d(Df, Q) = d(d, q). Now construct d' in the same way in Sk- By construction,(2) and (4) are satisfied. In what follows it is necessary to use triangle comparisons; ineach case, the comparisons are valid because the triangles or hinges involved are eithersmall enough or satisfy the inductive hypothesis. From A2 we get that d(e,b) ^ d{E, B).From this inequality and Al we get a(e; d, b) ^ a(E; D, B), and therefore a(e; q, d') ^a(E: Q, D'). From Al we now obtain that d(e, d') > d(E, D'), which implies (3). Fromd(e, d!) ^ d(E, D') and Al we get tx(d'e) > a{D'\ e, Q), which implies (1).

COROLLARY 47. IfX is a space of curvature ^ k > 0 then diam(X) ^ n/Vk. If in addi-tion X is locally compact, then X is compact.

3.5. The space of directions

In this section we assume that X is an inner metric space of curvature k. We now beginthe construction of the infinitesimal structure of a space of curvature bounded below. Atany point p we consider the space Sp of all unit geodesies starting at p. We identify geo-desies which agree on some initial interval; equivalently, for our space we can take all unitparameterized geodesies starting at p with maximal domain of definition. It is easy to ver-ify that the angle a between geodesies is a metric on Sp; Sp with this metric is called thespace of geodesic directions. We denote by £p the metric completion of Sp, which is calledthe space of directions at p. Note that in a Riemannian manifold, M", Zp = $'p = $* \and is identified with the unit sphere in the tangent space.

REMARK 48. We will abuse notation, writing у е Sp to mean simply that у is someunit parameterized minimal curve starting at p, such that y(0) — p. From now on, whenconsidering a minimal curve yah, we will represent, without further comment, the geodesicdirection corresponding to уаь (in either £« or S;,) by y.

PROPOSITION 49. For any p e X, ifEp is an inner metric space then all of Ep is a regionof curvature ^ I.

The essential argument for the above proposition may be found in Proposition 2.4 [72](the additional assumptions in [72] can be removed by using Corollary 38 in this chap-ter, rather than Lemma 2.3 in [72]). We prove the statement here under the simplify-ing assumption that every pair of points in Sp can be joined by a minimal curve; suchcurves can often be found (cf. Proposition 53). Let y\ m € Sp be such that Y\ , y2, y$


lie on a minimal curve in Sr. We also assume ctjj :— a{yi,yf) < n. Fix a base pointP in Sj and choose geodesies Г\,..., A starting from P such that а(Л,i" )) — «ty forall г ^ j except i = 3, j = 4. Since the directions Г, are isometrically embedded in thespace of directions Ep, which is isometric to Si, by monotonicity we can verify A2 byproving that а(Гз, A) > « м . Let f, -r* 0 and let о д : = /;(r,) and Ay :•• /(г,). Thend{aij,aik) = d(Atf»4j*) + o{/,). Let 0^ be minimal in X from a-,j to «,-t and Ф^ beminimal in S\ from At j to A^; then we have by A2 that <*(<#, | , Ф 2 4 ) > « C * l t > ^ 2 4 ^ + ° ^ ^ '

and so a(4>2V <p'24) ^ <*(#23> *2<t) + °('<)- ^ n o w follows from Al that d(A,-^, A,-4> 5^(aP3, a;4) + ofa), and therefore that ot(Ai A ) S «34.

In many important cases, which we will consider later, Sp is known to be an inner metricspace. As mis article was going to press we received a copy of the thesis of Stephanie Gloor(1998, Zurich), which contains an example of an inner metric space with curvature > кsuch that the space of directions at some point is not an inner metric space. This exampleleaves open the question of whether the space of directions must still have curvature > 1 inthe sense of Berestovskii-Wald, and whether (as is probably the case) the metric still hassome weaker geometric property related to being an inner metric.

The above proposition shows that the infinitesimal properties of spaces of curvature ^ кare intimately connected with the global properties of spaces of curvature ^ 1. We need afew more basic results.

LEMMA 50. Suppose p,q e X and p e Jq. Then forany q; -*• q and minimal curves ypqi,

To prove the lemma, note that by Lemma 38, Lemma 15 and Al it suffices to observethat if p' is such that a(p\q, p') is close to n then for i large, u (p; q,, p') is close to n.The next proposition is a generalization of Satz 8, Section 38 in [78].

PROPOSITION 51. Suppose p,q € X and p e Jq. For any minimal curve у starting at q,iffi is the {unique) unit minimal curve from q to p, thena(y, fi) = limt^^oek(q; y{t),p).

PROOF. If 0 and any segment of у together form a minimal curve, or if a(fi, y) = 0 thenthe proof is immediate; we assume the contrary. It is immediate from Al that a(/, 0) >]imr_>o£*£(#; y(t),p); we need to show the opposite inequality. Fix e > 0 and choose(using the definition of the angle) a := y(s) and b := j${s) / p such that otii(q;a,b) >ct(y,P) — e.Let D :—d(p,b) > 0 and fix positive t <s small enough that d(p, y(t)} > D.

We make the following claim: We can find a point и € Jp П Jfl n J4 arbitrarily close toy(t) with the following properties:

(1) For the point с on ypu such that d(c, p) = D,ctt,(q; u,c) ^ a t i q ; y(t), p).(2) ak(q\u,c)>Qtk(q\a,c)~e.To prove the claim, suppose we take a sequence Uj e Jp n Ja D Jq converging to y{t)

and the points Cj on yj := ypUi such that d(p, cj) = D. First, there exists an a> > 0 suchthat for all large /, «*(?; CJ, p) ^ u>. Suppose, on the contrary, щ($\ С$, р) ~* 0. Thenak(cj; p,q)^n anda(yC j P, yCjq) -ъ л. Thena(yC jUj, yC j q) -*• 0. If limsupd(c j, Uj) >d(b,q) then (taking a subsequence if necessary) for some q, on Ycjur qj ** »q- ByLemma 50, a(ypuj,0) -* 0 and, since y(t) ф q.a/dq; p, y(t)) -*• ТГ, a contradiction

842 С. Phut

to a ( ^ , y) ф п. A similar argument contradicts a(p, y) / 0 when limsupd(c;,H,) ^d(b,q).We now choose j large enough thatja*(</; uj, p) — ak(q; y(t),p)\ < шa ndu, isclose to y(t). Take representative triangles T(q, uj, Cj) and T(q,Cj, p) having a commonside between points q mdcj. By the triangle inequality, d(uj, p) ^d(uj,Cj)+d(£j, p) =d(Uj, p). In other words, to make a representative triangle of T(UJ, p, q), we need to ro-tate the segment between q and p to make the angle at q larger (or at least not smaller).Now we have that

,p) ^ <xk(q\ujt p)-a>^ak(q;UJ,Cj)+ak(q\cJtp)-a>

which proves part (1).To prove part (2), observe that АО implies that a^iq; y(t),с) ^ ak(q;a,c), so we can

satisfy (2) simply by taking uj close enough to y(t).To finish the proof of the proposition, choose r, -* 0, и,- and a satisfying (1) and (2)

above. By choosing U, close enough to y(f,) we can ensure that Щr -> q; Lemma 50and A2 now imply that c; -> b, and ctk(q;a,f;) ~»« Ar(?;a,i>), so for large enough i,en(q; я, с;) ^ a(y, ^) — 2E. Putting our inequalities together we get

Since the sequence {t,} and e were arbitrary, the proof is complete. D

The only place in the above proof where we used the fact that p € Jq was when weneeded to know that аф,y p U l ) -*• 0. Therefore the same proof (even a simpler one, sincewe don't have to pick the points иj) gives us the next proposition. We will need it somewhat

PROPOSITION 52. Let p Ф q in X, and let fi join p and q. Suppose that у is a minimalcurve starting at q and for some t -» 0, there exist minimal curves fit from p to у (t) such

PROPOSITION 53. Let p,a,heX and suppose that p e JaC\Jb. Then there exists a min-imal curve in Zp joining ypa and урь.

For each / > 0 we join y(t) and fi(t) by a minimal curve £f. Let Д and yt denotethe restrictions of the two curves to [0, t]. Our first claim is that the sum of the angles£(*) of tne triangle formed by # ,$ , ,& tends to Л as t -*• 0; that is, the triangle be-comes Euclidean in the limit. To see this, represent the hinge (y, ft) by (у, Д) in 5*,and let \t denote the minimal curve joining y(t) and Д(0- We give a sketch of the ar-gument, using some simplifying assumptions: we assume that every pair of points can bejoined by a minimal curve, and that у := ypa and $ := yph are in fact extendable be-yond p. In the proof of Proposition 51 we showed the kind of unpleasant extra detailsneeded if we do not use these simplifying assumptions. Also, we will limit our argu-ment to constructing a midpoint between /3 and y. Note that, by moving a and b along


y and ft, respectively, we can assume a, h e Jp. Then L(f>) — L{%t) = o(t). From Propo-sition 17, weseethatliminf,^oSU) S **- On the other hand, Proposition 51 implies that|</(a. 0(0) - d(a, 0(0)1 = o(r). From Proposition 17 and Al we see that ifa'(() denotesthe angle of the hinge complementary to (yt,$i) and a'(t) denotes the angle of the hingecomplementary to (£},&) then liminf,—o«'(O 3 =Нт,_»оа'(0- We immediately get thatlimsupf_,oa(}',,£,) < l im,_oa(r / .^)- Now liminf,_oa()'/,tr) > Hm»_»o«(>V.?t) fol-lows from Proposition 17 and Al. The same argument applies to linv_n«(&,&)- and wehave proved the claim.

Let and let mt,mt denote the midpoint of §,,|f and let %ущ be a minimal curvefrom p, p to m t ,m t , respectively. We complete the proof by showing lim,_na(fy. y) =a(y, /9)/2 (a similar argument shows that \imt->Qa(t}t,fi) = a(y,ft)/2).By Al and Propo-sition 17 we get liminfr_oa(^, y) ^ a(y, fi)/2. To get the opposite inequality, recallthat we have assumed that у is extendable (as a minimal curve) beyond p. Let с е Jp

be a point on that extension, and let <r be minimal from с to m t, with correspondingpoint с and curve <r in S*. Now, since lim^o<*(>',,£/) — 1т%_*оа(уь|/), we get fromA2 and Proposition 17 that Щ д - Щ;) ^ o(f)- That \Щ() - Щ()\ < o(f) can beobtained by observing that by Proposition 51 \d(c,/)(()) -d(c, Д(0)1 ™ <K0 and ap-plying АО the triangle with corners c, y(t), f${t). Proposition 17 and Al now imply thatИт,_,оа(^, ypc) ^ n -a(y,$)/2 = л -a(y,f i)/2, and the proof is complete by Corol-lary 38.

REMARK 54. When we refer to the unit sphere S" (or simply an и-sphere) we mean thesphere of curvature 1 when n ^ 2, a circle of length 2n when n = 1, and two points ofdistance ж when n = 0.

If yab is almost extendable past b, by Theorem 27 we can find a point с joined to b by aminimal curve, such that a(b\ a, c) is arbitrarily small. But then а&(й; а, с) is close to ж,so by Al, а{уьа, Ybc) is arbitrarily close to л. Given another such point c', we have fromLemma 38 that а{уЬс>,уь с) is small. In this way we can construct a Cauchy sequence ofdirections y, such thata(y, yt) -> Ж. We have proved:

PROPOSITION 55. Ifуаь is almost extendable past b, then у has a complement in Ep\i.e., a direction -y e Sj, such thata(y, -y) — n.

Suppose now that у e 5;, has a complement. First we restrict у to a segment yn whichis extendable past q. We then choose geodesic directions yi ->• y. Now by choosing pointsjt; sufficiently close to p along yiy we can ensure that ak{p;q,Xi) -¥ n. In this way wecan prove a converse to Proposition 55:

PROPOSITION 56. Ifу e Sp has a complement —y in Sp, then there is a restriction ofуthat is almost extendable past p (and is the unique minimal curve between its endpoints).

Ordinarily, the angle between minimal curves is only semicontinuous (cf. Lemma 36).Suppose that ft, щ start at p, and у, -• y, «, -• a, where y, a are minimal starting at

844 С. Plant

P = lim pi. Suppose that a is extendable past /), and denote by —upq a minimal curve start-ing at p, q e Jp, such that a(a, —a) — jr. Let ft be minimal from pi to q. By Lemma 36,a(aj,p,) = ТТ. Then by Lemma 38, a(y,a) ~it — ct(y, —a) ^ n — liminf a(K. A) =lim sup a^.Q

! ,) . Since we already know the opposite inequality, we have proved the con-tinuity of the angle. With a little more work one can show:

PROPOSITION 57. Let yi, ^ eSm, where peXh and p, -> p e X-//>;-• у e Sp, anda,; -¥• a e Sp and a is almost extendable past p thena(y,a) = lima(yi,ct().

3.6. Differentiability of the distance Junction

DEFINITION 58. Let X be a space of curvature ^ к and peX. Suppose у :[a,b] ->• X isunit minimal. We define dp:[a,b]-+ X by dp(t) = d(p, y(t)).

The differentiability properties of the distance function dp(t) in a space of curvature ^ кare of fundamental importance. In a Riemannian manifold, if there is a unique minimalcurve 0 from p to y(t), t e (a,b) then d'p(t) = -СО£«(И(М>}>Э) (in 5* this followseasily from the cosine laws). The second derivative of dp{t) is (unlike the first) dependenton curvature and the distance from p to уф). We will consider analytical properties ofthe distance function further in the next section. For the moment we limit ourselves to thefollowing results.

DEFINITION 59. Let x,v,ze X. For every e > 0, denote by a^ (y; x, z) the infimum of<*(>V> >Vv)< where the infimum is taken over all x',z such tha.td(x,x'),d(z,z') < e. Letoc(y; x, z) = linw

The proof of the following lemma is now immediate. It allows us to pick the points x'and z' in Jq, and we will use it without further reference,

LEMMA 60. Let X,?,Z e X. For every s > 0, denote by <x((y\x,z) the infimum of<x(yyx'>Yyz')> where the infimum is taken over all x1

\z! such that d{x,x'),d(z,z') < s andx\ z' € Jq. Then <x(y; x, z) = l i m ^ o a / ( y ; x, z).

PROPOSITION 61. Let рф ц <? € X and suppose that у е S(j and г ф р is in the interior ofy. Then dp(t) is differentiable from the right at t = 0 , and d'JO) = - c o s a ( p , y), wherea ( p , y) = \im)^oa/c(ci: p, y ( t ) } ^a(q; p, r).

PROOF. First note that the function f ( t ) = oikiq', p, Y(t)) is bounded and decreasing byAO and the second monotonicity, so the above limit exists. In St, consider a minimal curveFPQ such that L(F) = d{p, q). Now rotate Г, say, clockwise, at Q to a minimal curveT, such that а(Г, F,) = ak(q\ p, y(t)). Note that Го := l im ; ^ 0 П satisfies a(F(), F) -u{p, y) and d(P, r,(t))=dp(t). Then since a(r,, f® -*- 0,


Now

„ d(P, r o ( t)) -d(P, Q)lim = — co&a(p, Y)'-/-*O t

we get that d'p(O) = — cosa(p, y). The final inequality follows from the definition ofa(q; p,r) and Al. D

From Proposition 51 we get the following:

COROLLARY 62. Let р фц e X with p e Jq. For any minimal у starting at q, the rightderivative d'^iO) = —cosa(y,fi), where ft it the minimal curve joining p andq.

