Notes on Differential geometry

Koushik Viswanathan

August 5, 2015

Preface

These notes are an enlarged version of course notes for ‘Modern Differential Geometry’ that

I took at Purdue University. They are based largely on the book ‘An Introduction to Differ-

entiable Manifolds and Riemannian Geometry’ by William Boothby. Most of the figures are

also adapted from here. These notes are not a substitute for a text book or a shortcut to a

rigorous study of differential geometry. They merely expand on various topics not detailed

in standard texts. For this, some insights, based on books by Frankel and Dubrovin et al.

(see Bibliography), are also included. Logically, this is the first part of a complete course

on modern differential geometry, emphasizing on differential forms and manifolds. A second

part, with a focus on curvature, will hopefully also be completed soon.

Koushik Viswanathan

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1 The manifold Rn

In this chapter, we consider the oldest and most basic of all manifolds — n-dimensional

Euclidean space Rn. These spaces play a very fundamental role because all manifolds locally

appear Euclidean. After a brief review of the notions of continuity and differenitability,

some basic results in the linear algebra of maps and functions are introduced. Most of the

definitions are abstract, but because the object considered is Rn, one can easily visualize the

results in ordinary 2D or 3D space.

1.1 Why do we need abstraction?

A manifold is a generalization of the idea of a surface. Being abstract in their definitions,

manifolds can be defined for any dimension, and their properties can be studied using the

methods developed in modern differential geometry.

The basis for introducing differentiable manifolds is the intrinsic nature of certain geometric

properties of surfaces. Some surfaces, when visualized as embedded in R3 are not easily

identifiable as being ‘topologically equivalent.’ Two such surfaces can be deformed into one

another continuously without any tears. The most widely used example to illustrate this is the

equivalence between a doughnut and a coffee cup. Additional examples of such ‘equivalent’

2D surfaces in 3D space are shown in Fig. 1.1.

Clearly, one cannot at once claim that these surfaces are equivalent by looking at their rep-

resentations in the figure. Thus, our conventional notion of ‘embedding’ geometric objects

in some space (here in 3 dimensional Euclidean space R3) is not always advantageous. This

problem can be circumvented by studying the properties of these geometries as being some-

how intrinsic1, irrespective of their embedding in a space of higher dimension.

Surfaces can be cut/ pasted to generate new surfaces in R3. For instance, by taking a square

and identifying its edges in various ways, one can thus obtain a cylinder, Mobius band, torus

1A famous historical example of this is Gauss’ result that the entire surface of the earth cannot be mappedonto a piece of paper without any length or angle distortions.

2

Figure 1.1: Some topologically equivalent surfaces that appear completely different due totheir representation.

Figure 1.2: Four different surfaces obtained by pasting the edges of a square.

or a Klien bottle. This procedure is shown in Fig. 1.2. We add a further property that if

a unique surface normal can be defined for every point p ∈ M which varies continuously

with p, then the surface is orientable. Surfaces generated in such a manner often cannot be

embedded in 3D space, so applying direct visual geometric ideas is impossible.

As with any branch of pure mathematics, abstract ideas used in modern differential geometry

help us prove general results about surfaces. An illustration is the fundamental theorem of

2-manifolds that concerns all known 2D surfaces. Let us take the 2-sphere, cut out discs on

its surface and add cylinders joining two such discs, thus forming a handle. Then,

Fundamental theorem of 2-manifolds

Every compact, connected, orientable 2-manifold (a 2D surface) is homeomorphic to

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a sphere with handles added. Two such manifolds with the same number of handles

are homeomorphic and conversely, so that the number of handles (called the genus) is

the only topological invariant.

This says that all 2D surfaces (that are orientable, compact and connected) can be reshaped,

without tearing (homeomorphic) into a sphere with a certain number of handles added.

Clearly, the requirements for this theorem to hold are very minimal and so the result is very

powerful! It certainly helps us classify all 2D surfaces, irrespective of whether we can visu-

alize them correctly in 3D space.

While the above applications are secondary from a physical point of view, the fundamental

utility of differential geometry, as far as these notes are concerned, is in the study of continu-

ous fields on physical bodies. When codified in terms of vector or tensor fields on manifolds,

the tools afforded by modern differential geometry, particularly the language of forms, inte-

gration theory and the notion of metrics, enable very efficient modeling (and quantification)

of physical phenomona. It is with this application in mind that the resulting formalism is

presented.

1.2 Calculus — continuity and differentiability

In advanced calculus, the ideas of Continuity and Differentiability are introduced on topolog-

ical spaces . A topological space M is a set of points along with a collection τ that lists all

open subsets of M (including the null-set ∅ and the whole set M). τ is called the topology

of M .

Apart from continuity, which can be applied to functions between different topological spaces,

we consider differentiability only between Euclidean spaces. Furthermore, the words map and

function simply are conventions used when the range is some Rm or R respectively.

DEFINITION:

1. A map f : M → N between topological spaces M,N is continuous if every open set

V ∈ N , f−1(V ) = x ∈ M : f(x) ∈ V is an open subset of X. Thus f is a map

between elements of M,N but its continuity depends on the topologies of M,N .

2. Let f : Rn → R. We say that f(x) is differentiable at x = a if

f(x) = f(a) +n∑i=1

bi(xi − ai) + ‖x− a‖r(x, a) (1.1)

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where limx→a r(x, a) = 0. If f(x) is differentiable, then f(x) is continuous and

bi = ∂f∂xi

(a). Additionally, if ∂f/∂xi exist and are continuous on a neighbourhood of

a, then f is differentiable at a.

3. Suppose U ⊂ Rn is an open set, then f ∈ Cr(U), (r ≥ 1) if rth order partial deriva-

tives ∂rf∂xi1 ···∂xir exist and are continuous on U .

4. If f ∈ Cr(U)∀r then f ∈ C∞(U) and f is called a smooth function.

EXAMPLE: To illustrate point (ii) above, consider the function f(x, y) defined by the

following.

f(x, y) =xy

x2 + y2(x, y) 6= (0, 0)

and f(0, 0) = 0

then ∂f/∂x(0, 0) = 0 and ∂f/∂y(0, 0) = 0 (using the definition of partial derivatives as

limits). On the other hand, f(x, x) = 1/2 =⇒ f(x, y) is not continuous at (0, 0) as one

approaches the origin along the line x = y. Hence existence of partial derivatives is not

sufficient to ensure continuity in 2 dimensions or higher, unlike in 1D.

Now consider a map F : Rn → Rm. We write F (x) = (f1(x), f2(x), · · · , fm(x)) and we can

now carry over our earlier definitions (for functions) to each of the component functions.

F (x) is differentiable if all f1, f2, · · · , fm are differentiable.

Define the m× n Jacobian matrix DF by

DF =∂(f1, f2, · · · , fm)

∂(x1, x2, · · · , xn)=

∂f1/∂x

1 ∂f1/∂x2 · · · ∂f1/∂x

n

∂f2/∂x1 ∂f2/∂x

2 · · · ∂f2/∂xn

......

∂fm/∂x1 ∂fm/∂x

2 · · · ∂fm/∂xn

(1.2)

so that

F (x) = F (a) +DF (a)

x1

...

xn

+ ‖x− a‖r(x, a) (1.3)

where limx→a r(x, a) = 0.

If we have two maps F : Rn → Rm and G : Rm → Rp. We can define the composite map

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H = G F . The chain rule states that if F is differentiable at a and G is differentiable at

F (a), then H is differentiable at a and

DH(a) = DG(F (a)) ·DF (a) (1.4)

Further, by iteration, if F and G are Cr then H is also Cr.

EXAMPLE: Let U be an open subset of Rn and F : U → Rm, m ≤ n be a C1 mapping.

Suppose F is injective (one-to-one) F−1 : A→ U where A = F (U) is also of class C1. Then

we can show that m = n.

Consider the identity map I = F−1 F . The corresponding Jacobian matrices are related by

DI(a) = DF−1(F (a))DF (a)

the first being n× n, and second being n×m. If m < n, rank of DF (a) ≤ m < n. But rank

of DI(a) = n. Since rank(AB) ≤ min(rank(A), rank(B)), n ≤ m. This is only possible if

m = n.

A homeomorphism between two topological spaces is a continuous map with a continuous

inverse. This is clearly a more restrictive map than one that is merely continuous. The

following result quantifies this.

Brouwer’s invariance of domain theorem

There exists no homeomorphism from U ⊂ Rn into Rm, m < n.

Just like homeomorphisms deal with continuous functions, the notion of a diffeomorphism

corresponds to a differentiable homeomorphism. This idea makes sense only if differentiability

is defined, which we have done on subsets of Euclidean spaces Rn.

DEFINITION:

Let U ⊂ Rn and V ⊂ Rn be open. A mapping F : U → V is a Cr diffeomorphism if:

(i) F is a homeomorphism

(ii) Both F and F−1 are of class Cr, r ≥ 1.

This definition is a little redundant. It is not possible to have a homeomorphism between Eu-

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clidean spaces of different dimensions (Brouwer’s invariance of domain theorem), hence there

cannot be a diffeomorphism between spaces of different dimensions. The precise condition

under which one can define a diffeomorphism, given a differentiable map F , is enumerated

by the following result

Inverse function theorem

Let W be an open subset of Rn and F : W → Rn is a Cr mapping, r = 1, 2, · · · ,∞:

1. If p ∈ W and DF (p) is non-singular, then ∃ an open neighbourhood U of p such

that V = F (U) is open and F : U → V is a Cr diffeomorphism.

2. A necessary and sufficient condition for the C∞ map F to be a diffeomorphism

from W ⊂ Rn to F (W ) ⊂ Rn is that it be one-to-one and DF be nonsingular at

every point of W .

The second result above follows from the first.

We now move onto the notion of rank of a mapping. In linear algebra, there are three

equivalent definitions for the rank of an m×n matrix A — (i) The dimension of the subspace

of V n spanned by the rows, (ii) Dimension of the subspace of V m spanned by the columns

and (iii) Maximum order of any non-vanishing determinant. Clearly, rank(A) ≤ m,n. The

rank of a linear transformation is the rank of any matrix that represents it. This is also equal

to the dimension of the image.

Likewise, the rank of a mapping F : U → Rm from an open set U ⊂ Rn is defined at a

point p ∈ U as the rank of the Jacobian DF (a). If F is composed with diffeomorphisms, the

rank of the final composition is the rank of F because diffeomorphisms have non-vanishing

Jacobian.

Rank theorem

Let A0 ⊂ Rn, B0 ⊂ Rm be open sets, F : A0 → B0 be a Cr mapping so that DF (a)

has constant rank k on A0. If a ∈ A0 and b = F (a), then ∃ open sets A ⊂ A0 and

B ⊂ B0 with a ∈ A, b ∈ B and Cr diffeomorphisms G : A → U where U is open

in Rn and H : B → V where V is open in Rm so that H F G−1(U) ⊂ V and

H F G−1(x1, x2, · · · , xn) = (x1, · · · , xk, 0, · · · , 0) (with m− k zeros in the end).

The basic idea is that given a differentiable map F with a constant rank k on a domain that is

some subset of Rn (k ≤ n by definition of rank), we can construct a projection diffeomorphism

onto the subspace Rk. So although F need not be a diffeomorphism (continuous inverse), we

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Figure 1.3: Illustration for rank theorem

can get a diffeomorphism that is restricted to some subspace or hyperplane of dimension k

in Rn in the first place, in some subset of the domain of F .

In closing, it might be noted that the inverse function theorem is implied by the rank theorem

as a special case.

1.3 Tangent space of Rn

We now precisly understand the terms continuity and differentiability of functions (and maps)

defined on an Rn. Our first real foray into abstract differential geometry begins by considering

the notion of vectors in this space. Conventionally, our notion of geometry has centered

around equating the collection of all vectors in Rn to Rn itself. This is a special property of

the space, and we now start by disentangling the two ideas. Our final aim is to define vectors

without any reference to an embedding space or to the space in which they exist. Subsequent

chapters will generalize this notion to other abstract manifolds.

Loosely speaking, the tangent space is the collection of all vectors tangent to a surface at a

given point. The exact definition can be provided in four separate ways. Three ways that

we will not follow are first illustrated below.

Let a ∈ Rn. The tangent space of Rn at a can be defined using either of the following

(a) Geometric : Look at arrows with base point a. Add vectors by the parallelogram rule.

By the usual methods of scalar multiplication and the parallelogram rule, we obtain

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a vector space structure at a. Now we can identify TaRn ' TbRn at a and b by rigid

motion (translation). i.e. two the vector spaces TaRn and TbRn are equivalent if for

any vectors vb ∈ TbRn, va ∈ TaRn, the result vb − va = r is some constant vector. This

approach is intuitive and can be worked for dimensions n = 2, 3.

(b) Algebraic : This is the approach taken in coordinate geometry. Choose a coordinate

system so that ~a = (a1, · · · , an) and ~x = (x1, · · · , xn). Identify ~ax ' (x1− a1, · · · , xn−an) as the vector from the ‘origin’ a. This reduces to defining ToRn. By the usual

rules, addition and scalar multiplication are component-wise giving rise to the vector

space structure. This method implicitly relies upon the existence of standard ‘pre-

ferred’coordinates.

(c) Equivalence of curves : Consider C1 curves in Rn passing through a. Write them

as x(t), with x(0) = a. Define an equivalence relation x(t) ∼ y(t) if x′(0) = y′(0).

TaRn then consists of the equivalence classes of curves. It can be easily seen that each

equivalence class contains a unique straight line ~x(t) = ~a+ t~v in Rn. This definition is

often generalized when defining the tangent spaces of general manifolds.

In addition to these, a fourth motivation exists, which we will use in the remainder of these

notes.

Take a point a ∈ Rn. Given f ∈ C∞(U1) and g ∈ C∞(U2) where U1 and U2 are open subsets

of Rn containing a, we say that f ∼ g if ∃V ⊂ U1∩U2 so that f(x) = g(x) for x ∈ V . The set

of equivalent classes are called germs of C∞ functions on a neighbourhood of a and denoted

C∞(a). Basically, C∞(a) collects all ‘distinct’ functions in a neighborhood of a.

DEFINITION:

Define the set D of derivations at a, which are linear functionals D : C∞(a)→ R satisfy-

ing:

(i) D(αf + βg) = αDf + βDg, α, β ∈ R

(ii) D(fg) = f(a)D(g) + g(a)D(f) (product rule)

Then the tangent space TaRn is isomorphic to D(a)

While this definition is extremely abstract, it is the most transparent when doing calculations.

It is also easily generalized to arbitrary manifolds. After the properties of D(a) are established

below, it will be seen to reproduce what we know of tangent spaces from either of the earlier

definitions.

