INTRODUCTION TODIFFERENTIABLE

MANIFOLDS

spring 2009

Course by Prof. dr. R.C.A.M. van der VorstSolution manual Dr. G.J. Ridderbos

1

2

Introduction to differentiable manifoldsLecture notes version 2.1, February 16, 2009

This is a self contained set of lecture notes. The notes were written by Rob van derVorst. The solution manual is written by Guit-Jan Ridderbos. We follow the book‘Introduction to Smooth Manifolds’ by John M. Lee as a reference text.

This document was produced in LATEX and the pdf-file of these notes is availableon the following website

www.few.vu.nl/˜vdvorst

They are meant to be freely available for non-commercial use, in the sense that“free software” is free. More precisely:

This work is licensed under the Creative CommonsAttribution-NonCommercial-ShareAlike License.To view a copy of this license, visithttp://creativecommons.org/licenses/by-nc-sa/2.0/

or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California94305, USA.

CONTENTS

I. Manifolds 51. Topological manifolds 52. Differentiable manifolds and differentiable structures 113. Immersions, submersions and embeddings 19

II. Tangent and cotangent spaces 314. Tangent spaces 315. Cotangent spaces 376. Vector bundles 396.1. The tangent bundle and vector fields 436.2. The cotangent bundle and differential 1-forms 45

III. Tensors and differential forms 487. Tensors and tensor products 488. Symmetric and alternating tensors 539. Tensor bundles and tensor fields 5910. Differential forms 6311. Orientations 67

IV. Integration on manifolds 73

3

12. Integrating m-forms on Rm 73

13. Partitions of unity 7414. Integration on of m-forms on m-dimensional manifolds. 7915. The exterior derivative 8516. Stokes’ Theorem 90

V. De Rham cohomology 9517. Definition of De Rham cohomology 9518. Homotopy invariance of cohomology 96

VI. Exercises 100

ManifoldsTopological Manifolds 100Differentiable manifolds 101Immersion, submersion and embeddings 102

Tangent and cotangent spacesTangent spaces 104Cotangent spaces 105Vector bundles 106

TensorsTensors and tensor products 107Symmetric and alternating tensors 107Tensor bundles and tensor fields 108Differential forms 109Orientations 109

Integration on manifoldsIntegrating m-forms on R

m 110Partitions of unity 110Integration of m-forms on m-dimensional manifolds 110The exterior derivative 111Stokes’ Theorem 112Extra’s 113

De Rham cohomologyDefinition of De Rham cohomology 115

4

VII. Solutions 116

5

I. Manifolds

1. Topological manifolds

Basically an m-dimensional (topological) manifold is a topological space Mwhich is locally homeomorphic to R

m. A more precise definition is:

Definition 1.1.1 A topological space M is called an m-dimensional (topological)

manifold, if the following conditions hold:

(i) M is a Hausdorff space,(ii) for any p∈M there exists a neighborhood2 U of p which is homeomorphic

to an open subset V ⊂ Rm, and

(iii) M has a countable basis of open sets.

Axiom (ii) is equivalent to saying that p ∈ M has a open neighborhood U phomeomorphic to the open disc Dm in R

m. We say M is locally homeomorphic toR

m. Axiom (iii) says that M can be covered by countably many of such neighbor-hoods.

FIGURE 1. Coordinate charts (U,ϕ).

1See Lee, pg. 3.2Usually an open neighborhood U of a point p ∈M is an open set containing p. A neighborhood

of p is a set containing an open neighborhood containing p. Here we will always assume that aneighborhood is an open set.

6

FIGURE 2. The transition maps ϕi j.

Recall some notions from topology: A topological space M is called Hausdorff

if for any pair p,q ∈ M, there exist (open) neighborhoods U p, and U q suchthat U ∩U = ∅. For a topological space M with topology τ, a collection β ⊂ τ isa basis if and only if each U ∈ τ can be written as union of sets in β. A basis iscalled a countable basis if β is a countable set.

Figure 1 displays coordinate charts (U,ϕ), where U are coordinate neighbor-hoods, or charts, and ϕ are (coordinate) homeomorphisms. Transitions betweendifferent choices of coordinates are called transitions maps ϕi j = ϕ j ϕ−1

i , whichare again homeomorphisms by definition. We usually write x = ϕ(p), ϕ : U →V ⊂R

n, as coordinates for U , see Figure 2, and p = ϕ−1(x), ϕ−1 : V →U ⊂M, asa parametrization of U . A collection A = (ϕi,Ui)i∈I of coordinate charts withM = ∪iUi, is called an atlas for M.

The following Theorem gives a number of useful characteristics of topologicalmanifolds.

Theorem 1.4.3 A manifold is locally connected, locally compact, and the union of

countably many compact subsets. Moreover, a manifold is normal and metrizable.

1.5 Example. M = Rm; the axioms (i) and (iii) are obviously satisfied. As for (ii)

we take U = Rm, and ϕ the identity map.

1.6 Example. M = S1 = p = (p1, p2) ∈R2 | p2

1 + p22 = 1; as before (i) and (iii)

are satisfied. As for (ii) we can choose many different atlases.

3Lee, 1.6, 1.8, and Boothby.

7

(a): Consider the sets

U1 = p ∈ S1 | p2 > 0, U2 = p ∈ S1 | p2 < 0,

U3 = p ∈ S1 | p1 < 0, U4 = p ∈ S1 | p1 > 0.

The associated coordinate maps are ϕ1(p) = p1, ϕ2(p) = p1, ϕ3(p) = p2, andϕ4(p) = p2. For instance

ϕ−11 (x) =

x,

1− x2

,

and the domain is V1 = (−1,1). It is clear that ϕi and ϕ−1i are continuous, and

therefore the maps ϕi are homeomorphisms. With these choices we have found anatlas for S1 consisting of four charts.

(b): (Stereographic projection) Consider the two charts

U1 = S1\(0,1), and U2 = S1\(0,−1).

The coordinate mappings are given by

ϕ1(p) =2p1

1− p2, and ϕ2(p) =

2p1

1+ p2,

which are continuous maps from Ui to R. For example

ϕ−11 (x) =

4xx2 +4

,x2−4x2 +4

,

which is continuous from R to U1.

FIGURE 3. The stereographic projection describing ϕ1.

(c): (Polar coordinates) Consider the two charts U1 = S1\(1,0), and U2 =S1\(−1,0). The homeomorphism are ϕ1(p) = θ ∈ (0,2π) (polar angle counterclockwise rotation), and ϕ2(p) = θ ∈ (−π,π) (complex argument). For exampleϕ−1

1 (θ) = (cos(θ),sin(θ)).

8

1.8 Example. M = Sn = p = (p1, · · · pn+1) ∈ Rn+1 | |p|2 = 1. Obvious exten-

sion of the choices for S1.

1.9 Example. (see Lee) Let U ⊂Rn be an open set and g : U →R

m a continuousfunction. Define

M = Γ(g) = (x,y) ∈ Rn×R

m : x ∈U, y = g(x),

endowed with the subspace topology.4 This topological space is an n-dimensionalmanifold. Indeed, define π : R

n×Rm → R

n as π(x,y) = x, and ϕ = π|Γ(g), whichis continuous onto U . The inverse ϕ−1(x) = (x,g(x)) is also continuous. Therefore(Γ(g),ϕ) is an appropriate atlas. The manifold Γ(g) is homeomorphic to U .

1.10 Example. M = PRn, the real projective spaces. Consider the following

equivalence relation on points on Rn+1\0: For any two x,y ∈ R

n+1\0 define

x∼ y if there exists a λ = 0, such that x = λy.

Define PRn = [x] : x ∈ R

n+1\0 as the set of equivalence classes. One canthink of PR

n as the set of lines through the origin in Rn+1. Consider the natural

map π : Rn+1\0→ PR

n, via π(x) = [x]. A set U ⊂ PRn is open if π−1(U) is

open in Rn+1\0. This makes PR

n a compact Hausdorff space. In order to verifythat we are dealing with an n-dimensional manifold we need to describe an atlasfor PR

n. For i = 1, · · ·n+1, define Vi ⊂ Rn+1\0 as the set of points x for which

xi = 0, and define Ui = π(Vi). Furthermore, for any [x] ∈Ui define

ϕi([x]) =x1

xi, · · · , xi−1

xi,xi+1

xi, · · · , xn+1

xi

,

which is continuous, and its continuous inverse is given by

ϕ−1i (z1, · · · ,zn) =

(z1, · · · ,zi−1,1,zi, · · · ,zn)

.

These charts Ui cover PRn. In dimension n = 1 we have that PR ∼= S1, and in the

dimension n = 2 we obtain an immersed surface PR2 as shown in Figure 4.

The examples 1.6 and 1.8 above are special in the sense that they are subsets ofsome R

m, which, as topological spaces, are given a manifold structure.

Define

Hm = (x1, · · · ,xm) | xm ≥ 0,

as the standard Euclidean half-space.

4For a definition see the end of this section.

9

FIGURE 4. Identification of the curves indicated above yields animmersion of PR

2 into R3.

Definition 1.12. A topological space M is called an m-dimensional (topological)

manifold with boundary ∂M ⊂M, if the following conditions hold:

(i) M is a Hausdorff space,(ii) for any point p ∈ M there exists a neighborhood U of p, which is homeo-

morphic to an open subset V ⊂Hm, and

(iii) M has a countable basis of open sets.

Axiom (ii) can be rephrased as follows, any point p ∈M is contained in a neigh-borhood U , which either homeomorphic to Dm, or to Dm∩H

m. The set M is locallyhomeomorphic to R

m, or Hm. The boundary ∂M ⊂M is a subset of M which con-

sists of points p for which any neighborhood cannot be homeomorphic to an opensubset of int (Hm). In other words point p ∈ ∂M are points that lie in the inverseimage of V ∩ ∂H

m for some chart (U,ϕ). Points p ∈ ∂M are mapped to points on∂H

m and ∂M is an (m−1)-dimensional topological manifold. 1.14 Example. Consider the bounded cone

M = C = p = (p1, p2, p3) ∈ R3 | p2

1 + p22 = p2

3, 0≤ p3 ≤ 1,

with boundary ∂C = p ∈ C | p3 = 1. We can describe the cone via an atlasconsisting of three charts;

U1 = p ∈C | p3 < 1,

with x = (x1,x2) = ϕ1(p) = (p1, p2 +1), and

ϕ−11 (x) =

x1,x2−1,

x2

1 +(x2−1)2,

10

FIGURE 5. Coordinate maps for boundary points.

The other charts are given by

U2 = p ∈C | 12

< p3 ≤ 1, (p1, p2) = (0, p3),

U3 = p ∈C | 12

< p3 ≤ 1, (p1, p2) = (0,−p3).

For instance ϕ2 can be constructed as follows. Define

q = ψ(p) = p1

p3,

p2

p3, p3

, σ(q) =

2q1

1−q2,1−q3

,

and x = ϕ2(p) = (σ ψ)(p), ϕ2(U2) = R× [0, 12) ⊂ H

2. The map ϕ3 is definedsimilarly. The boundary is given by ∂C = ϕ−1

2 (R×0)∪ϕ−13 (R×0), see Figure

6.

1.16 Example. The open cone M = C = p = (p1, p2, p3) | p21 + p2

2 = p23, 0 ≤

p3 < 1, can be described by one coordinate chart, see U1 above, and is therefore amanifold without boundary (non-compact), and C is homeomorphic to D2, or R

2,see definition below.

So far we have seen manifolds and manifolds with boundary. A manifold canbe either compact or non-compact, which we refer to as closed, or open manifoldsrespectively. Manifolds with boundary are also either compact, or non-compact. Inboth cases the boundary can be compact. Open subsets of a topological manifoldare called open submanifolds, and can be given a manifold structure again.

Let N and M be manifolds, and let f : N → M be a continuous mapping. Amapping f is called a homeomorphism between N and M if f is continuous andhas a continuous inverse f−1 : M → N. In this case the manifolds N and M are said

11

FIGURE 6. Coordinate maps for the cone C.

to be homeomorphic. Using charts (U,ϕ), and (V,ψ) for N and M respectively,we can give a coordinate expression for f , i.e. f = ψ f ϕ−1.