The following result is useful for computing angles when the points in question are notmoving along some fixed minimal curve. Its proof can be found in [74].

PROPOSITION 63. Let p e X and suppose that yl>x is the unique minimal curve betweenits endpoints. Suppose that а, ,Ь,—*р, у; is minimal from a{ to x and fij is minimal fromaj tobj. Then a = lim(_vtx,a(}';, Д) exists if and only if

d(x,bi)-d(x,ai)L- Inn

exists. If L and a exist, then L = — cos a.

From Proposition 61 and Lemma 38, we get the following:

LEMMA 64. Suppose p e X andy: [—t,t] is a geodesic with p^y(0). Thenifу(0) is лlocal minimum for dp(t) :=d{p,y(t)) then for any minimal ft from y(0) to p, ffi(y, j3) =7 Г / 2 .

4. Constructions

The importance of a lower curvature bound is partly due to the fact that it is preserved un-der a great variety of constructions, many of which, such as cones and joins, generally failto preserve the property of being a Riemannian manifold - or even a topological manifold.The basic constructions given here were noticed by various people, sometimes indepen-dent of one another. Gluings, and scaling, and products are "classical" constructions; coneconstructions were first discovered by Berestovskii, who used them in his solution of Bor-suk's problem of metrizing polyhedra [9]. We remind the reader of our exclusion of one-dimensional spaces (cf. Remark 41), which often require separate but trivial treatment. Forexample, Propositions 87 and 90 are only true as stated if the cone is not 1-dimensional. Ifthe cone is 1-dimensional then X is a one or two point space.

846 С. Plant

4,1, Induced metrics, gluing, scaling

DEFINITION 65. X is called pre-'mner if every pair of points in X can be joined by arectifiable curve, and for every p e X and 5 > 0 there exists an e > 0 such that if у еB(p, e) then у is joined to p by a curve of length less than S.

A pre-inner metric space (X,d) has an induced inner metric dj, where dj(x,y) —inf{L(c): с joins X and y]. It follows from Definition 65 that d/ (which always satisfiesd/ ^ d)has the same topology as d. A special case is when У is a subset of an inner metricspace X so that the subspace metric on Y is pre-inner; we refer to the inner metric inducedby the subspace metric on Y as simply the induced inner metric d\ on Y, We do not knowof any purely topological conditions that are necessary and sufficient for the existence ofan inner metric on a topological space.

DEFINITION 66. If У is a subset of X, then Y h caUed metrically embedded if the restric-tion of the metric of X to Y is an inner metric on Y'.

Let {X | . d\) and {X2, di) be metric spaces and Y\ с X\, Y2 С Xi be closed subsets suchthat (with the subspace metric) there is an isometry /: Y\ -* УЬ. We "glue" X\ to Xjusing/ in the following way. First, we take the disjoint union X \ U Xj and define a space X byidentifying the points in у\ g Y\ and y2 e Y2 such that l(y\) = У2- We have a quotient mapq: X, UX2 - X; we denote by Г the set q{Y\) = <?№), and identify X; with <?(*/). Thenthe space X; U/ X2 is obtained by putting on X the following metric: d(a,b) = dj(a,b) ifo and b both lie X,- for fixed ( = 1,2, or d(a,fc) = inf>iey{rf|((7,y) + d2(y, b)} if a e X|and £> e X2. It is not hard to verify d is a well-defined metric, and the subspace metric onXi is the same as d,-. The following proposition is not hard.

PROPOSITION 67. Let X\, X2 be inner metric spaces, Yi c X\, Y2C X2 be closed sub-sets, and I: Y\ -v Y2 be an isometry. Then X\ U/ X2 is an inner metric space, and X,- ismetrically embedded in X\ U/ Xj.

Generally, gluing sets with a lower curvature bound does not result in a space withcurvature bounded below, even when the sets Yt are metrically embedded. For example,one can glue two triangles along a side of equal length, identifying two vertices with angles> JT/2. Then the glued edges form a minimal curve that branches along the two remainingedges at each endpoint, so the glued space cannot have a lower curvature bound.

If (Y, d) is a metric space, we can scale Y by the positive factor с to produce the newmetric space {Y, cd). Clearly this operation preserves the property of being an inner metricspace. Scaling a space form S^ by a factor of с results in the space form S" 2, and soit follows from Definition 22 that scaling a space of curvature > к results in a space ofcurvature ^ k7


4.2. Gromov-Hausdorff convergence

Recall that if A, В are subsets of a metric space Z, then the Hausdorff distance between Aand В is denned to be

dH(A,B) = M\e>Q: А с BS{B) and В с B £ ( A ) } ,

where B,(A) = U € Z: d(jt, A) < e}.

DEFINITION 68. Let W,У be compact metric spaces. The Gromov-Hausdorff distancebetween W and Y is defined to be

where the infimum is taken over all metric spaces Z with isometric embeddings ф\у : W —>Zandфу:Y-> Z.

The Gromov-Hausdorff metric is a honafide metric on the space of all compact metricspaces, if isometric spaces are considered equivalent. In general, the Gromov-Hausdorffmetric is very weak, and says little about the structure of close spaces. However, in classesof metric spaces with sufficient bounds on geometric quantities, the Gromov-Hausdorffmetric achieves two important properties: it is totally bounded (i.e., has compact metriccompletion, or is "precompact"), and spaces close in the Gromov-Hausdorff metric aretopologically "close". We will discuss these properties later in this article.

We will be interested in convergent sequence of metric spaces in the Gromov-Hausdorffmetric, and will now establish our notation. From now on, X, -> Xo will denote that themetric spaces X\ converge to Xo in the Gromov-Hausdorff metric. By the results of [32]we can assume that the spaces X, are all subspaces of a fixed metric space Z, and conver-gence is in the original Hausdorff sense. It is easy to show that for each p, q € XQ thereexist points pt, q, € X, such that pi —* p, qt —> q, and given any such sequences p, —• pand qs -> ц, dii.Pi, яд ~* doip.q). From these facts we immediately have the followinglemma:

LEMMA 69. Let X, --> Xo (X, compact metric spaces).(1) If each Xj, i > 0, is an inner metric space then XQ is an inner metric space.(2) If B(pi,r) С Xt is a region of curvature > к for all i > 0 and pi ^> p e Xo, then

B{p, r) is a region of curvature > к in Xo.

REMARK 70. It follows from Theorem 43 and the above lemma that Xo has curvature^ к if each X, has curvature >£ . In other words, we see that the class of all compact innermetric spaces of curvature > к and diam( X) ^ D (for fixed к and D) is closed under theGromov-Hausdorff metric.

Given curves ^;: [a, b] ~* X, С Z, we will write fa -* ^o if fa converges uniformlyto ^o in (the ambient metric space) Z. Clearly, if each fa is minimal, then so is fa). More

848 С. Plaut

generally, it follows from the way in which lengths are measured that liminfZ,(^) ^L(^o)-It is not always possible to approximate minimal curves in XQby minima] curves inXt. A simple counter example can be constructed as follows: Let S be a circle and a \ andai be antipodal points. Fix a semicircle E determined by a\ and «2, and points aFJ -> aj(} — 1,2) in E. Form inner metric spaces S, by attaching segments o\,ai of fixed length toan, i, an, respectively. In the limit we attach 07 to a/, and there are two minimal curves fromthe midpoint of o\ to the midpoint of 02, °ne passing through E and one not. The latterminimal curve cannot be approximated by minimal curves in the sequence Si. However,if Xo has curvature > k, then by Corollary 39, each minimal curve in Xo has a restrictionthat is the unique minimal curve у between its end points, say p and #. But then choosingPi -• p and m -> q and y< = Ур-,щ. it follows from uniqueness that у, ~* у. With a littlemore care (cf. [74]) we obtain the following:

LEMMA 71. Let X, -v Xo, and suppose XQ has curvature ^ k. Then for each уаь, Уис inX{) there exist a,- -> a and yaihi, Yma in Xf such that y(llb: -* yllh and ya.a -* yat.

DEFINITION 72. The dilatation of a mapping f: A -* В between metric spaces A and Вis defined to be

di l(/) :=logI supd(x,y)

Note that a mapping is distance nonincreasing if and only if it has nonpositive dilatation.

DEFINITION 73. If Л and В are homeomorphic metric spaces, the Lipschitz, distance be-tween them is defined to be

where the infimum is over all homeomorphisms between A and B. If A and В are nothomeomorphic, we define di(A, B) = 00.

Note that for compact metric spaces A, B, dL(A, B) — 0 if and only if A and В areisometric. It is not hard, then, to verify that dL is a bona-fide metric on metric spaces(with possibly infinite value). The following proposition, whose proof can be found in[32], shows one connection between Lipschitz and Gromov-Hausdorff convergence.

PROPOSITION 74. Suppose {Г,} is a sequence of compact metric spaces having uniformlybounded diameters. If Y is a metric space such that for every s-net N in Y there existSi -nets Mi in Y, such that {M} converges to N in the Lipschitz metric and e; -> s, then Y;converges to Y in the Gromov-Hausdorff metric.

The converse to the above proposition, that Gromov-Hausdorff convergence of F, to Yimplies that nets in Y can be approximated by nets in Yt in the Lipschitz metric, is alsoessentially true (cf. [32]).


We need to be able to study convergence of locally compact inner metric spaces, whichneed not be compact.

DEFINITION 75. A sequence (Г,, pi) of pairs, where Г, is a locally compact inner metricspace and p, € lj_, is said to be (pointed) Gromov-Hausdorff convergent to a pair (Y, p) iffor every r > 0 , B(p;,r) is Gromov-Hausdorff convergent to B{p,r).

Clearly pointed Gromov-Hausdorff convergence preserves the same metric propertiesas the nonpointed version, and the limit space Y is a locally compact inner metric space.

Finally, we give Gromov's Precom pact ness Criterion [32]:

THEOREM 76. Let X be a collection of compact metric spaces. Then X is precompact{totally bounded) in the Gromov-Hausdorff metric if for every e > 0 there exists an N(s)such that every X £ X has an e-net with at most N(s) elements.

4.3. Products

Given metric spaces (Xi,d\), (Х2,^г), we define the product metric on X\ x X2 to be themetric D((X\ , X2), (>'i,У2У) = yfd\ (x\, y\ )2 + di(x2, Уг)2

- It is easy to see that (*i, x2) andCzi , zi) have a midpoint (y\, уг) if and only if yj is a midpoint of x\ and ц , and y2 is amidpoint of X2 and г?. By adding ens to the argument and applying Proposition 7 we easilyobtain that the product of inner metric spaces is an inner metric space. Note that, if M\and Mi are Riemannian manifolds and M\ x Mj is given the product Riemannian metric,then the corresponding Riemannian distance in M\ x M2 is the same as the metric product(in the above sense) of the Riemannian distances in M\ and M2. Furthermore, it is well-known in Riemannian geometry that S», x S^ has sectional curvature ^ min{0, k\, ki\.Therefore, if X\ has curvature ^ £1 and Xj has curvature ^ k2, then every quadruple inX\ x X2 can be embedded in S*, x 5^2, and hence in some Sm with m > m i n { 0 , y t | Д г ) -

We have proved:

PROPOSITION 77. If X\, X2 are spaces of curvature ^ к\, кг, respectively, then X\ x Xjis a space of curvature ^ min{0, k\, ki}-

We next consider the space of directions at a point (p,q) e X| X X2. It follows from ourprevious discussion of midpoints that a curve y(t) — (y$t), yi{t)) is minimal if and onlyif 71 it) and KJ(O are minimal. More precisely, 7 is a unit geodesic if and only if yi is unitforI = 1,2, and у{t) в {y\{k\t), yntkg}), where kj + k^ = 1 and y\, yj are unit; we writey = k] y\ + &2K2- In other words, E{P#) is the set of all linear combinations of elementsof Et> and Sq whose coefficients are square summable to 1. By definition, the set £(p,q)is the join Ep * Sq. If у = k\y[ + km e £{P.q) and ft — тф\ + m202 e E(P,q) thencosa(y, fi) = k\m\ cosa(yi,fi$ + k2m2 cosafy:, рг)- (This is the same formula one getsfor the angle between two vectors in Euclidean space, written as a linear sum). From thisformulation it is easy to see that if Ep and Fq are inner metric spaces, then so isWe summarize the properties of Z{p,q) in the following proposition:

850 С. Phut

PROPOSITION 78. Let K\, Xi be spaces of curvature bounded below. For any p e X\,q e X2, £{p,q) consists of orthogonal, isometric copies of Ep and Zq, together with aunique segment of length л /2 joining each element of Up to each element of Eq.

Proposition 77 can be easily extended to finite products. Now suppose (X,,<£), i =1, 2 , . . . , are metric spaces whose diameters are square summable. Then we can define theproduct metric on X1 x X2 x - • • by

It is easy to see that X = X\ x Xj x • • • is the Gromov-Hausdorff limit of the finiteproducts Xk = X] x ••• x Хь since each Xk isometricallу embeds as a slice in Xk+l.Combining this with our earlier results in this section we obtain the following:

PROPOSITION 79. Let (Xt,di) be spaces of curvature ^ к whose diameters are squaresummable. Then X } x X% x - • • with the product metric is an inner metric space of curvature^min{k,0}.

4.4. Submetries, quotients

DEFINITION 80. Let W, Y be metric spaces. A mapping / : W -> Y is called a weaksubmetry (respectively submetry) if for each metric hall B(x,r) (respectively closed metricball B(JT,r» in W, f(B(x,r)) = B(f{x),r) (respectively /(B(x,r)) = B(jr, f ( r ) ) .

The above definition is due to Berestovskii [11], and generalizes the notion of Rie-mannian submersion. Weak submetries are characterized by the property that for everye > 0 and yuyzG Y, there exist x, e fl(yi) such that \d{xuxz) - d{yuyz)\ < s. Forstrong submetries we can take e = 0 . The following is now not difficult to prove (cf. [16]);

PROPOSITION 81. ff f; X ~* Y is a weak submetry and X is an inner metric space ofcurvature ^ k, then Y is an inner space of curvature ^ k.

Let G beagroupof isometries of X. The quotient space X/G is the set of all equivalenceclasses [x] of x € X, were x = у if and only if there exists a ^ e C such that g(x) = y.The orbit of a point x € X is the set G(x) = {y: v = x] = {g(x): g e G]. If each orbitG(x) is closed, then for any x, у e X, d(x, G(y)) > 0 if and only if x £ G(y). For anyz 6 G ( X ) , there is an isometry g € G such that g(x) = 2. If w e G(y) and D(X, W) isclose to realizing d(x, G(>')), then g(w) e G(y) and d(z, g(w)) — d(xt w ). It follows that

, G(y)) = d(z, G(y)) = dH{G(x),

DEFINITION 82. LetG be a group of isometries of X such that the orbits of G are closed.We define the quotient metric on X/G by d([x], [y]) = dH{G{x), G(y)).