Having defined D(a) as a colletion of all derivations Ds, the following result holds

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PROPOSITION 1: D(a) is a real vector space with the definition (D1, D2 ∈ D(a),

α, β ∈ R) (αD1 + βD2)f = αD1f + βD2f .

Proof. To prove this, we have to verify that (αD1 + βD2)f = αD1f + βD2f obeys the

properties of derivations (i) and (ii) above, if D1 and D2 do. This can be done by

following the definitions and using linearity.

An example of a derivation is the directional derivative Df

Df =n∑i=1

bi∂f

∂xi(a) b1, b2, · · · , bn ∈ R

Directional derivatives are actually special in that every derivation can be represented in

terms of directional derivatives, as shown by the following result.

THEOREM 1: Every element of D(a) is a directional derivative. In particular

dim(D(a)) = n = dim(Rn).

Proof. Let f ∈ C∞(a). Think of f as a function, technically it is an equivalence class,

but we can work with a representative. Consider x in the neighbourhood of a and the

line sx + (1− s)a ‘interpolating’ between x and a (for 0 < s < 1). By the fundamental

theorem of calculus,

f(x)− f(a) =

∫ 1

0

d

dsf(sx+ (1− s)a)ds

=n∑i=1

∫ 1

0

∂f

∂xi(sx+ (1− s)a)(xi − ai)ds using the chain rule

=n∑i=1

(xi − ai)gi(x)

where we define gi(x) =∫ 1

0∂f∂xi

(sx + (1 − s)a)ds. Now let D ∈ D(a) and apply D on

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both sides above2,

Df =n∑i=1

D((xi − ai)gi(x))

=n∑i=1

gi(a)D(xi) by the product rule (Leibniz)

=n∑i=1

∂f

∂xi(a)D(xi) using the definition of gi(x) above

So we see that Df is a directional derivative, with some bi = D(xi).

REMARK: The proof given above (for finite dimensionality of D(a)) is valid only for C∞(a).

If f ∈ Ck(a), then g ∈ Ck−1(a) so we cannot apply D (which is of order k) to g when using

the product rule. In fact, dim D(a) = ∞ for germs of C1 functions, even if n = 1. A basic

example of this is the function f(x) = |x|α for 1 < α < 2. f(0) = 0 and df/dx(0) = 0. But

|x|α is not a product of C1 functions vanishing at 0.

Now it is clear that there is a one-to-one correspondence between vectors at a point and

directional derivatives. In conventional geometry, vectors determine directional derivatives

by defining the tangent to a curve along which to evaluate the derivative. However, we

must notice that now we have worked backwards. Defining directional derivatives (by only

asserting linearity and the Leibniz rule) first allows us to identify each vector as some linear

combination of directional derivatives in a coordinate system. This approach will be readily

generalized to general manifolds in the next chapter.

Finally, the above definitions can be extended to define vector fields — A vector field X

on an open subset U ⊂ Rn is a function which assigns to p ∈ U a tangent vector Xp ∈ TpRn.

We say X is a smooth vector field if X =∑n

i=1Xi(p)ei where X i are smooth functions. Here

ei are the standard basis for Rn, as derivations ei = ∂∂xi

.

It is clear that the vector field X is a map C∞(U)→ C∞(U) that takes f → Xf , Xf being

defined by its action on a point p ∈ U as (Xf)(p) = Xpf in terms of the tangent vector at p.

X is itself also a derivation (because at every point in U it is a derivation). It is customary

to use the notion of a derivation on an algebra, instead of how it has been introduced here.

A derivation D : A→ A is a linear map that satisfies the product rule. Here A is C∞(U).

2Note that D(1) = D(1.1) = 2D(1) by the chain rule, hence D(1) = 0. By extension, D(c) = 0 for someconstant c = f(a).

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1.4 An aside — some results from topology

The next step in trying to introduce geometric concepts is to transition from a space (a

collection) to a manifold (with additional properties). Being abstract enough to only include

lists of open sets, a topological space can have multiple separation properties. Two most

commonly used separation axioms are:

1. Hausdorff: Different points can be separated by disjoint open sets

2. Normal: We can separate disjoint closed sets C1, C2 by disjoint open sets U1, U2

Roughly, these separation properties determine how two distinct points can be ‘separated

from each other’. These notions are only included here for completeness. If a topological

space is normal, then,

Urysohn’s lemma

A topological space is normal if and only if, given two disjoint closed subsets C1, C2,

∃f(x) such that f(x) = 0 for x ∈ C1 and f(x) = 1 for x ∈ C2.

It will be useful, when studying partitions of unity and in integration theory of differential

forms, to be able to extend Urysohn’s lemma to differentiable functions f , given the sepa-

ration and topological properties of the spaces C1, C2. A restricted version of this result is

presented below:

PROPOSITION 2: Let F ⊂ Rn be a closed set and K ⊂ Rn be a compact3 set.

Assume F ∩K = ∅. There there exists a C∞ function σ(x) defined on all of Rn whose

range is in the closed interval [0, 1] such that

(i) σ(x) = 1 on K, i.e. x ∈ K

(ii) σ(x) = 0 on F , i.e. x ∈ F

Proof. Since K is a compact set, we can cover K by a finite collection of open balls whose

doubles (same center but double radius) are disjoint from F . Define bump functions

φ ∈ C∞(Rn), x ∈ Rn by φ(x) = 1 for |x| ≤ 1 and φ(x) = 0 for |x| ≥ 2, such that they

have compact support.

3Note the difference here between this result and Urysohn’s lemma above. Now one of the sets is compactwhereas in the latter case both sets were just closed.

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If pi ∈ K and ri > 0, define gi(x) = φ(‖x−pi‖ri

)where ‖.‖ denotes the Euclidean distance

between x and pi. Now take a finite collection of open balls B(pr, ri), i = 1, · · · , l with

centre and radius given by pi and ri.

Define σ(x) = 1 −l∏

i=1

(1− gi(x)). Now if x ∈ K, then gi(x) = 1 for some i and so

σ(x) = 1 for x ∈ K. On the other hand, if x ∈ F , all double balls are disjoint from

F =⇒ gi(x) = 0∀i. Hence σ(x) = 0 for x ∈ F and σ(x) is the required function.

REMARK: This proof is not valid for K just closed (not compact). However, the theorem

is true nonetheless. The corresponding function σ(x) must be constructed differently. Now

that we know such a function σ exists, we can take any function on a subset U of Rn and

extend it to all over Rn. This is illustrated by the following result.

PROPOSITION 3: Let f ∈ C∞(U), p ∈ U , where U ⊂ Rn. Then ∃ an open set

V ⊂ U with p ∈ V and f ∈ C∞(Rn) such that

f(x) =

f(x) x ∈ V

0 x /∈ U(1.5)

Hence f(Rn) is like an extension of f(U) to the whole of Rn which agrees with f only

on V ⊂ U .

Proof. This follows from the previous proposition. Let us choose V to be a ball of radius

r centered at p so that the ball of radius 2r centered at p is contained in U . Define

σ(x) = φ(‖x−p‖r

), where φ(x) is the standard cut-off function on R. Define f(x) by

f(x) =

σ(x)f(x) x ∈ U

0 x /∈ U(1.6)

which is a C∞(Rn) function because σ(x) = 0 outside the ball of radius 2r anyway which

is contained in U . So it continuous and differentiable across the boundary of U .

From this it is clear that any smooth function f defined on a subset of Rn can be extended

to the entire space Rn via the function f . The possibility of such a construction will find

utility in defining partitions of unity.

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1.5 Topological manifolds

Finally, having briefly seen what separation criteria are and how they affect the extensibility

of functions, we define a topological manifold . This is a precursor to the definition of a

differentiable manifold, which is the fundamental basis of the structures introduced in these

notes.

DEFINITION:

A topological manifold of dimension n is a topological space M satisfying

1. M is locally Euclidean of dimension n. That is, each p ∈M has a neighbourhood Up

which is homeomorphic to an open subset of Rn, for some fixed n

2. M is Hausdorff

3. M has a countable basis of open sets — In Euclidean space, balls with rational centre

and radius form a basis

Additionally, a topological manifold with a boundary is a Hausdorff space M with a count-

able basis of open sets which has the property that each p ∈ M is either contained in an

open set U with a homeomorphism φ to either

1. an open set U ′ of Hn − ∂Hn or

2. an open set U ′ of Hn with φ(p) ∈ ∂Hn, i.e. a boundary point of Hn.

Here Hn is the half plane, a subspace of Rn defined by Hn = (x1, x2, · · · , xn) ∈ Rn|xn ≥ 0and ∂Hn is the boundary of Hn given by xn = 0.

REMARK: A topological manifold is locally connected, locally compact, a countable union

of compact sets, normal and metrizable.

EXAMPLES:

Let us consider some examples of topological manifolds to illustrate the point. For each of

these cases, one can check that the above 3 conditions are satisfied.

1. Open subsets of Rn.

2. Sn ∈ Rn+1 with the subspace topology — Take Euclidean space open sets intersected

with the sphere x21 + x2

2 + · · ·+ x2n+1 = 1.

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3. Quotient space (or covering space) construction: T n = Rn/Zn, where Zn = (i1, i2, · · · , in),

ij ∈ Z (integers). This is the space formed by identifying points x ∈ Rn using the equiv-

alence relation ∼, given by x ∼ x + z, z ∈ Zn. For instance T 2 is the 2 dimensional

torus.

4. It can be shown that every p ∈ M belongs to one class or the other. The first type

are interior points of M while the second are boundary points of M . Manifolds with

boundary will play an important role in the integration theory of differential forms.

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2 Differentiable manifolds

2.1 Introduction and examples

Let M be a topological manifold. We recall that for a topological space to be a topological

manifold, it has to have the property that each p ∈ M has a neighbourhood that is locally

homeomorphic to an open subset of Rn. Now consider the pair (U, φ), where U is the open

set (neighbourhood) of p ∈M and φ is the homeomorphism to Rn. This is called a coordinate

neighbourhood .

To p ∈ U , we can assign n-coordinates (x1(p), · · · , xn(p)) of its image φ(p) in Rn, each

xi(p) being a real-valued function on U , called the ith-coordinate function. Let us say that p

also lies in a second coordinate neighbourhood (V, ψ), and it gets assigned coordinates (say)

(yi(p), · · · , yn(p)). Since φ and ψ are homeomorphisms, this defines the homeomorphism

ψ φ−1 : φ(U ∩ V )→ ψ(U ∩ V ). This sequence of maps is illustrated in Fig. 2.1

To be useful, these maps must satisfy some notion of differentiability and must be mutually

compatible (as defined below). We then can define, in an abstract manner, what a manifold

really is.

Figure 2.1: Schematic of a manifold with compatible charts

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DEFINITION:

A differentiable (or C∞ or smooth) manifold is a topological manifold M with a family

U = Uα, φα of coordinate neighbourhoods such that

(i) The Uα cover M

(ii) Any pair (Uα, φα) and (Uβ, φβ) are C∞ compatible. We say that (U, φ) and (V, ψ) are

C∞-compatible if φψ−1 : ψ(U ∩V )→ φ(U ∩V ) and ψ φ−1 : φ(U ∩V )→ ψ(U ∩V )

are diffeomorphisms. If Uα ∩ Uβ = ∅, they are considered trivially compatible.

(iii) Any coordinate neighbourhood (V, ψ) compatible with every Uα, φα belongs to the

family U .

The last condition is a maximality criterion in order to incorporate all possible compatible

charts.

REMARK: As a precursor to the following examples, we define a quotient space. Let

X be a topological space and ∼ an equivalence relation on X. Denote the equivalence class

of x ∈ X by [x] = y ∈ X|y ∼ x. We denote the set of all equivalence classes by X/ ∼,

which is called the quotient space of X relative to the equivalence relation ∼.

EXAMPLES:

We consider some typical examples of manifolds

1. Euclidean space : Rn, open subsets of Rn are manifolds, with global coordinate charts.

2. Unit sphere : The unit n-sphere Sn ⊂ Rn+1. Consider the case n = 2 in 3D space.

The defining equation (x, y, z being coordinates in R3) is x2 + y2 + z2 = 1. If z = 0 is

omitted, then there are charts z = ±√

1− x2 − y2 for x2 + y2 < 1. Similarly, if x 6= 0,

x = ±√

1− y2 − z2 and y 6= 0, y = ±√

1− x2 − z2. These 6 charts cover the sphere

S2. An alternative maximal chart is to consider stereographic projections omitting the

north N and south S poles. One can then cover the 2-sphere by 2 charts.

3. Projective space : RP n is the n-dimensional real projective space, defined as the quo-

tient space of Sn under identification of antipodal points : (x1, · · · , xn+1) ∼ (−x1, · · · ,−xn+1).

It cannot be defined as a subset of Rn+1. The quotient space inherits a differentiable

structure from Sn and can be identified with the space of all lines through the origin in

Rn+1.

4. Grassman manifold : G(k, n) is the manifold of k planes through the origin in Rn(1 ≤

17

k ≤ n). We also note that G(1, n + 1) = RP n. It can be shown that the Grassmann

manifold has dimension k(n− k).

2.2 Abstracting from Rn — continuity and differentia-

bility on manifolds

We know that on a topological manifold, the notion of a continuous map makes sense (because

of the topology itself). However, if we need to define differentiable maps or diffeomorphisms,

we need to be more careful because the notion of differentiability has so far been discussed

only for Euclidean spaces. The general idea, however, is to use the locally Euclidean property

of differentiable manifolds to compose maps and define differentiability on this composition.

This can be done because the coordinate functions themselves are diffeomorphisms. We must

formalize this notion before it can be of use. In what follows, if not mentioned, it is assumed

that M represents a differentiable manifold.

DEFINITION:

As with Rn the only difference between maps and functions on a manifold is in the number

of arguments.

1. f : M → R. is a differentiable function if for each p ∈M , there exists a chart (U, φ)

with p ∈ U so that f φ−1 : φ(U) → R is differentiable. If (V, ψ) is another chart,

then f φ−1 = (f ψ−1) (ψ φ−1) on φ(U ∩ V ) and hence f ψ−1 is differentiable

on U ∩ V .

2. If M and N are differentiable manifolds then F : M → N is a differentiable mapping

if for each p ∈ M and F (p) ∈ N , there exist charts (U, φ) on M with p ∈ U and

(V, ψ) on N with F (p) ∈ V so that ψ F φ−1 : φ(U)→ ψ(V ) is differentiable.