Recall the subspace topology. Let X be a topological space and let S ⊂ X beany subset, then the subspace, or relative topology on S (induced by the topologyon X) is defined as follows. A subset U ⊂ S is open if there exists an open setV ⊂ X such that U = V ∩S. In this case S is called a (topological) subspace of X .A (topological) embedding is a continuous injective mapping f : X → Y , which isa homeomorphism onto its image f (X)⊂Y with respect to the subspace topology.Let f : N →M be an embedding, then its image f (N)⊂M is called a submanifold

of M. Notice that an open submanifold is the special case when f = i : U →M isan inclusion mapping.

2. Differentiable manifolds and differentiable structures

A topological manifold M for which the transition maps ϕi j = ϕ j ϕ−1i for all

pairs ϕi,ϕ j in the atlas are diffeomorphisms is called a differentiable, or smooth

manifold. The transition maps are mappings between open subsets of Rm. Dif-

feomorphisms between open subsets of Rm are C∞-maps, whose inverses are also

C∞-maps. For two charts Ui and Uj the transitions maps are mappings:

ϕi j = ϕ j ϕ−1i : ϕi(Ui∩Uj)→ ϕ j(Ui∩Uj),

and as such are homeomorphisms between these open subsets of Rm.

12

FIGURE 7. Differentiability of ϕ ϕ−1 is achieved via the map-pings ϕ (ϕ)−1, and ϕ ϕ−1, which are diffeomorphisms sinceA∼A

, and A ∼A

by assumption. This establishes the equiva-lence A∼A

.

Definition 2.1. A C∞-atlas is a set of charts A = (U,ϕi)i∈I such that

(i) M = ∪i∈IUi,(ii) the transition maps ϕi j are diffeomorphisms between ϕi(Ui ∩Uj) and

ϕ j(Ui∩Uj), for all i = j (see Figure 2 ).

The charts in a C∞-atlas are said to be C∞-compatible. Two C∞-atlases A and A

are equivalent if A∪A is again a C∞-atlas, which defines a equivalence relation on

C∞-atlases. An equivalence class of this equivalence relation is called a differen-

tiable structure D on M. The collection of all atlases associated with D, denotedAD, is called the maximal atlas for the differentiable structure. Figure 7 showswhy compatibility of atlases defines an equivalence relation.

Definition 2.3. Let M be a topological manifold, and let D be a differentiablestructure on M with maximal atlas AD. Then the pair (M,AD) is called a (C∞

-

)differentiable manifold.

13

Basically, a manifold M with a C∞-atlas, defines a differentiable structure onM. The notion of a differentiable manifold is intuitively a difficult concept. Indimensions 1 through 3 all topological manifolds allow a differentiable structure(only one up to diffeomorphisms). Dimension 4 is the first occurrence of manifoldswithout a differentiable structure. Also in higher dimensions uniqueness of differ-entiable structures is no longer the case as the famous example by Milnor shows;S7 has 28 different (non-diffeomorphic) differentiable structures. The first exampleof a manifold that allows many non-diffeomorphic differentiable structures occursin dimension 4; exotic R

4’s. One can also consider Cr-differentiable structures andmanifolds. Smoothness will be used here for the C∞-case.

Theorem 2.4.5 Let M be a topological manifold with a C∞-atlas A. Then there

exists a unique differentiable structure D containing A, i.e. A⊂AD.

Proof: Let A be the collection of charts that are C∞-compatible with A. By thesame reasoning as in Figure 7 we prove that all charts in A are C∞-compatible witheachother proving that A is a smooth altas. Now any chart that is C∞-compatiblewith any chart in A is C∞-compatible with any chart in A and is thus in A and A isa maximal atlas. Clearly any other maximal altas is contained in A and thereforeA = AD.

2.5 Remark. In our definition of of n-dimensional differentiable manifold weuse the local model over standard R

n, i.e. on the level of coordinates we expressdifferentiability with respect to the standard differentiable structure on R

n, or dif-feomorphic to the standard differentiable structure on R

n.

2.6 Example. The cone M = C⊂R3 as in the previous section is a differentiable

manifold whose differentiable structure is generated by the one-chart atlas (C,ϕ)as described in the previous section. As we will see later on the cone is not asmoothly embedded submanifold.

2.7 Example. M = S1 (or more generally M = Sn). As in Section 1 considerS1 with two charts via stereographic projection. We have the overlap U1 ∩U2 =S1\(0,±1), and the transition map ϕ12 = ϕ2 ϕ−1

1 given by y = ϕ12(x) = 4x ,

x = 0,±∞, and y = 0,±∞. Clearly, ϕ12 and its inverse are differentiable functionsfrom ϕ1(U1∩U2) = R\0 to ϕ2(U1∩U2) = R\0.

2.9 Example. The real projective spaces PRn, see exercises Chapter VI.

2.10 Example. The generalizations of projective spaces PRn, the so-called (k,n)-

Grassmannians GkR

n are examples of smooth manifolds.

5Lee, Lemma 1.10.

14

FIGURE 8. The transition maps for the stereographic projection.

Theorem 2.11.6 Given a set M, a collection Uαα∈A of subsets, and injective

mappings ϕα : Uα → Rm, such that the following conditions are satisfied:

(i) ϕα(Uα)⊂ Rm is open for all α;

(ii) ϕα(Uα∩Uβ) and ϕβ(Uα∩Uβ) are open in Rm for any pair α,β ∈ A;

(iii) for Uα∩Uβ = ∅, the mappings ϕα ϕ−1β : ϕβ(Uα∩Uβ)→ ϕα(Uα∩Uβ) are

diffeomorphisms for any pair α,β ∈ A;(iv) countably many sets Uα cover M.(v) for any pair p = q ∈ M, either p,q ∈Uα, or there are disjoint sets Uα,Uβ

such that p ∈Uα and q ∈Uβ.

Then M has a unique differentiable manifold structure and (Uα,ϕα) are smoothcharts.

Proof: Let us give a sketch of the proof. Let sets ϕ−1α (V ), V ⊂ R

n open, form abasis for a topology on M. Indeed, if p ∈ ϕ−1

α (V )∩ϕ−1β (W ) then the latter is again

of the same form by (ii) and (iii). Combining this with (i) and (iv) we establisha topological manifold over R

m. Finally from (iii) we derive that (Uα,ϕα) is asmooth atlas.

Let N and M be smooth manifolds (dimensions n and m respectively). Let f :N →M be a mapping from N to M.

Definition 2.12. A mapping f : N → M is said to be C∞, or smooth if for everyp ∈ N there exist charts (U,ϕ) of p and (V,ψ) of f (p), with f (U) ⊂ V , such thatf = ψ f ϕ−1 : ϕ(U)→ ψ(V ) is a C∞-mapping (from R

n to Rm).

The above definition also holds true for mappings defined on open subsets ofN, i.e. let W ⊂ N is an open subset, and f : W ⊂ N → M, then smoothness on W

6See Lee, Lemma 1.23.

15

FIGURE 9. Coordinate representation for f , with f (U)⊂V .

is defined as above by restricting to pionts p ∈W . With this definition coordinatemaps ϕ : U → R

m are smooth maps.The definition of smooth mappings allows one to introduce the notion of dif-

ferentiable homeomorphism, of diffeomorphism between manifolds. Let N and Mbe smooth manifolds. A C∞-mapping f : N → M, is called a diffeomorphism ifit is a homeomorphism and also f−1 is a smooth mapping, in which case N andM are said to be diffeomorphic. The associated differentiable structures are alsocalled diffeomorphic. Diffeomorphic manifolds define an equivalence relation. Inthe definition of diffeomorphism is suffices require that f is a differentiable bijec-tive mapping with smooth inverse (see Theorem 2.15). A mapping f : N → M iscalled a local diffeomorphism if for every p ∈ N there exists a neighborhood U ,with f (U) open in M, such that f : U → f (U) is a diffeomorphism. A mappingf : N →M is a diffeomorphism if and only if it is a bijective local diffeomorphism. 2.14 Example. Consider N = R with atlas (R,ϕ(p) = p), and M = R with atlas(R,ψ(q) = q3). Clearly these define different differentiable structures. BetweenN and M we consider the mapping f (p) = p1/3, which is a homeomorphism be-tween N and M. The claim is that f is also a diffeomorphism. Take U = V = R,then ψ f ϕ−1(p) = (p1/3)3 = p is the identity and thus C∞ on R, and the samefor ϕ f−1 ψ−1(q) = (q3)1/3 = q. The associated differentiable structures arediffeomorphic. In fact the above described differentiable structures correspond todefining the differential quotient via limh→0

f 3(p+h)− f 3(p)h .

16

Theorem 2.15.7 Let N,M be smooth manifolds with atlases AN and AM respec-

tively. For continuous maps the following properties hold.

(i) Given smooth maps fα : Uα → M, for all Uα ∈ AN, with fα|Uα∩Uβ =fβ|Uα∩Uβ for all α,β. Then there exists a unique smooth map f : N → Msuch that f |Uα = fα.

(ii) A smooth map f : N →M between smooth manifolds is continuous.(iii) Let f : N→M be continuous, and suppose that the maps fαβ = ψβ f ϕ−1

α ,for charts (Uα,ϕα)∈AN, and (Vβ,ψβ)∈AM, are smooth on their domainsfor all α,β. Then f is smooth.

Proof: Define f by f |Uα = fα, which is well-defined by the overlap conditioins.Given p∈M, there exists a chart Uα p and f = fα which is smooth by definition,and thus f is smooth.

For any p ∈U (chart) and choose f (p) ∈V (chart), then f is a smooth map andf |U = ψ−1 f ϕ : U →V is continuous. Continuity holds for each neighborhoodof p ∈M.

Let p ∈ Uα and f (p) ∈ Vβ and set U = f−1(Vβ)∩Uα ⊂ Uα which is open bycontinuity of f . Now f = fαβ on the charts (U,ϕα|U) and (Vβ,ψβ) which provesthe differentiability of f .

FIGURE 10. Coordinate diffeomorphisms and their local repre-sentation as smooth transition maps.

7See Lee Lemmas 2.1, 2.2 and 2.3.

17

With this theorem at hand we can verify differentiability locally. In order toshow that f is a diffeomorphism we need that f is a homeomorphism, or a bijectionthat satisfies the above local smoothness conditions, as well as for the inverse. 2.17 Example. Let N = M = PR

1 and points in PR1 are equivalence classes

[x] = [(x1,x2)]. Define the mapping f : PR1 → PR

1 as [(x1,x2)] → [(x21,x

22)]. Con-

sider the charts U1 = x1 = 0, and U2 = x2 = 0, with for example ϕ1(x) =tan−1(x2/x1), then ϕ−1(θ1) =

[cos(θ1),sin(θ1)]

, with θ1 ∈V1 = (−π/2,π/2). In

local coordinates on V1 we have

f (θ1) = tan−1sin2(θ1)/cos2(θ1),

and a similar expression for V2. Clearly, f is continuous and the local expressionsalso prove that f is a differentiable mapping using Theorem 2.15.

The coordinate maps ϕ in a chart (U,ϕ) are diffeomorphisms. Indeed, ϕ : U ⊂M→R

n, then if (V,ψ) is any other chart with U ∩V = ∅, then by the definition thetransition maps ψϕ−1 and ϕψ−1 are smooth. For ϕ we then have ϕ = ϕψ−1

and ϕ−1 = ψϕ−1, which proves that ϕ is a diffeomorphism, see Figure 10. Thisjustifies the terminology smooth charts. 2.18 Remark. For arbitrary subsets U ⊂ R

m and V ⊂ Rn a map f : U ⊂ R

m

→ V ⊂ Rn is said to be smooth at p ∈ U , if there exists an open set U† ⊂ R

m

containing p, and a smooth map f † : U† → Rm, such that f and f † coincide on

U ∩U†. The latter is called an extension of f . A mapping f : U → V betweenarbitrary subsets U ⊂ R

m and V ⊂ Rn is called a diffeomorphism if f maps U

homeomorphically onto V and both f and f−1 are is smooth (as just describedabove). In this case U and V are said to be diffeomorphic.