Using the previous discussion it is not hard to verify that the ahove definition gives abonafide metric. We also quickly obtain the following results:

PROPOSITION 83. IfG is a group of isometrics ofX such that the orbits ofG are closed,then the natural quotient map q: X —> X/G is a weak submetry when X/G has the quo-tient metric.

COROLLARY 84. Let G be a group of isometries of X such that the orbits ofG are closed.Suppose that X/G is given the quotient metric.

(1)IfX is an inner metric space then X/G is an inner metric space.(2) ffX has curvature ^ к then X/G has curvature > k.

REMARK 85. The orbits of G acting on X are closed if and only if the topological spaceX/G is Hausdorff.

4,5. Cones, joins and simplicial complexes

DEFINITION 86. Let X have diameter ^ jr . The Euclidean cone on X is the quotient spacecX = X x [0, oo)/ =, where (x, t) = (>\ s) if and only if s = t = 0, with the followingmetric:

d((x, s), (y, t)) = ^s2 + t2- 2stcosd(x, y).

In the above definition we abuse notation slightly, to simplify the formulation. Anotherway to compute d((x, s), (y, t)) is to take two lines in the plane with angle d{x, y) at 0,and measure length s to a point xs on one line, and t to a point yt on the other line. Thend((x, s), (>',/)) = d(xx, y,). It follows from elementary trigonometry that if X is an innermetric space then cX is an inner metric space. For example, if x and у have a midpoint zin X, then (x, s) and (y, t), for s, t > 0 and d(x, y) < я, have a midpoint (z, r), where ris the distance from 0 to the intersection of the segment from xs to y, with a bisector ofthe angle between the lines. Note that if £ > 1 then cS£ is a Euclidean cone with angle thediameter of S". We claim that cS£ has curvature ^ 0. To see this, let p denote the apex ofthe cone, and suppose, for simplicity, that n = 1. By slicing along a radial line from p andexamining the behavior of straight lines in the resulting flat triangle, we see that no shortestcurve between points away from p passes through p. Now consider a hinge (уаь, уас) incS£. We can assume that b ф с (for then A2 is trivial). By continuity, we can assume thatc^ p. If we slice along a radial line from p through c, we see that (уаь, Уве) correspondsto a hinge (У~ЛД, ГАС) in the flat triangle, having equal side length and the same angle.However, since we are making identifications, d(b, c) < d(B, С), А2 is satisfied.

Now a quadruple (a,h,c,d) in a X can be isometrically mapped onto a quadruple(a\b',c',d') in Sm for m > I if and only if the quadruple ((a,r), (b,s), (c,(), (d,w»in cX is isometric to the quadruple (<я', г), (bf, s), (c', t), (d', u)) in cSm. Since cSm isnonnegatively curved, we immediately obtain:

852 C. Plaut

PROPOSITION 87. X is an inner metric space of curvature > 1 if and only if cX is aninner metric space of curvature ^ 0.

The following is an immediate consequence of the various definitions:

PROPOSITION 88. If X is an inner metric space of curvature ^ 1 and p is the apex of cX,then Sp = Ep is isometric to X.

One problem with the Euclidean cone is that the subspace (X, 1) is not generally a met-rically embedded subspace. For this and other reasons it is useful to consider the sphericalcone c\X, which is analogous to the Euclidean cone, except that it uses the cosine lawfor S\.

DETTNITION 89. Let X have diameter ^ n. The spherical cone on X is the quotient spacec\ X = X x [0, RR/2]/ =, where (x, t) = (y, s) if and only if s = t = 0, with the followingmetric:

d[(x,s),(>',()) = cos-1 (cost cos s — sinГsinscosd(л:,у)).

Note that the spherical cone on S" is a hemisphere in S"+l. For к > 1, we can use anargument analogous to the one above for the Euclidean cone, to see that C|S£ has curvature^ 1. For simplicity, we take n = 1. First note that if p is the "apex" of CiS\, then atany point away from p, there is a neighborhood that is isometric to a neighborhood of ahemisphere Я in S^. In fact, we can take the neighborhood to be a "slice" S(x, x') betweenradial geodesies from p to antipodal points x, x' on S[. Thus the complement of {p} hasa lower curvature bound, so geodesies do not bifurcate. Now suppose that у is a minimalcurve starting at a point a = (x, t) / p. If у is not a radial geodesic toward p, it cannotintersect p as it moves across a slice S{x,x') (because this slice is isometric to a slicein H). Either у runs into the boundary S[ x [тт/2] and stops, or it runs into the far sideof the slice, to a point (xr, s). Since there is another geodesic "going around the back"from (x, t) to (x', s), у cannot continue beyond (x1, s) as a minimal curve. In other words,any two points distinct from p are joined by a minimal curve not passing through p. Tocomplete the proof as in the proof for the Euclidean cone, note that when we make a radialcut in ci $1 we obtain a space isometric to H with a section between two radial geodesiesremoved, and apply the same argument. As above we obtain the following results:

PROPOSITION 90. X is an inner metric space of curvature ^ 1 if and only if с\X is aninner metric space of curvature > 1• Furthermore, X is metrically embedded in c\X as theslice X x { Т Г / 2 } .

PROPOSITION 91. If X is an inner metric space of curvature > 1 and p is the apex ofc\ (X), then Sp = Sp is isometric to X.

If X has curvature ^ 1 then it is not hard to show that c\ X U/ c\ X, where I : X x{JT/2} —> X x {JT/2} is the identity, is a space of curvature ^ 1, called the spherical sus-pension E\ X. We now generalize this notion to that of the metric join:


DEFINITION 92. Let X, Y be inner metric spaces of curvature > I. Let p be the apex ofcX and q be the apex of cY. Then the metric join X * Y of X and К is defined to be thespace of directions S\p,q) ofcY x cY.

The properties of the metric join can be seen in Proposition 78. We only emphasize thatX * Y is again a space of curvature > 1. If we take Y = S°, then it is not hard to see thatX * Y is isometric to Z\ X. In addition, S? * 5f is isometric to 5"+ffl+1.

DEHNITION 93. Let X bean inner metric space of curvature > 1. Then the q-foldsphe r-ical suspension ЩХ X is defined to be S p 1 * X, which is isometric to q iterated sphericalsuspensions E\E\••• E\ •

Clearly Si X is again a space of curvature ^ 1. There are related cone constructions(cf. [19, Section 4.31). The elliptic cone produces a space of curvature ^ - 1 from one ofcurvature ^ 1; the parabolic cone, a space of curvature ^ - 1 from one of curvature ^ 0;and the hyperbolic cone, a space of curvature ^ - 1 from one of curvature > - 1 .

A simplex in S" is a convex subset of S% homeomorphic to a standard topological sim-plex of dimension n. For example, a simplex in S\ is a triangle (with its interior) whosesides are minimal curves, which is entirely contained in a closed hemisphere. Let С be asimplicial complex formed by gluing together finitely many simplices from a fixed spaceS£, so that each codimension-1 face meets two simplices and the sum of the dihedral an-gles at any codimension-2 face is ^ In. Then С has curvature >&. Note that we have onlyto verify the curvature bound locally. First, С is locally a cone (of one of the above types)on another finite simplicial complex whose simplices are in S"~', The result can now beproved by induction on dimension.

5. Examples

EXAMPLE 94. Let M be a Riemannian manifold. Recall that this means that M is asmooth manifold, together with a smooth assignment of an inner product to the tangentspace at each point. We measure the lengms of piecewise differentiable curves by integrat-ing the lengths of its velocity vectors. The distance between two points is denned to be theinfimum of the lengms of curves joining the points. It can be shown that this distance isinner. In fact, shortest curves are always geodesies in the sense of differential geometry:curves whose acceleration (in terms of the Riemannian connection) is 0. Equipped withthis metric, a Riemannian manifold M is a space of curvature > k, if к is 3 lower boundfor the sectional curvature of M. This follows from the Rauch Comparison Theorem (cf.[231). Conversely, if M has curvature ^ к in the sense of Definition 22, then M has sec-tional curvature ^ k. The space of directions Sp is always the unit sphere in the tangentspace; M is metrically complete if and only if M is geodesically complete.

EXAMPLE 95. Let lZ{n,k, D) denote the space of all rc-dimensional Riemannian mani-folds having sectional curvature > к and diameter < D. Then the closure K(n,k, D) inthe Gromov-Hausdorff metric consists of inner metric spaces having curvature ^ к and

854 С Plant

Hausdorff dimension < п. We remark that collapse is certainly possible: The flat torusT2 = s\ x Si is Riemannian, with sectional curvature 0. By letting к -> oo we obtain asequence of Riemannian manifolds with constant curvature 0 and bounded diameter con-verging in the Gromov-Hausdorff metric to S\ . For a discussion of collapse and con-vergence of Riemannian manifolds, see, e.g., [241. If one smoothly rounds the apex of aEuclidean cone with angle < jr. one can approximate the cone by Riemannian manifoldswith nonnegative curvature. This shows that geometric singularities can arise in Gromov-Hausdorff convergence of Riemannian manifolds. It is also possible to construct a sequenceof positively curved topological 2-spheres converging in the Gromov-Hausdorff metric toa line segment; thus topological singularities can also arise. Convergence of Riemannianmanifolds will be discussed in greater detail later.

EXAMPLE 96. Let M be a closed Riemannian manifold of sectional curvature ^ k. andG be a compact group of isometries of M. We have already seen (Propositions 83 and 81)that M/G has curvature > k.

EXAMPLE 97. Every convex surface (see the introduction) is a space of curvature ^ 0.A particularly "bad" convex surface can be constructed in the following way (cf. [5]);Begin with the boundary of a simplex EQ in R3. Take a barycentric subdivision of eachface of Ец, and "push" the barycenterof each face outwards by a small amount, to obtain anew surface E\. Subdivide E\, and again push out the barycenters slightly - always doingso by an amount small enough that the surface remains convex, and no faces are at anglen with one another. Continuing this procedure, one obtains a space В such that the spaceof directions at a dense set of points has diameter strictly less than n - that is, there is adense set of "singular" points. Note that this surface (and indeed all convex surfaces) canbe approximated by polyhedra [2].

EXAMPLE 98. The Alexandrov's rfowWe dM Q is obtained by gluing two Euclidean disksalong their boundaries. D can be obtained as a limit of convex surfaces, and so has cur-vature ^ 0. In fact, Q is usually considered as a kind of degenerate convex surface. Thearea of Q is maximal among all convex surfaces having the same diameter. It is a long-standing conjecture of Alexandrov that any convex surface of maxima] area (relative to itsdiameter) must be isometric to Q. Note that Q is geodesically complete. In fact, geodesiesare straight lines in the interior of either disk, then "flip" back and forth across the "rim"(identified boundaries) of Q. The rim itself is not a geodesic; it is shorter to follow oneof the straight lines on either side, to join any pair of points on the rim, Q can also beapproximated by Riemannian manifolds of positive curvature, spheres that are graduallyflattened; the sectional curvature tends to +oo along the equator that becomes the rim inthe limit.

EXAMPLE 99. Spaces of curvature bounded below that are definitely not manifolds ormanifolds with boundary can be obtained by cones and suspensions, or quotients. Forexample, real projective space RP", being a covered by S", admits a Riemannian metric ofconstant sectional curvature 1. Forл > 1, Z\RP" has curvature > 1, but it is well knownthat this suspension is not a topological manifold.


EXAMPLE 100. Consider the Poincare homology 3-sphere X, which, being covered byS3, also admits a Riemannian metric of constant sectional curvature 1. Edwards' DoubleSuspension Theorem asserts that Ef X must be a manifold, and hence a sphere. By Propo-sitions 91,90, E^ X has curvature > 1, but at the two cone points the space of directionsis isometric to E^X, which is definitely not a manifold, let alone a sphere. This exam-ple makes the important point that even when X is a topological manifold, the space ofdirections need not be a topological sphere.

EXAMPLE 101. Let { S,}) be a sequence of circles of having square summable diameters.Then the metric product T™ = S] x S2 X • • • is I compact abelian group with an innermetric of curvature > 0 , Clearly the metric is miwianf in terms of the group structure; thatis, left (and right, since Г°0 is abelian) translation is an isometry. Note that Г х is infinitedimensional, but not an infinite dimensional manifold. T™ is a Gromov-Hausdorff limitof finite dimensional flat tori. The fundamental group of Г°° is Z x Z x • • •; in particularit has an infinite number of generators. This shows that dimension restrictions are requiredto bound the topology of a Gromov-Hausdorff sequence of spaces, even if the curvatureis bounded above and below. Since any compact Lie group admits a bi-invariant metricof curvature ^ 0, this same construction can be carried out with any sequence of compactLie groups. The resulting product group can have extremely complicated local topology.More generally, every locally compact, metrizable, arcwise connected topological groupadmits a compatible left-invariant metric of curvature ^к for some к [77,16]. These kindsof examples will be discussed further in Section 13.

6. Nonnegatively curved spaces

DEFINITION 102. A ray starting at a e X is an isometry />: [0, со) ->• X such that p(0) =a. A line (centered a ta) in X is an isometry Л:(— ос,со)-> X such that A.(0) = a.

The condition that p or к be an isometry is equivalent to p or к being unit parameterizedand minimizing when restricted to any finite subinterval. The next theorem is known as theToponogov Splitting Theorem. It was proved by Toponogov for Riemannian manifolds,and for locally compact inner metric spaces of curvature ^ 0in f39].

THEOREM 103. Let X be a locally compact inner metric space of curvature > 0. If Xcontains a line к then X splits as a metric direct product X — k>cY, where Y is a metricallyembedded inner metric space of curvature ^ 0.

To prove the theorem, let к be a line centered at a in X. For any point 6 e X we constructa unique "line parallel to A" through b, k/,, by taking a sequence of minimal curves yt

+

from b to k(i), i e 1,2,..., which converges uniformly on every finite interval to a rayp+ starting at b. Likewise we can construct a ray p~ as a limit of minimal curves y~from b to —~i. We claim that p^ and p~ fit together to form a line. In fact, consider thepoints p := p+(t) and q \— p~(t). These are the limits of the points y*(t) and y.(f),respectively. Now basic Euclidean geometry shows that ao(}/.+(r), ^(~(f)) -> n. It follows

856 С. Plant

from АО that d{p, q) — \imd(y*(t), y~(t)) ~ 2t. Uniqueness of кь follows from the fact

that, if we were to choose another subsequence «I ", we again obtain that, as i a n d j become

large, an (ait (;), y~(t)) -*• ж, so а(ол~, yt) -* 7 Г and a(a>t, y,+) -> 0. We could apply

the same argument to another subsequence a>7 to get uniqueness of the line.Let У denote the set of all у e X such that у is a closest point to a on the unique line Xy

through у parallel to #. Suppose that z e X is arbitrary, and let у be a point on A; closestto а. We claim that

</(«. z f = d(a, y) 2 + d(y, z)2. (6)

If у is minimal from у to a, a(y,kz) — nji by Lemma 64, and d(a, г)2 ^ d(a ,y) 2 +d{y, z)2 by A2. To get the opposite inequality, suppose, for definiteness, that z =p+(t), where we are using the notation of the previous paragraph. Then d{y,a) =limrf(7,"(f),a)), and d(y^{t),a)) is, by АО, greater than or equal to the correspond-ing distance on the Euclidean triangle representing the triple of points (k(t); a, q). Since«o(MO: o, q)) -t> JT/2, the opposite inequality is obtained.