The sequence of maps corresponding to M and N above is illustrated in Fig. 2.2

We now note that the map F : M → N is a diffeomorphism if F is a homeomorphism and

F−1 is C∞. Hence, it also follows that if M and N are diffeomorphic (i.e. there exists a

diffeomorphism F : M → N) then dimM = dimN .

EXAMPLE: F : R → R, F (x) = x3 so that F−1(y) = y1/3. Clearly, F is a differentiable

map, F is also a homeomorphism. However, F−1 is not C∞ and so F is not a diffeomorphism.

18

Figure 2.2: Differentiability of maps

Once the basic idea of composing with coordinate charts is grasped, we can extend most of

the results from Rn to manifolds. As an example, consider the following extension of the

smooth function result (See. Sec. 3).

PROPOSITION 4: Let U be an open subset of M , p ∈ U and f ∈ C∞(U). Then

there exists a neighbourhood V of p, with V ⊂ U and a C∞ function g with f = g on V

and g = 0 outside U .

Proof. The proof is essentially the same as that in Sec.3, only now applied to coordinate

charts composed with U and V .

We can hence extend all the linear algebra results (Sec. 1.2) to manifolds. Assume M and

N are differentiable manifolds. F is a differentiable map. The layout with the corresponding

charts is shown in Fig. 2.2. Intuitively, we define the rank of a map on differentiable manifolds

by converting it to a map between Euclidean spaces by using compositions with the coordinate

charts.

DEFINITION:

The rank of a differentiable map F : M → N between C∞ manifolds at p ∈ M is defined

to be the rank of F = ψ φ−1 at φ(p) i.e. the rank of the matrix D(ψ F φ−1)(φ(p)).

This means that if in local coordinates F (x1, · · · , xm) = (f 1(x1, · · · , xm), · · · , fn(x1, · · · , xm)),

19

then the rank of F at p is the rank of the Jacobian matrix∂f1

∂x1· · · ∂f1

∂xm

......

∂fn

∂x1· · · ∂fn

∂xm

(2.1)

evaluated at p. This definition, it turns out, is independent of the coordinate charts.

REMARK: The Rank theorem can now be easily extended to the case of manifolds. Let

F : M → N and let n = dimN,m = dimM . If the rank of F (p), p ∈ M is k, then

there exist coordinate neighbourhoods U, φ and V, ψ as above such that φ(p) = (0, · · · , 0),

ψ(F (p)) = (0, · · · , 0) and F = ψ F φ−1 is given by F (x1, · · · , xm) = (x1, · · · , xk, · · · ).Moreover we may take φ(U) = Cm

ε (0) and ψ(V ) = Cnε (0) to be the cubes of side ε centered

around the origin in Rn or Rm. It follows from the above remark that the necessary condition

for F to be a diffeomorphism is that dimN = dimM = rankF .

2.3 Immersions and submanifolds

We intuitively what it means for a 2D surface to be imbedded in R3. But there is a formal

definition for this operation — not every map results in an imbedded surface. Immersions

and submersions are weaker versions. The following notions will make these concepts precise.

DEFINITION:

F : M → N is said to be an immersion if rankF = m = dimM . If rankF = n = dimN

then it is a submersion. If F is an injective (1-1) immersion then F (M) = N ⊂ N is a

submanifold of N and F : M → N is a diffeomorphism.

We note that rankF is always ≤ max(m,n). So, for an immersion, m ≤ n while for a

submersion m ≥ n. One can intuitively think of this as being differentiable maps from a

manifold to a higher dimensional manifold and vice versa.

With regards to these three different types of maps between manifolds, it is important to

note the following:

1. An immersion need not be 1-1 globally even if it is locally so. For example, the map

F : R→ R2 given by F (t) = (2 cos(t− π/2), sin 2(t− π/2)) represents a figure of eight.

This maps the points 0,±2π,±4π, · · · to the same point in R2.

2. Even when an immersion is 1-1, it need not be a homeomorphism to its image. For

instance, in the above example, if we use g(t) = π + 2 tan−1 t instead of t on the RHS

20

Figure 2.3: Submanifold property of a subset

(i.e. F (g(t))) the origin in R2 is traversed only once. This maps R to the figure eight,

asymptotically reaching the origin (0, 0) as t → ±∞. Clearly, this is a map taking a

non-compact manifold R to a compact one in R2 and cannot be a diffeomorphism.

Now that we know of these weaker maps, we can define what is meant by an imbedding,

which is exactly what our usual notion of a 2D surface in R3 corresponds to. Additionally,

we can also answer the question — when does a subset of a manifold become a submanifold?

DEFINITION:

Imbeddings and submanifolds:

1. An imbedding is a one-to-one immersion F : M → N which is a homeomorphism

from M → N = F (M) ⊂ N . The image F (M) is an imbedded submanifold

Consqeuently, if F : M → N is an immersion, then each p ∈M has a neighbourhood

U ⊂M such that F |U is an imbedding of U in N .

2. A subset M of a differentiable manifold N has the submanifold property if for

each p ∈ M , there is a coordinate neighbourhood (U, φ) on N with local coordinates

x1, x2, · · ·xn such that

(a) φ(p) = (0, · · · , 0)

(b) φ(U) = Cnε

(c) φ(U ∩M) = Cmε where m ≤ n and φ(U ∩M) = x ∈ Cn

ε |xm+1 = · · · = xn = 0.

This idea of the submanifold property is illustrated in Fig. 2.3. Clearly, m = dimM .

21

If M has the submanifold property, then the inclusion map i : M → N, i(x) = x is an

imbedding (it is a trivial homeomorphism and is a 1-1 immersion too). Conversely, if F :

M → N is an imbedding, then F (M) has the submanifold property. Any subspace M of

N which has the submanifold property hence has the inclusion map as an imbedding and is

called a regular submanifold .

Hence we have two types of submanifolds:

1. Immersed submanifolds, obtained from 1-1 immersions F : M → N .

2. Imbedded or regular submanifolds for which the map is a 1-1 immersion and in addition

a homeomorphism F : M → N ⊂ N where N = F (M).

As a special case, if F : M → N is a 1-1 immersion and M is compact, then F is an imbedding

and N = F (M) ⊂ N is a regular submanifold. Thus, a general result is that a submanifold

of N , if compact, is regular. Such is the case for spheres, ellipsoids and other similar surfaces

in R3.

Finally, it often occurs that many common manifolds occur as examples of submanifolds and

so we state the following result without proof (See II.5 of Boothby [Boo86])

PROPOSITION 5: Let M,N be differentiable manifolds of dimensions m,n and

F : M → N a C∞ mapping. If F has constant rank k on M and q ∈ F (M). Then

F−1(q) is a closed regular submanifold of M of dimension m− k.

This is a consequence of the theorem on rank and the implicit function theorem, as generalized

to manifolds above.

EXAMPLE: Consider the map F : Rn+1 → R given by F (x1, · · · , xn+1) = (x1)2 + · · · +(xn+1)2. The gradient ∇F = (2x1, · · · , 2xn+1) and is non-zero except at the origin. So

F : Rn+1 − 0 → R has constant rank 1. F−1(1) is the unit sphere (x1)2 + · · · (xn+1)2 = 1,

which is a regular submanifold of Rn+1.

Such level–set type definitions are common ways to define many manifolds (such as the

sphere, as we have just seen). The above proposition hence tells us that it is possible to

obtain manifolds by algebraic means.

22

2.4 Tangent space of a differentiable manifold

It must be noticed from the preceeding discussion that we have not spoken about vectors,

lengths or angles at all. We have, so far in our study of manifolds, restricted ourselves to maps

between manifolds and their properties (continuity, differentiability, rank). Having logically

generalized the idea of surfaces to define manifolds, we now define vectors on manifolds.

The idea of vectors ‘sharing their abode’ with the space itself is an artifact of Euclidean

geometry. For instance in 2D, what do we mean when we talk of vectors on a 2-sphere? We

imply that the vectors are on the surface, i.e. they are tangential to it, when viewed embedded

in 3D Euclidean space. Hence, just as in the case of manifolds, we need an intrinsic definition

of something being tangent to a surface. This leads to the concept of a tangent space, which

is the fundamental structure housing all vectors on the surface.

Take a differentiable manifold M and a point p ∈M . If U is an open set containing p, then U

is itself a differentiable manifold and C∞(U) has been defined. Consequently, C∞(p) germs

at p are defined.

DEFINITION:

The tangent space TpM of a manifold M consists of all mappings Xp : C∞(p) → R such

that:

1. Xp(αf + βg) = αXp(f) + βXp(g) for α, β ∈ R

2. Xp(fg) = f(p)Xp(g) + g(p)Xp(f)

The first property is an expression for linearity. The second one is an abstraction of the

Liebniz rule for ordinary differentiation.

From our discussion of the tangent space of Rn, we found that if we took the set of all C∞

linear functions, it forms an infinite dimensional vector space. However, restricting this set

by enforcing the second property makes it finite dimensional and isomorphic to the space of

differential operators. The same situation holds for manifolds too.

PROPOSITION 6: TpM is a vector space with the following vector space operations:

1. (Xp + Yp)(f) = Xp(f) + Yp(f)

2. (αXp)(f) = α(Xpf) for α ∈ R.

Proof. The conditions needed for vector space can easily be checked, by applying the

definition of Xp or Yp, i.e. that they are linear operators and obey the Liebniz rule.

23

We have so far studied the properties of maps between manifolds. Suppose F : M → N is a

C∞ map. We can compose maps between manifolds — let g be a function from N to some

other manifold V . This can be pulled back to M by composing it with F i.e. F ∗g = gF . This

construction respects equivalence classes of germs. The pullback F ∗ : C∞(F (p))→ C∞(p) is

defined at each point p ∈ M . Using this composition, we can define a mapping between the

tangent spaces of M and N induced by F , given by F∗ : TpM → TF (p)N . If V ∈ TpM then

F∗V ∈ TF (p)N is defined by (F∗V )(g) = V (F ∗g) = V (g F ) where g ∈ C∞(F (p)).

Other notations for the induced map F∗ include F ′, DF, dF and is often referred to as the

differential map. If F is only a C∞ map betwen M,N then F∗ is a vector space homomor-

phism (respects vector addition and scalar multiplication but is not necessarily 1-1 and onto).

Also, (G F )∗ = G∗ F∗.

When the map F is a diffeomorphism (as opposed to be merely smooth), the differential

map F∗ has an important property:

PROPOSITION 7: If F : M → N is a diffeomorphism between differentiable

manifolds M,N then F∗ : TpM → TF (p)N is a vector space isomorphism.

Proof. To prove this proposition, consider the composite map F−1 F and its induced

map. This should have a maximum rank. In particular if (U, φ) is a coordinate chart,

then φ∗ : TpM → TpRn is an isomorphism. Consequently, dimTpM = dimTφ(p)Rn =

n.

This result hence leads to an important property for coordinate charts on manifolds. Since

each coordinate chart φ on a manifold is a diffeomorphism, we have an isomorphism between

the tangent space TpM at a point p ∈ M and the corresponding tangent space at its image

point in the coordinate chart. The latter has a natural basis, so this isomorphism gives us a

natural basis for the tangent space of a manifold.

COROLLARY: To each coordinate neighbourhood U of M , there corresponds a nat-

ural basis E1p, · · · , Enp of TpM for every p ∈ U . If f is a function defined in a neigh-

bourhood of p, and its coordinate expression is f = f φ−1 relative to (U, φ). Then

Eipf = (φ−1∗ (∂/∂xi))(f) = ∂/∂xi(f φ−1) = ∂/∂xi(f).

It is clear from above that if xi(p) is a coordinate function on U , the components of Xp in

this coordinate chart is given by Xp =∑

i(Xpxi)Epi.

Going back to the more general case of a smooth map between manifolds, consider a smooth

24

map F : M → N with U, φ, V, ψ coordinate neighbourhoods on M,N respectively and

F (U) ⊂ V . If in these local coordinates yi = f i(x1, · · · , xm), given m = dimM,n = dimN ,

1 ≤ i ≤ n, then the bases transform according to the following rule

F∗(Eip) =

(∂yj

∂xi

)EjF (p) (2.2)

where as before Eip = φ−1∗ (∂/∂xi) and EjF (p) = ψ−1

∗ (∂/∂yj) are the bases at p and F (p) on

M and N respectively, defined using the local charts.

Even more specifically, if we have a vector Xp = αiEip, then under the map F∗, it maps

into F∗(Xp) = βjEjF (p) where the components βj, αi are related by the Jacobian matrix :

αi = (∂yi/∂xj)βj. It must be remarked that vectors are often defined precisely by these

transformation rules (for the particular case when F is a diffeomorphism). Now we have

derived these transformation rules as a consequence of our systematic construction.

A final note on the rank of the smooth map F : M → N is needed at this junction. The

rank of F at p ∈M is exactly the dimension 1 of the image F∗(TpM). Additionally, F∗ is an

isomorphism into if and only if the rank is equal to dimM and is onto if it is equal to dimN .

EXAMPLES:

1. Using the above formalism, we can make sense of TpM as a collection of tangent vectors

to curves2. Consider a curve c : I → M where I = (a, b) ⊂ R. Since I as defined is a

1D manifold, it has only 1 basis vector. Consider a point t0 ∈ I, the basis vector at this

point is (d/dt)t0 . Under the map c, it becomes c∗(d/dt) at p = c(t0) ∈ M . Using the

definition of the induced map, the vector in M has components given by c∗(d/dt)(xi)

if xi is a coordinate chart on M . So by Eq. (2.2) we can argue that under F∗, d/dt

becomes3

F∗

(d

dt

)xi =

dxi(F (t))

dtEip

which coincides with the conventional definition. This notion of a tangent vector to a

curve is made more intuitive when we consider vector fields and 1-parameter groups on

1recall the definition of rank of a map given earlier, it is the rank of the matrix DF , which is nothing butF∗.

2The idea that all vectors are tangent to some curves on the manifold M is often used as a basis fordefining TpM .

3Note that in Eq. (2.2) F was just a C∞ map not a diffeomorphism, so it is applicable in this case.

25

a manifold.

2. Consider a surface M , which is a submanifold of R3. Let a rectangle W ⊂ R2, be

parameterized by coordinates (u, v). Take an imbedding θ such that, θ(W ) ⊂M . Now

θ−1 becomes a coordinate map (M being 2-dimensional). Using the general coordinates

(x, y, z) on R3, we have

x = f(u, v) y = g(u, v) z = h(u, v)

Since θ is an imbedding, the Jacobian ∂(f, g, h)/∂(u, v) has rank 2 at each point on W .