2.19 Example. An important example of a class of differentiable manifolds areappropriately chosen subsets of Euclidean space that can be given a smooth mani-fold structure. Let M ⊂ R

be a subset such that every p ∈ M has a neighborhoodU p in M (open in the subspace topology, see Section 3) which is diffeomorphicto an open subset V ⊂R

m (or, equivalently an open disc Dm ⊂Rm). In this case the

set M is a smooth m-dimensional manifold. Its topology is the subsapce topologyand the smooth structure is inherited from the standard smooth structure on R

,which can be described as follows. By definition a coordinate map ϕ is a diffeo-morphisms which means that ϕ : U →V = ϕ(U) is a smooth mapping (smoothnessvia a smooth map ϕ†), and for which also ϕ−1 is a smooth map. This then directlyimplies that the transition maps ψ ϕ−1 and ϕ ψ−1 are smooth mappings, seeFigure 11. In some books this construction is used as the definition of a smoothmanifold. In Section ?? we will refer to the above class of differentiable manifolds

18

FIGURE 11. The transitions maps ψ ϕ−1 and ϕ ψ−1 smoothmappings establishing the smooth structure on M.

as manifolds in Euclidean space, which are smoothly embedded submanifolds inEuclidean space.

2.21 Example. Let us consider the cone M = C described in Example 1 (see alsoExample 2 in Section 1). We already established that C is manifold homeomrphicto R

2, and moreover C is a differentiable manifold, whose smooth structure isdefined via a one-chart atlas. However, C is not a smooth manifold with respect tothe induced smooth structure as subset of R

3. Indeed, following the definition in theabove remark, we have U = C, and coordinate homeomorphism ϕ(p) = (p1, p2) =x. By the definition of smooth maps it easily follows that ϕ is smooth. The inverseis given by ϕ−1(x) =

x1,x2,

x2

1 + x22, which is clearly not differentiable at the

cone-top (0,0,0). The cone C is not a smoothly embedded submainfold of R3

(topological embedding).

Let U,V and W be open subsets of Rn, R

k and Rm respectively, and let f : U →V

and g : V →W be smooth maps with y = f (x), and z = g(y). Then the Jacobians

are

J f |x =

∂ f1∂x1

· · · ∂ f1∂xn

.... . .

...∂ fk∂x1

· · · ∂ fk∂xn

, Jg|y =

∂g1∂y1

· · · ∂g1∂yk

.... . .

...∂gm∂y1

· · · ∂gm∂yk

,

and J(g f )|x = Jg|y= f (x) · J f |x (chain-rule). The commutative diagram for themaps f , g and g f yields a commutative diagram for the Jacobians:

19

V

U W

g

f

g f

Rk

Rn

Rm

Jg|y

J f |x

J(g f )|x

For diffeomorphisms between open sets in Rn we have a number of important

properties. Let U ⊂Rn and V ⊂R

m be open sets, and let f : U →V is a diffeomor-phism, then n = m, and J f |x is invertible for any x ∈U . Using the above describedcommutative diagrams we have that f−1 f yields that J( f−1)|y= f (x) · J f |x is theidentity on R

n, and f f−1 yields that J f |x · J( f−1)|y= f (x) is the identity on Rm.

Thus J fx has an inverse and consequently n = m. Conversely, if f : U → Rn,

U ⊂ Rn, open, then we have the Inverse Function Theorem;

Theorem 2.22. If J f |x : Rn →R

n is invertible, then f is a diffeomorphism betweensufficiently small neighborhoods U and f (U ) of x and y respectively.

3. Immersions, submersions and embeddings

Let N and M be smooth manifolds of dimensions n and m respectively, and letf : N → M be a smooth mapping. In local coordinates f = ψ f ϕ−1 : ϕ(U)→ψ(V ), with respects to charts (U,ϕ) and (V,ψ). The rank of f at p ∈ N is definedas the rank of f at ϕ(p), i.e. rk( f )|p = rk(J f )|ϕ(p), where J f |ϕ(p) is the Jacobianof f at p:

J f |x=ϕ(p) =

∂ f1∂x1

· · · ∂ f1∂xn

.... . .

...∂ fm∂x1

· · · ∂ fm∂xn

This definition is independent of the chosen charts, see Figure 12. Via the commu-tative diagram in Figure 12 we see that ˜f = (ψ ψ−1) f (ϕ ϕ−1)−1, and by thechain rule J ˜f |x = J(ψ ψ−1)|y ·J f |x ·J(ϕ ϕ−1)−1|x . Since ψ ψ−1 and ϕ ϕ−1

are diffeomorphisms it easily follows that rk(J f )|x = rk(J ˜f )|x , which shows thatour notion of rank is well-defined. If a map has constant rank for all p ∈N we sim-ply write rk( f ). These are called constant rank mappings. Let us now considerthe various types of constant rank mappings between manifolds.

20

FIGURE 12. Representations of f via different coordinate charts.

Definition 3.2. A mapping f : N → M is called an immersion if rk( f ) = n, and asubmersion if rk( f ) = m. An immersion that is injective,8 or 1-1, and is a home-omorphism onto (surjective mapping9) its image f (N) ⊂ M, with respect to thesubspace topology, is called a smooth embedding.

A smooth embedding is an (injective) immersion that is a topological embed-ding.

Theorem 3.3.10 Let f : N → M be smooth with constant rank rk( f ) = k. Then

for each p ∈ N, and f (p) ∈ M, there exist coordinates (U,ϕ) for p and (V,ψ) forf (p), with f (U)⊂V , such that

(ψ f ϕ−1)(x1, · · ·xk,xk+1, · · ·xn) = (x1, · · ·xk,0, · · · ,0).

3.4 Example. Let N = (−π2 , 3π

2 ), and M = R2, and the mapping f is given by

f (t) = (sin(2t),cos(t)). In Figure 13 we displayed the image of f . The Jacobian isgiven by

J f |t =

2cos(2t)−sin(t)

Clearly, rk(J f |t) = 1 for all t ∈ N, and f is an injective immersion. Since N isan open manifold and f (N) ⊂ M is a compact set with respect to the subspace

8A mapping f : N →M is injective if for all p, p ∈N for which f (p) = f (p) it holds that p = p.9A mapping f : N →M is surjective if f (N) = M, or equivalently for every q ∈M there exists a

p ∈ N such that q = f (p).10See Lee, Theorem 7.8 and 7.13.

21

topology, it follows f is not a homeomorphism onto f (N), and therefore is not anembedding.

FIGURE 13. Injective parametrization of the figure eight.

3.6 Example. Let N = S1 be defined via the atlas A = (Ui,ϕi), with ϕ−11 (t) =

((cos(t),sin(t)), and ϕ−12 (t) = ((sin(t),cos(t)), and t ∈ (−π

2 , 3π2 ). Furthermore, let

M = R2, and the mapping f : N → M is given in local coordinates; in U1 as in

Example 1. Restricted to S1 ⊂ R2 the map f can also be described by

f (x,y) = (2xy,x).

This then yields for U2 that f (t) = (sin(2t),sin(t)). As before rk( f ) = 1, whichshows that f is an immersion of S1. However, this immersion is not injective atthe origin in R

2, indicating the subtle differences between these two examples, seeFigures 13 and 14.

FIGURE 14. Non-injective immersion of the circle.

3.8 Example. Let N = R, M = R2, and f : N →M defined by f (t) = (t2, t3). We

can see in Figure 15 that the origin is a special point. Indeed, rk(J f )|t = 1 for allt = 0, and rk(J f )|t = 0 for t = 0, and therefore f is not an immersion.

3.10 Example. Consider M = PRn. We established PR

n as smooth manifolds.For n = 1 we can construct an embedding f : PR

1 → R2 as depicted in Figure 16.

For n = 2 we find an immersion f : PR2 → R

3 as depicted in Figure 17.

22

FIGURE 15. The map f fails to be an immersion at the origin.

FIGURE 16. PR is diffeomorphic to S1.

FIGURE 17. Identifications for PR2 giving an immersed non-

orientable surface in R3.

3.13 Example. Let N = R2 and M = R, then the projection mapping f (x,y) = x

is a submersion. Indeed, J f |(x,y) = (1 0), and rk(J f |(x,y)) = 1.

23

3.14 Example. Let N = M = R2 and consider the mapping f (x,y) = (x2,y)t . The

Jacobian is

J f |(x,y) =

2x 00 1

,

and rk(J f )|(x,y) = 2 for x = 0, and rk(J f )|(x,y) = 1 for x = 0. See Figure 18.

FIGURE 18. The projection (submersion), and the folding of theplane (not a submersion).

3.16 Example. We alter Example 2 slightly, i.e. N = S1 ⊂ R2, and M = R

2, andagain use the atlas A. We now consider a different map f , which, for instance onU1, is given by f (t) = (2cos(t),sin(t)) (or globally by f (x,y) = (2x,y)). It is clearthat f is an injective immersion, and since S1 is compact it follows from Lemma3.18 below that S1 is homeomorphic to it image f (S1), which show that f is asmooth embedding (see also Appendix in Lee).

3.17 Example. Let N = R, M = R2, and consider the mapping f (t) =

(2cos(t),sin(t)). As in the previous example f is an immersion, not injective how-ever. Also f (R) = f (S1) in the previous example. The manifold N is the universalcovering of S1 and the immersion f descends to a smooth embedding of S1.

Lemma 3.18.11 Let f : N →M be an injective immersion. If

(i) N is compact, or if(ii) f is a proper map,12

then f is a smooth embedding.

Proof: For (i) we argue as follows. Since N is compact any closed subset X ⊂ Nis compact, and therefore f (X) ⊂ M is compact and thus closed; f is a closed

11Lee, Prop. 7.4, and pg. 47.12A(ny) map f : N →M is called proper if for any compact K ⊂M, it holds that also f−1(K)⊂N

is compact. In particular, when N is compact, continuous maps are proper.

24

mapping.13 We are done if we show that f is a topological embedding. By theassumption of injectivity, f : N → f (N) is a bijection. Let X ⊂ N be closed, then( f−1)−1(X) = f (X) ⊂ f (N) is closed with respect to the subspace topology, andthus f−1 : f (N)→ N is continuous, which proves that f is a homeomorphism ontoits image and thus a topological embedding.

A straightforward limiting argument shows that proper continuous mappingsbewteen manifolds are closed mappings (see Exercises).

Let us start with defining the notion of embedded submanifolds.

Definition 3.19. A subset N ⊂ M is called a smooth embedded n-dimensional

submanifold in M if for every p ∈ N, there exists a chart (U,ϕ) for M, with p ∈U,such that

ϕ(U ∩N) = ϕ(U)∩R

n×0

= x ∈ ϕ(U) : xn+1 = · · · = xm = 0.

The co-dimension of N is defined as codimN = dimM−dimN.

The set W = U ∩N in N is called a n-dimensional slice, or n-slice of U , seeFigure 19, and (U,ϕ) a slice chart. The associated coordinates x = (x1, · · · ,xn) arecalled slice coordinates. Embedded submanifolds can be characterized in terms ofembeddings.

FIGURE 19. Take a k-slice W . On the left the image ϕ(W ) corre-sponds to the Eucliden subspace R

k ⊂ Rm.

Theorem 3.21.14 Let N ⊂ M be a smooth embedded n-submanifold. Endowed

with the subspace topology, N is a n-dimensional manifold with a unique (induced)smooth structure such that the inclusion map i : N →M is an embedding (smooth).

13 A mapping f : N → M is called closed if f (X) is closed in M for any closed set X ⊂ N.Similarly, f is called open if f (X) is open in M for every open set X ⊂ N.

14See Lee, Thm’s 8.2

25

To get an idea let us show that N is a topological manifold. Axioms (i) and(iii) are of course satisfied. Consider the projection π : R

m → Rn, and an inclusion

j : Rn → R

m defined by

π(x1, · · · ,xn,xn+1, · · · ,xm) = (x1, · · · ,xn),

j(x1, · · · ,xn) = (x1, · · · ,xn,0, · · · ,0).

Now set Z = (π ϕ)(W ) ⊂ Rn, and ϕ = (π ϕ)|W , then ϕ−1 = (ϕ−1 j)|Z , and

ϕ : W → Z is a homeomorphism. Therefore, pairs (W, ϕ) are charts for N, whichform an atlas for N. The inclusion i : N →M is a topological embedding.