Formula (6) has two basic consequences:(1) On each line parallel to X, there is a unique point closest to a. Henceforth we will

only denote a line parallel to A. by кь, where b is the point on k\, closest to u.(2) If кь is the line parallel to к through b, then к is the unique line parallel to kf,

through a, and a is the closest point to b on k.It now follows that formula (6) is valid if a is replaced by any point in Y. It remains

only to show that Y is metrically embedded (Definition 66), for then it is immediate that Yis nonnegatively curved. Suppose that а, у s Y and let x be a midpoint between a and y.Then л lies on some kc, withe g F. If сфх then d(c, a) ^d(x,a) and d(c, y)which violates the fact that x was a midpoint.

7. Analytical tools

In this section we present a few more technical tools needed to understand the structureof spaces of curvature bounded below. Throughout this section, X will denote a completeinner metric space of curvature ^ к, and p e X.

7.1. The tangent cone

We now discuss the tangent cone, exponential map, and cut radius map.

DEFINITION 104. The tangent cone Tp at p is defined to be cEp. The exponential mapexpp is defined to be the mapping from a subset of Tp into X defined by ехр^(гу) — y(t),where у is a geodesic direction, and / is small enough that the definition makes sense.

REMARK 105. We will often refer to elements of Tp as "vectors" and adopt some vectornotation. For И e Tp, и is the equivalence class [r, y], where у e Sp and r ^ 0, and this


representation is unique unless r = 0. We write u = ry, and define "scalar multiplication"by cu =[cr, y]. We denote [0, y] simply by 0. We let \u\ :=r = d(u,0). For и — ry,v =cp€ Tp we let £*(«,*/) :=a(y,P) and j« - v]1 = d(u, v)1 = M 2 + \v\2 - 2)wllulcosa.Finally, we let

. . \u\2 + \v\2 - \u - v\2

(u, v) = — |w||t)|cosa.

Certainly {rtt, v) ={u,rv) = r{u,v) for any r >O and |и|2 = {и,и). Note also that а(и,v)is the angle between the minimal curves tu and tv from 0 to и and v.

DEFINITION 106. Two vectors и,ь е Tp are called polar (respectively opposite) if for allw e Tp, (и, to) + (w, v) ^ 0 (respectively (к, w) + {w, v) = 0).

To see the geometric meaning of these terms, note that и and v are polar if and onlyif their directions are polar, so we can assume и and v are unit. Then if w is also unit,{и, w) + {w, v) > 0 is equivalent to a(u, w) + a(w, v) ^л. In other words, every directionis within angle ж/2 from either и or v. Two (possibly not unit) vectors и and v are oppositeif and only if \u - v\ = 2|к| = 2ju|, which is equivalent to a(u, v) = Я (i.e., w and v arecomplementary in Up) and И and v having the same length.

Note that by Proposition 87, if Ep is an inner metric space, then Tp is an inner metricspace of curvature ^ 0. If Ep = S%' . thea ТР = Ж". In a Riemannian manifold, the expo-nential map is denned on all of Tp = W, and is a diffeomorphism on some open set nearthe origin. In a space of curvature bounded below, the situation generally is not nearly sonice. To make this precise we need a few geometric notions.

DEFINITION 107. The cut radius map is the mapping Cp: Ep -> K+ и со given byCp(y) = sup{r: y\w,t] is defined and minimal} if у e Sp and Cp(y) = 0 otherwise.

The cut radius map measures how far out a geodesic extends as a minimizing curve.There are two possibilities: The geodesic continues past the cut radius as a geodesic (whichalways happens in the geodesically complete case, or it stops, as does a geodesic that hitsthe apex of a Euclidean cone (with angle < >R}.

DEFINITION 108. A point p e X is called a geodesic terminal if there exists a geodesicyqp which cannot be extended past p.

Note that X is geodesically complete if and only if it has no geodesic terminals. It isinstructive to consider the simple case of Q (Example 98). If p is a point on the rim, thenTp is isomorphic to the Euclidean plane. Then expf, is defined on the plane with a line(say, the x-axis) removed, except for the origin. The removed half-lines point in directions"tangent" to the rim. A geodesic starting at p is minimizing until it strikes the rim again;that is where the cut radius occurs, since another shortest path can be found by travellingalong the other side of Q. In this case the cut radius is continuous.

858 С Ptaut

It is not hard to see that, in general, the cut radius map is semicontinuous. For a Rie-mannian manifold it is continuous, but in general it is not. For example, the product spaceSl x Q is geodesically complete, but Cp is not continuous in the directions tangent to anS1 slice. In fact, it is possible for Cp to have no points of continuity on Sp (it is alwayscontinuous at nongeodesic directions):

EXAMPLE 109. Consider the "bad sphere" В of Example 97. This convex surface hasa dense set of singular points, each of which is a geodesic terminal; more strongly, nogeodesic passes through such a singular point. We claim that there is no function /:Ep—*Ш+ such that / ^ Cp, f > 0 on Sp and / is continuous at some у e Sp. If there weresuch a function, with f ( y ) = e > 0, then any geodesic a near у (in angle) would satisfyCp(ot) ^ f ( a ) > e/2 > 0. But then there would be no singular points near у(s/2), since allpoints near у (E/2) would have at least one geodesic passing through them. This contradictsthe fact that the singular points are dense. It follows that, in this case, the domain of expp

does not contain an open set. The fact that the domain of expp may not contain an open setmakes it much more difficult to understand the local structure of spaces of curvature ^ k,because exp; j is not a local homeomorphism as it is for Riemannian manifolds.

7,2, Concave functions

We now recall a few elementary facts from analysis. A function / :Y ->• Z between metricspaces is said ta Ъе k-Lipschttz if for every x. y e Y, d(f(x), /(>')) < \d(x, y). The num-ber k is called the Lipschitz constant for / . When it is unimportant, we do not mentionA. specifically. A Lipschitz function is obviously uniformly (and absolutely) continuous.If Y = [a, b] and Z = [c, d] then /, being absolutely continuous, is differentiable almosteverywhere (a.e.), and f(x) = f{a) + f* /'(0df. A Lipschitz curve с in a metric space isobviously rectifiabk, with L(c) ^ Mb — a). The next lemma is an immediate consequenceof the triangle inequality.

LEMMA 110. If c: [a,b] -* Z is X-Lipschitz then for any p e Z, the function f(t) =d(p, c(t)) is X-Lipschitz from [a,b] into R+.

Given any rectiflable curve c: [a,b] ->- Y, the length L(t) = L(c|^.,j) is monotone in-

creasing, and so differentiable a.e. Then L{c) = Ja L'(t) dt. For any ( e (a, b),

d(cU),c(t + &)) K\L(t + S) - L(t)\

for any 8 > 0 near 0, and so the limit ]\т&^оd(c(t),c(t + S))/\S\ exists a.e.; we denotethis limit by |c'(f)l- Ь follows from the definition of length that He) = f% \c'(t)\ dt. Notethat if с is unit, then \c'(t)\ = 1 a.e. If/ is Lipschitz and 1-1 a n d / J is also Lipschitz, wesay that / is Ы-Lipschitz; clearly / is a homeomorphism in this case.

Recall that a function / : \a, b] -> К is called concave if the segment between any pairof points on the graph of / lies below (not necessarily strictly) the graph. The followingstatements are equivalent:

Metric spaces of curvature ^ k 859

(1) / is concave,(2) / is continuous and for any x, y, f(^) ^ / U ) t / ( > ' - , and(3) / is the integral of a monotone decreasing function / ' .If / is concave then the secant slope ^lZ.{U) is a decreasing function of both s and t<<)• It follows that the left and right derivatives

r « B and /+(f)s-*r s — t s-n+ s —t

exist for al! r, and are approached in a monotone way. (Since/ isLipschitz, f (t) = / + ( f)a.e.) We now adopt the notation of [65].

DEHNITION 111. Let ф, f, F: [a, b] -> R be continuous functions such that F" = f. Wewrite ф" ^ / if ф - F is concave.

If ф is twice differentiable, then ф" < / has two (equivalent) meanings. Note that afunction ф is concave if and only if ф" ^ 0. Also, if ф" ^ / then a.e. ф' exists and is equalto the sum of a monotone decreasing functionh(t), and the differentiable function ¥'. Forthe convenience of the reader we state a few basic results connected with Definition 111.These follow from the basic properties of concave functions that were discussed above,plus elementary facts like: concave functions are closed under certain algebraic operationsand inf s.

LEMMA 112. Let ф, ф, •, f, F, g, G: [a, b] -> R be continuous functions such that F" = /and G" — g. Then

(1) Iff; $ f апй фЧ < g then (0i + ф7)" ^ f + g.(2) I/ф" </andc>0 then (сф)" < cf.(3) //0" ^fandf^g then ф" ^ g.(4) // 0, -* 4> on [a. b] and tf ^ / for alt i. /fen 0" ^ /.(5) / / # ' < fforalli, then (inf&)" ^ /.(6) 1/ф" ^ / and g is concave, then (g оф)" ^ snp{^'}/.(7) Цф" < / then ф~ it) ап( 1ф+ (1) exist for all t and are equal a.e.(8) //0" ^ / /Ae« 0 + ( O ^ f ' J>^"» _ maxTe|,..tJ | / ( r ) | ( * -t) whenevers > /.(9) Finally, ф" < / ifand only if {ф + G)" <

LEMMA 113. Цф4>" ^ f on[a,c] and[c,b]for some с e (a,b) then ф" < / on [a,b]ifand only ifф~(c) ^ ф+(с).

The above lemma allows us to "glue" together concave functions.If 0" < /, then for any fixed / we can write, for some constant K,

860 С. Plaut

Here h is the decreasing function mentioned above Lemma 112; we immediately get

+T- max {\f(s)-f(t)\},—• \S f| T

where the last term is О(Т2), since / is continuous. The converse statement is easy, and wehave;

LEMMA 114. For any ф, f: [a, b] -> R, ф" / if and only if for some constant К,

> + o(r 2 ) .

7.3, Development of curves and quasigeodesics

We have seen (and Example 109 emphasizes) that the usefulness of geodesies is morelimited for general spaces of curvature > к than it is for Riemannian manifolds. Quasigeo-desics were introduced by Alexandrov [1] in order to overcome some of those limitations.We use here an "invariant" definition that comes from Proposition 1.7 in [65], and does notneed a curvature bound.

DEFINITION 115. Let Y be a metric space and n: [a, b] -> К be a unit curve. Then n iscalled a quasigeodesic if for every q e Y there exists a function / such that lim.f_,o j^- =0, and (d4(t)2 - 1 2 ) " < f{dq(t)) for all t s \a, b]. We write (dQ(t)2 -t2)" ^ оЦ^СО) forshort.

Our next task is to show that quasigeodesics, while generally not geodesies (as we willsee later), have the same "comparison" properties. If p is a point in Euclidean space, у is aline, and a is the angle between у and the line from p to y(f) then lettingds :=d(p, y(s)),we can write d2+T =d2 — (2dt cosa)r + т 2 . In a space of curvature ^ 0 , we know that thecorresponding distance dT+T is smaller than dl+T (if dt = d, and we use the same angle a),and we can write df+T < d2 - (2d, cosa)T + r2. Using the same arguments and the cosinelaws for space forms, one can prove:

PROPOSITION 116. Let peX and у bea geodesic in X, andlet dt —dip,уif)). Let

ifк>0,d4 ifk = 0,

Then f" ^ 1 - kf.

Metric spaces of curvature > к 861

Conversely, a locally compact inner metric space in which the conclusion of Proposi-tion 116 always holds (for fixed £), for every geodesic, must have curvature ^ k. To see thisin the case when к — 0, consider the expression df+T < df + КТ + т 2 + о(г2). Argue thatX must have some lower curvature bound k; this determines the value of К = Id, cos »,and the result follows. However, we will not need this result.

In Sk we fix a point p; the space of directions Sp is a circle of diameter N ; we fix amonotone reparameterization C{9) of Ep (i.e., С(в) = (cos(/(#)), sin(/(0)) for somemonotone surjection f: R -* R). In other words, C(9) moves, say, counterclockwise,without turning back. Then every point q e Sk has C-polar coordinates (r,9)c, wherer — d(p, q), and в is such that С(в) is the direction of a minimal curve from p to q. If0 < d(p,q) < n/^/k then 9 is uniquely defined in the interval /"'([0, 2JT)). Polar coor-dinates always uniquely determine a point, although they are not denned when r > n/*/k.Note that a curve c{t) = (r(t),O{t)) is rectifiable if and only if each of r(t) and 9(t) isrectifiable. When / is the identity map, we simply write (r, 9) for (r, 9)c (these are thestandard polar coordinates in Euclidean space).

DEFINITION 117. Let У be a metric space and с: [a, b] -> Y be a Lipschitz curve and/>G Y. The -development of cat pis the curve с :[а,Ь]-+ St whose C-polar coordinatesare (d(p, c(t)), t)c, where the parameterization С is chosen so that с is unit parameterized.

From Lemma 110 and the above comments, we see that the curve (d(p, c(t)), t) is atleast rectifiable; we can then choose a unique mapping / to make (d(p, c(t)), t)c unitparameterized (cf. [65]). That is, as long as d(p, c(t)) < л/Vk, we have a unique (up toisometric motion of S*) unit ^-development of с at p; we implicitly assume d(p, c(t)) <jr/v^t when discussing the -development in the future. The k-development of a curve wasdiscovered by Alexandrov f 1 ]. The k -development с of a curve с is called convex if thefollowing holds for any t and sufficiently small s > 0. Let x = (г, 0)c He on the minimalcurve у joining c(t) and c(t + s ) , with/ < 9 < Г + e . T h e n r ^ d(p,c(9)). In other words,the "section" determined by с in [t,t + s] is convex in St. (The word "convex" is greatlyover-used; in this case we say a subset S of an inner metric space is convex if every pairof points in S is joined by a minimal curve lying in S.) We now have yet another usefulway to characterize curvature ^ k. We give only one direction here; if one assumes localcompactness (or the existence of sufficiently many minimal curves), then one obtains theconverse. The proof of this theorem is not hard. In fact, the convexity of a it-developmentis equivalent to geodesies satisfying the conditions of Proposition 116.

THEOREM 118. For any p e X andgeodesicу:[a,b]-*- X, the k-development of у at pis convex.

DEFINITION 119. If У is a metric space, a curve щ:[a,b]-> Y is called k-convex if forevery p € Y, the ^-development of q at p is convex.

Note that a curve с which is ^-convex is t'-convex for any k' ^ k.