We might ask now what happens to ∂/∂u and ∂/∂v, which form a basis for R2 and

hence W , under the map θ. Define

(Xu)0 = θ∗

(∂

∂u

)=∂x

∂u

∂

∂x+∂y

∂u

∂

∂y+∂z

∂u

∂

∂z

(Xv)0 = θ∗

(∂

∂v

)=∂x

∂v

∂

∂x+∂y

∂v

∂

∂y+∂z

∂v

∂

∂z

The vectors Xu and Xv span a 2-dimensional subspace of R3, and locally they are vector

fields on M . Their components are given by the corresponding entries in the Jacobian

of the parameterization of the surface.

2.5 Lie groups and their actions

There exist some special manifolds, whose constituent points also obey group properties.

Practically, they arise as transformation groups of other manifolds. Let us recall that to

define a manifold, we (loosely speaking) need some parametrization. The minimum number

of parameters needed defines the dimension of the manifold. For instance, if we parametrize

the orientation of a rigid body in 3D space, we would need 3 parameters (Euler angles) to

completely specify its orientation. These parameters obey the properties of a group and each

of them is a real number. Thus, the set of all such orientations forms a continuous or Lie

group.

This notion can be generalized and formalized as follows.

DEFINITION:

Suppose G is both a differentiable manifold and a group. We say that G is a Lie group if

the group operations are C∞ mappings, i.e. x→ x−1, (x, y)→ xy are C∞. (x, y) ∈ G×Gwhich is also a differentiable manifold.

26

This definition ensures compatibility between the group structure and the differentiable stru-

ture of the manifold. Being groups, Lie groups have operations of left and right translations

which are nothing but group multiplications by a group element.

EXAMPLES:

Let us first consider some common examples of Lie groups

1. GL(n,R) — the general linear group, consists of n× n matrices A with real entries. If

A ∈ GL(n,R) then detA 6= 0. GL(n,R) ⊂ Rn2is an open subset of a Euclidean space

2. C∗ = C − 0 consisting of the complex plane without the origin is an open subset

of R2. C∗ is a group with respect to multiplication of complex numbers. Consider S1

consisting of all the points |z| = 1, z = eiθ. S1 forms a Lie group, a subgroup of C∗.

Since a Lie group is also a manifold, what about its submanifolds? Are they Lie groups too?

Suppose H is a subgroup of a Lie group G. If H is an imbedded submanifold, then H is a

Lie group.

So if we have a subgroup which by itself is a regular or imbedded submanifold, we can

generate more Lie groups. In particular, we can use the result of Prop. 5 to generate regular

submanifolds if we can define suitable maps of constant rank.

EXAMPLE: SL(n,R) = X ∈ GL(n,R)| detX = 1 is a subgroup and a submanifold.

Consider the map F : GL(n,R)→ R−0 by F (X) = detX. F is a map from GL(n,R)→GL(1,R) and can be shown to have constant rank. This is done as follows:

F (AX) = det(AX) = det(A) det(X) = det(A)F (X) for A ∈ GL(n,R). So if we denote the

left-translation of X by A by AX = LAX, this becomes (F LA)(X) = (LdetA F )(X) =

(detA)F (X).

Taking the differential map, DF DLA(X) = detADF (X). So,

rank(DF (AX)) = rank(DF ).DLA(X) = rank(DF (X))

Hence F has constant rank = 1. So by Prop. 5, SL(n,R) = F−1(1) is a regular submanifold

of GL(n,R).

A map between two (general) groups F : G1 → G2 is called a homomorphism if it respects the

product rule. An extended notion exists for Lie groups, which in addition have differentiable

structure. Let F : G1 → G2 be a (general group) homomorphism between Lie groups G1 and

27

G2. If F is also a C∞ mapping, it is called a Lie group homomorphism

Historically, groups arose as sets of 1-1 permutations of objects of a set X within themselves.

Much later on, there was an association between groups and geometry — certain groups

consisted of transformations (‘permutations’ of the points) in a certain geometry obeying a

certain constraint. From a knowledge of these transformation groups, the properties of the

underlying geometry could be obtained.

EXAMPLE: For example, if G is the group of Euclidean geometry E then T ∈ G only

if the distances (metric) d(x, y) = d(Tx, Ty)∀x, y ∈ E. In this example, the group has a

‘representation’ which allows us to multiply group elements with elements of the underlying

geometry (such as using matrix multiplication by representing points x, y as vectors and so

on). So for instance, we could say if two figures in a plane are congruent — they are if there

is a σ ∈ G which takes one figure to the other.

This primitive notion of a Lie group (T ∈ G in the example above) acting on a manifold

(Rn or E in the example) is first generalized to the concept of group actions on sets (and

then Lie group actions on manifolds).

DEFINITION:

Let G be a group and X a set. G acts on X to the left if there is a mapping θ : G×X → X

satisfying:

1. θ(e, x) = x for the identity e ∈ G, ∀x ∈ X

2. If g1, g2 ∈ G then θ(g1, θ(g2, x)) = θ(g1g2, x) ∀x ∈ X.

the mapping θ is the action of the Lie group G on the manifold M .

When G is a topological group, X is a topological space and θ is continuous. When G is a

Lie group and X is a C∞ manifold, θ is a C∞ action of G on X. Exactly analogous to the

left action, we can define a right action too with corresponding conditions θ(x, e) = x and

θ(θ(x, g1), g2) = θ(x, g1g2).

Let us take S(X) to be the set of all permutations of X. The notion of permutations is

clear if X is discrete; for the continuous case, the permutations are provided by a fixed g

such that θ(g, .) ≡ θg : X → X ‘permutes’ the elements of X. Then the map θg : X → X is

obtained from a fixed g. This g → θg is a homomorphism of G into S(X). Conversely, any

28

such homomorphism determines θg ≡ θ(g, .).

Three important properties of actions are:

1. Effective — A group G has a homomorphism taking g ∈ G to θg : M → M . If this

homomorphism is injective, then the action θ is effective. If G is a Lie group, then

the action θ is a homeomorphism. Stated differently, an action θ is effective if the unit

element e ∈ G is unique in determining the trivial action, i.e. if θ(g, p) = p, ∀p ∈ Mthen g = e

2. Transitive — An action θ is transitive if for some x ∈M , θg(x) covers all x ∈M∀g ∈ G.

In other words, for any p1, p2 ∈M , ∃g ∈ G such that θ(g, p1) = p2

3. Free — If the action is free, then the following property holds : If ∃g ∈ G : gx = x for

x ∈M , then g = e.

Additional terms describing the action θ are the orbit of x ∈M , which is the set of all points

y ∈M such that y = θ(g, x)∀g ∈ G. If θ(g, x) = x∀g ∈ G, then x is a fixed point of G.

If G and X are as above, the isotropy group Gx is defined by Gx = g ∈ G|θ(g, x) = x.Sometimes the action θ(g, x) is abbreviated by θ(g, x) = gx. For a free group action, the

isotropy group just contains the identity e.

EXAMPLE: A natural example of Lie group actions on manifolds is the action of GL(n,R)

on Rn. Let A ∈ GL(n,R) and x ∈ Rn we can define θ(A, x) = Ax via matrix multiplication

in Rn. This is a C∞ action as the requirements can be checked easily.

To further understand why Lie groups are considered groups of transformations of a manifold,

we can consider an alternate view of a Lie group action on a manifold (p. 42 of Ref. [DFN85]):

We can say that a Lie group G is represented as a group of transformations of the manifold

M (or has a Lie action on M), if there is associated with each g ∈ G a diffeomorphism

θg : M → M given by x ∈ M → θg(x) ∈ M such that θgh = θgθh for all g, h ∈ G4. In this

context, one can see for example, that the group GL(n,Rn) acts on Euclidean space Rn in

an intuitive way, i.e. via matrix multiplication. This is an example of a natural action of the

Lie group GL(n,Rn) on Rn. Note that in this case, it is actually acting on a local copy of

Rn at each point x which defines the tangent space TxRn.

4Recall from the theory of finite groups, that a representation of a group is a linear operator on some vectorspace. Specifically, an n-dimensional representation of a group G is a set of linear operators, correspondingto the group elements and not necessarily one-to-one, acting on vectors in some n-dimensional vector space.This view of the Lie group action is exactly analogous in that it is a map that takes x ∈M to x′ ∈M .

29

We conclude this section by noting that the action of a Lie group is not apriori unique. By

its definition, it only has to satisfy the two properties outlined in the definition above. The

reason that one speaks of natural actions is that often these actions can be represented in a

canonical manner. For instance, we will see later that the action of the Lie group R can be

represented in some local coordinate system as a translation of points on the manifold.

30

3 Smooth vector fields

Going from vectors defined at a single point to vectors at each point in a manifold is a non-

trivial step. In fact the possiblilty of doing is also linked to the topology of the manifold.

However, once we know that vector fields can be defined globally on a manifold, there are an

infinite variety of vector fields, each differing in description. Finally, these are very important

as they result in a natural Lie group action on the manifold.

3.1 Vector fields and one parameter groups

We can define a vector field x formally as assigning a tangent vector Xp ∈ TpM to every

p ∈ M . This is hence a function from M → TM where TM =⋃p∈M TpM is the tangent

bundle of M .

DEFINITION:

If M is a differentiable manifold, with p ∈ M,TpM being the tangent space, a vector field

is a function X : p → Xp ∈ TpM so that for each f ∈ C∞(M), (Xf)(p) = Xpf is a C∞

function on M .

EXAMPLES:

1. Given a manifold M and coordinate charts U, φ containing p ∈M , the natural basis at

TpM is φ−1∗ (∂/∂xi). These have components αi ≡ δji , which are trivially C∞ functions.

So the coordinate basis vectors form a set of C∞ vector fields on M . This is often called

the coordinate frame field. Similarly a set of k vector fields is often referred to as a

field of k-frames on M . If k = dimM , then they form a basis at each point. We must

remember that it might not always be possible to find such global fields on M .

2. On S2 we cannot find even a single global C∞ vector field. Intuitively, one can think of

this result as follows. Let a vector field have tangent vectors lying on the latitudes of

31

Figure 3.1: Examples of maps where F∗(X) does not lead to a vector field. Top row showsF which is not 1-1 and the bottom is an example of F which is not onto.

S2, then naturally the poles will be points where the latitudes are not defined. Hence

there are points at which the vector field is not defined.

3. A general version of the above result is that on even dimensional spheres, we cannot find

a single continuous C∞ vector field. On odd dimensional spheres, we can find atleast

one. Yet another intriguing result is that we can find a basis of vector fields1 only for

S1, S3, S7. Manifolds on which we can find globally defined coordinate frame fields are

said to be parallelizable2

Now suppose we have two manifolds M,N and a C∞ map F between them, F : M → N . A

natural question to ask is what happens to a vector field X on M under F∗ : TpM → FF (p)N .

In general, if X is a vector field on M , then F∗(X) is not well defined as a vector field, even

though F∗ exists at each point on the manifold. This is because either F need not be 1-1 or

it may not be onto. Two examples are illustrated in Fig. 3.1 The figure on the left shows

an example where F is not 1-1 and the resulting F∗(X) is not uniquely determined at the

node point. The figure on the right shows an example where F is not onto, thus there is no

natural extension of the immersion to define a global vector field on R2.

If, however, on the other hand, F is 1-1 and onto, then we can define vector fields related by

F as follows.

1By this we mean that a coordinate frame field can be determined globally on the manifold.2A more trivial example of a parallelizable manifold is a coordinate neighbourhood, which being an open

set is a submanifold. By definition of the coordinate charts, it contains a global coordinate frame field.

32

DEFINITION:

The notion of F -related vector fields can be defined as follows:

1. If X and Y are vector fields on M and N and F : M → N is a C∞ map, then X

and Y are F -related if F∗(Xp) = YF (p) for each p ∈M .

2. If F : M → N is a diffeomorphism, then for each vector field X on M , there is a

unique F -related vector field Y on N .

3. Furthermore, if F : M →M is a diffeomorphism, and X is a C∞ vector field on M

such that F∗(X) = X, then X is said to be invariant with respect to F .

The notion of F -relatedness finds particular use in the theory of Lie groups. As we’ve ob-

served earlier, left translation Lg on a Lie group G is a diffeomorphism Lg : G → G. If we

have a vector field X on G, and g ∈ G, then (Lg)∗X = X implies that the vector field is left

invariant. With a little more imaginiation, we can argue that each such vector field can be

determined by its value at TeG, and by performing left translation, using the map (Lg)∗ for

all g ∈ G. Conversely, given Xe ∈ TeG, we can determine a unique left-invariant vector field

on G by repeatedly applying left translation.

Given these notions of vector fields, we can now appreciate the role of actions of 1-parameter

groups. Recalling the definition of the action of a Lie group, we say that any such group

action ‘generates’ a vector field on M . This means, infinitesimally, the image of the action

(the orbit) is composed of uniquely determined tangent vectors. This intimate relationship is

reflected in the fact 1-parameter group actions determine curves on M and vectors determine

tangents to these curves. This definition might seem vague, but we can make it precise in

the following.

Recall that an action of R on M is a C∞ map θ : R×M →M which satisfies:

1. θ0(p) = p for all p ∈M

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

For each t ∈ R, θt is a diffeomorphism, so a vector Vt is naturally extended to a unique Vt+∆t

by (θt)∗.

DEFINITION:

33

1. The infinitesimal generator of a 1-parameter group, via the action θt is the vector

field X defined at each point p ∈M by

Xpf = lim∆t→0

1

∆t[f(θ∆t(p))− f(p)] =

d

dt(f θt(p))|t=0 (3.1)

2. If θ : G ×M → M is the action of a group G on M , then the vector field X on M

is said to be G-invariant if (θg)∗X = X for all the diffeomorphisms θg∀g ∈ G.

This definition is defined only locally near t = 0. So the infinitesimal generator is defined

only at a point p ∈M . This also applies to the 1-parameter group action, as we shall see at

the end of this section.

PROPOSITION 8:

1. If Xp = 0 for some p, then for each q in the orbit of p under the 1-parameter group

action θ, Xq = 0.

2. The orbit of p, under the action θ is either a single point p or an immersion of Rin M by the map t→ θt(p), depending on whether or not Xp = 0.

Proof. It can be easily seen from the above definitions that the infinitesimal generators

of the 1-parameter group action θ are invariant under θ, i.e. (θt)∗Xp = Xθt(p). The two

results above follow from this relation.

This now brings us to a very important notion concerning curves. We first note that t→ θt(p)

is a map from R→M . So for a fixed p, we can take varying values of t and obtain a curve.

This is exactly analogous to the definition of solutions of ordinary differential equations,

wherein we need a map (which determines the derivative or the right hand side) and an

initial condition to fully specify the solution. If we keep the initial condition constant, and

vary t, we traverse a particular solution curve3. We denote this variation by F (t), i.e. p now

plays the role of a starting point.