Given slice charts (U,ϕ) and (U ,ϕ) and associated charts (W, ϕ) and (W , ϕ)for N. For the transitions maps it holds that ϕ ϕ−1 = π ϕ ϕ−1 j, which arediffeomorphisms, which defines a smooth atlas for N. The inclusion i : N →M canbe expressed in local coordinates;

i = ϕ i ϕ−1, (x1, · · · ,xn) → (x1, · · · ,xn,0, · · · ,0),

which is an injective immersion. Since i is also a topological embedding it is thusa smooth embedding. It remains to prove that the smooth structure is unique, seeLee Theorem 8.2.Theorem 3.22.

15 The image of an embedding is a smooth embedded submanifold.

Proof: By assumption rk( f ) = n and thus for any p∈N it follows from Theorem3.3 that

f (x1, · · · ,xn) = (x1, · · · ,xn,0, · · · ,0),

for appropriate coordinates (U,ϕ) for p and (V,ψ) for f (p), with f (U)⊂V . Con-sequently, f (U) is a slice in V , since ψ( f (U) satisfies Definition 3.19. By assump-tion f (U) is open in f (N) and thus f (U) = A∩ f (N) for some open set A⊂M. Byreplacing V by V = A∩V and restricting ψ to V , (V ,ψ) is a slice chart with sliceV ∩ f (N) = V ∩ f (U).

Summarizing we conclude that embedded submanifolds are the images ofsmooth embeddings.

A famous result by Whitney says that considering embeddings into Rm is not

not really a restriction for defining smooth manifolds.Theorem 3.23.

16 Any smooth n-dimensional manifold M can be (smoothly) em-bedded into R

2n+1.

A subset N ⊂ M is called an immersed submanifold if N is a smooth n-dimensional manifold, and the mapping i : N →M is a (smooth) immersion. Thismeans that we can endow N with an appropriate manifold topology and smooth

15See Lee, Thm’s 8.316See Lee, Ch. 10.

26

structure, such that the natural inclusion of N into M is an immersion. If f : N →Mis an injective immersion we can endow f (N) with a topology and unique smoothstructure; a set U ⊂ f (N) is open if and only if f−1(U) ⊂ N is open, and the(smooth) coordinate maps are taken to be ϕ f−1, where ϕ’s are coordinate mapsfor N. This way f : N → f (N) is a diffeomorphism, and i : f (N) →M an injectiveimmersion via the composition f (N)→ N →M. This proves:Theorem 3.24.

17 Immersed submanifolds are exactly the images of injective im-mersions.

We should point out that embedded submanifolds are examples of immersedsubmanifolds, but not the other way around. For any immersion f : N → M, theimage f (N) is called an immersed manifold in M.

In this setting Whitney established some improvements of Theorem 3.23.Namely, for dimension n > 0, any smooth n-dimensional manifold can be em-bedded into R

2n (e.g. the embedding of curves in R2). Also, for n > 1 any smooth

n-dimensional manifold can be immersed into R2n−1 (e.g. the Klein bottle). In

this course we will often think of smooth manifolds as embedded submanifolds ofR

m. An important tool thereby is the general version of Inverse Function Theorem,

FIGURE 20. An embedding of R [left], and an immersion of S2

[right] called the Klein bottle.

which can easily be derived from the ‘Euclidean’ version 2.22.Theorem 3.26.

18 Let N,M be smooth manifolds, and f : N → M is a smoothmapping. If, at some point p ∈ N, it holds that J f |ϕ(p)=x is an invertible matrix,then there exist sufficiently small neighborhoods U0 p, and V0 ∈ f (p) such thatf |U0 : U0 →V0 is a diffeomorphism.

As a direct consequence of this result we have that if f : N →M, with dimN =dimM, is an immersion, or submersion, then f is a local diffeomorphism. If f is abijection, then f is a (global) diffeomorphism.

17See Lee, Theorem 8.16.18See Lee, Thm’s 7.6 and 7.10.

27

Theorem 3.27. Let f : N → M be a constant rank mapping with rk( f ) = k. Thenfor each q ∈ f (N), the level set S = f−1(q) is an embedded submanifold in N withco-dimension equal to k.

Proof: Clearly, by continuity S is closed in N. By Theorem 3.3 there are coor-dinates (U,ϕ) of p ∈ S and (V,ψ) of f (p) = q, such that

f (x1, · · · ,xk,xk+1, · · · ,xn) = (x1, · · · ,xk,0, · · · ,0) = ψ(q) = 0.

The points in S∩U are characterized by ϕ(S∩U) = x | (x1, · · · ,xk) = 0. There-fore, S∩U is a (n−k)-slice, and thus S is a smooth submanifold in M of codimen-sion k.

In particular, when f : N → M is a submersion, then for each q ∈ f (N), thelevel set S = f−1(q) is an embedded submanifold of co-dimension codimS = m =dimM. In the case of maximal rank this statement can be restricted to just onelevel. A point p ∈ N is called a regular point if rk( f )|p = m = dimM, otherwisea point is called a critical point. A level q ∈ f (N) is called a regular level if allpoints p ∈ f−1(q) are regular points, otherwise a level is called a critical level.

Theorem 3.28. Let f : N → M be a smooth map. If q ∈ f (N) is a regular value,then S = f−1(q) is an embedded submanifold of co-dimension equal to dimM.

Proof: Let us illustrate the last result for the important case N = Rn and M = R

m.For any p ∈ S = f−1(q) the Jacobian J f |p is surjective by assumption. Denote thekernel of J f |p by kerJ f |p ⊂ R

n, which has dimension n−m. Define

g : N = Rn → R

n−m×Rm ∼= R

n,

by g(ξ) = (Lξ, f (ξ)− q)t , where L : N = Rn → R

n−m is any linear map whichis invertible on the subspace kerJ f |p ⊂ R

n. Clearly, Jg|p = L⊕ J f |p, which, byconstruction, is an invertible (linear) map on R

n. Applying Theorem 2.22 (InverseFunction Theorem) to g we conclude that a sufficiently small neighborhood of Uof p maps diffeomorphically onto a neighborhood V of (L(p),0). Since g is adiffeomorphism it holds that g−1 maps

R

n−m×0∩V onto f−1(q)∩U (the 0∈

Rm corresponds to q). This exactly says that every point p ∈ S allows an (n−m)-

slice and is therefore an (n−m)-dimensional submanifold in N = Rn (codimS =

m).

3.30 Example. Let us start with an explicit illustration of the above proof. LetN = R

2, M = R, and f (p1, p2) = p21 + p2

2. Consider the regular value q = 2, thenJ f |p = (2p1 2p2), and f−1(2) = p : p2

1 + p22 = 2, the circle with radius

√2.

We have kerJ f |p = span(p1,−p2)t, and is always isomorphic to R. For example

28

FIGURE 21. The map g yields 1-slices for the set S.

fix the point (1,1) ∈ S, then kerJ f |p = span(1,−1)t and define

g(ξ) =

L(ξ)

f (ξ)−2

=

ξ1−ξ2

ξ21 +ξ2

2−2

, Jg|p =

1 −12 2

where the linear map is L = (1 − 1). The map g is a local diffeomorphism andon S∩U this map is given by

g(ξ1,

2−ξ21) =

ξ1−

2−ξ2

1

0

,

with ξ1 ∈ (1− ε,1 + ε). The first component has derivative 1 + ξ1√2−ξ2

1, and there-

fore S∩U is mapped onto a set of the formR×0

∩V . This procedure can be

carried out for any point p ∈ S, see Figure 21.

3.31 Example. Let N = R2\(0,0)×R = R

3\(0,0,λ), M = (−1,∞)×(0,∞),and

f (x,y,z) =

x2 + y2−1

1

, with J f |(x,y,z) =

2x 2y 00 0 0

We see immediately that rk( f ) = 1 on N. This map is not a submersion, but is ofconstant rank, and therefore for any value q ∈ f (N) ⊂ M it holds that S = f−1(q)is a embedded submanifold, see Figure 22.

3.33 Example. Let N,M as before, but now take

f (x,y,z) =

x2 + y2−1

z

, with J f |(x,y,z) =

2x 2y 00 0 1

Now rk( f ) = 2, and f is a submersion. For every q ∈ M, the set S = f−1(q) is aembedded submanifold, see Figure 23.

3.35 Example. Let N = R2, M = R, and f (x,y) = 1

4(x2−1)2 + 12 y2. The Jacobian

is J f |(x,y) = (x(x2−1) y). This is not a constant rank, nor a submersion. However,

29

FIGURE 22. An embedding of a cylinder via a constant rank map-ping.

FIGURE 23. An embedding circle of an submersion.

any q > 0 is a regular value since then the rank is equal to 1. Figure 24 shows thelevel set f−1(0) (not an embedded manifold), and f−1(1) (an embedded circle)

FIGURE 24. A regular and critical level set.

Theorem 3.37.19 Let S ⊂M be a subset of a smooth m-dimensional manifold M.

Then S is a k-dimensional smooth embedded submanifold if and if for every p ∈ S

19See Lee, Prop.8.12.

30

there exists a neighborhood U p such that U ∩S is the level set of a submersionf : U → R

m−k.

Proof: Assume that S ⊂ M is a smooth embedded manifold. Then for eachp ∈ S there exists a chart (U,ϕ), p ∈U , such that ϕ(S∩U) = x : xk+1 = · · · =xm = 0. Clearly, S∩U is the sublevel set of f : U → R

m−k at 0, given by f (x) =(xk+1, · · · ,xm), which is a submersion.

Conversely, if S∩U = f−1(0), for some submersion f : U → Rm−k, then by

Theorem 3.28, S∩U is an embedded submanifold of U . This clearly shows that Sis an embedded submanifold in M.

31

II. Tangent and cotangentspaces

4. Tangent spaces

For a(n) (embedded) manifold M ⊂ R the tangent space TpM at a point p ∈

M can be pictured as a hyperplane tangent to M. In Figure 25 we consider theparametrizations x+tei in R

m. These parametrizations yield curves γi(t) = ϕ−1(x+tei) on M whose velocity vectors are given by

γ(0) =ddt

ϕ−1(x+ tei)t=0

= Jϕ−1|x(ei).

The vectors p+γ(0) are tangent to M at p and span an m-dimensional affine linearsubspace Mp of R

. Since the vectors Jϕ−1|x(ei) span TpM the affine subspace isgiven by

Mp := p+TpM ⊂ R,

which is tangent to M at p.

FIGURE 25. Velocity vectors of curves of M span the ‘tangentspace’.

32

The considerations are rather intuitive in the sense that we consider only em-bedded manifolds, and so the tangent spaces are tangent m-dimensional affine sub-spaces of R

. One can also define the notion of tangent space for abstract smoothmanifolds. There are many ways to do this. Let us describe one possible way (seee.g. Lee, or Abraham, Marsden and Ratiu) which is based on the above considera-tions.

Let a < 0 < b and consider a smooth mapping γ : I = (a,b)⊂R→M, such thatγ(0) = p. This mapping is called a (smooth) curve on M, and is parametrized byt ∈ I. If the mapping γ (between the manifolds N = I, and M) is an immersion,then γ is called an immersed curve. For such curves the ‘velocity vector’ Jγ|t =(ϕ γ)(t) in R

m is nowhere zero. We build the concept of tangent spaces in orderto define the notion velocity vector to a curve γ.

Let (U,ϕ) be a chart at p. Then, two curves γ and γ are equivalent, γ∼ γ, if

γ(0) = γ(0) = p, and (ϕ γ)(0) = (ϕ γ)(0).

The equivalence class of a curve γ through p ∈M is denoted by [γ].

Definition 4.2.20 At a p ∈ M define the tangent space TpM as the space of all

equivalence classes [γ] of curves γ through p. A tangent vector Xp, as the equiva-lence class of curves, is given by

Xp := [γ] =

γ : γ(0) = γ(0) = p, (ϕ γ)(0) = (ϕ γ)(0),

which is an element of TpM.

FIGURE 26. Immersed curves and velocity vectors in Rm.