PROPOSITION 120. A curve с in X is a quasigeodesic if and only if it is k-convex and unitparameterized.

862 C Phut

The "only if" direction follows from the formulas in Proposition 116, and Lemma 114.{Use the Maclaurin's series when k / 0.) The converse involves a little more analysis (cf.[65], Proposition 1.7).

7.4. Differential and gradient

Yet another kind of curve that will be of use to us is gradient curves. We give a slightlymodified version of the definitions and results of [65J. Recall that the gradient V/ of asmooth function / on a Riemannian manifold measures the direction and magnitude max-imal increase of / . Such a direction exists due to the smoothness of / and the compactnessof the space of directions at every point. We begin with the following:

DEFINITION 121. Let У be a metric space and / ; U С Y - * Ж b e A-Lipschitz, where Uis open. The absolute gradient of / at p € U is defined to be

. „ , . . . L .. f(Pi) fip)\ . ,|V/|(/>) = max{ 0, hm sup —— -*— ) < X.

I Pi^P d(p,Pi) |

A point p is called critical for / if | V/|(/>) = 0.

DEFINITION 122. For any АсX and p e X, we denote by YP,\ a minimal curve from pto a point in A such that L{YPA) = d(p,A) := mmx ^ A d(p, x). Let A'p denote the subsetof Ep corresponding to curves УРЛ - When the reference to p is clear we will eliminate thesubscript to simplify notation. When A = [q}, we will simply write q'.

Note that A' is always closed. It follows easily from Al that, for the distance functiondA, the above notion of critical point is equivalent to the extremely useful one introducedin [44] for Riemannian manifolds (it can be stated in the current generality).

PROPOSITION 123. Let A be a subset ofX. A point q is a critical point of dA if and onlyif A'q forms a nji-net in Eq.

In other words, it is impossible to move from q in a direction that points away from A inthe sense that its angle with some minimal curve from q to A is more than я/2. As may beexpected the notion of absolute gradient is only useful with functions having fairly strongproperties.

DEFINITION 124. Let Y be an inner metric space. A Lipschitz function / : U с Y -> К issaid to be к-concave if for every geodesic у ia U, (/ оу)" ^К. The function / is calledsemiconcave if for every pG U there exists a tc(p) > 0 such that / is ic(p) concave near p.

Note that our definition differs by a constant multiple from the definition of "k-convexity" in [65]. It follows from Lemma 112 that a function is К-concave if and onlyif there exists a bounded function g such that for every geodesic у in U, (/ о у)" ^ g.


Note that for t = 0, the formula of Proposition 116 is {d})" ^ 2. By using the Maclaurin'sexpansion of cos( and coshf and Lemma 114 one can show that d} is semiconcave forit Ф 0 as well, where the local parameter К depends continuously on the distance from p.Since D& = inf^eAf^K we get from Lemma 112 the following proposition.

PROPOSITION 125. For any closed А С X, dA is semiconcave on X\A, and d\ is semi-concave on X.

We will need a technical lemma, which follows from the definition of concavity (of

4

LEMMA ] 26. Suppose f is a real function such that f" ^ К for some constant к. Thenfor any O<s<t, f(s) > /(0) + f {f(t) - /(0)) - \K\s(t - s).

DEFINITION 127. If Y is a metric space, /:UсY-^Ш.is semiconcave and у is a unitgeodesic starting at p, p e Y, we denote (/ о y)+ by Dy(f).

The number Dy(f) is the "directional derivative" of / in the direction of y. The fol-lowing proposition is immediate from Lemma 33:

PROPOSITION 128, Let f:X -+Ж K be semiconcave with Lipschitz constant k. Thenforany y\, YI £ Sp, P € X,

In particular, the mapping D*{f) :S/, —*• R is continuous and has a unique continuousextension to Ep and also to Tfl.

DEFINITION 129. We denote the continuous extension of £>*(/) to Tp by dfp, or simplydf\ it is called the differential of / at p.

Obviously, |V/|(/>) > sup{£>*(/)}. From Lemma 112 we get the first part of the fol-lowing proposition. The second part follows from the first part and Theorem 27.

PROPOSITION 130. If Y is a metric space, f:UСY-*•Шis к-concave, and у is ageodesic starting at peУ, then

whenever t > 0 is small enough that у restricted to [0, t] lies in U. Suppose, furthermore,that Y is a complete inner metric space of curvature > k and U is a metric ball B(p, r),r > 0 . Then for all q e U,

864 C. Plaut

COROLLARY 131. If f;U С X -> Ж is semiconcave then |V/| is semicontinuous; thatis, limin f w _ p > \Vf$pi)

COROLLARY 132. If f :U с X-> R is semiconcave then the set of all noncritical pointsof f is open in U.

COROLLARY 1 33. If f : U С X -* R « .semiconcave then the function dp has no criticalpoints in a neighborhood of p e U.

COROLLARY 134. I f f . U C X -• R « semiconcave then |V/ |(p) = sup{D*(/)}.

To prove Corollaries 131 and 134, fix positive e < 1/2 and choose p ' such that

f(p') - fip)dip,]/)

>\Vf\(p)-e

a.n& d(p, p') < e. Let q be close enough to p that f(p') - f (q) ^ O - e)(f(p') - f(p))and d(p', $ ) < (1 + s')d(p', p) < 2s. Then by Proposition 130 (choosing к by the semi-concavity),

Letting e -»• 0 proves the first corollary. For the latter one, by moving p' slightly, wecan assume there is a minimal curve у from p to p'. Again apply Proposition 130 to getDy(f) > IV/K/i) - (1 + \K$£. U t e -> 0.

DEFINITION 135. Suppose that / : V с X ->• R is A--concave. We make the followingdefinition for any p, q e X:

PROPOSITION 136. Suppose that f:UсX—*Rй к-twicave, яи^ ( /й bounded. Forevery e > 0 fftere exists a $ikt К) such thatfor any p,q\,q^, if

for i = l,2, then ak(p; q\, qi) < e.

This proposition says essentially that points which come close to realizing the gradientmust be nearly "lined up". We prove it as follows: First, by moving the points qt slightly


we can assume that there are unit minimal curves у := yl)qi and p1 := ур^, as well asunit minimal curves ft from л:,- := у(ТГ') to qi. Setting >v : = # ( 2 ~ ' ) , we obtain fromLemma 126 that

t*. 92)

If the formula in Proposition 136 holds for some 5, then for j large enough we get

(/(<72) - /(*<)) ,

d(Xi,q2)

and then

Likewise,

/(*,) - fip) > dip,XtXl ~2&)\Vf\(p).

Now from Proposition 130 we get

Adding and subtracting /(X,) to the right side of the last inequality, and combining it withthe previous two inequalities gets us

From Lemma 16 it now follows that at{Xj;yj.p) is close to л, ьаak{xi\q2,p) is close ton, and «t <JR,; 2 i^i) is close to O. Letting.^ ^^ /?completes the proof. The next corollarywas proved in the finite dimensional case in [65].

COROLLARY 137. Suppose that f: U с X —* R is к-concave and without critical pointsin U. Then for every peU there exists a unique direction у e Ep such that |V/|(/>) =

PROOF. Suppose that y\, y2 e Sfl satisfy df(yt) > (1-8/4){Vf\(p) for some $ > 0. Thenforr > 0 sufficiently small,

866 С. Plaut

Given в > 0, Proposition 136 now implies that for S small enough, a i p; /i (0> У 2 (0) < £for all small t. In other words, a(yi,yn) < 8 , and the corollary follows. D

DEFINITION 138. We denote by V / ( p ) e Tp the vector |V/|(/>) • y, where 7 is thedirection given by the above corollary, if |V/ | (p ) > 0; otherwise we let V / (p ) := 0. Thevector V/(/J) is called the gradient of / at p.

I f / has a local maximum at p then Definition 121 obviously implies that V / ( p ) = 0 ,If V/(/J) > 0 then we can find a geodesic direction у such that dfp{y) > 0, and so / doesnot have a local minimum at p. We have proved:

PROPOSITION 139. Suppose that f; U С X -• К is к-concave. Then peV is extremal(i.e., is either a local max or a local miri) only if V f(p) — 0.

7,5. Gradient curves

The construction which follows is roughly from [65], which in turn is similar to that givenin [79, Theorem 3].

DEFINITION 140. Let У be a metric space and / : ( / с У - > - R b e a Lipschitz function.A curve с : (я,b) -> U is called an f-gradient curve if

That is, an /-gradient curve is one along which one can measure the gradient of / .Note that being an /-gradient curve is independent of (orientation preserving, monotone)reparameterization. If U has no critical points of / , then the function h{t) = f{cit))is strictly increasing. If we reparameterize, letting b{t) := c(h ~ ] ( t )) then we get thatf{b{t)) = t. A curve b satisfying f(b(t)) = t will be called f-parameterized. Clearly the/-parameterization is unique. The next proposition, whose proof is now easy, reconcilesour definition with the one in [65].

PROPOSITION 141. Let Y be a metric space and f :U dY -f'Rbe a Lipschitzfunctionwithout critical points. A curve c:(a,b)—> U is an f-gradient curve if and only ifс hasan f-parameterization such that

,. d(b(s),b(t)) 1hm — -

Let us pause for a moment to see what these definitions mean in a Riemamiian manifold,in the case when с and / are smooth and / is without critical points (i.e., V/ never


vanishes) in U.To compute |V/|(p) in theRiemannian sense we can choose a geodesic у

such that y'(0) = Zl\^l a n d compute

* * ' » ~ * * ' » = l i r o

On the other hand, for any p,•, -• p , so that Шо?-*ое '^$п~'У e xi s t s and equals somenumber £, then by choosing a subsequence if necessary we can assume that the directions

converge to у e Sp. We see that for some t ; -• 0,

It follows that the two notions of | V/|, and hence of critical point, coincide. Now let с besuch that c'00 = V/(c(0). Then for some real function g, /(c(f(f))) = I if and only if

2

Now

s-i

In other words, с is an /-gradient curve, and every /-gradient curve is simply a reparame-terization of c.

PROPOSITION 142. Let f be semiconcave and without critical points in U С X, Thenfor each p e U there exists a unique (up to reparameterization), complete gradient curvestarting at p.

A proof of the above proposition is sketched in the appendix of [65]. It is based on thefollowing extension of Proposition 136.

PROPOSITION 143. Let f be К-convex and without critical points in U. Then there existsa constant С such that for any sufficiently small S > 0 the following holds: Let p, a, b e Usuch that A(p, a)>(1- S2)\Vf\(p). Then

To construct a gradient curve, one first constructs a kind of approximating gradi-ent sequence. (We roughly follow the sketch in [651, with a few more details in somespots.) Fix a small 8 > 0. Begin with p, choose a point q such that d(p,q) < 8 and

868 С. Plaut

, я) ^ U - 52)I^7/I(/J) f°r some small 5 > 0. If the construction leads to an accu-mulation point, we begin a new sequence. An application of Zorn's Lemma allows us tocontinue this process until we no longer have an accumulation point in U, At this point wehave constructed a possibly infinite sequence indexed over a well-ordered subset of the ra-tionals which itself has possibly countably many accumulation points. Suppose xa is suchan accumulation point. We choose a point xp with ft < a very close to xu, and removeall points with indices between a and ft. We call xp and exceptional point. We carry thisprocedure out at (countably many) accumulation points so that the sum of the distances be-tween exceptional points and their successors is finite and small. We re-index the sequenceover the integers. Given two such sequences {„*;} and {>',} we can estimate rate of changeof the distance between two points *, andyj such fhatO ^ /(>';) — f(xi) < S2, and JT, andyj are not exceptional. By the above proposition.

Either d(x,, Vj) < 5 or d(xi, yj) ^ S and

A(Xi, yj) - A(Xi,xi+\) > s \

From this last inequality it follows that for some constant C ,

This is essentially a difference quotient; using an inductive argument is possible to showthat pairs such that | / ( j y ) — /<х*)| < b2 satisfy d{x{, у/) < C"<5, as long as neither ele-ment is exceptional. But the exceptional points were chosen so that their contribution tothe pairwise distances is small. These estimates allow one to show that any family of suchsequences, with 8 —> 0, must converge to some curve. By the semicontinuity of the gradi-ent, the resulting curve is a gradient curve. Uniqueness follows from the fact that a givengradient curve contains a sequence of the above type.

As we have observed before, the distance function is semiconcave. We will be mostlyinterested in gradient curves of the function dp. Note that if у е Sp then for any r, thesegment of у starting at t and going away from p is the gradient curve for dp startingat y(t). Now let q be in the interior of y. Then у is a ^-gradient curve, and there is aunique ^-gradient curve о starting at q. By uniqueness, у and о togemer must form ad;,-gradient curve. We have proved:

PROPOSITION 144. Every minimal curve starting at p e X extends to a complete dp-gradient curve in any neighborhood in which dp has no critical points.

We next state the result from [65], that representative angles have a similar monotonicityalong gradient curves to that along geodesies.

Metric spaces of curvature ^i t 869

DEFINITION 145. Given p e X and a curve c, let cj%(p; c(t{),cfe)) denote the an-gle at P of the triangle (P,C\,C2) in Sk such that d(P,C\) = d{p,c\(r)),d{P,C2) =d(.p, c{t2)), and d(C\, Ci) = \t\ - is). If c\ andсз are curves such that ci(O) = <72(0) = />,let a^{p;c|((|),C2{/2)) be the angle at p of the triangle (Р, К\,Кг) in S* such that

In other words, the above representatives use the parameter of the curve instead of dis-tance, when any two points on the triple lie on one of the specified curves.

PROPOSITION 146. Let у\,уг be dp-gradient curves starting at p. Then there exists a"proper" monotone reparameteriz,ation ofy\,Y2. on [0, oo) such that

(1) For any q e M, щ{рчд, yi(t)) is nonincreasing in t.(2) The function a^ip: y\ (s), nit)) is nonincreasing in s, t.

In other words, the same triangle comparisons hold for gradient curves as for geodesies ifthe parameter is used instead of the distance; the advantage is that gradient curves continuepast the cut locus -until a critical point of dp is met.

The parameterization of a dp-gradient curve у is obtained by beginning with a dp-parameterization (cf. comments after Definition 140), then considering у о р~\, wherep{t) = exp(/(/)), where

l = nwp)]

We remark that this parameterization slows down rapidly as the curve approaches a pointwhere Vdp — 0; otherwise, the curve continues. In addition, p(t)/t -> 1 as t -• 0, so theinitial derivative of the gradient curve is unchanged. We will need gradient curves tan-gent to more general directions. The following proposition can be proved using Proposi-tions 146, 144, and Corollary 133.

PROPOSITION 147. For any direction у е Zp there exists a unique dp-gradient curve уstarting at p such that у' (0) = у. Furthermore, on any compact subset С of E p there existsana > 0 such that the gradient curves tangent to elements ofС are uniformly Lipschitz on

The latter notation /'(О) = у means that shortest curves y, from p to y(t) converge inangle to y. Thus when Sp is compact (which, as we will see, is true in the finite dimen-sional case), we can always construct a "gradient exponential map" that is locally surjectiveand distance decreasing (if Tp is given the metric of curvature > к via the hyperbolic cone,given at the end of Section 4.5). Note that gradient curves do "bifurcate". For example, ifthere are two minimal curves from p to q then the two minimal curves extend as gradientcurves; but by uniqueness, the two must coincide beginning at q.