Having seen that θt(p) : R→M defines an immersion of R in M , we can now make sense of

why 1-parameter groups determine curves on manifolds. These curves are the images of the

immersion and form a submanifold locally at each point of M . Using our notions of regular

submanifolds, we can say then that non-intersecting curves form regular submanifolds of M ,

of dimension 1.

3This particular curve is nothing but the orbit of p under θ.

34

So what does it mean to define a tangent vector? In the framework above, a tangent vector

then naturally becomes the image under F∗ of the only basis vector of R, considered at the

point t0. But we’ve already defined the tangent vector infinitesimally as the generator of the

1-parameter group action. So this then leads to the relation (denote by F (t) the image of Rin M for a fixed p ∈M , i.e. F (t) ≡ θt(p)).

In essence, all we’re doing when we use the action θt to act on the fixed p ∈ M , is following

the vectors at each point Xp (constituting a vector field X) to reach an adjacent point. Two

things are clear from this — that the action θt is hence determined by a particular vector

field, and that we can hence have an infinite number of actions θt of R on M . When defining

a Lie group action in an abstract manner, this latter fact is not always clear.

F∗

(d

dt

)= Xθt0 (p) = XF (t0)

Alternative ways of denoting the LHS of the above equation are (dF/dt)t0 and F (t0). If we

had a different parameterization, say t = f(s) so that G(s) = F (f(s)), then

G∗

(d

ds

)= F∗ f∗

(d

ds

)= F∗

(dt

ds

d

dt

)where the last relation is again obtained by the Jacobian expression.

Given a vector field X on M , we say that a map F (t) : t → F (t) ⊂ M , defined (in the

sense above) on an open interval I ⊂ R, is an integral curve of X if F∗(d/dt) = XF (t).

The idea of global group actions is fairly restrictive for most purposes. Like everything else,

these notions always make sense locally, which motivates the following definition

DEFINITION:

A local 1-parameter group action or flow on a manifold M is a C∞ map θ : W ⊂ R×M →M which satisfies the conditions:

1. θ0(p) = p for all p ∈M

2. If (s, p) ∈ W , there exists a suitable interval where θt+s(p) is defined and θs θt(p) =

θt+s(p).

It might be recognized that the definitions above are essentially dealing with solutions of

ordinary differential equations. Naturally, we can then ask when do solutions to an ordinary

differential equation exist?

35

PROPOSITION 9: Given a vector field X and p ∈M , there exists a unique integral

curve satisfying F (0) = p and defined for |t| < ε for some ε > 0

Proof. Choose local coordinates xi centered at p. In these coordinatesXq =∑n

i=1 ξi(q)Eqi,

equivalently written with the induced map on the coordinate chart supressed as Xq =∑ni=1 ξ

i(q) ∂∂xi

.

If in local coordinates, F (t) = (x1(t), · · · , xn(t)), then the above relation between

F∗(d/dt) and X gives us

F∗

(d

dt

)xi ≡ dxi(t)

dt= XF (t)x

i = ξi(q) ≡ ξi(x)

In order to have a unique integral, defined by unique xi(t) for a given ξi(x), the above

equation must have a unique solution in the neighbourhood of p. This is guaranteed by

the existence and uniqueness properties of ordinary differential equations.

So locally, near a point, for a non-zero vector field X, we can find a unique integral curve to

traverse along. We can string such local integral curves together to obtain a curve defined

for a suitably long time.

A related question then, is one fundamental to vector fields — if one has a vector field on a

manifold, is the resulting integral curve unique?

PROPOSITION 10: Every vector field generates a unique local one parameter group

θt satisfying θt+s(p) = θt(θs(p)) for |s|, |t| < ε for some ε > 0.

Proof. Define θs(p) to be the integral curve of X starting at p. Note that θs+t(p) and

θs(θt(p)) both are integral curves of X that start at the point θt(p). If we regard t as

fixed and s as the parameter of the integral curve, then

dθs+t(p)

ds≡ dθu(p)

du

du

ds|u=s+t =

dθu(p)

du= Xθs+t(p)

At s = 0, θs+t(p) = θt(p). By uniqueness of the previous proposition, θs+t(p) = θs(θt(p))

is a unique 1-parameter curve if |s| < ε and |t| < ε.

Finally, we end this section by mentioning a very important result. As we’ve seen, there

are infinity of one–parameter group actions on any manifold. However, all of them can be

reduced to a reduced form.

36

Local translation theorem

Consider a 1-parameter group θt, and its generating vector field X. If Xp 6= 0, then

there exist local coordinates (y1, y2, · · · , yn) centered at p so that

θt(y1, y2, · · · , yn) = (y1 + t, y2, · · · , yn)

for |t| < ε, for some suitable ε > 0.

So the action of the Lie group R on a manifold M can be viewed as a translation of any 1

coordinate by a real number, the others remaining constant.

3.2 Applications, relation to subgroups of Lie groups

The idea of a 1-parameter group of transformations, i.e. an action of R on a manifold

M occurs naturally in the theory of ordinary differential equations. As we’ve seen in the

previous section, an ordinary differential equation can be thought of as the expression, in

local coordinates, of the induced map F (t)∗ from Tt0R→ TF (t)M . Also, as per the results at

the end of the previous section, we know that a vector field either generates a unique local

1-parameter group, or it is 0. This implies that we cannot generate an orbit from a point

Xp = 0 by the action of a 1-parameter group. Such a point p ∈ M where Xp = 0 is called a

singular point (or a singularity) of the vector field X. If Xq 6= 0 then q is a regular point.

Figure 3.2: Some typical vector fields in the vicinity of a singular point (in 2D).

37

In the neighbourhood of a regular point, the integral curves are families of parallel curves in

Rn — cf. the local translation theorem. However, near an isolated singularity this is not true

and the vector field pattern can take on many forms. Singularities of vector fields abound

in the theory of dynamical systems, where the behaviour of an ODE is often visualized as a

motion in phase space.

Some visual examples of singular points are shown in Fig. 3.2, for 2D manifolds. This appears

often, in the theory of nonlinear second order differential equations. Interestingly, these vector

fields in the vicinity of a singular point have a property that reflects the ‘strength’ of the

singularity.

DEFINITION:

For M of dimension 2, the index ip of a vector field X at an isolated singular point p is

defined as follows. Let C be any curve surrounding p and no other singular point. Let θq

denote the angle between Xq and a fixed direction. The change in angle θq as q traverses

C is then an integral multiple 2πip, with sign depending on the direction of angle change.

When M is a compact manifold (unlike the case of R2), the index ip is related to the Euler

characteristic of M , and is so independent of C and X! A consequence of this relation is that

any vector field on S2 must have atleast 1 singular point, a fact that we’ve already noted

when talking about vector fields on S2.

Generalizations of this result to manifolds of dimension greater than 2, but still compact,

need the notion of the degree of a map and can be found in Refs. [DFN85], p.102 and [Arn73],

Sec.36.7.

Finally, we must understand that vector fields that define global 1-parameter groups are

unique. A vector field X is termed complete if it generates a global 1-parameter group

θ : R×M →M .

PROPOSITION 11: If M is compact then every vector field is complete.

Proof. Let F (t) be a maximal integral curve starting at p = F (0). Suppose F (t) is only

defined for −a < t < b where b <∞. Let bi be an increasing sequence bi < b and bi → b

as i→∞.

By compactness of M , after passing a subsequence, we may assume F (bi) → q ∈ M as

i → ∞. By uniform dependence on parameters in existence and uniqueness of ODEs,

∃ ε > 0 and a neighbourhood U of q and θtq is defined for all q ∈ U and |t| < ε.

So for i > i0, |bi − b| < ε2. Now we can extend F (t) by setting F (t) = θt−bi(F (bi)) for

t < bi + ε and keep going, but this contradicts b < ∞. Hence, every vector field on a

38

compact manifold M is complete.

It can be shown that a left invariant vector field X on a Lie group G is complete (see Theorem

IV.5.7 in Boothby [Boo86]).

The action of Lie groups on manifolds has been defined earlier. It turns out that 1-parameter

groups can be considered as subgroups of Lie groups, and in the process we can naturally

relate them to the corresponding Lie algebras. This will be the motivation in the remainder

of this section.

DEFINITION:

If R is the additive group of real numbers, considered as a Lie group and G is an arbitrary

Lie group, a 1-parameter subgroup H of G is the homomorphic image H = F (R) of a

homomorphism F : R→ G.

This is a sort of ‘embedding’ of the Lie group R in G via the homomorphism F . Conversely,

if a Lie group G acts on a manifold M via an action φ : G ×M → M along with the ho-

momorphism F , then it automatically defines an action θ : R× →M by θ(t, p) = φ(F (t), p).

Hence for this homomorphism and action pair, we have an associated local generator, integral

curves etc.

EXAMPLES:

1. Let G be GL(n,R) and φ its action on R3 by matrix multiplication. Consider the

homomorphism F given by the matrix

F (t) =

eat 0 0

0 ebt 0

0 0 ect

This is a homomorphism because when acted on t + s the exponential arguments add

up too. Additionally, the action of φ : G × R3 → R3 is just matrix multiplication. It

has the infinitesimal generator X = ax1 ∂∂x1

+ bx2 ∂∂x2

+ cx3 ∂∂x3

, which can be obtained at

once from the definition in Eq. 3.1. The integral curves are radial lines from the origin

because X is nothing but the radius vector from the origin.

2. Let the Lie group G = SO(3) and again take the matrix action φ : SO(3) × R3 → R3,

39

and restrict it to the sphere S2. The homomorphism F defined by

F (t) =

cos at sin at 0

− sin at cos at 0

0 0 1

leads to a subgroup of SO(3). The corresponding action θ(t, x1, x2, x3) consists of all ro-

tations about the x3 axis — a vector (x1, x2, x3) is taken to (x1 cos at+x2 sin at,−x1 sin at+

x2 cos at, x3). Again, using the definition of a generator (Eq. 3.1), it is found to be nor-

mal to the radial line, parallel to the (x1, x2) plane, except for the poles where the vector

fields are 0. The integral curves (and orbits) of this action are hence lines of latitude.

Just as we have used left–translations for a Lie group, one can also define right translations.

A Lie group G can act on itself on the right by right translations. If G isn’t Abelian, then

the two are different.

DEFINITION:

Let F : R → G be a one parameter subgroup of the Lie group G and X the left–invariant

vector field on G defined by Xe = F (0). Then θ(t, g) = RF (t)(g) defines an action θ :

R × G → G of R on G with X as infinitesimal generator. Conversely, if X were a left–

invariant vector field and θ : R × G → G the corresponding action, then F (t) = θ(t, e) is

a one parameter subgroup of G and θ(t, g) = RF (t)(g).

The above is actually a result that can be proved from the definitions of left invariant vector

fields and right translation (see Ref. [Boo86], IV.5.13 for a proof). The idea behind this

result is that left and right translations are universally equal only at the identity e. If we

take a left–invariant vector field’s value at e, we can continue it by right translation using

F (t) which then generates a subgroup of G. There is a direct one-to-one correspondence

between the elements of Te(G) and 1-parameter groups of G.

3.3 Lie algebra and Lie derivative of vector fields

DEFINITION:

Consider any vector space V over R. We say that it is a Lie algebra if in addition to its

vector space properties, it possesses a product V × V → V taking X, Y ∈ V to [X, Y ] ∈ Vwhich has the properties

1. [., .] is bilinear in R

40

2. It is anti–commutative: [X, Y ] = −[Y,X]

3. It satisfies the Jacobi identity: [X, [Y, Z]] + [Y, [Z,X]] + [Z, [X, Y ]] = 0

In going from a general vector space to the tangent space of a manifold, we note the following.

Let X, Y be C∞ vector fields on M , i.e. X, Y ∈ X(M). Then X, Y : C∞(M) → C∞(M)

and XY : C∞(M)→ C∞(M) but XY /∈ X(M). However, Zp = (XY − Y X)p ∈ TpM and so

Z ∈ X(M). This can be seen by applying Zp to f, g at p ∈ M and seeing if the Leibniz rule

holds.

(XY − Y X)p(fg) = (X(Y (fg)))p − (Y (X(fg)))p

= (X(fY (g) + gY (f)))p − (Y (gX(f) + fX(g)))p

= f(p)(X(Y (g)))p +Xp(f)Yp(g) + g(p)(X(Y (f)))p +Xp(g)Yp(f)

− g(p)(Y (X(f)))p − Yp(g)Xp(f)− f(p)(Y (X(g)))p − Yp(f)Xp(g)

= f(p) (XY (g)− Y X(g))p + g(p) (XY (f)− Y X(f))p

Hence X(M) with [X, Y ] = XY − Y X forms a Lie algebra. The properties above can be

easily verified for this definition.

PROPOSITION 12: Suppose F : M → N is a differentiable map. Let X1, X2 be

vector fields on M which are F -related to Y1, Y2 on N . Then [X1, X2] is F -related to

[Y1, Y2].

Proof. This follows from the definitions of a Lie algebra and F -relatedness of vector

fields.

We have so far developed the notion of the directional derivative of a function f as Xpf . Now

we want to determine the ‘rate of change’ of one vector field along another. For instance,

suppose we wanted to know how the earth’s magnetic field varied across its surface — we’d

need to evaluate the rate of change of the magnetic field vector along the (say) latitudes of

the earth, which are generated by an entirely different vector field.

DEFINITION:

Suppose X, Y ∈ X(M). LXY denotes the rate of change of Y as we traverse the integral

curves of X, called the Lie derivative of Y with respect to X and is given by

(LXY )p = limt→0

1

t

[θ−t∗(Yθt(p))− Yp

](3.2)

41

the idea being that we can take the field Y at θt(p) ∈ M and ‘bring it back’ to p ∈ M

using the differential map (remember that θt(p) is a diffeomorphism and can be inverted).

Once we have the two vectors in the same tangent space, we can subtract them and evaluate

the change. Notice, however, that LXY needs the specification of a vector field X. Being

dependent on the action θt generated by X, it naturally depends on the values of X in a

neighbourhood of p. Thus it mixes up the variation of Y with the variation of X, thereby

being dependent on both fields and not a property of one in particular 4.

Hadamard’s Lemma

Let X ∈ X(M) and θ : W×M →M with W ⊂ R. Given p ∈M and f ∈ C∞(U), U an

open set containing p, then for δ > 0 and V ⊂ U containing p such that θ(Iδ×V ) ⊂ U ,

there is a C∞ function g(t, q) defined on Iδ × V such that ∀ q ∈ V, t ∈ Tδ,

f(θ(t, q)) = f(q) + tg(t, q) and Xqf = g(0, q)

Using this result, one can prove the following relation :

PROPOSITION 13: For X, Y ∈ X(M),

LXY = [X, Y ]

Proof. See Ref. [Boo86], IV.7.8.