20 Lee, Ch. 3.

33

The above definition does not depend on the choice of charts at p ∈ M. Let(U ,ϕ) be another chart at p ∈ M. Then, using that (ϕ γ)(0) = (ϕ γ)(0), for(ϕ γ)(0) we have

(ϕ γ)(0) =(ϕ ϕ−1) (ϕ γ)

(0)

= J(ϕ ϕ−1)x(ϕ γ)(0)

= J(ϕ ϕ−1)x(ϕ γ)(0)

=(ϕ ϕ−1) (ϕ γ)

(0) = (ϕ γ)(0),

which proves that the equivalence relation does not depend on the particular choiceof charts at p ∈M.

One can prove that TpM ∼= Rm. Indeed, TpM can be given a linear structure as

follows; given two equivalence classes [γ1] and [γ2], then

[γ1]+ [γ2] :=

γ : (ϕ γ)(0) = (ϕ γ1)(0)+(ϕ γ2)(0),

λ[γ1] :=

γ : (ϕ γ)(0) = λ(ϕ γ1)(0).

The above argument shows that these operation are well-defined, i.e. indepen-dent of the chosen chart at p ∈M, and the operations yield non-empty equivalenceclasses. The mapping

τϕ : TpM → Rm, τϕ

[γ]

= (ϕ γ)(0),

is a linear isomorphism and τϕ = J(ϕ ϕ−1)|x τϕ. Indeed, by considering curvesγi(x) = ϕ−1(x+ tei), i = 1, ...,m, it follows that [γi] = [γ j], i = j, since

(ϕ γi)(0) = ei = e j = (ϕ γ j)(0).

This proves the surjectivity of τϕ. As for injectivity one argues as follows. Suppose,(ϕ γ)(0) = (ϕγ)(0), then by definition [γ] = [γ], proving injectivity.

Given a smooth mapping f : N →M we can define how tangent vectors in TpNare mapped to tangent vectors in TqM, with q = f (p). Choose charts (U,ϕ) forp ∈ N, and (V,ψ) for q ∈ M. We define the tangent map or pushforward of f asfollows, see Figure 27. For a given tangent vector Xp = [γ] ∈ TpN,

d fp = f∗ : TpN → TqM, f∗([γ]) = [ f γ].

The following commutative diagram shows that f∗ is a linear map and its definitiondoes not depend on the charts chosen at p ∈ N, or q ∈M.

TpNf∗−−−−→ TqM

τϕ

τψ

Rn J(ψ fϕ−1)|x−−−−−−−→ R

m

34

FIGURE 27. Tangent vectors in Xp ∈ TpN yield tangent vectorsf∗Xp ∈ TqM under the pushforward of f .

Indeed, a velocity vector (ϕ γ)(0) is mapped to (ψ f (γ))(0), and

(ψ f (γ))(0) = (ψ f ϕ−1 ϕ γ)(0) = J fx·(ϕ γ)(0).

Clearly, this mapping is linear and independent of the charts chosen.If we apply the definition of pushforward to the coordinate mapping ϕ : N →R

n,then τϕ can be identified with ϕ∗, and J(ψ f ϕ−1)|x with (ψ f ϕ−1)∗. Indeed,τϕ([γ]) = (ϕ γ)(0) and ϕ∗([γ]) = [ϕ γ], and in R

n the equivalence class can belabeled by (ϕ γ)(0). The labeling map is given as follows

τid([ϕ γ]) = (ϕ γ)(0),

and is an isomorphism, and satisfies the relations

τid ϕ∗ = τϕ, ϕ∗ = τ−1id τϕ.

From now one we identify TxRn with R

n by identifying ϕ∗ and τϕ. This justifiesthe notation

ϕ∗([γ]) = [ϕ γ] := (ϕ γ)(0).

Properties of the pushforward can be summarized as follows:

35

Lemma 4.5.21 Let f : N →M, and g : M → P be smooth mappings, and let p∈M,

then

(i) f∗ : TpN → Tf (p)M, and g∗ : Tf (p)M → T(g f )(p)P are linear maps (homo-morphisms),

(ii) (g f )∗ = g∗ · f∗ : TpN → T(g f )(p)P,(iii) (id)∗ = id : TpN → TpN,(iv) if f is a diffeomorphism, then the pushforward f∗ is a isomorphism from

TpN to Tf (p)M.

Proof: We have that f∗([γ]) = [ f γ], and g∗([ f γ]) = [g f γ], which definesthe mapping (g f )∗([γ]) = [g f γ]. Now

[g f γ] = [g ( f γ)] = g∗([ f γ]) = g∗( f∗([γ])),

which shows that (g f )∗ = g∗ · f∗.

A parametrization ϕ−1 : Rm → M coming from a chart (U,ϕ) is a local dif-

feomorphism, and can be used to find a canonical basis for TpM. Choosing localcoordinates x = (x1, · · · ,xn) = ϕ(p), and the standard basis vectors ei for R

m, wedefine

∂∂xi

p:= ϕ−1

∗ (ei).

By definition ∂∂xi

p∈ TpM, and since the vectors ei form a basis for R

m, the vectors∂

∂xi

p form a basis for TpM. An arbitrary tangent vector Xp ∈ TpM can now be

written with respect to the basis ∂∂xi

p:

Xp = ϕ−1∗ (Xiei) = Xi

∂∂xi

p.

where the notation Xi∂

∂xi

p= ∑i Xi

∂∂xi

p denotes the Einstein summation convention,

and (Xi) is a vector in Rm!

We now define the directional derivative of a smooth function h : M → R in thedirection of Xp ∈ TpM by

Xph := h∗Xp = [h γ].

In fact we have that Xph = (h γ)(0), with Xp = [γ]. For the basis vectors of TpMthis yields the following. Let γi(t) = ϕ−1(x+ tei), and Xp = ∂

∂xi

p, then

(2) Xph =∂

∂xi

ph =

(hϕ−1) (ϕ γi)

(0) =∂h∂xi

,

in local coordinates, which explains the notation for tangent vectors. In particular,for general tangent vectors Xp, Xph = Xi

∂h∂xi

.

21Lee, Lemma 3.5.

36

Let go back to the curve γ : N = (a,b)→M and express velocity vectors. Con-sider the chart (N, id) for N, with coordinates t, then

ddt

t=0

:= id−1∗ (1).

We have the following commuting diagrams:

Nγ−−−−→ M

id

ϕ

Rγ=ϕγ−−−−→ R

m

TtNγ∗−−−−→ TpM

id∗

ϕ∗

Rγ∗−−−−→ R

m

We now define

γ(0) = γ∗ d

dt

t=0

= ϕ−1

∗(ϕ γ)(0)

∈ TpM,

by using the second commuting diagram.If we take a closer look at Figure ?? we can see that using different charts at

p ∈M, or equivalently, considering a change of coordinates leads to the followingrelation. For the charts (U,ϕ) and (U ,ϕ) we have local coordinates x = ϕ(p)and x = ϕ(p), and p ∈U ∩U . This yields two different basis for TpM, namely ∂

∂xi

p

, and

∂∂xi

p

. Consider the identity mapping f = id : M→M, and the push-

forward yields the identity on TpM. If we use the different choices of coordinatesas described above we obtain

id∗

∂∂xi

p

h = (ϕ ϕ−1)∗

∂h∂xj

=∂xj∂xi

∂h∂xj

.

In terms of the different basis for TpM this gives

∂∂xi

p=

∂xj∂xi

∂∂xj

p.

Let us prove this formula by looking a slightly more general situation. Let N,Mbe smooth manifolds and f : N → M a smooth mapping. Let q = f (p) and (V,ψ)is a chart for M containing q. The vectors ∂

∂y j

q form a basis for TqM. If we write

Xp = Xi∂

∂xi

p, then for the basis vectors ∂

∂xi

p we have

f∗

∂∂xi

p

h =

∂∂xi

p(h f ) =

∂∂xi

h f

=∂

∂xi

h f ϕ−1

=∂

∂xi

hψ f ϕ−1 =

∂∂xi

h f

=∂h∂y j

∂ f j

∂xi=

∂ f j

∂xi

∂∂y j

q

h

37

which implies that TpN is mapped to TqM under the map f∗. In general a tangentvector Xi

∂∂xi

p is pushed forward to a tangent vector

(3) Yj∂

∂y j

q=

∂ f j

∂xiXi

∂

∂y j

q∈ TqM,

expressed in local coordinates. By taking N = M and f = id we obtain the abovechange of variables formula.

5. Cotangent spaces

In linear algebra it is often useful to study the space of linear functions on agiven vector space V . This space is denoted by V ∗ and called the dual vector space

to V — again a linear vector space. So the elements of V ∗ are linear functionsθ : V → R. As opposed to vectors v ∈V , the elements, or vectors in V ∗ are calledcovectors.

Lemma 5.1. Let V be a n-dimensional vector space with basis v1, · · · ,vn, thenthe there exist covectors θ1, · · · ,θn such that

θi · v j := θi(v j) = δij, Kronecker delta,

and the covectors θ1, · · · ,θn form a basis for V ∗.

This procedure can also be applied to the tangent spaces TpM described in theprevious chapter.

Definition 5.2. Let M be a smooth m-dimensional manifold, and let TpM be thetangent space at some p ∈M. The the cotangent space T ∗p M is defined as the dualvector space of TpM, i.e.

T ∗p M := (TpM)∗.

By definition the cotangent space T ∗p M is also m-dimensional and it has a canon-ical basis as described in Lemma 5.1. As before have the canonical basis vectors

∂∂xi

|p for TpM, the associated basis vectors for T ∗p M are denoted dxi|p. Let us nowdescribe this dual basis for T ∗p M and explain the notation.

The covectors dxi|p are called differentials and we show now that these areindeed related dhp. Let h : M → R be a smooth function, then h∗ : TpM → R

and h∗ ∈ T ∗p M. Since the differentials dxi|p form a basis of T ∗p M we have thath∗ = λidxi|p, and therefore

h∗∂

∂xi

p= λ jdx j|p ·

∂∂xi

p= λ jδ j

i = λi =∂h∂xi

,

38

and thus

(4) dhp = h∗ =∂h∂xi

dxip.

Choose h such that h∗ satisfies the identity in Lemma 5.1, i.e. let h = xi ( h = xi ϕ= ϕ,ei). These linear functions h of course span T ∗p M, and

h∗ = (xi ϕ)∗ = d(xi ϕ)p = dxi|p.

Cotangent vectors are of the form

θp = θidxip.

The pairing between a tangent vector Xp and a cotangent vector θp is expressedcomponent wise as follows:

θp ·Xp = θiXjδij = θiXi.

In the case of tangent spaces a mapping f : N → M pushes forward to a linearf∗;TpN → TqM for each p ∈ N. For cotangent spaces one expects a similar con-struction. Let q = f (p), then for a given cotangent vector θq ∈ T ∗q M define

( f ∗θq) ·Xp = θq · ( f∗Xp) ∈ T ∗p N,

for any tangent vector Xp ∈ TpN. The homomorphism f ∗ : T ∗q M→ T ∗p N, defined byθq → f ∗θq is called the pullback of f at p. It is a straightforward consequence fromlinear algebra that f ∗ defined above is indeed the dual homomorphism of f∗, alsocalled the adjoint, or transpose (see Lee, Ch. 6, for more details, and compare thethe definition of the transpose of a matrix). If we expand the definition of pullbackin local coordinates, using (3), we obtain

f ∗dy j

q

·Xi

∂∂xi

p

= dy jq· f∗

Xi

∂∂xi

p

= dy jq

∂ f j

∂xiXi

∂

∂y j

q=

∂ f j

∂xiXi

Using this relation we obtain that

f ∗σ jdy jq

·Xi

∂∂xi

p= σ j ∂ f j

∂xiXi = σ j ∂ f j

∂xidxi

pXi∂

∂xi

p,

which produces the local formula

(5) f ∗σ jdy jq=

σ j ∂ f j

∂xi

dxi

p=

σ j f

∂ f j

∂xi

x

dxip.