870 С. Plaut

8. Dimension

Throughout this section, X denotes an inner metric space of curvature ^ к for some /:.

8.1. Spherical sets

DEFINITION 148. A set of n points {a\, ...,an) in a metric space is called Hemispheri-cal if det[cosd(a,, o/)] > 0. A set of 2n points \a\, b\,...,an, bn} is called spherical if[a\ ,..., «„} is semi spherical andd(a,-, />,-) — Л for all I,

In other words, a spherical set can be isometrically embedded in S" С Rn + 1 asendpointsat of unit vectors forming a basis of K"+ 1, together with their antipodal points fe,-.

THEOREM 149. If X has curvature > 1 and contains a spherical set Е of2(n+ 1) points,then there is a subset SofX isometric to the unit sphere S" such that £ С 5.

The proof is a somewhat involved induction on n. We sketch some details. For n = 0,the proof is trivial. For n = 1 we essentially need to find a closed geodesic of length 2тгthrough the four points such that each segment of length less than ж is minimizing. First,we can join a\ and b\ by a minimal curve through ят as follows. Using Theorem 27 wechoose points Xj -* ai joined to a\,b\ by minimal curves щ, fi;, respectively. But then Alimplies (since we are comparing with the unit sphere) that o((flfj/),•)—» n. It follows fromthe second monotonicity that the sequence has a uniform limit, which is the desired curve.We then complete the construction joining a \, b\ by a minimal curve through hi - The factthat the resulting geodesic is an isometrically embedded circle follows from the observationthat segments of it (considered as hinges of angle я ) satisfy EA2 (see Proposition 35). Thestep from n — 1 to n = 2 is representative of the higher dimensional cases, and can bepatterned after the proof of Proposition 35. We construct an embedded circle С containinga\, a2, b\, bj using the inductive step. We then consider all minimal curves (and these canbe constructed without local compactness, as above) from a?,, b$ to C. Then EA2 can beverified for all hinges made of these curves; thus we have "filled in" a unit 2-sphere.

The standard hemisphere shows that not a single point can be removed from the sphericalset without making the theorem false. This is in contrast to the situation for Riemannianmanifolds, where Toponogov's Rigidity Theorem says that a Riemannian м-manifold ofsectional curvature ^ 1 and diameter я must be isometric to S". It is interesting thatsmoothness is not a necessary assumption; the theorem can be proved using only geodesiccompleteness [74].

8.2. Regularity

We now use Theorems 27 and 149 to construct spheres in the space of directions at almostevery point in X. The proof, in the present generality, is quite technical (see [76, Section 5]).We will attempt to make simplifications in the argument that do not obscure the general


idea. For this argument (but not later!), the term "almost everywhere" will mean in a densed set. Note that if у(,ь is almost extendable beyond b, then by Proposition 55 there isa O-sphere \p, -,6} in Ef,. Thus by Theorem 27 there is a (unit) O-sphere in the spateof directions of almost every point (and one of the two directions in the O-sphere is ageodesic direction). Now suppose there is a direction a € Ef, distinct from fi and —fi.Since Sb is dense in Еь, we can assume that a is a geodesic direction. It may be that ahas no complement in Еь, so there may not be a spherical set of four elements in Еь.However, we can find such a spherical set in nearby spaces of directions as follows. Fix/ > 0 smail enough that both /J and a restricted to [0, /] are almost extendable past b(cf. Proposition 56). By Corollary 30 there is a dense G$ set of points p joined to bothx = f}(t) andу =a(t) by minimal curves almost extendable pastp. But by Proposition 57,<x(Ypx, Yi)v) must be close to a(p", a) when p is close to b; hence we obtain a spherical set

Now we are faced with the problem that Ep may not be an inner metric space, so we donot automatically get a 1 -sphere in Ep from Theorem 149. However, from Proposition 53we can find a minimal curve in Ep joining these two directions. In this way we can con-struct an inner metric space containing the spherical set as a convex subset, and completethe proof. By this argument, the given spherical set is contained in a sphere S С Ер. Ei-ther S = Ep, or there is another geodesic direction у not in S. By Proposition 57 and thecontinuity of the determinant function, we can find a spherical set of 2(n + 1) points (andhence an п-sphere) in Ep for points q near p.

In light of the above construction, the following definition makes sense:

DEFINITION 150. For any p € X, we define the regularity of p to be R(p) = sup{«: Ep

contains a subset isometric to S"" 1}. (We consider Sj"1 = 0 and S^ to be two points ofdistance N) . We let R"(X) = {p € X: R(p) ^ n]. We define the local dimension of Xto be ldim(X) — sup{n: R"(X) / 0}. A point p is said to be regular (of dimension n) ifEP = S1~1.

The results of our construction can now be stated as follows:

THEOREM 151. Let p e X, and suppose that R(p) = n. If p is not regular, thenRn+](X) П B(p, S) contains a dense Gs for all small 5 > 0.

8.3. Equivalence of Hausdorff and covering dimension

Our next task is to relate local dimension to two other standard notions of dimension:Hausdorff dimension and covering dimension. See [47] for the definitions of these twodimensions. We restrict ourselves to the following comments: We will denote Hausdorffdimension by hdim and covering dimension by dim. One can define the «-dimensionalHausdorff measure vn of a metric space; и„ is decreased by distance decreasing functions.Then /idim(X) = sup{m: vm(X) > 0}. It follows that distance nonincreasing maps do notincrease Hausdorff dimension. Also, Л dim depends on the metric, not just the topology, ofX. The following lemma is a consequence of the definitions:

872 C. Plaul

LEMMA 152, If Y is a metric space then(1) Scaling the metric of Y does not change its Hausdorff dimension.(2) Л dim У = и if and only if h dim cY = n + 1.(3) ffY is the Gromov-Hausdorff limit of metric spaces У, - , ^ /idimV^

On the other hand, dim is defined for topological spaces, and is a topological invari-ant. Any topological space Y containing an open subset homeomorphic to an open sub-space of R" has dimУ > п. For any metric space Y, dim Y < hdimY. Indeed, dimУ =inf{/idim(F,d)}, where the infimum is overall metrics d compatible with the given topol-ogy. Every n-dimensional Riemannian manifold has Hausdorff dimension n with respectto the distance induced by the Riemannian metric, and each finite metric ball has finiten-dimensional Hausdorff measure.

If ld'xm(X) = n then by Theorem 151 we can find a regular point p, i.e., Tp = W.For any £, we can put on 5(0, n/Vk) с R" a Riemannian metric of constant sectionalcurvature kt such that the radial lines are still geodesies, using one of the cone constructionsat the end of Section 4.5. Equipped with such a metric, it is immediate from Theorem 43and A2 that exp^ is distance decreasing. Since it is distance decreasing, we can extend itcontinuously to a mapping e\pp defined on the closure С of the domain of expp. If welet Cr be the compact set С П S(0, r) then exp^(Cr) is compact, and equals B{p, r ) . Itfollows that X is locally compact, and therefore, by Theorem 8 expf) is surjective. Sincethe domain of expp is contained in the space Tp, which has Hausdorff dimension s, we seethat h dim(X) ^ n. More strongly, we have proved:

PROPOSITION 153. IfldimX — n, then X is locally compact, and every compact subsetC ofX has u,i(C) < 00,

To show that regular points are manifold points, we need to use the distance coordinates,which were discovered by Berestovskii [6].

DEFINITION 154. Let x\,..., xn be points in X. We define the distance coordinatesassociated with {jti *„} to be the (obviously continuous) mapping Л : X -* W given

We are interested in determining when the distance coordinates are indeed a coordinatesystem near some points (i.e., a homeomorphism onto its image). In the case Berestovskiiconsidered, he already knew that X was a manifold by other means; having shown thatA was 1-1 was sufficient, by invariance of domain, to conclude that Л was a homeomor-phism. In the present case, our argument for surjectivity is based on that from [ 19, Theorem5.4].

Suppose that /dim(X) - n, and let p € X be such that Ep — S"~ '. Then for any smallS > 0 and all 1 ^ i ^ n we can find unit geodesies yi = Ypx, and Д- = ypyi of fixed lengthr > 0 such that \a(yt,yf) — ~NJ2\ < Ь if i / j and \ct(y, ,fii) — N\ < S. Furthermore, wesuppose that each >*J,$ is almost extendable past p and is the unique minimal curve be-tween its endpoints (cf. Proposition 56). This special kind of "almost" spherical set wascalled an (n,<5)-strainerin [18], and an (H,<5)-explosion in [191. We claim that the distance


coordinates Л associated with {x u ...,xn) are a homeomorphism of a small open set aboutp onto its image in Ш". First, suppose Л is not locally 1-1. Then there exist aj,bj -• psuch that d(fl;,*,•) = d(bj,Xj) forall/, j . lM- ji be minimal from Й; to*;,|,;/ be minimalfrom aj to y,, and r\j be minimal from UJ to bj. Then on the one hand it is immediate fromProposition 57 that if a j is near p, the curves #y are a semispherical set in E a ] (i.e., theyare nearly orthogonal). By Proposition 63, l ima(^y, rjj) = Я / 2 , SO i?n together with щform a semispherical set E of n + 1 elements. If the elements of S all had complements(and they might not), we would have a contradiction to Theorem 151. However, note thata($ij,fyj) can be made arbitrarily close to я — S. Then using a technical modification ofProposition 57 it is possible to prove that we can find, in the space of directions at pointsnear «;, a spherical set of 2(n + 1) elements, which contradicts Theorem 151. Thus Л islocally 1-1.

To see why A is locally surjective, letx be close to p and suppose that у = (yi ,..., уи)is close to A(x). We need to find a point*' close to x such that ф(х') = у. First, \d(xj,x)-yt| is maximal for some i. Join x to XJ by a minimal curve KJ and x to yy by a minimalcurve kj. If x is close enough to p then |«(«•,,Kj) — nj1\ < 2<5 if i ф j and \a(Kj, k j) —7t\ < 28. Now we move along either KI or kj until we have reached a point z\ such thatd(z [,Xi) — yi (this can be done by the Intermediate Value Theorem). In doing so, since weare moving along a curve that is almost orthogonal to KJ for / ф i, we see thatd(z),xj)is only slightly changed from d{x,Xj). We continue in this way, constructing a Cauchysequence {г, \ which converges to the desired point x'. We have now proved:

PROPOSITION 155. /f/dini(Ar) = n and £p = S"~l then a small neighborhood of pis homeomorphic to an open subset ofW, via the distance coordinates of appropriatelychosen points near p.

The final result of this section shows that all notions of dimension are equivalent forspaces of curvature ^ *. The key remaining part of the proof, that finite topological dimen-sion implies finite Hausdorff dimension, is from 165].

THEOREM 156. Let X be a space of curvature 3= k. Then for any n, the following areequivalent:

(1) /d imX=/ t ,(2) hdimX=n,(3) dimX=n,(4) for some point p € X, Up = S"~,(5) there exists a dense Gg set of points p€X such that Sp = S"~',(6) X has a dense set of n-manifold points.

Furthermore, when the above conditions hold for some n, X is locally compact.

So far we know the following: From Theorem 151 weget(1)=>• (4). From the proof ofProposition 153 we get

n. (7)

874 С. Plaut

From Proposition 155 we get

«. (8)

Putting (7) and (8) together we get (4) => (2) and (3). Now if h dim X < n then dim X < nso (8) implies /dimX ф п. If hdiraX > и then we reach the same conclusion from (7)and we have proved (1) ^ (2) o (4). Now (5) a+ (4) is obvious. To prove the oppositeimplication, note first that by Proposition 153 and (1), X is locally compact, and henceseparable. Choose a countable dense subset Y of X, and let К = f]yGy Jy Certainly К isa dense Gs- An easy modification of the proof of Theorem 151 shows that Hp = S f ~ l forall p e K. (5) =$• (6) is an immediate consequence of Proposition 155. We need only provetwo more implications: (3) =s- (1) and (6) => (1) .

Both are proved in the same way. As was pointed out in [65], the dimension argumentof Proposition 2.10, [72] can now be carried out by using the "gradient exponential map"rather than the usual exponential map. Suppose /dim X > m, and let p e X be such that Ep

contains a copy S of £j"~. Then by Proposition 147, there exists a numbera > 0and, ineach direction у of >?, a corresponding proper gradient curve gy :[0, o ) ^ X. Note that wedo not know if Sp D S is dense in S, However, we can find finite subsets Y, of Sp which arebetter and better approximations of S; the union Y :=\J Fj is precompact, and 5 С Y. Nowchoose a finite e-net Ee = {yj } of elements of Y. Since E is finite, the cut radius C r hasa positive lower bound JR^ on Ee.lfe is small, then Proposition 146 implies that gradientcurves tangent to vectors in Y, and hence in S, do not meet on (0, Re) unless their angleis small. In other words, if <pp : B(0, а) c Km -* X is defined by 0p(fr) = gv(P*), thenthe inverse images of points are uniformly small for p small. By [47, IV.5.A], this meansdimX > dimB(0. a) = m. In fact, the dimension of any small ball nearp has dimension> m. So if dimX = « then n > lAimX = huimX ^ dimX = n, and we have proved(3) => (1). We can apply the same argument to an л-manifold point to get (6) => (1).

9. Alexandrov spaces

Finite dimensional spaces of curvature ^ к have become known simply as "AlexandrovSpaces". From the last section we know that "finite dimensional" means that the topologi-cal, Hausdorff, or local dimension is finite. We know that such a space is locally compact,has a dense Gs subset of points in which the space of directions is a sphere of one dimen-sion lower, and has an open dense subset of manifold points. In this chapter we discussfurtherthe structure of Alexandrov spaces. Throughout this section, X denotes an Alexan-drov space of dimension n.

9.1. Basic properties

THEOREM 157. Ifn ^ l then for any p e X, SP is an (n - I)-dimensional Alexandrovspace of curvature ^ 1.


To prove this theorem, first observe that the space of directions at every point is compact.This follows from the fact that the £p is a sphere at a dense set of points and the semi-continuity of the angle (Lemma 36). In fact, if Ep were not compact at some point p, wecould find a sequence {y;} in Sp such that «(7,,yj) > в for all i,j. Let p ; be the endpointof Yi different from p, yy -> p be such that Sy. is isometric to S"~ l, and fij be minimalfrom yj to pi. Then by Lemma 36 for any /,*, a(y,j,уц) > e/2 for large enough i. Inother words, we can construct in S"~] a finite sequence of points with arbitrarily manyelements having pairwise distance > e/2. This is impossible, since S"~ ] is compact. Wenow know that Up is compact for all p € X. The proof of Theorem 157 is now finished byProposition 87, Lemma 152 and the following proposition.