It follows that if θt and σs are 1–parameter groups with generators X and Y respectively,

then [X, Y ] = 0 ⇐⇒ σs θt = θt σs. Furthermore, Using the definition [X, Y ] = XY −Y Xof the Lie bracket, we can show that the coordinate frames Ei have vanishing Lie bracket

[Ei, Ej] = 0.

Coordinate frame fields are also special and deserve some attention. They consist of n inde-

pendent vector fields on a manifold of dimensionM . In order to be a part of the coordinate

frame field on a part of M , a vector field must be derivable from a coordinate. This means it

should be a curve resulting from an action of R on M , i.e. an integral curve, passing through

4The covariant derivative fulfils this latter task.

42

the current point p ∈ M . This is actually a strong restriction on the nature of the vector

field itself.

Let us consider the map φ : Ip ⊂ R → M as a single such coordinate curve, say with pa-

rameter xi. The tangent vector, which is one of the coordinate basis vectors is φ∗(∂∂xi

). If

any two such coordinates xi and xj are chosen independently, the Lie bracket between the

corresponding tangent vectors must vanish.

Finally, the case of Lie groups is very important and recurs in the study of bundles.

DEFINITION:

If G is a Lie group, then the left–invariant vector fields on G form a Lie algebra g with the

product [X, Y ] and dim(g) = dim(G). Further g is isomorphic to Te(G). If F : G1 → G2

is a homomorphism of Lie groups then F∗ : g1 → g2 is a homomorphism of Lie algebras.

We see that if H is a subgroup of a Lie group G, then i∗(h) is a subalgebra of g consisting of

elements of Te(G) that are tangent to H and its cosets gH.5

5If X is a left–invariant vector field tangent to H, then by left–invariance, g∗X = X is tangent to gH.

43

4 Covectors, tensor fields and the ex-

terior algebra

Having defined the tangent space of a manifold M as a finite–dimensional vector space,

we can now talk about the existence of duals or covectors. These form the covector space

and there exist, as one would imagine, covector fields on the manifold. This can be further

extended to include multilinear maps, thereby defining tensors, and tensor fields. A special

algebra is also introduced, which has very important implications when attempting to extend

the theory of integration to manifolds.

4.1 Tangent covectors

First, we recall some standard concepts.

1. For a finite dimensional vector space V , its dual space V ∗ (the space of covectors)

consists of linear functionals σ ∈ V ∗ where σ : V → R, also denoted by 〈v, σ〉 ∈ R for

v ∈ V .

2. If S is a subspace of V , then S⊥ = σ ∈ V ∗|σ(v) = 0 ∀ v ∈ S. S⊥ is called the

annihilator of S.

3. Suppose T∗ : V → W is a linear transformation, then T ∗ : W ∗ → V ∗ is the induced

dual space map and is defined by T ∗(σ)(v) = σ(T∗v), where T∗v ∈ W

4. (RangeT∗)⊥ = Ker(T ∗) are both subspaces of W ∗. Consequently, T ∗ is injective if and

only if T∗ is surjective. Moreover, T∗ is injective if and only if T ∗ is surjective.

For a differentiable manifold M , let V = TpM for p ∈ M and V ∗ = T ∗pM in the above

definitions. The concept of covector fields carries over naturally.

σ : M → ∪p∈MT ∗pM satisfying σ(p) ∈ T ∗pM is a covector field provided σ(X) is a differentiable

function for every smooth vector field X on M . Equivalently, in local coordinates (U, φ),

σ((φ−1)∗

∂∂xi

)must be differentiable on U .

44

DEFINITION:

If f is a C∞ function on M and X a vector field, it defines a C∞–covector field, denoted

by df , by the formula

〈Xp, dfp〉 = Xpf inC∞(M) (4.1)

By the above, it is clear that the coordinate frames, which are vector fields, have a natural

dual basis of coframe fields. For instance, the field ∂∂xi

has a dual basis dxi which returns

dxi( ∂∂xj

) = ∂xi

∂xj= δij. In effect, for a general vector field Xa at a point a ∈ M , dxi(Xa)

determines the ith component of Xa at the point a. This implies that it determines the

change in coordinate of a point a as it moves infinitesimally along Xa, which is the usual

interpretation when such a quantity is introduced in usual calculus treatments.

dx1, · · · , dxn are coframe fields dual to the vector fields ∂∂x1, · · · , ∂

∂xn, and in components, the

general covector field df is given by

df =∂f

∂x1dx1 + · · ·+ ∂f

∂xn(4.2)

DEFINITION:

Consider a differentiable map F : M → N between two manifolds M,N . If p ∈ M ,

then F∗ : TpM → TF (p)N and the induced dual map (called the pullback) is automatically

defined by F ∗ : T ∗F (p)N → T ∗pM by F ∗(σ)(v) = σ(F∗v) for v ∈ TpM .

Further, if σ ∈ T ∗F (p)N is a smooth covector field, then F ∗σ is also a smooth covector field

(on M).

This is actually subtly different from the case of vector fields. For a covector field σ, F ∗σ

is also a covector field. However, for a vector field X, F∗X is, in general, not a vector field

unless F is a diffeomorphism. This is because, locally F∗ might be one–one but if F maps 2

points of M to the same point of N then there will be an ambiguity in F∗ at this point on

N . This then (in a global sense) violates the requirements for vector fields to have unique

vectors at each point on the manifold. However, the covector field does not suffer from

this deficiency because it only uses the local expression of F∗. Already, the importance of

covectors is becoming evident. This will be particularly evident when discussing forms and

the exterior algebra below.

Like in the case of vector fields, we can use the above definition to obtain the ‘transformation

law’ for covector fields as follows. If yi form a coordinate chart on N and ωi ≡ dyi form

45

a basis for T ∗F (p)N with Ejp, EiF (p) a basis for TpM and TF (p)N , then

F ∗(ωi)(Ejp) =n∑k=1

ωi(∂yk

∂xjEkF (p)

)=∂yi

∂xj(4.3)

Likewise, for F : M → N , and σ =∑n

i=1 αiωi ∈ T ∗F (p)N , if F ∗(σ) =

∑mj=1 βjω

j ∈ T ∗pM

with covector bases ωj and ωj on M and N , the components β and α are related by

βj =∑n

i=1∂yi

∂xjαi.

4.2 Bilinear forms, the Riemannian metric

Extending the concept of the dual V ∗ of a vector space V , a bilinear form Φ is a map

Φ : V × V → R which takes Φ(v, w) ∈ R and is linear with respect to both v, w ∈ V . It has

the following properties:

1. If e1, · · · , en is a basis for V , then Φ is completely determined by the matrix Φ(ei, ej)

and conversely.

2. Any bilinear form Φ can be either symmetric, if Φ(v, w) = Φ(w, v) or antisymmetric if

Φ(v, w) = −Φ(w, v) or neither. However, every bilinear form Φ(., .) can be split into a

sum of symmetric and antisymmetric parts.

3. If Φ(v, v) ≥ 0, with equality holding if and only if v = 0, then the bilinear form

is positive–definite. A positive definite bilinear form is said to determine an inner

product 〈., .〉 on V .

The extension to a differentiable manifold M is again done by identifying V ≡ TpM for

p ∈M . Then a bilinear form field is a map Φp : TpM × TpM → R at each p ∈M .

As in the case of covector fields, the pull back of a bilinear form field generates another

bilinear form field. Suppose F : M → N is a differentiable mapping and Φ is a C∞ field of

bilinear forms on N , then F ∗(Φ)p(v, w) = ΦF (p)(F∗v, F∗w) for v, w ∈ TpM . Again, we don’t

need F to be a diffeomorphism for F ∗Φ to be a bilinear form field. Further, if Φ is symmetric

(or antisymmetric) F ∗Φ is also symmetric (or antisymmetric).

DEFINITION:

For a differentiable manifold M , a Riemannian metric is a symmetric, positive definite

bilinear form on M . A Riemannian manifold is a differentiable manifold endowed with a

specific Riemannian metric Φ.

46

This is perhaps the first instance where the usefulness of ‘covariant tensors’ (i.e. multilinear

maps of V to R) becomes evident. The above fact that the pullback of a field of bilinear

forms also results in a field of bilinear forms is useful for ensuring that we can pull back

metrics in a very consistent manner.

PROPOSITION 14: If F : M → Rk, with k > m = dim(M) is an immersion (i.e.

rank(F ) = m, then F ∗Φ is a Riemannian metric on M , Φ being the usual Riemannian

metric on Rk, given by the Euclidean inner product.

Proof. F ∗Φ(v, w) = Φ(F∗v, F∗w) = (F∗v) · (F∗w). So if F ∗Φ(v, v) = 0 then, by the

property of the Euclidean inner product, this implies v = 0. This satisfies all the

properties of the bilinear form outlined above, as can be easily checked.

This result implies that every M immersed in some Rk is endowed with a Riemannian metric,

obtained via the pull back F ∗ of the immersion, from Rk. So, for instance, a surface immersed

in R3 has a Riemannian metric.

Whitney’s embedding theorem

Every differentiable manifold of dimension n can be immersed in R2n.

COROLLARY: Every differentiable manifold admits a Riemannian metric.

Recall the notion of the rank of a map (cf. Secs. 2.2 and 1.2) between spaces — here V × Vand R (for metric fields, V is the tangent space of the manifold). For a bilinear form Φ

on V , if rank(Φ) = dim(V ), then Φ : V × V → R determines a map φ : V → V ∗ i.e.

〈φ(v), w〉 ≡ Φ(v, w), which is hence an isomorphism between V and V ∗. This is often called

musical notation wherebywe can associate vectors with covectors and vice versa. This is also

the same operation as ‘raising’ and ‘lowering’ indices on tensors using the metric tensor, in

traditional tensor calculus.

If (M, g) be a connected Riemannian manifold with Riemannian metric g. Then the covector

X[ associated with a vector X and the vector ω] associated with the covector ω are defined

by

X[(Y ) ≡ g(X, Y ) ω(Y ) ≡ g(ω], Y ) (4.4)

where Y is any vector ∈ TpM .

47

Going back to the idea of induced metrics, we can now clearly understand what the no-

tion of arc length exactly means. For a Riemannian (M, g), if γ(t) : (a, b) ⊂ R → M is a

curve in M . The length L of γ(t) is given by

L(γ) =

∫ b

a

[g

(dγ

dt,dγ

dt

)] 12

dt wheredγ

dt≡ γ∗

(∂

∂t

)(4.5)

This can be rewritten in terms of the pullback of the metric g on M as

L(γ) =

∫ b

a

γ∗(g)

(∂

∂t,∂

∂t

) 12

dt (4.6)

and reinterpreted in terms of the ‘integral of a volume form’, inline with the interpretation

of integrals on manifolds defined in a subsequent chapter. The integral here is done over a

curve γ(t) so an integral of a 1-form on a manifold is the integral of the ‘pulled–back’ volume

form as we will see later. Note that with this interpretation, we have the integral of a 1-form

over the 1D manifold R, whose (only) component is determined by the metric induced on Rby the metric g and the map γ. This is the most natural interpretation of length of a curve

given by a line integral.

REMARK: A comment on the terminology ‘metric’ for a symmetric bilinear form is in

order. Given two points p, q ∈ M , define d(p, q) = infγL(γ) where infinimum is taken over

all piecewise C1 curves joining p, q. It can be shown (see Ref. [Boo86] V.3.1) that d(., .) so

defined satisfies the properties of a metric

1. d(p, q) = d(q, p)

2. d(p, q) ≤ d(p, r) + d(r, q)

3. d(p, q) ≥ 0

thus making M also a metric space.

When dealing with bilinear forms, it is tempting to think in terms of a matrix of values.

However, this is not entirely justified, in a coordinate free manner — bilinear forms correspond

to (0, 2) tensors, while matrices represent linear transformations i.e. (1, 1) tensors. For

instance, let A be a linear transformation on V , A : V → V . If a metric g is specified on V ,

then A can be identified with a unique (0, 2) tensor A′ by

A′(u, v) = g(u,A(v)) ∀u, v ∈ V (4.7)

48

In particular, an eigenvalue problem A(v) = λv makes sense for A but not for A′ because the

resulting eigenvalues will depend on the basis chosen! Since eigenvalues, are by definition,

properties of a matrix and not the basis vectors, this is clearly unacceptable. However, we

can define eigenvalues of a bilinear form, but only in reference to a particular metric or

inner product g. Then a one-to-one correspondence between A and A′ is established and no

inconsistencies arise.

4.3 Tensor fields and their symmetry

Given that V ∗ is the dual space of a vector space V , we can extend the definition to multi-

linear maps of V and V ∗ to obtain higher–order tensors. A tensor Φ is a multilinear map

Φ : V × · · · × V × V ∗ × · · · × V ∗ → R with r V and s V ∗ entries, also known as an (r, s)

tensor, Φ ∈ T rs (V ). For r = 1, s = 0, we get covectors, while r = 2, s = 0 corresponds to

bilinear forms.

Since there are, for the r and s copies of V and V ∗ respectively, n basis vectors each, we

naturally expect a basis of T rs (V ) to be constructed from each copy of these n basis vectors.

Hence dim(T rs (V )) = nr+s where n = dim(V ). In other words, any Φ ∈ T r

s (V ) is completely

determined by its values on the bases of each V and V ∗.

In the remainder of this chapter, we set s = 0. The definition above considers only a

vector space V and its dual V ∗. Just like we did for the cases of covectors (r = 1) and

bilinear forms (r = 2), we can extend the above ideas to manifolds. A tensor field is a map

p → Φp ∈ T r(TpM) taking p ∈ M to a specific r–linear map1 over TpM . We only require

that if X1, · · · , Xr are vector fields, then Φ(X1, · · · , Xr) be a smooth function.

If F : M → N is a differentiable map, then F ∗ : T r(TF (p)N) → T r(TpM) given by

F ∗Φ(X1p, · · · , Xrp) = Φ(F∗(X1p), · · · , F∗(Xrp)) defines a smooth tensor field on M , given

a smooth tensor field Φ on N .

Extending the idea of symmetry and antisymmetry from bilinear forms, a tensor Φ ∈ T r(V )

is symmetric if Φ(vσ(1), · · · , vσ(r)) = Φ(v1, · · · , vr) for any permutation σ of the indices

1, 2, · · · , r. Likewise, it is antisymmetric if Φ(vσ(1), · · · , vσ(r)) = sgn(σ) Φ(v1, · · · , vr).The collection of all symmetric and antisymmetric tensors form the subspaces Σr(V ),Λr(V ) ⊂

1This notation can be potentially confusing. Note that T r(V ) is the collection of all r–linear maps overthe vector space V . Here, V ≡ TpM ; In order to denote fields themselves, the notation T r(M) is often used,indicating that the linear maps are defined at every point on the manifold M . Note that this does not implythat M is a vector space by any means!