Lemma 5.3. Let f : N →M, and g : M → P be smooth mappings, and let p ∈M,then

39

(i) f ∗ : T ∗f (p)M → T ∗p N, and g∗ : T ∗(g f )(p)P→ T ∗f (p)M are linear maps (homo-morphisms),

(ii) (g f )∗ = f ∗ ·g∗ : T ∗(g f )(p)P→ T ∗p N,(iii) (id)∗ = id : T ∗p N → T ∗p N,(iv) if f is a diffeomorphism, then the pullback f ∗ is a isomorphism from T ∗f (p)M

to T ∗p N.

Proof: By definition of the pullbacks of f and g we have

f ∗θq ·Xp = θq · f∗(Xp), g∗ωg(q)Yq = ωg(q) ·g∗(Yq).

For the composition g f it holds that (g f )∗ω(g f )(p) ·Xp = ω(g f )(p) ·(g f )∗(Xp).Using Lemma 4.5 we obtain

(g f )∗ω(g f )(p) ·Xp = ω(g f )(p) ·g∗

f∗(Xp)

= g∗ω(g f )(p) · f∗(Xp) = f ∗g∗ω(g f )(p)

·Xp,

which proves that (g f )∗ = f ∗ ·g∗.

Now consider the coordinate mapping ϕ : U ⊂ M → Rm, which is local diffeo-

morphism. Using Lemma 5.3 we then obtain an isomorphism ϕ∗ : Rm → T ∗p M,

which justifies the notation

dxi|p = ϕ∗(ei).

6. Vector bundles

The abstract notion of vector bundle consists of topological spaces E (totalspace) and M (the base) and a projection π : E → M (surjective). To be moreprecise:

Definition 6.1. A triple (E,M,π) is called a real vector bundle of rank k over M if

(i) for each p ∈ M, the set Ep = π−1(p) is a real k-dimensional linear vectorspace, called the fiber over p, such that

(ii) for every p∈M there exists a open neighborhood U p, and a homeomor-phism Φ : π−1(U)→U×R

k;(a)

πΦ−1(p,ξ) = p, for all ξ ∈ R

k;(b) ξ →Φ−1(p,ξ) is a vector space isomorphism between R

k and Ep.

The homeomorphism Φ is called a local trivialization of the bundle.

40

FIGURE 28. Charts in a vector bundle E over M.

If there is no ambiguity about the base space M we often denote a vector bundleby E for short. Another way to denote a vector bundle that is common in theliterature is π : E →M. It is clear from the above definition that if M is a topologicalmanifold then so is E. Indeed, via the homeomorphisms Φ it follows that Eis Hausdorff and has a countable basis of open set. Define ϕ =

ϕ× IdRk

Φ,

and ϕ : π−1(U)→ V ×Rk is a homeomorphism. Figure 28 explains the choice of

bundle charts for E related to charts for M.For two trivializations Φ : π−1(U)→U×R

k and Ψ : π−1(V )→V×Rk, we have

that the transition map ΨΦ−1 : (U ∩V )×Rk → (U ∩V )×R

k has the followingform

ΨΦ−1(p,ξ) = (p,τ(p)ξ)),

where τ : U ∩V → Gl(k,R) is continuous. It is clear from the assumptions thatΨ Φ−1(p,ξ) = (p,σ(p,ξ)). By assumption we also have that ξ → Φ−1(p,ξ =A(p)ξ, and ξ →Ψ−1(p,ξ) = B(p)ξ. Now,

ΨΦ−1(p,ξ) = Ψ(A(p)ξ) = (p,σ(p,ξ)),

and Ψ−1(p,σ) = A(p)ξ = B(p)σ. Therefore σ(p,ξ) = B−1Aξ =: τ(p)ξ. Continuityis clear from the assumptions in Definition 6.1.

If both E and M are smooth manifolds and π is a smooth projection, such that thelocal trivializations can be chosen to be diffeomorphisms, then (E,M,π) is calleda smooth vector bundle. In this case the maps τ are smooth. The following resultallows us to construct smooth vector bundles and is important for the special vectorbundles used in this course.

41

FIGURE 29. Transition mappings for local trivializations.

Theorem 6.4.22 Let M be a smooth manifold, and let Epp∈M be a family of k-

dimensional real vector spaces parametrized by p ∈M. Define E =F

p∈M Ep, andπ : E →M as the mapping that maps Ep to p ∈M. Assume there exists

(i) an open covering Uαα∈A for M;(ii) for each α ∈ A, a bijection Φα : π−1(Uα) → Uα ×R

k, such that ξ →Φ−1

α (p,ξ) is a vector space isomorphism between Rk and Ep;

(iii) for each α,β ∈ I, with Uα ∩Uβ = ∅, smooth mappings ταβ : Uα ∩Uβ →Gl(k,R) such that

Φβ Φ−1α (p,ξ) = (p,ταβ(p)ξ).

Then E has a unique differentiable (manifold) structure making (E,M,π) a smoothvector bundle of rank k over M.

Proof: The proof of this theorem goes by verifying the hypotheses in Theorem2.11. Let us propose charts for E. Let (Vp,ϕp), Vp ⊂Uα, be a smooth chart forM containing p. As before define ϕp : π−1(Vp)→ Vp×R

k. We show now, usingTheorem 2.11, that (π−1(Vp),ϕp), p ∈M are smooth charts, which gives a uniquedifferentiable manifold structure.

22See Lee, Lemma 5.5.

42

6.5 Example. Consider the set E defined as

E =(p1, p2,ξ1,ξ2) | p1 = cos(θ), p2 = sin(θ), cos(θ/2)ξ1 + sin(θ/2)ξ2 = 0

.

Clearly, E is a smooth vector bundle over S1 of rank 1, embedded in R4. For

example if U = S1\(1,0) (θ ∈ (0,2π)), then Φ(π−1(U)) = (p1, p2,ξ1) is a localtrivialization. This bundle is called the Mobius strip.

To show that E is a smooth vector bundle we need to verify the conditions inTheorem 6.4. Consider a second chart U = S1\(−1,0) (θ ∈ (−π,π)), and thetrivialization Ψ(π−1(U )) = (p1, p2,ξ2). For the Φ we have that

Φ−1(p1, p2,ξ1) =

p1, p2,ξ1,−cos(θ/2)sin(θ/2)

ξ1

,

and thus

ΨΦ−1(p1, p2,ξ1) =

p1, p2,−cos(θ/2)sin(θ/2)

ξ1

,

which gives that τ(p) is represented by τ =− cos(θ/2)sin(θ/2) , for θ ∈ (0,π), which invert-

ible and smooth in p.

Mappings between smooth vector bundles are called bundle maps, and are de-fined as follows. Given vector bundles π : E → M and π : E → M and smoothmappings F : E → E and f : M →M, such that the following diagram commutes

E F−−−−→ E

π

π

Mf−−−−→ M

and F |Ep : Ep → E f (p) is linear, then the pair (F, f ) is a called a smooth bundlemap.

A section, or cross section of a bundle E is a continuous mapping σ : M → E,such that πσ = IdM. A section is smooth if σ is a smooth mapping. The space of

FIGURE 30. A cross section σ in a bundle E.

43

smooth section in E is denoted by E(M). The zero section is a mapping σ : M → Esuch that σ(p) = 0 ∈ Ep for all p ∈M. 6.7 Remark. By identifying M with the trivial bundle E0 = M×0, a section isa bundle map from E0 = M×0 to E. Indeed, define σ : E0 → E by σ(p,0) =σ(p), and let f = IdM. Then πσ = πσ = IdM.

6.1. The tangent bundle and vector fieldsThe disjoint union of tangent spaces

T M :=G

p∈MTpM

is called the tangent bundle of M. We show now that T M is a fact a smooth vectorbundle over M.

Theorem 6.8. The tangent bundle T M is smooth vector bundle over M of rank m,and as such T M is a smooth 2m-dimensional manifold.

Proof: Let (U,ϕ) be a smooth chart for p ∈M. Define

Φ

Xi∂

∂xi

p

=

p,(Xi)

,

which clearly is a bijective map between π−1(U) and U ×Rm. Moreover, Xp =

(Xi) → Φ−1(p,Xp) = Xi∂

∂xi

p

is vector space isomorphism. Let (Uα,ϕα) be a cov-

ering of smooth charts of M. Let x = ϕα(p), and x = ϕβ(p). From our previousconsiderations we easily see that

Φβ Φ−1α (p,Xp) = (p,ταβ(p)Xp),

where ταβ : Uα∩Uβ → Gl(m,R). It remains to show that ταβ is smooth. We haveϕβ ϕ−1

α = (ϕβ × Id) Φβ Φ−1α (ϕ−1

α × Id). Using the change of coordinatesformula derived in Section 4 we have that

ϕβ ϕ−1α

x,(Xi)

=

x1(x), · · · ,xm(x),

∂xj∂xi

Xj

,

which proves the smoothness of ταβ. Theorem 6.4 can be applied now showing thatT M is asmooth vector bundle over M. From the charts (π−1(U),ϕ) we concludethat T M is a smooth 2m-dimensional manifold.

44

Definition 6.9. A smooth (tangent) vector field is a smooth mapping

X : M → T M,

with the property that πX = idM. In other words X is a smooth (cross) section inthe vector bundle T M, see Figure 31. The space of smooth vector fields on M isdenoted by F(M).

FIGURE 31. A smooth vector field X on a manifold M [right],and as a ‘curve’, or section in the vector bundle T M [left].

For a chart (U,ϕ) a vector field X can be expressed as follows

X = Xi∂

∂xi

p,

where Xi : U → R. Smoothness of vector fields can be described in terms of thecomponent functions Xi.

Lemma 6.11. A mapping X : M → T M is a smooth vector field at p ∈U if andonly if the coordinate functions Xi : U → R are smooth.

Proof: In standard coordinates X is given by

X(x) = (x1, · · · ,xm, X1(x), · · · , Xm(x)),

which immediately shows that smoothness of X is equivalent to the smoothness ofthe coordinate functions Xi.

In Section 4 we introduced the notion of push forward of a mapping f : N →M.Under submersions and immersion a vector field X : N → T N does not necessarilypush forward to a vector field on M. If f is a diffeomorphism, then f∗X = Y is avector field on M. We remark that the definition of f∗ also allows use to restatedefinitions about the rank of a map in terms of the differential, or pushforward f∗at a point p ∈ N.

45

6.2. The cotangent bundle and differential 1-formsThe disjoint union

T ∗M :=G

p∈MT ∗p M

is called cotangent bundle of M.

Theorem 6.12.23 The cotangent bundle T ∗M is a smooth vector bundle over M of

rank m, and as such T ∗M is a smooth 2m-dimensional manifold.

Proof: The proof is more identical to the proof for T M, and is left to the readeras an exercise.

The differential dh : M → T ∗M is an example of a smooth function. The aboveconsideration give the coordinate wise expression for dh.

Definition 6.13. A smooth covector field is a smooth mapping

θ : M → T ∗M,

with the property that π θ = idM. In order words θ is a smooth section in T ∗M.The space of smooth covector fields on M is denoted by F

∗(M). Usually the spaceF∗(M) is denoted by Γ1(M), and covector fields are referred to a (smooth) differ-

ential 1-form on M.

For a chart (U,ϕ) a covector field θ can be expressed as follows

θ = θidxip,

where θi : U → R. Smoothness of a covector fields can be described in terms ofthe component functions θi.

Lemma 6.14. A covector field θ is smooth at p ∈U if and only if the coordinatefunctions θi : U → R are smooth.

Proof: See proof of Lemma 6.11.

We saw in the previous section that for arbitrary mappings f : N →M, a vectorfield X ∈ F(N) does not necessarily push forward to a vector on M under f∗. Thereason is that surjectivity and injectivity are both needed to guarantee this, whichrequires f to be a diffeomorphism. In the case of covector fields or differential 1-forms the situation is completely opposite, because the pullback acts in the oppositedirection and is therefore onto the target space and uniquely defined. To be moreprecise; given a 1-form θ ∈ Λ1(M) we define a 1-form f ∗θ ∈ Λ1(N) as follows

( f ∗θ)p = f ∗θ f (p).

23See Lee, Proposition 6.5.

46

Theorem 6.15.24 The above defined pullback f ∗θ of θ under a smooth mapping

f : N →M is a smooth covector field, or differential 1-form on N.