PROPOSITION 158. Forany p<=X, (Tp,0) is the pointed Gromov-Hausdorfflimit ofthescaled metric spaces (Xn, p), where Xn is X with the scaled metric nd, as n -*• 00.

The proof of the above proposition uses Proposition 74 and the compactness of Ep,proved above. In fact, this makes B(0,г) С Tp compact for any r > 0. By moving itslightly, we can assume that an £-net M in B(0, r) is of the form s;/,-, where each у, е Sp

and ST r, and i e 1,...,m. Now for any large k,

is uniformly small (cf. Lemma 33), and so in the scaled metric, {y,(s;/A;)} is Lipschitzclose to ЛЛ

Theorem 157 allows the useful technique of making definitions and proving theoremsby induction on the dimension of the space.

We can now improve on the "first variation formula", Proposition 61, in the finite di-mensional case. Note that for an Alexandrov space A' (Definition 122) is compact. Herewe are using the notation of Section 7.4.

PROPOSITION 159. Let p^qinX. Thenfar any у е Щ, didp^iy) = - c o s a ( y , q ' ) .

In particular, if у is a geodesic direction then, by definition, the (right-hand) derivativeof dp(y(t)) at t —Q is equal to -cosmin(a(y.^>: p joins p and q\. Proposition 61 im-plies that, for у e Sp, d(d,,)q(,y) ^ — Qosa{y,q'). On the other hand, if we choose anygeodesies fit from p to y{t), these, by the compactness of the space of directions, mustconverge in angle to some minimal curve joining p and q. Applying Proposition 52 andProposition 61 gets the opposite inequality.

The last result in this section concerns the measure of regular points, and was proved in[601. We defined regular point in Definition 150.

DEFINITION 160, A point which is not regular is called singular. The set of singularpoints in X is denoted by Sx.

876 С. Plaut

This theorem is a strengthening of the fact nonregular, or singular, points are nowheredense.

THEOREM 161. The set S ofsingular points in К has/tdim(5) ^ и — 1.

9.2. e-Open maps

Here we recount some definitions and results from [19], Sections 5 and 11, about e-openmaps. Note that if Y, Z are metric spaces and U с Y is open, a mapping ф: U -> Z is openif and only if for every x eU there exists a p > 0 such that if j £ B{<f>(x),p) then thereexists an x' e U such that ф(х') = v. The next definition clearly strengthens the notion ofopen map.

DEHNITION 162. Let Y, Z be metric spaces, U с У be open and x e {/. We de-fine p(x,U) : = sup{r: fi(x,r) с f/}. A mapping ф-.U -> Z is called s-open if forevery y g В(ф(х),ep(x, Uj), there exists an x' e X such that 0(x') = у and d(x,x') ^

Note that if Z is an inner metric space and Y is locally compact then we can replaceВ(ф(х), ep(x, U)) with 8 ( Ф ( Х ) , sp(x, U)) in the above definition (cf. comments prior toTheorem 8). The proofs of the next lemmas are obvious:

LEMMA 163. Let W, Y be metric spaces, U с W be open and ф: U -* Y bes-open. Thenfor all x e V. B(4>(x),£ft(x, U))C<f>(B(x,p(x, (/))).

LEMMA 164. Let W, Y he metric spaces, U С W be open and ф-.U -> Y he 1-1, con-tinuous, ands-open. Then for all x e W, the mapping ф^1 : В(ф(х),ер(р,U)) -5» U is aLipschitz homeomorphism (onto its image) with Lipschitz. constant 1 /e.

DEFINITION 165. A Lipschitz function f:U с X -> R h called directionally differen-tiable if for each p e U there exists a continuous function dfr : EP ~* Ж such that forevery у e Sp, ( f o y ) ^ o e j " s t s a n d i s equal t° dfp{y).

Note that dfp can be extended "linearly" to Tp. From Proposition 128 we know thatsemiconcave functions are directionally differentiable. As in the case of semiconcave func-tions we will omit the subscript "p" when no confusion will result. We are now in a positionto extend Proposition 155.

PROPOSITION 166. Let f ] f m : U С X -• K be a collection of directionally dif-ferentiable functions. Suppose that there exist S.s with 0 < nS < e such that for anyp e U there exist directions y^~, уГ € Spfori = 1,...,m with the following property:

for i^j e {l,..., m}, d f {y+) > e, d/dyf) < -s, and WiiYp\ <&• ™en the mapf := (/), ..., fm) is a-open, where a := &^, with respect to the norm \\v\\ = 521^1inW.


We remind the reader that in the case of Proposition 155 we were dealing with distancefunctions, so we were able to use an almost spherical set to prove the surjectivity of themapping. In the present case the idea is similar: the directions yr- and the values of df\ inthese directions force the functions to map onto an open region in Rm. To prove it, forx eX, \ct В = B~(f(x),crp(x,U)) and let А С В be the set of all y e B such that there existsan x' such that d(x, xr) ^ ^d(f(x), y) and /(*') — y. Then A is nonempty (x e A) andclosed. If the proposition were false, then for some x there would exist a z € B\A. Let уbe the element of A closest to z and *' be such that f(x') = у andd(x, x') ^ ^d(f(x),y).Suppose, for definiteness, that z' < v'. Now by the assumptions of the proposition we canfind a geodesic direction у at x' near some particular y^; i.e., so that dft(y) > e andf j ( y ) \ < &• Now by moving a short way along the geodesic у we get a point л" whoseimage y" = f(x") contradicts the choice of у as the closest point in A to z-

9.3. Stratification and local cone structure

Two fundamental theorems about the topological structure of Alexandrov spaces are dueto Perelman [62,63]. We will sketch the arguments in subsequent sections.

THEOREM 167. A small open sphere at any point in X is homeomorphic to the cone onits boundary.

THEOREM 168. X possesses a canonical stratification with strata homeomorphic to topo-logical manifolds.

These two theorems are proved using a kind of generalized Morse theory for the distancefunction. Morse theory for the distance function was discovered in the now famous paperof Grove and Shiohama [44], see also [35]. The first theorem of Morse theory is that if Mis a Riemannian manifold and / : M -*• R is smooth, proper (i.e., preimages of compactsets are compact) and without critical points (see Definition 121 and the comments beforeProposition 142) in the region / ~ ' ( [ a , b ] ) then Ma : = f ~ [ ( ( — oo,a]) is diffeomorphic toMh := / ' ((—oc, b]), and the former is a deformation retract of the latter. The idea ofthe proof is to retract Mb to Ma along the gradient curves of / . Another way to state thisresult is that / is a trivial bundle map on f ~ [ ( [ a , b]):

DEHNITION 169. If У is a topological space, a map / : F —* К is a (locally trivial) bundlemap if for each p e Y, there is a neighborhood U of p homeomorphic to a product /([/) xC such that / restricted to U is the projection onto f (U) . A bundle map / is called trivialif we can take U = Y.

The problem with applying Morse theory to the distance function, even for Riemannianmanifolds, lies in the fact that the distance function dp is not smooth at points in the cutlocus of p. In an Alexandrov space, the problem is manifested in the behavior of the gra-dient curves of the distance function which, as we observed at the end of Section 7.4, maybifurcate.

878 С. Plaut

9.4. DER functions

In this subsection we denote by E an Alexandrov space of curvature 1. In order to extendthe first theorem of classical Morse theory to Alexandrov spaces, Perelman introducesin [63] a class of functions, called admissible maps, which include distance functions.Admissible maps satisfy a slightly stronger condition than к-convexity (Definition 124),and have differentials on Sp which can be written as finite formal sums

(9)

where as usual dAi denotes the distance from the compact set A,-, and щ are non-negativeconstants such that £ я ; ^ 1.

DEHNITION 170. Expressions of the form (9) on E are called functions of class DER orsimply DER functions. When no confusion will occur, we will not explicitly mention thespace E. Note that a given function may be expressed as a DER function in more than oneway.

We list several properties of DER functions, which can be proved by direct computation:

PROPOSITION 171. // / = £,• -a/ cos dAi is a DER function then for allx,y,z.eE suchthat у is between x and z,

(1) |/Cx)l^l,(2) \f(x)-f(y)\£d(x,y),(3) sind(x,y)f(y) >sinrf(*,y)f(z) + sind(y,z)f(x),(4) dfx is the DER function J2 it4sind(x, Ai)co$dA>.

The last part of the proposition allows one to prove statements about this class of func-tions by induction.

It is useful to define the following "scalar product" on DER. Given / = £ , —Щ c o s d ,and g = £ • —hjCosdBJ,]& (f, g) = J^tjOtbfOOsd{A{, Bj). The scalar product is bilin-ear (although one should keep in mind that DER is not closed under addition and positivescalar multiplication due to the restriction J 2a> ^ ' ) . If one defines the characteristic func-tions Xp : ~ —cosdp and XA '•= -cosd^,forany p e E or А с Е, one immediately has

and (f,XA)>~f(4) f o r a l l ? e A . (10)

We now can state the following lemma (in it, and in the rest of this section, we consideronly the differential restricted to Sp).

LEMMA 172. // /,g €DER then (df,dg) ^ ( / , g ) - f(p)g(p).

The lemma is proved by applying bilinearity and the following lemma, which is animmediate consequence of the definitions, formula (1), and A2.


LEMMA 173. For any А, В с S,

cosd(A, B) ^ msd(p, A)cosd(p, B) + sind(p, A) s,md(p, A) cosd(A', B'

Questions about DER functions can often be reduced to ones about characteristicfunctions. We make some observations that will facilitate this reduction. First, if / =£ ; —Щ cos<i,4- and g = £ ; - —bj cosdnj, then picking pi e A:, we immediately have that(Hi -BiXpi. 8) < (/• g)- More strongly,

LEMMA 174. Given any DER function on E there exists a function axP such thatifg) > {a xP,g) for any DER function g on E.

The observation before the lemma allows one to reduce the proof to the case / —ai Xpt +a2Xfii' where p\ ф pi- if d{p\, рг) = я then assuming, say, a\ ^ «2. one can pickp = p\ and a —a\ — at- Otherwise, Proposition 171(3) implies that the point p on Yp\P2and a such that a/sind(p\, pi) = a\ l ^md{p, p2) =a2/sind(p, p]) win work. From theabove lemma we can now finish the properties of the scalar product:

PROPOSITION 175. The scalar product (*,*} on the space of DER functions on E is asymmetric; bilinear operator such that

for every: DER function f.

The main result about DER functions is the following:

PROPOSITION 176. Let E be an n-dimensional Alexandrov space of curvature > 1,n > 0. Let fj be DER functions on E for i = 0,...,k + I for some k ^ 0. If s :=mino</,t./<*+iM/. fj)\ >0 then

(1) there exists a q e E such that f(q) >sforall0^ i ^ к,(2) к < и, and(3) there exists a p € E such that f(p) = 0for 1 i^ i t, foip) ^ s and fk+\(p) ^

—ft.

To prove the first part note that by Lemma 174 and formula (10) we can find a functionaxij such that fj(q) — — {aXq, fj) £ —(Л+ьfj) > * • The second part can be proved byinduction, noting that the derivatives of functions satisfying the hypotheses of the propo-sition again satisfy the hypotheses. The point p can be found as a point where fy attainsits maximum value among all points where fi > 0 fori = 0 , ,.., k. If one of the functions/,, i = 1 к , doesn't vanish at p, Lemma 172 implies that the first part of the presentproposition (proved above) can be applied to the differentials {dfi} at p, i — 0, . . . , k. Wethen find a geodesic direction у such that dfi(y) > 0 for all i = 0, . . . , k. Moving along уthen contradicts the maximally of fo(p). The condition on /i+i can be proved similarly.

880 С. Ptaut

9.5. Admissible functions

DEFINITION 177. A directionally differenti able map / : V С X - * К is called admissibleif dfp is a DER function on Ep for all p, and there exists а A e R such that for everyp.qeU,

/(<?) < f(p) ~ (df. X^pdip, q) ~ Wp. q)2. (11)

Note that K-convexity implies, by Lemma 112(8), f(q) ^ f(p) + df(y)d(p, q) +\ic\d(p,q)2 for every у e q'. Since ~{df, xq!) ^ cf/(y) by Lemma 10 we see that Def-inition 177 strengthens К-convexity. The main examples of admissible functions are thedistance functions. If А С X, an easy generalization of Proposition 159 is that, givenp € X and у € Sp if / := d& then, dfp = ЦА>. In other words, by formula (10),- {df, Xq')p = -co&(x(A'p, q'). Since a(A'p, q') = m i n { a ( ^ , Л'р)}, formula( 11) reducesto л:-convexity. We have proved:

PROPOSITION 178. For any closed Ас X, the function dA is admissible, and d{dj£)p =

DEFINITION 179. A map g\ X -> Wk is called admissible in U if g = G о {/j ft)where each // is admissible in the sense of Definition 177 and С is a Lipschitz homeo-morphism of open sets of R*. A point peU is called в-regular for g if / := (/;,.. ., /A)satisfies the following properties:

(1) If q is sufficiently close to p and г Ф j then (/•', f ' - ) q < — s, and(2) s y . = { y e E p : d M y ) > * f o c a l l 1 < i ^ i } # f t

A point p is called a regular point of g if p is e-regular for some £ > 0. If № e R*has the property that every /~'(w) is nonempty and contains only regular points then x iscalled a regular value off.

REMARK 180. We will say that every mapping / ; X -> R° is admissible without criticalpoints. Note that if k — 1 then a point p is a regular point of g if and only if it is not acritical point of the admissible map f\. This follows from Corollary 134.

Note that, since an admissible function is К-convex, condition (2) above implies that£q ф 0 for all q sufficiently close to p. It follows that the set of regular points of anadmissible function is open. However, the set of regular points of g may be empty. Thefunctions idfi)q and %y, where у 6 Щ, satisfy the conditions of Lemma 176, so we canfind directions y^ € Eq such that dfjiyf1) > s md dfiiyf) = 0 whenever i / j . ApplyingProposition 166 we now get:

PROPOSITION 181. If p is an e-regular point of an admissible map g then g is s-open ina neighborhood of p.


DEFINITION 182. A function g: X -• K* is called incomplementable at a regular pointp if p is not a regular point of any admissible map (g,gk+\) '• V -# Ht

+1, where U is aneighborhood of p.

PROPOSITION 183. Suppose g: X -> Rk, к ^n, is an admissible map in a neighborhoodof a regular point p, which is incomplementable at a regular point p. Then there exists acontinuous nonpositive function gk+\: X —> К which vanishes at p, and a neighborhoodU of p, such that к :=(g, gt+i) ; X —»- Mk +l is admissible in U and

(1) for small enough p > 0 the set Kp := U n g^ftO, -2р]) Л g-] (B(g(p), p)) iscompact,

(2) ifw Eg(Kp), then Kp n ^ J , ( 0 ) r \ g ~ l ( w ) is a point, and(3) every point of К p\g^}(0) is a regular point of k.

The proof of this proposition may be found in [631.

REMARK 184. Note that K;i\g^(0) ф 0 if к < n . Otherwise, by condition (2) above,g: Kp —• Rk would be 1 - 1 , which is impossible given the difference in dimension.