49

T r(V ).

DEFINITION:

We define the following operators on T r.

1. The symmetrizer S : T r(V )→ Σr(V ) is defined by

S Φ(v1, · · · , vr) =1

r!

∑σ∈Sr

Φ(vσ(1), · · · , vσ(r)) (4.8)

2. The antisymmetrizer A : T r(V )→ Λr(V ) is given by

A Φ(v1, · · · , vr) =1

r!

∑σ∈Sr

sgn(σ)Φ(vσ(1), · · · , vσ(r)) (4.9)

In the above, some remarks are warranted. Firstly, Sr is the set containing all permutations of

r objects. σ ∈ Sr means that we take one particular permutation of the indices 1, 2, · · · , r.Additionally, sgn(σ) is the sign of the permutation, i.e. the number of shifts needed to return

it to the ordered set 1, 2, · · · , r.It can be shown that S and A obey the following properties:

1. A 2 = A and S 2 = S

2. A and S are surjective — A (T rV ) = Λr(V ) and S (T rV ) = Σr(V )

3. Φ is alternating (symmetric) iff A Φ = Φ (S Φ = Φ)

4. If F∗ : V → W is a linear map, then under the mapping F ∗, symmetric/ alternating

nature is preserved (i.e. A and S commute with F ∗).

For manifolds, these properties are valid for linear maps on the tangent space at each point,

just as in the generalization introduced for covectors and bilinear forms.

4.4 Tensor products and exterior algebra

Just as in the other cases introduced above, we start with the case of simple vector spaces and

finally generalize all the results to manifolds. Suppose V is a vector space and φ ∈ T r(V ),

ψ ∈ T s(V ), then we define the tensor product as the operator ⊗ : T r(V ) × T s(V ) →T r+s(V ) given by

φ⊗ ψ(v1, · · · , vr, vr+1, · · · , vr+s) = φ(v1, · · · , vr)ψ(vr+1, · · · , vr+s) (4.10)

50

where the right hand side is just a product of two real numbers. Using this definition, the

following properties can be again verified

1. ⊗ is bilinear and associative — this follows from the fact that each argument of the

product is an argument of the individual tensors, which are themselves linear2.

2. If ωi1 , · · · , ωin is a basis of V ∗ ≡ T 1(V ), then ωi1 ⊗ · · · ⊗ ωir is a basis for T r(V )

over all 1 ≤ i1, · · · , ir ≤ n.

3. If F∗ : W → V is a linear map, then F ∗(φ⊗ ψ) = (F ∗φ)⊗ (F ∗ψ)

The generalization to tensor fields on a manifold M is straightforward. The correponding

map T rM ×T sM → T r+sM is bilinear and associative. Likewise, if ωi1 , · · · , ωin is a basis

of T 1(M), (i.e. are a set of covector fields defined all over M), then every element of T r(M)

is a linear combination, with C∞ coefficients, of ωi1 ⊗ · · · ⊗ ωir. Finally, if F : N → M is

a map between manifolds and ω ∈ T r(M), ψ ∈ T s(M) then F ∗(ψ ⊗ φ) = (F ∗ψ) ⊗ (F ∗φ)

forms a tensor field on N .

REMARK: Note that we do not, in general, have globally a defined basis for T 1(M).

However, these exist in Rn and can hence be pulled back to each neighbourhood U of M .

Hence each φ ∈ T r(U) has a unique expression of the form

φ =∑i1

· · ·∑ir

ai1···irωi1 ⊗ · · · ⊗ ωir (4.11)

requiring at most nr coefficients, which are smooth functions on U . At each point p ∈ U ⊂M ,

ωip being the dual basis to Eip, it follows that ai1···ir = φ(Ei1p, · · · , Eirp)

For a vector space V , we have seen that for each r > 0, we have Λr(V ) ⊂ T r(V ) con-

sisting of all alternating tensors.3 The direct sum Λ(V ) of all the spaces Λr(V ) is contained

in the direct sum T (V ) of the spaces T r(V ).

Λ(V ) = Λ0(V )⊕ Λ1(V )⊕ · · · ⊂ T 0V ⊕T 1V ⊕ · · · (4.12)

Defined this way, a crucial difference between T (V ) and Λ(V ) is the following — T (V )

is infinite dimensional — each T rV has dimension nr and this goes on for any r. The

alternating tensors have the important property that for r > n, they are zero, forcing the

2In this context, bilinearity refers to the two arguments of ⊗. If φ1, φ2 ∈ T rV and ψ ∈ T sV then forreal numbers α, β, (αφ1 + βφ2)⊗ ψ = αφ1 ⊗ ψ + βφ2 ⊗ ψ

3This is the image of T r(V ) under the mapping A

51

direct sum to actually trivially terminate, depending on the dimension of the underlying

space V . We also see that Λ(V ) is a subset of T (V ) but not a subalgebra. This means that

if φ ∈ Λr(V ), ψ ∈ Λs(V ) then φ⊗ψ /∈ Λr+s(V ) necessarily. The motivation to close the Λ(V )

as an algebra results in the definition of another product on Λ(V ).

DEFINITION:

The wedge product or exterior product is the map ∧ : Λr(V )× Λs(V )→ Λr+s(V ) defined

by

(φ, ψ)→ φ ∧ ψ =(r + s)!

r! s!A (φ⊗ ψ) (4.13)

REMARK: There are some differences in notation with respect to the exterior product.

The definition given here contains r! in the denominator. Some authors restrict (for eg.

Ref. [Fra11]) the summation to include terms with ordered indices ~I = (i1, i2, · · · , ir) with

i1 < i2 < · · · < ir. Clearly allowing all permutations (via A ) amounts to r! duplicates,

each with a ±1 in front. Some other authors (for eg. Refs. [Fla12, Dar94]) define ∧ on the

tangent spaces and treat alternating forms as covectors on the space obtained by using ∧ on

r vectors. This latter notation can be quite confusing.

The exterior product defined in this manner is both bilinear and associative. Associativ-

ity means that the relation φ∧ (ψ∧ θ) = (φ∧ψ)∧ θ holds for alternating tensors of arbitrary

order, and can be proved using the property of the antisymmetrizer map:

A (φ⊗ ψ ⊗ θ) = A (A (φ⊗ ψ)⊗ θ) = A (φ⊗A (ψ ⊗ θ)) (4.14)

We can hence write the expression for the multiple wedge product. If φ ∈ Λri(V ), i = 1, · · · , kthen

φ1 ∧ φ2 ∧ · · · ∧ φk =(r1 + r2 + · · ·+ rk)!

r1! r2! · · · rk!A (φ1 ⊗ φ2 ⊗ · · · ⊗ φk) (4.15)

Two important properties of the wedge product are

1. (φ ∧ ψ) ∧ θ = φ ∧ (ψ ∧ θ)

2. φ ∧ ψ = (−1)rs(ψ ∧ φ) for φ ∈ Λr(V ) and ψ ∈ Λs(V ).

The second result above implies that φ ∧ φ = 0 for forms of odd order, NOT forms of even

order. To drive home this point, it is important to realize that the most common example

used to illustrate the algebra of forms is R3. Here, the only non-trivial forms of even order

52

are 2-forms. But their exterior product is zero for an entirely different reason, because the

dimension of the space is 3. Hence it will help to commit this property to memory so that

it may be used in general cases. Also, another important consequence is that even forms

commute and odd-forms anti-commute.

If r > n = dim(V ) then Λr(V ) = 0. For 0 ≤ r ≤ n, dim(Λr(V )) =n Cr. If ω1, · · · , ωn is a

basis for Λ1(V ), then

ωi1 ∧ · · · ∧ ωir |1 ≤ i1 < i2 < · · · < ir ≤ n

is a basis for Λr(V ) and dim(Λ(V )) = 2n. As a result, any ordered set of indices ik for

k = 1, · · · , r determines a basis for Λr(V ) in terms of covectors ωi. The other ordered sets

are linear combinations (with coefficient ±1 that result from the interchange of indices).

Grassman algebra

The direct sum Λ(V ) = Λ0(V )⊕Λ1(V )⊕ · · · along with the exterior product ∧ forms

an associative algebra over R. For each φ = φ1 + φ2 + · · ·+ φk where φi ∈ Λri(V ) and

ψ = ψ1 + ψ2 + · · ·+ ψl where ψi ∈ Λsi(V ), with φ, ψ ∈ Λ(V ), we have

φ ∧ ψ =k∑i=1

l∑j=1

φi ∧ ψj ∈ Λ(V ) (4.16)

The double summation in the above is a result of the fact that the exterior product is dis-

tributive over addition, which in turn stems from the bilinear nature of the exterior product.

As expected, the generalization to manifolds is done in a straightforward manner. Λ(M)

denotes the exterior algebra of differential forms on a manifold M , consisting of all Λr(M)

the space of r-form fields on M and the wedge product ∧.

4.5 Volume form and orientation

In the spirit of the chapter so far, let us start with a real vector space V , with dim(V ) = n.

We have earlier seen that dim(Λn(V )) = 1. Suppose e1, · · · , en and f1, · · · , fn are bases

of V . Write ei =∑n

j=1 αjifj where αji is invertible so its determinant 6= 0. Define an

equivalence relation on the set of bases such that two bases ei and fk are equivalent if

det(αji ) > 0. Thie equivalence relation hence splits all possible bases into two equivalence

53

classes thus determining an orientation of a vector space.

Now suppose Ω 6= 0 ∈ Λn(V ). if vi =∑n

j=1 βji ej are another set of linearly independent

vectors, then Ω(v1, · · · , vn) = det(βji )Ω(e1, · · · , en). Hence if ei is an orthonormal ba-

sis, i.e. Ω(e1, · · · , en) = 1 by definition, then another orthonormal basis vj results in

Ω(v1, · · · , vn) = ±1, depending on the sign of the determinant of β.

So the equivalence of basis, as defined above, is seen in the equivalence of the effect of acting

on the basis vectors by an n-form. Since the very idea of orthonormality on the vector space

is made possible by a bilinear form or an inner product, Ω = ±1 is uniquely possible when

such an inner product is available, as for example from a Riemannian metric on a manifold.

Hence, we can naturally define the orientability of a manifold.

DEFINITION:

A manifold M is orientable if it is possible to define a smooth n-form Ω which is not zero

anywhere on M . Such a choice of Ω choses an orientation for M .

REMARK: For an orientable Riemannian manifold (M,Φ), ∃ a unique n-form Ω that

determines the orientation of M such that Ω(F1, · · · , Fn) = 1 for any orthonormal basis

F1, · · · , Fn of the tangent space. For concreteness, let p ∈ M and Fjp be a set of

orthonormal basis vectors of TpM . For any other basis vectors Eip, determined by an

arbitrary coordinate chart, Φ(Eip, Ejp) ≡ gij. If Eip =∑n

j=1 αjiFjp then

gij =n∑

l,k=1

αki αljδkl =⇒ (gij) = ATA (4.17)

which in turn implies that det(A) =√g, g = det(gij) and so the effect of Ω on the oriented

basis Ejp is Ω(E1p, · · · , Enp) =√g.

So we see that on a Riemannian manifold (M, g), a unique n-form field Ω can be defined

which returns ±1 on orthonormal bases. This is hence the unique volume element on M and

plays a central role in the theory of integration on manifolds, as will be shown later. Often

(particularly in the physics literature), this is replaced by a pseudo–form that changes sign

depending on which of the two orientations (members of the equivalence classes) is chosen

as the standard orientation of M . However, it is a simple matter of accounting for the sign

changes in this definition and will not be frequently alluded to henceforth, except in specific

applications.

54

4.6 Exterior, Lie derivatives of forms and the interior

product

So far, to develop the concepts of differential forms, tensor fields, the exterior algebra etc.,

we started by considering a single vector space. The generalization to manifolds has then

proceeded in the usual way — identify at each p ∈M , the vector space with TpM and extend

the definitions trivially. However, in such ‘point–wise’ definitions, the notion of derivatives

is not natural. Hence, the ideas introduced in this section have no parallel in the case of

general vector spaces and need a smooth manifold structure to make any sense.

DEFINITION:

If M is a differentiable manifold, Λ(M) its algebra of differential forms, then there exists

a unique map dM : Λ(M)→ Λ(M) satisfying

1. If f ∈ Λ0(M) = C∞(M) then dMf = df as defined earlier.

2. If θ ∈ Λr(M), σ ∈ Λs(M) then dM(θ ∧ σ) = dMθ ∧ σ + (−1)rθ ∧ dMσ

3. d2M = 0

dM is the exterior derivative on M

The exterior derivative, so defined, obeys the following relations:

1. If F : M → N is a differentiable map and θ ∈ ΛkN then F ∗dθ = dF ∗θ, i.e. d commutes

with F ∗.

2. If ω ∈ Λ1(M) and X, Y are vector fields on M , then

dω(X, Y ) = X(ω(Y ))− Y (ω(X))− ω([X, Y ]) (4.18)

one can also think of this relation as follows: X(ω(Y )) being the action of a vector

(differential operator) on a smooth function ω(Y ), obeys some sort of product rule.

The above relation, with only this term on the right, spells out such a rule.

3. A generalization of the above formula to p-forms φ is given by:

dφ(X1, · · · , Xp+1) =

p+1∑i=1

(−1)i−1Xi(φ(X1, · · · , Xi, · · · , Xp+1))

+∑i<j

(−1)i+jφ([Xi, Xj], X1, · · · , Xi, · · · , Xj, · · · , Xp+1) (4.19)

55

where Xk denotes that Xk is not present in the arguments.

The above relations can be verified by choosing a basis for the forms and using the definition

of d above. For instance, the exterior derivative of a one form ω = widxi is given by

dω = d(widxi) = dwi ∧ dxi (4.20)

in accordance with the property d2 = 0. Formulae for higher order forms are obtained in a

similar manner.

A differential r-form φ on M is closed if dφ = 0 and it is exact if ∃ an r − 1-form θ on

M such that φ = dθ. Clearly, all exact forms are closed but not vice versa.

In the presence of a C∞ vector field X, two other ‘derivative-esque’ operators can be defined

on the space of forms Λ(M).