Proof: Write θ ∈ Λ1(M) is local coordinates; θ = θidyiq. By Formula (5) we

then obtain

f ∗θidyif (p)=

θi

y∂ fi

∂x j

dx j

p=

θi f

∂ fi

∂x j

x

dx jp,

which proves that f ∗θ ∈ Λ1(N).

Given a mapping g : M → R, the differential, or push forward g∗ = dg definesan element in T ∗p M. In local coordinates we have dg

p=

∂g∂xi dxi

p, and thus defines

a smooth covector field on M; dg ∈ Λ1(M). Given a smooth mapping f : N →M,it follows from (5) that

f ∗dg = f ∗∂g∂xi

dxif (p)=

∂g∂xi f ∂ fi

∂x jdy j

p.

By applying the same to the 1-form d(g f ) we obtain the following identities

(6) f ∗dg = d(g f ), f ∗(gθ) = (g f ) f ∗θ.

Using the formulas in (6) we can also obtain (5) in a rather straightforward way.Let g = ψ j = ψ,e j= y j, and ω = dg = dy j|q in local coordinates, then

f ∗(σ j ψ)ω

=

σ j ψ f

f ∗dg =

σ j ψ f

d(g f ) =

σ j f

∂ f j

∂xi

xdxi

p,

where the last step follows from (4).

Definition 6.16. A differential 1-form θ ∈Λ1(N) is called an exact 1-form if thereexists a smooth function g : N → R, such that θ = dg.

The notation for tangent vectors was motivated by the fact that functions on amanifold can be differentiated in tangent directions. The notation for the cotangentvectors was partly motivated as the ‘reciprocal’ of the partial derivative. The in-troduction of line integral will give an even better motivation for the notation forcotangent vectors. Let N = R, and θ a 1-form on N given in local coordinates byθt = h(t)dt, which can be identified with a function h. The notation makes sensebecause θ can be integrated over any interval [a,b]⊂ R:

Z

[a,b]θ :=

Z b

ah(t)dt.

24Lee, Proposition 6.13.

47

Let M = R, and consider a mapping f : M = R→N = R, which satisfies f (t) > 0.Then t = f (s) is an appropriate change of variables. Let [c,d] = f ([a,b]), then

Z

[c,d]f ∗θ =

Z d

ch( f (s)) f (s)ds =

Z b

ah(t)dt =

Z

[a,b]θ,

which is the change of variables formula for integrals. We can use this now todefine the line integral over a curve γ on a manifold N.

Definition 6.17. Let γ : [a,b] ⊂ R → N, and let θ be a 1-form on N and γ∗θ thepullback of θ, which is a 1-form on R. Denote the image of γ in N also by γ, then

Z

γθ :=

Z

[a,b]γ∗θ =

Z

[a,b]θi(γ(t))γi(t)dt, 25

the expression in local coordinates.

The latter can be seen by combining some of the notion introduced above:

γ∗θ · ddt

= (γ∗θ)t = θγ(t) · γ∗ddt

= θγ(t) · γ(t).

Therefore, γ∗θ = (γ∗θ)tdt = θγ(t) · γ(t)dt = θi(γ(t))γi(t)dt, andZ

γθ =

Z

[a,b]γ∗θ =

Z

[a,b]θγ(t) · γ(t)dt =

Z

[a,b]θi(γ(t))γi(t)dt.

If γ is nowhere zero then the map γ : [a,b]→ N is either an immersion or em-bedding. For example in the embedded case this gives an embedded submanifoldγ⊂ N with boundary ∂γ = γ(a),γ(b). Let θ = dg be an exact 1-form, then

Z

γdg = g

∂γ= g(γ(b))−g(γ(a)).

Indeed,Z

γdg =

Z

[a,b]γ∗dg =

Z

[a,b]d(g γ) =

Z b

a(g γ)(t)dt = g(γ(b))−g(γ(a)).

This identity is called the Fundamental Theorem for Line Integrals and is a spe-cial case of the Stokes Theorem (see Section 16).

25The expressions γi(t) ∈ R are the components of γ(t) ∈ Tγ(t)M.

48

III. Tensors and differentialforms

7. Tensors and tensor products

In the previous chapter we encountered linear functions on vector spaces, linearfunctions on tangent spaces to be precise. In this chapter we extend to notion oflinear functions on vector spaces to multilinear functions.

Definition 7.1. Let V1, · · · ,Vr, and W be real vector spaces. A mapping T : V1×· · ·×Vr →W is called a multilinear mapping if

T (v1, · · · ,λvi +µvi, · · · ,vr) = λT (v1, · · · ,vi, · · ·vr)+µT (v1, · · · ,vi, · · ·vr), ∀i,

and for all λ,µ ∈ R i.e. f is linear in each variable vi separately.

Now consider the special case that W = R, then T becomes a multilinear func-

tion, or form, and a generalization of linear functions. If in addition V1 = · · · =Vr = V , then

T : V × · · ·×V → R,

is a multilinear function on V , and is called a covariant r-tensor on V . The numberof copies r is called the rank of T . The space of covariant r-tensors on V is denotedby T r(V ), which clearly is a real vector space using the multilinearity property inDefinition 7.1. In particular we have that T 0(V ) ∼= R, T 1(V ) = V ∗, and T 2(V ) isthe space of bilinear forms on V . If we consider the case V1 = · · · = Vr = V ∗, then

T : V ∗ × · · ·×V ∗ → R,

is a multilinear function on V ∗, and is called a contravariant r-tensor on V . Thespace of contravariant r-tensors on V is denoted by Tr(V ). Here we have thatT0(V )∼= R, and T1(V ) = (V ∗)∗ ∼= V . 7.2 Example. The cross product on R

3 is an example of a multilinear (bilinear)function mapping not to R to R

3. Let x,y ∈ R3, then

T (x,y) = x× y ∈ R3,

which clearly is a bilinear function on R3.

49

Since multilinear functions on V can be multiplied, i.e. given vector spaces V,Wand tensors T ∈ T r(V ), and S ∈ T s(W ), the multilinear function

R(v1, · · ·vr,w1, · · · ,ws) = T (v1, · · ·vr)S(w1, · · · ,ws)

is well defined and is a multilnear function on V r ×W s. This brings us to thefollowing definition. Let T ∈ T r(V ), and S ∈ T s(W ), then

T ⊗S : V r×W s → R,

is given by

T ⊗S(v1, · · · ,vr,w1, · · ·ws) = T (v1, · · · ,vr)S(w1, · · ·ws).

This product is called the tensor product. By taking V = W , T ⊗ S is a covariant(r + s)-tensor on V , which is a element of the space T r+s(V ) and ⊗ : T r(V )×T s(V )→ T r+s(V ).Lemma 7.3. Let T ∈ T r(V ), S,S ∈ T s(V ), and R ∈ T t(V ), then

(i) (T ⊗S)⊗R = T ⊗ (S⊗R) (associative),(ii) T ⊗ (S +S) = T ⊗S +T ⊗S (distributive),

(iii) T ⊗S = S⊗T (non-commutative).

The tensor product is also defined for contravariant tensors and mixed tensors.As a special case of the latter we also have the product between covariant andcontravariant tensors. 7.4 Example. The last property can easily be seen by the following example. LetV = R

2, and T,S ∈ T 1(R2), given by T (v) = v1 + v2, and S(w) = w1−w2, then

T ⊗S(1,1,1,0) = 2 = 0 = S⊗T (1,1,1,0),

which shows that ⊗ is not commutative in general. The following theorem shows that the tensor product can be used to build the

tensor space T r(V ) from elementary building blocks.

Theorem 7.5.26 Let v1, · · · ,vn be a basis for V , and let θ1, · · · ,θn be the dual

basis for V ∗. Then the set

B =

θi1 ⊗ · · ·⊗θir : 1≤ i1, · · · , ir ≤ n,

is a basis for the nr-dimensional vector space T r(V ).Proof: Compute

Ti1···ir θi1 ⊗ · · ·⊗θir(v j1 , · · · ,v jr) = Ti1···ir θi1(v j1) · · ·θir(v jr)

= Ti1···ir δj1j1 · · ·δ

jrjr = Tj1··· jr

= T (v j1 , · · · ,v jr),

26See Lee, Prop. 11.2.

50

which shows by using the multilinearity of tensors that T can be expanded in thebasis B as follows;

T = Ti1···ir θi1 ⊗ · · ·⊗θir ,

where Ti1···ir = T (vi1 , · · · ,vir), the components of the tensor T . Linear independencefollows form the same calculation.

7.6 Example. Consider the the 2-tensors T (x,y) = x1y1 +x2y2, T (x,y) = x1y2 +x2y2 and T = x1y1 +x2y2 +x1y2 on R

2. With respect to the standard bases θ1(x) =x1, θ2(x) = x1, and

θ1⊗θ1(x,y) = x1y1, θ1⊗θ2(x,y) = x1y2,

θ1⊗θ2(x,y) = x2y1, and θ2⊗θ2(x,y) = x2y2.

Using this the components of T are given by T11 = 1, T12 = 0, T21 = 0, and T22 = 1.Also notice that T = S⊗ S, where S(x) = x1 + x2, and S(y) = y2. Observe thatnot every tensor T ∈ T 2(R2) is of the form T = S⊗S. For example T = S⊗S,for any S,S ∈ T 1(R2).

In Lee, Ch. 11, the notion of tensor product between arbitrary vector spaces isexplained. Here we will discuss a simplified version of the abstract theory. LetV and W be two (finite dimensional) real vector spaces, with bases v1, · · ·vnand w1, · · ·wm respectively, and for their dual spaces V ∗ and W ∗ we have thedual bases θ1, · · · ,θn and σ1, · · · ,σm respectively. If we use the identificationV ∗∗ ∼= V , and W ∗∗ ∼= W we can define V ⊗W as follows:

Definition 7.7. The tensor product of V and W is the real vector space of (finite)linear combinations

V ⊗W :=

λi jvi⊗w j : λi j ∈ R

=

vi⊗w j

i, j

,

where vi⊗w j(v∗,w∗) := v∗(vi)w∗(w j), using the identification vi(v∗) := v∗(vi), andw j(w∗) := w∗(w j), with (v∗,w∗) ∈V ∗ ×W ∗.

To get a feeling of what the tensor product of two vector spaces represents con-sider the tensor product of the dual spaces V ∗ and W ∗. We obtain the real vectorspace of (finite) linear combinations

V ∗ ⊗W ∗ :=

λi jθi⊗σ j : λi j ∈ R

=

θi⊗σ j

i, j

,

where θi⊗σ j(v,w) = θi(v)σ j(w) for any (v,w) ∈V ×W . One can show that V ∗ ⊗W ∗ is isomorphic to space of bilinear maps from V×W to R. In particular elementsv∗ ⊗w∗ all lie in V ∗ ⊗W ∗, but not all elements in V ∗ ⊗W ∗ are of this form. Theisomorphism is easily seen as follows. Let v = ξivi, and w = η jw j, then for a givenbilinear form b it holds that b(v,w) = ξiη jb(vi,w j). By definition of dual basis

51

we have that ξiη j = θi(v)σ j(w) = θi⊗σ j(v,w), which shows the isomorphism bysetting λi j = b(vi,w j).

In the case V ∗ ⊗W the tensors represent linear maps from V to W . Indeed,from the previous we know that elements in V ∗ ⊗W represent bilinear maps fromV ×W ∗ to R. For an element b∈V ∗⊗W this means that b(v, ·) : W ∗ →R, and thusb(v, ·) ∈ (W ∗)∗ ∼= W . 7.8 Example. Consider vectors a ∈ V and b∗ ∈W , then a∗ ⊗ (b∗)∗ can be iden-tified with a matrix, i.e a∗ ⊗ (b∗)∗(v, ·) = a∗(v)(b∗)∗(·) ∼= a∗(v)b. For example leta∗(v) = a1v1 +a2v2 +a3v3, and

Av = a∗(v)b =

a1b1v1 +a2b1v2 +a3b1v3

a1b2v1 +a2b2v2 +a3b2v3

=

a1b1 a2b1 a3b1

a1b2 a2b2 a3b2

v1

v2

v3

.

Symbolically we can write

A = a∗ ⊗b =

a1b1 a2b1 a3b1

a1b2 a2b2 a3b2

=

a1 a2 a3

⊗

b1

b2

,

which shows how a vector and covector can be ‘tensored’ to become a matrix. Notethat it also holds that A = (a ·b∗)∗ = b ·a∗.