9.6. Topological arguments

This section finishes the proofs of Theorems 168 and 167. In this section we consider theopen cone on the empty set И to be a point. A conical neighborhood of a point p in a topo-logical space Y is an open set U containing p which is (pointed) isomorphic to the opencone cZ on a topological space Z; i.e., U is homeomorphic to cZ via a homeomorphism hsuch that h(p) is the apex 0 of the cone. Conical neighborhoods are unique up to pointedhomeomorphism [48].

DEFINITION 185. A (-l)-dimensional MCS-space is defined to be the empty set. Ifn ^ 0 then an и -dimensional MCS-space is a metric space, each of whose points has aconical neighborhood, pointed homeomorphic to the open cone over a compact (и — 1)-dimensional MCS-space.

For example, a O-dimensional MCS-space is discrete and a 1-dimensional MCS-spaceis a graph. Clearly the cone or suspension of an n-dimensional MCS-space is an (n +I )-dimensional MCS-space. MCS-spaces are stratified by manifolds: The m-dimensionalstratum Z m consists of those points whose conical neighborhood splits as W x cZ, whereZ is a compact MCS-space, and m is the largest number so that such a splitting exists.To see why Zm is a manifold, note that the mapping фг:Rm x cZ -> Rm x cZ given by4>t(s,x) — (s + t,x) is a homeomorphism. Therefore any two points (s,0) and (i,0) haveconical neighborhoods isomorphic to Rm x cZ, and points not in the slice of 0 have conicalneighborhoods isomorphic to M m + I x Z. It follows that if Л: е Zm then the points on theR'" slice of the conical neighborhood of x are also in Z m . The remaining points near x arein Zk for к > т. In particular, MCS-spaces are locally connected WCS-sets in the sense of[83, Definition s.1].

882 С. Plaut

THEOREM 186. Let g: X-*• Rk he an admissible map, к ^ и .(1) g isa trivial bundle map in some neighborhood of any regular point p,(2) if w eUk is a regular value for g then g~] {w) is an (n — k)-dimensional MCS-

space,(3) ifg is proper and without critical points in U then f is a bundle map on U.

Recall that a map is proper if the preimage of every compact set is compact; a closedmap is proper if the preimages of points are compact. The proof of the theorem is by reverseinduction in к; we denote the above three statements by If/:), 2(k), and 3(k). For any k,1(k) and 2(&), together with the facts that MCS-spaces are locally connected WCS-spaces(so [83, Theorem 5.4], can be applied) and Lipschitz maps are closed, allow Corollary6.14 of [83] to be applied, and 3(k) follows. Statements l (n) and 2(n) follow from Propo-sition 181. We suppose that conditions l(k + 1), 2(k + 1), and 3(k + 1) hold for somek + 1 n. II g is complementable then there is no problem. If g is incomplementableat a regular point p, consider die map gk+] and the compact set Ka given by Proposi-tion 183. Since every point of Ka is a regular point of {g,gk+\) applying part 3(k + 1)to this map we get a mapping ф:М x B(g(p),p) x \_-2p,0) -s» £p- gj^j(0), whereM := KP П g-^ ( -p) П#-' (g(p)). But since KpП g ' ^ (0) Пg~l(u) is a single point forall v € g(Kp), we can extend this trivialization to cM x I —> Kp. This implies 1(£) and

Theorem 168 now follows from the statement 2(0). Since dp has no critical points near/>(cf. Corollary 133).

9,7, Extremal sets

In this section we mention some results from [66] about extremal sets. Extremal sets wereintroduced to further refine the stratification of Alexandrov spaces.

DEFINITION 187. A closed subset F o f X is called extremal if for every q e X,ifdq hasa local minimum at p when restricted to F, then pis a critical point of d4 on A\ If X haspositive curvature (scaled to curvature > 1) we assume that B(F, тг/2) = X.

REMARK 188. It is possible in many cases to avoid the latter assumption (and simplify thedefinition) by working with the tangent cone instead of the space of directions (cf.

DEFINITION 189. Let p e X. We define K(p) = [q e X; q has a conical neighborhoodisomorphic to a conical neighborhood of p\.

PROPOSITION 190. Let F be a closed subset of X such that if p € F then for some Ucontaining p, K (p) C\ U С F. Then F is extremal.

In fact, if p € F is not a critical point of / = dq then p is a regular point of / (cf.Remark 180). Then by Theorem 186, / is a trivial bundle map in a neighborhood of p.By moving along the slice of R through p we can then find points a e K(p) (and hence


in F) arbitrarily close to p such that f(a) < f(p); so p cannot be a local minimum of /on F, Since the strata of an MCS-space are determined by the topologicat type of conicalneighborhoods of their points, we immediately have:

COROLLARY 191. The strata of the stratification ofX jmm Theorem 168 are extremal.

Extremal sets have the following useful property. Note that there are variations in theliterature on the following definition and notation:

DEFINITION 192. Let F be a closed subset of X and p e F. The closure in Ep of alldirections of minimal curves from p to F i s denoted by EPF.

PROPOSITION 193. A closed set F in X is extremal if and only if Zp F is extremal for allp € F. {However, the "if" part may not be valid if we include the additional conditions forpositive curvature in Definition 187.)

One of the points of [66] is to show that Theorem 186 remains valid for the restrictions ofadmissible functions to extremal sets, and here a key point is that the observation made inRemark 184 is no longer valid. This problem is overcome by introducing the more generalclass of MCS-spaces, where the conical neighborhoods are permitted to be isomorphic tocones over compact MCS-spaces of dimension < n— 1 instead of dimension equal to и — 1.Tn consequence, the Theorems 168 and 167 also hold for extremal sets.

PROPOSITION 194. For every compact set К С X there exists an e > 0 such that if F С Кis extremal and dp denotes the induced inner metric on F (Section 4.1), then for everyx,yeF dF(x,y)^^

A natural question to ask at this point is whether an extremal set, with the induced innermetric, is again an Alexandrov space with the same curvature bound. A counterexample tothis statement for a codimension 3 extremal subset is given in [70]. The question remainsopen for smaller codimension. The next statements are also proved in [66J (the first part ofthe next proposition is obvious from the definition):

PROPOSITION 195. // F and G are extremal subsets of X with FфС then F U G , FC\Gand F — G are all extremal sets.

DEFINITION 196. An extremal subset F in X is called primitive if for any other extremalset G in X, F П G is not open in F.

PROPOSITION 197. IfF is extremal in X then F can be written uniquely as a locally finiteunion of primitive subsets each having nonempty intersection with F.

Extremal sets occur naturally in connection with quotients (cf. Section 4.4).

884 C Plant

PROPOSITION 198. Suppose Г is a compact group acting on X by isometries, andn: X —> G/F is the quotient map. Then

(1) if F is extremal in X thenn(F) is extremal in G/Г, and(2) if Г' is a closed subgroup of Г and F is the set ofall fixed points of'/" thenn(F)

is extremal in G/F.

We mention here a useful construction in the proof of this proposition, namely the con-vex hull of a compact set. We state a special case of this result, which is given a separate(and simpler) proof in [461.

PROPOSITION 199. For every x e X there are a neighborhood U of x and a strictlyconcave function f:U—»• R so that x is the maximum of f and the inverse images ofpoints are compact.

The construction of / is such that it is "stable" in a sense under Gromov-Hausdorffconvergence.

9.8. Quasigeodesics and gradient curves

Note that X itself is obviously extremal; therefore the following statements apply to X asa whole. The existence of quasigeodesics in an Alexandrov space was proved in [65J.

THEOREM 200. Let F С X be extremal, p € F, andy € SPF, and f be К-convex withoutcritical points on a neighborhood of F. Then

C1) any gradient curve which begins in F remains in F,(2) there exists a quasigeodesic x defined on Ж+ starting at p, with х'Ф)—у.

In fact, |^y € SpF whenever p e F. We remind the reader of our comments afterProposition 147, which describe the construction of a "gradient exponential map".

The following has been called a "Generalized Lieberman Lemma", after the result in[521. It was proved first in [661, and again with a simpler proof in [70].

THEOREM 201. Let F с X be extremal and let df denote the induced inner metric on F.Then minimal curves in dp are quasigeodesics in X,

9.9. Boundary

DEFINITION 202. If X is a 1-dimensional Alexandrov space, then the boundary pointsBX of X are the endpoints of X if X is an interval; otherwise X has no boundary. If X isan Alexandrov space of dimension и ^ 2 then p e X is a boundary point if and only if Ep

contains boundary points. A point which is not a boundary point is called an interior point.


DEFINITION 203. We now establish some notation we will use from now on for Alexan-drov spaces of curvature ^ 1, which will always denote by E. If E is O-dimensional, then27 is a point if дЕ ф $, and a pair of points at distance я otherwise. If IT is 1-dimensionalthen £ is a closed interval of length я if дE ф0, and a circle of diameter ж otherwise.

Then next theorem was proved in [62], by induction on the dimensions of the two spaces.

THEOREM 204. If E, E ' are Alexandrov spaces of curvature > 1 (and possibly differentdimension) such that K* x c E is homeomorphic to Rm x c.E' for some m, k, then E iswith boundary if and only if E ' is with boundary.

COROLLARY 205. A point p e X is a boundary point if and only if X has a conicalneighborhood isomorphic to the cone on an Alexandrov space with boundary. In particular,the boundary is closed.

From Proposition 190 we now get:

COROLLARY 206. The boundary of an Alexandrov space is extremal.

COROLLARY 207. If p e ЭХ then p has a conical neighborhood in dX isomorphic to

A following "Gluing Theorem" was proved originally in the 2-dimensional case byAlexandrov. Perelman [62] proved a related "Doubling Theorem" for doubling an Alexan-drov space with boundary. The generalization here is due to Petrunin [70].

THEOREM 208. Let Xi and Хг be Alexandrov spaces of curvature > к such that there isan isometryф : дХ\ -» ЭХ2 (where each boundary is given the induced inner metric). Thenthe space obtained by gluing X \ and X2 together along their boundaries via ф naturallyhas the structure of an Alexandrov space of curvature ^ к having no boundary.

In particular, one can "double" a space with boundary by using two copies of it and theidentity map on the boundary.

10. Differentia hlc structures

Throughout this section X denotes an Alexandrov space of dimension n. This applies inparticular to our comments about spaces with curvature above. Although it is possibleto consider an upper curvature bound without a lower curvature bound {and this is evenessential for, say, the theory of nonpositively curved groups), such an approach is outsidethe scope of this article. A good reference for the upper curvature bound condition is [4].

886 C. Ptaut

10.1. Curvature bounded below and above

As mentioned in the introduction, Alexandrov's problem of synthetic differential geom-etry was solved by Berestovskii and Nikolaev [6,55-571. Their solution involved spaceswith curvature bounded both above and below. Curvature bounded above in an Alexan-drov space can be defined by reversing the inequalities in any of the three comparisonconditions Al, A2, and A3 (cf. Section 3.3), and assuming that one (hence all) of theseconditions is true locally. Despite the analogous conditions, there are significant differ-ences between curvature bounded above and curvature bounded below. For example, theGlobal Comparison Theorem (Theorem 43) is not valid for curvature bounded above.For example, the flat torus S 1 x S l has curvature locally bounded above and below by0 (so curvature = 0) locally, but not globally. The easiest way to see this is to note thefollowing fundamental property of curvature ^ K. Since the space forms SK have theproperty that geodesies starting at a point p of length less than ТГ/*/К do not meet, diesame must be true in a region of curvature ^ К in X. Thus no compact space can haveglobally curvature ^ 0. It also follows that for any point p, the cut radius map cp (De-finition 107) has a positive lower bound. If in addition space is geodesically complete(Definition 25) then following result is immediate from our previous work on Alexandrovspaces:

THEOREM 209. Suppose X has curvature ^ К and is geodesically complete. Then forevery p e X

(1) p is regular,(2) TP=R",(3) expp is a local homeomorphism.

From the third part of the above statement it follows that X is a topological manifold.Berestovskii introduced the distance coordinates to prove that X is a differentiable mani-fold with continuous Riemannian metric. Later, Nikolaev synthetically constructed a notionof parallel transport, which he used to build so-called harmonic coordinates. He proved thefollowing theorem. Note that finite dimensionality is not a required assumption.

THEOREM 210. IfУ is a geodesically complete inner metric space of curvature boundedabove and below then Y has the structure of a smooth manifold, and the distance d on Y isinduced by a Riemannian metric on Y, which has class c ' 1" with respect to the harmoniccoordinates.

Nikolaev later showed [ 13] that the smoothness of the Riemannian metric cannot be im-proved; that is, the above theorem is the best possible solution to Alexandrov's problem ofsynthetic differential geometry that can be obtained using curvature bounds. Nikolaev alsoproved [58] that every geodesically complete space of curvature bounded above and belowcan be approximated Riemannian manifolds having nearly the same (sectional) curvaturebounds and the same dimension. The approximation is in the Gromov-Hausdorff sense,and in the stronger sense of bi-Lipschitz homeomorphisms.


The following theorem generalizes Berestovskii's result to spaces which are not geo-desically complete [73J. Note that many issues already arise even when only the geodesiccompleteness condition is dropped, in particular the problem of finite dimensionality. Forexample, the Hilbert cube fO,! 1] x [0,1 /2] x • • • clearly is compact and has curvature = 0,but is infinite dimensional.

THEOREM 211. Suppose Y is an inner metric space of curvature bounded above andbelow. Then the following are equivalent:

(1) the set T of geodesic terminals (Definition 108) in Y is nowhere dense,(2) dimУ <оо,(3) Y is homeomorphic to a smooth manifold with boundary, and dY = T.

Since a smooth manifold with boundary admits a Riemannian metric such that the in-duced metric has curvature bounded below, including the boundary, one has:

COROLLARY 212. A topological space M admits the structure of a smooth manifold withboundary if and only if it is finite dimensional and has a metric of curvature bounded aboveand below.

10,2. Other differentiable structures

As mentioned in the previous section, curvature bounded above implies that the cut radiusat every point has a positive lower bound. One can therefore weaken the notion of anupper curvature bound by requiring that the space have positive cut radius {in Riemanniangeometry this means the same thing as having positive "injectivity radius"). Note that adifferentiable manifold may have several nondiffeomorphic differentiable structures. Thenext theorem was proved in [74"|:

THEOREM 213. If X positive cut radius then X is a smooth manifold having a uniquedifferentiable structure such that dp is differentiable near [but not at) p.

In more general cases there are differentiable structures, but not uniqueness results. Thenext theorem was proved in [60] and [591- We refer the reader to those papers for theprecise definitions. The general theory of differentiable structures on Alexandrov spacesseems still to be in a state of development.

THEOREM 214. Let X be an Alexandrov space.(1) There exists a set Xo such that X — XO has (N -dimensional) measure 0 and contains

the set Sx of singular points in X.(2) The metric on X comes from a cl /~-Riemannian metric g on XO.(3) g extends continuously to X - Sx.(4) X has an almost everywhere approximately second differentiable structure in the

sense ofStolz.

chapter 16 metric spaces of curvature ^ i conrad plaut department of mathematics, university

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