DEFINITION:

The interior product iX : Λr(M)→ Λr−1(M) is given by

iX(φ)(X1, · · · , Xr−1) = φ(X,X1, · · · , Xr−1) if r ≥ 0 and iX = 0 on Λ0(M) (4.21)

The interior product has the following properties:

1. iX : Λ(M) → Λ(M) is C∞(M)-linear for fixed X and X → iXφ is also C∞(M)-linear

for fixed φ

2. i2X = 0 and if φ ∈ Λr(M), ψ ∈ Λs(M) then

iX(φ ∧ ψ) = (iXφ) ∧ ψ + (−1)rφ ∧ (iXψ) (4.22)

We have seen the Lie derivative of vector fields (Sec. 3.3), which measured non–closure of

integral curves of a pair of vector fields. Is there a corresponding version for forms? And

what is its interpretation?

DEFINITION:

The Lie derivative LX : Λr(M)→ Λr(M) of a form at p ∈M is given by

(LXφ)p = lim∆t→0

1

∆t

[θ∗∆t(φθ(∆t,p))− φp

](4.23)

where θt(p) is the local one–parameter group action generated by X.

56

Further, for r ≥ 1,

LXφ = iXdφ+ diXφ (4.24)

and for r = 0, LXf ≡ X(f).

The idea behind the Lie derivative of a form is the same as that behind the Lie derivative of

a vector field. Take a form field φ, at some point along the flow of X, starting from p ∈M so

that φθt(p) ∈ T ∗θt(p)M . Now pull back the form so that it lies in T ∗pM and compare the effect

of these two forms on any vector. Take the difference, divide by t and take the limit t→ 0.

This is what we did for vector fields earlier (cf. Eq. 3.2).

The Lie derivative LX has the following properties

1. LX : Λ(M)→ Λ(M) is R linear.

2. LXd = dLX or LX commutes with d.

3. If φ ∈ Λr(M) and ψ ∈ Λs(M) then

LX(φ ∧ ψ) = (LXφ) ∧ ψ + φ ∧ (LXψ) (4.25)

This is the general form of the Liebniz rule for a derivation. Note that this guarantees

that LX is only R-linear in both X and ψ and not C∞-linear in either.

4. LfXφ = df ∧ iXφ+ fLXφ for f ∈ C∞(M).

5. The Lie derivative of an r-form φ obeys

LXφ(X1, · · · , Xr) = X(φ(X1, · · · , Xr))−r∑i=1

φ(X1, · · · , LXXi, · · · , Xr) (4.26)

Again, this reminds us of the product rule for the action of a differential operator X

on the smooth function φ(X1, · · · , Xr). All such expressions must be equivalent and

hence there exist relations between iX , d and LX .

The Lie derivative LX of a form is a derivation, since it obeys the Liebniz rule for differential

operators. However, iX and d are antiderivations because they obey iX(α ∧ β) = (iXα) ∧β + (−1)rα ∧ (iXβ) and likewise for d.

As noted above, LX , iX and d obey the following relations, some of which are also mentioned

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earlier (X, Y are vector fields)

LX = iX d + d iX (4.27)

i[X,Y ] = LX iY − iY LX (4.28)

LX d = d LX (4.29)

In contrast to the three operators on forms we’ve introduced above, we have so far only

encountered the Lie derivative for vector fields. iX and d act on a particular form and either

decrease or increase its order respectively. Only LX is a map between forms of the same

order. An additional derivative operator for vector fields will be introduced at a later time.

Finally, a note on a physical interpretation of the Lie derivative. If Ω is a volume ele-

ment on a Riemannian manifold M , then in local coordinates, we have seen that it is given

by

Ω =√g dx1 ∧ · · · ∧ dxn (4.30)

Naturally, the Lie derivative of Ω, given by LXΩ, measures how volumes are changing under

the flow θt, the 1–parameter group of diffeomorphisms introduced by X. In this sense, LXΩ

is actually describing a property of the flow X, namely how it is ‘deforming’ the volume

element along M .

4.7 Time dependent fields

So far, when dealing with vector fields and forms, we have considered them as being intrinsic

to the manifold, and not dependent on an ‘external ’ parameter. Hence the fields themselves

generate a flow on the manifold M , which cannot be altered by changing an external pa-

rameter. A case very commonly encountered in practical problems is that where the fields

depend continously on an external parameter, which we generically here denote by time t.

Caution must be exercised to prevent confusion with a representation of the one-parameter

group of a vector field.

In contrast to what we have said about vector fields in general, a time–dependent vector

field on M does not necessarily generate a flow. Consider for instance, the vector field X(t)

at two instances t0 and t1. At a point p ∈ M , the vector field X(t0) has an integral curve

through p while X(t1) has an, entirely different, integral curve through p. This clearly vi-

olates the theorem on existence of integral curves for vector fields. This is because at two

different instances of time, the vector fields X that actually exist on M are actually different.

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So in this sense, one has to be careful when defining and dealing with flows generated by

time–dependent vector fields.

One way to completely ensure the existence of (local) flows is to extend the space, and instead

consider the manifold R×M . Such a construct is visualized in Fig. 4.1

Figure 4.1: Extending the domain of a time–dependent vector field to include I ⊂ R.

Now at a particular ‘slice’ ti, we have a vector field on M and everything works as usual.

In this regard, a time–dependent vector field X(t) on M , has at the point (t, p), where

t ∈ I ⊂ R and p ∈M , the components with respect to a coordinate chart on M , given by

X(t) =∂

∂t+X i ∂

∂xiwhere X i ≡ X(t)(xi) (4.31)

Other formulae are modified in a similar manner on this manifold R×M . For instance, the

exterior derivative d is given by

d(bi1i2,··· ,ikdxi1 ∧· · ·∧dxik) =∂bi1,i2,··· ,ik

∂tdt∧dxi1 ∧· · ·∧dxik +

∂bi1i2,··· ,ik∂xj

dxj ∧dxi1 ∧· · ·∧dxik

(4.32)

Two things of note in the formula above — firstly, any time–dependent form α(t) on M ,

when extended to R ×M does not have components along dt. We did this for the case of

vector fields in order to make sense of integral curves. The exterior derivative, however, acts

on R×M and will hence introduce the dt term automatically.

4.8 Applications to vector analysis

Having developed the formalism of differential forms, we now see how the operations of vector

calculus in Euclidean space can be expressed in terms of derivations of differential forms. It

59

will be evident that oftentimes simple relations between forms are expressed as complicated

relationships between vectors in Rn by subconsciously using the Euclidean metric. This can

be avoided if the exterior algebra is used throughout and the necessary identifications made

when comparing with actual physical quantities and coordinate systems.

We consider the case of R3 first, with Cartesian coordinates (x, y, z). Denote the usual

Euclidean metric on R3 by g and let Ω be the volume form. Let A,B,C be vector fields —

they can just be constant vectors at a point or all over R3 it doesn’t matter in this context —

with the usual cross and dot products in R3. Using g, we can uniquely identify the following

associated fields:

Ω = dx ∧ dy ∧ dz

α = g(., A) β = g(., B) γ = g(., C)

α = iAΩ β = iBΩ γ = iCΩ

With this notation, the following identifications can then be made4 for the common operations

of vector calculus in R3:

α ∧ γ = iA×CΩ

α ∧ β = (A ·B) Ω

iCα = C · A

g(., B × C) = iC β

g(.,∇f) = df

dα = i∇×A Ω

dβ = (∇ ·B) Ω

di∇fΩ = (∇2f) Ω

We can easily use the relations above to establish some standard identities used in vector

analysis — the simplest of these results from d2 = 0 and reads ∇ · (∇×) = 0 when applied

to a 1-form and ∇×∇ = 0 when applied to a function.

4∇f is considered a vector with the usual components as defined in vector calculus

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5 Integration of Differential Forms

Having defined exterior differential forms and the various operations on them, we now see

how this formalism can be put to use. We first start off by recalling some formal definitions

of integration on R3, following which it will be seen that integrals are natural operations on

forms alone and that, in passing, one can talk of integrals of functions and vector fields over

curves, surfaces etc. after certain identifications are made, some of which might include the

Riemannian metric on the manifold, and the (generalized) Stokes’ theorem unifies all known

theorems on integrals to corollaries. The entire formalism of differential forms on manifolds

shows its elegance and power in this regard. Pseudoforms are then mentioned and their

physical significance explained.

5.1 Integration on Rn

We first establish some formal notions of integration on Rn which, when generalized to man-

ifolds, will help rationalize some subtle points. First a set A ⊂ R3 has Jordan content zero

i.e. C(A) = 0 if for any ε > 0, there exists a finite collection of cubes Ci covering A such that∑i volume(Ci) < ε. One can think of content zero like measure zero i.e. if f is discontinuous

on ∂D, it does not contribute to an integral.

A bounded subset D ⊂ Rn is a domain of integration if ∂D has content zero. A func-

tion is almost continuous if the set of points at which it fails to be continuous has content

zero. Finally, we say that if D is a domain of integration and f is bounded and almost

continuous on D then∫DfdV exists and is bounded, denoting the Riemann integral of f .

The Riemann integral has the following properties:

1. If C(D) = 0 then∫DfdV = 0

2.∫D1∪D2

f dV =∫D1f dV +

∫D2f dV −

∫D1∩D2

f dV

3.∫D

(af + bg) dV = a∫Df dV + b

∫Dg dV for a, b ∈ R

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4. If f ≥ 0 on D then∫Df dV ≥ 0. Equality holds if and only if f = 0 except on a set of

content zero.

Suppose G : D → D′ is a diffeomorphism between D ⊂ U and D′ ⊂ U ′. Let f(x) = f ′(G(x))

i.e. f(x1, · · · , xn) = f ′(G1(x), · · · , Gn(x)). Then∫D′ f

′(y) dV ′ =∫Df(x)|∆G| dV with y =

G(x) and |∆G| denoting the determinant of G.

PROPOSITION 15: Let A be a compact subset of Rn and F : A → Rm, n‘m be a

C1 mapping. Then F (A) has content zero if n < m or if n = m and A has content zero.

Roughly, the above proposition states that any map of a lower dimensional object when

viewed in a higher–dimensional space has content zero. Immediate examples are maps of

lines or surfaces in R3 which have zero volume.

5.2 Integration on manifolds

Suppose M is an oriented manifold with dim(M) = n. If ω ∈ Λn(M) has compact support,

we will define the integral∫Mω. More generally, let Ω be a non–banishing n-form with

ω = hΩ where h is integrable, as defined earlier. Let the local expression for ω be:

(φ−1)∗ω = f(x)dx1 ∧ · · · ∧ dxn (5.1)

A special neighbourhood of a point is a cube. Formally, we can define it as follows:

DEFINITION:

A subset Q ⊂ M is a cube if it lies in the domain of an associate chart (U, φ) such that

φ(Q) is a cube in Rn

If ω is supported in a cube Q ⊂M , then, denoting C = φ(Q) the image, under the coordinate

chart φ, of Q we define the integral of ω as∫Q

ω =

∫C

(φ−1)∗ω =

∫C

fdx1 ∧ · · · ∧ dxn (5.2)

Using then the idea of a partition of unity, we can piece together the integrals over all such

cubes to define the integral of ω as the Euclidean integral over Rn obtained via the coordinate

charts. We can also show that this integral is independent of the coordinate chart used to

evaluate it. The integral so defined has the following properties:

1. If the orientation of M is reversed,∫Mω → −

∫Mω

62

2.∫Mαω1 + βω2 = α

∫Mω1 + β

∫Mω2

3. If ω = gΩ where Ω is the volume form on M with g ≥ 0 then∫MgΩ ≥ 0 with equality

if and only if g = 0 except on a set of content 0

4. If F : M → N is a diffeomorphism then∫MF ∗ω = ±

∫Nω where ± depends on whether

F preserves orientation or reverses orientation of a set of bases1

So the notion of the integral of an n-form on a manifold with dimension n is clear — being a

1D space, Λn(M) is spanned by the volume form Ω. Any n-form is then a multiple of Ω with

the coefficient a C∞ function on M . So when we speak of the integral of a function f over

a volume on a manifold M , we are speaking of the integral of the n-form whose coefficient

is f , as defined above. Once an orientation is picked for M ,∫Mω is just the n-dimensional

integral of f , evaluated by usual means of integral calculus.

Now let us consider the integration of k-forms on M . Consider a map F : U → M where

U is a k-dimensional manifold (note necessarily a submanifold of M), then the integral of

α ∈ Λk(M) is given by ∫M

α =

∫U

F ∗α (5.3)

And since for any differentiable map F , F ∗ is defined, the above integral can be evaluated for

a given U and F . The value of this integral hence depends critically on both. For example,

F (U) can be a curve (k = 1) or a surface (k = 2) on M . The corresponding integral are

familiar line and surface integrals, and they depend on the curve or the surface over which

they are evaluated, i.e. the map F . Note, however, that∫UF ∗α is independent of coordinates

on U , i.e. parameterization of the surface, just as∫Mω is independent for the integral of an

n-form on M .

Example :

Using this formalism of defining integrals of p-forms on a manifold M with dimension n, we

can reinterpret some conventional integrals in R3. Take a vector field B ∈ X(M), where

M ⊂ R3 and of dimension 3. We want to compute, in the Euclidean sense, the ‘area flux’

of B through a surface S, which (using conventional notation of vector calculus) amounts to

evaluating the following integral:

I =

∫S

B · n dS (5.4)

1Recall that F being a diffeomorphism ensures that M and N have the same direction. It thus makessense to compare the orientations of the two manifolds in terms of orientations of their bases.

63

Now consider the 2–form β corresponding to B given in terms of the volume form Ω on R3

by β = iBΩ. The integral of this two form, when considering the surface S : U → M for

U ⊂ R2, is given by

I =

∫M

β =

∫U

S∗β (5.5)

Using coordinates xi on M , if β = bij(x)dxi∧dxj, then the integral above can be evaluated

as ∫U

S∗β =

∫U

S∗(bijdxi ∧ dxj) =

∫U

β

(S∗

∂

∂u, S∗

∂

∂v

)du ∧ dv (5.6)

with a given parameterization (u, v) of S. When expanded, this is the correct expression for

B · n once the appropriate identifications are made (cf. Sec. 4.8)

64

Bibliography

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[Boo86] W. M. Boothby. An introduction to differentiable manifolds and Riemannian ge-

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[Dar94] R. W. R. Darling. Differential forms and connections. Cambridge University Press,

1994.

[DFN85] B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov. Modern geometry—methods

and applications: The geometry and topology of manifolds, volume 2. Springer,

1985.

[Fla12] H. Flanders. Differential forms with applications to the physical sciences. Courier

Corporation, 2012.

[Fra11] T. Frankel. The geometry of physics: an introduction. Cambridge University Press,

2011.

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