Lemma 7.9. We have that

(i) V ⊗W and W ⊗V are isomorphic;(ii) (U⊗V )⊗W and U⊗ (V ⊗W ) are isomorphic.

With the notion of tensor product of vector spaces at hand we now conclude thatthe above describe tensor spaces T r(V ) and Vr(V ) are given as follows;

T r(V ) = V ∗ ⊗ · · ·⊗V ∗

r times

, Tr(V ) = V ⊗ · · ·⊗V r times

.

By considering tensor products of V ’s and V ∗’s we obtain the tensor space ofmixed tensors;

T rs (V ) := V ∗ ⊗ · · ·⊗V ∗

r times

⊗V ⊗ · · ·⊗V s times

.

Elements in this space are called (r,s)-mixed tensors on V — r copies of V ∗, and scopies of V . Of course the tensor product described above is defined in general fortensors T ∈ T r

s (V ), and S ∈ T rs (V ):

⊗ : T rs (V )×T r

s (V )→ T r+rs+s (V ).

52

The analogue of Theorem 7.5 can also be established for mixed tensors. In the nextsections we will see various special classes of covariant, contravariant and mixedtensors. 7.10 Example. The inner product on a vector space V is an example of a covariant2-tensor. This is also an example of a symmetric tensor.

7.11 Example. The determinant of n vectors in Rn is an example of covariant n-

tensor on Rn. The determinant is skew-symmetric, and an example of an alternating

tensor.

If f : V →W is a linear mapping between vector spaces and T is an covarianttensor on W we can define concept of pullback of T . Let T ∈ T r(W ), then f ∗T ∈T r(V ) is defined as follows:

f ∗T (v1, · · ·vr) = T ( f (v1), · · · , f (vr)),

and f ∗ : T r(W )→ T r(V ) is a linear mapping. Indeed, f ∗(T +S) = T f +S f =f ∗T + f ∗S, and f ∗λT = λT f = λ f ∗T . If we represent f by a matrix A withrespect to bases vi and w j for V and W respectively, then the matrix for thelinear f ∗ is given by

A∗ ⊗ · · ·⊗A∗ r times

,

with respect to the bases θi1⊗ · · ·⊗θir and σ j1⊗ · · ·⊗σ jr for T r(W ) and T r(V )respectively. 7.12 Remark. The direct sum

T ∗(V ) =∞M

r=0T r(V ),

consisting of finite sums of covariant tensors is called the covariant tensor algebra

of V with multiplication given by the tensor product ⊗. Similarly, one defines thecontravariant tensor algebra

T∗(V ) =∞M

r=0Tr(V ).

For mixed tensors we have

T (V ) =∞M

r,s=0T r

s (V ),

which is called the tensor algebra of mixed tensor of V . Clearly, T ∗(V ) and T∗(V )subalgebras of T (V ).

53

8. Symmetric and alternating tensors

There are two special classes of tensors which play an important role in theanalysis of differentiable manifolds. The first class we describe are symmetrictensors. We restrict here to covariant tensors.

Definition 8.1. A covariant r-tensor T on a vector space V is called symmetric if

T (v1, · · · ,vi, · · · ,v j, · · · ,vr) = T (v1, · · · ,v j, · · · ,vi, · · · ,vr),

for any pair of indices i ≤ j. The set of symmetric covariant r-tensors on V isdenoted by Σr(V )⊂ T r(V ), which is a (vector) subspace of T r(V ).

If a ∈ Sr is a permutation, then define

aT (v1, · · · ,vr) = T (va(1), · · · ,va(r)),

where a(1, · · · ,r) = a(1), · · · ,a(r). From this notation we have that for twopermutations a,b ∈ Sr, b(aT ) = baT . Define

Sym T =1r! ∑

a∈Sr

aT.

It is straightforward to see that for any tensor T ∈ T r(V ), Sym T is a symmetric.Moreover, a tensor T is symmetric if and only if Sym T = T . For that reasonSym T is called the (tensor) symmetrization. 8.2 Example. Let T,T ∈ T 2(R2) be defined as follows: T (x,y) = x1y2, andT (x,y) = x1y1. Clearly, T is not symmetric and T is. We have that

Sym T (x,y) =12

T (x,y)+12

T (y,x)

=12

x1y2 +12

y1x2,

which clearly is symmetric. If we do the same thing for T we obtain:

Sym T (x,y) =12

T (x,y)+12

T (y,x)

=12

x1y1 +12

y1x1 = T (x,y),

showing that operation Sym applied to symmetric tensors produces the same ten-sor again.

Using symmetrization we can define the symmetric product. Let S ∈ Σr(V ) andT ∈ Σs(V ) be symmetric tensors, then

S ·T = Sym (S⊗T ).

54

The symmetric product of symmetric tensors is commutative which follows di-rectly from the definition:

S ·T (v1, · · · ,vr+s) =1

(r + s)! ∑a∈Sr+s

S(va(1), · · · ,va(r))T (va(r+1), · · · ,va(r+s)).

8.3 Example. Consider the 2-tensors S(x) = x1 + x2, and T (y) = y2. Now S⊗T (x,y) = x1y2 + x2y2, and T ⊗ S(x,y) = x2y1 + x2y2, which clearly gives that S⊗T = T ⊗S. Now compute

Sym (S⊗T )(x,y) =12

x1y2 +12

x2y2 +12

y1x2 +12

x2y2

=12

x1y2 +12

x2y1 + x2y2 = S ·T (x,y).

Similarly,

Sym (T ⊗S)(x,y) =12

x2y1 +12

x2y2 +12

y2x1 +12

x2y2

=12

x1y2 +12

x2y1 + x2y2 = T ·S(x,y),

which gives that S ·T = T ·S.

Lemma 8.4. Let v1, · · · ,vn be a basis for V , and let θ1, · · · ,θn be the dualbasis for V ∗. Then the set

BΣ =

θi1 · · ·θir : 1≤ i1 ≤ · · ·≤ ir ≤ n,

is a basis for the (sub)space Σr(V ) of symmetric r-tensors. Moreover, dimΣr(V ) =n+ r−1

r

= (n+r−1)!

r!(n−1)! .

Proof: Proving that BΣ is a basis follows from Theorem 7.5, see also Lemma8.10. It remains to establish the dimension of Σr(V ). Note that the elements in thebasis are given by multi-indices (i1, · · · , ir), satisfying the property that 1 ≤ i1 ≤· · · ≤ ir ≤ n. This means choosing r integers satisfying this restriction. To do thisredefine jk := ik +k−1. The integers j range from 1 through n+r−1. Now choose

r integers out of n+ r−1, i.e.

n+ r−1

r

combinations ( j1, · · · , jr), which are

in one-to-one correspondence with (i1, · · · , ir).

Another important class of tensors are alternating tensors and are defined asfollows.

55

Definition 8.5. A covariant r-tensor T on a vector space V is called alternating if

T (v1, · · · ,vi, · · · ,v j, · · · ,vr) =−T (v1, · · · ,v j, · · · ,vi, · · · ,vr),

for any pair of indices i ≤ j. The set of alternating covariant r-tensors on V isdenoted by Λr(V )⊂ T r(V ), which is a (vector) subspace of T r(V ).

As before we defineAlt T =

1r! ∑

a∈Sr

(−1)a aT,

where (−1)a is +1 for even permutations, and −1 for odd permutations. We saythat Alt T is the alternating projection of a tensor T , and Alt T is of course aalternating tensor. 8.6 Example. Let T,T ∈ T 2(R2) be defined as follows: T (x,y) = x1y2, andT (x,y) = x1y2 − x2y1. Clearly, T is not alternating and T (x,y) = −T (y,x) isalternating. We have that

Alt T (x,y) =12

T (x,y)− 12

T (y,x)

=12

x1y2−12

y1x2 =12

T (x,y),

which clearly is alternating. If we do the same thing for T we obtain:

Alt T (x,y) =12

T (x,y)− 12

T (y,x)

=12

x1y2−12

x2y1−12

y1x2 +12

y2x1 = T (x,y),

showing that operation Alt applied to alternating tensors produces the same tensoragain. Notice that T (x,y) = det(x,y).

This brings us to the fundamental product of alternating tensors called the wedge

product. Let S ∈ Λr(V ) and T ∈ Λs(V ) be symmetric tensors, then

S∧T =(r + s)!

r!s!Alt (S⊗T ).

The wedge product of alternating tensors is anti-commutative which follows di-rectly from the definition:

S∧T (v1, · · · ,vr+s) =1

r!s! ∑a∈Sr+s

(−1)aS(va(1), · · · ,va(r))T (va(r+1), · · · ,va(r+s)).

In the special case of the wedge of two covectors θ,ω ∈V ∗ gives

θ∧ω = θ⊗ω−ω⊗θ.

In particular we have that

56

(i) (T ∧S)∧R = T ∧ (S∧R);(ii) (T +T )∧S = T ∧S +T ∧S;

(iii) T ∧S = (−1)rsS∧T , for T ∈ Λr(V ) and S ∈ Λs(V );(iv) T ∧T = 0.

The latter is a direct consequence of the definition of Alt . In order to prove theseproperties we have the following lemma.

Lemma 8.7. Let T ∈ T r(V ) and S ∈ T s(V ), then

Alt (T ⊗S) = Alt ((Alt T )⊗S) = Alt (T ⊗Alt S).

Proof: Let G ∼= Sr be the subgroup of Sr+s consisting of permutations that onlypermute the element 1, · · · ,r. For a ∈ G, we have a

∈ Sr. Now a(T ⊗ S) =aT ⊗S, and thus

1r! ∑

a∈G(−1)a a(T ⊗S) = (Alt T )⊗S.

For the right cosets ba : a ∈ G we have

∑a∈G

(−1)ba ba(T ⊗S) = (−1)b b

∑a∈G

(−1)a a(T ⊗S)

= r!(−1)b b

(Alt T )⊗S

.

Taking the sum over all right cosets with the factor 1(r+s)! gives

Alt (T ⊗S) =1

(r + s)! ∑b

∑a∈G

(−1)ba ba(T ⊗S)

=r!

(r + s)! ∑b

(−1)b b

(Alt T )⊗S

= Alt ((Alt T )⊗S),

where the latter equality is due to the fact that r! terms are identical under thedefinition of Alt ((Alt T )⊗S).

Property (i) can now be proved as follows. Clearly Alt (Alt (T⊗S)−T⊗S) = 0,and thus from Lemma 8.7 we have that

0 = Alt ((Alt (T ⊗S)−T ⊗S)⊗R) = Alt (Alt (T ⊗S)⊗R)−Alt (T ⊗S⊗R).

By definition

(T ∧S)∧R =(r + s+ t)!(r + s)!t!

Alt ((T ∧S)⊗R)

=(r + s+ t)!(r + s)!t!

Alt(r + s)!

r!s!Alt (T ⊗S)

⊗R

=(r + s+ t)!

r!s!t!Alt (T ⊗S⊗R).

57

The same formula holds for T ∧ (S∧R), which prove associativity. More generallyit holds that for Ti ∈ Λri(V )

T1∧ · · ·∧Tk =(r1 + · · ·+ rk)!

r1! · · ·rk!Alt (T1⊗ · · ·⊗Tk).

Property (iii) can be seen as follows. Each term in T ∧S can be found in S∧T .This can be done by linking the permutations a and a

. To be more precise, howmany permutation of two elements are needed to change

a↔ (i1, · · · , ir, jr+1, · · · , jr+s) into a ↔ ( jr+1, · · · , jr+s, i1, · · · , ir).

This clearly requires rs permutations of two elements, which shows Property (iii). 8.8 Example. Consider the 2-tensors S(x) = x1 + x2, and T (y) = y2. As beforeS⊗T (x,y) = x1y2 + x2y2, and T ⊗S(x,y) = x2y1 + x2y2. Now compute

Alt (S⊗T )(x,y) =12

x1y2 +12

x2y2−12

y1x2−12

x2y2

=12

x1y2−12

x2y1 = 2

S∧T (x,y).

Similarly,

Alt (T ⊗S)(x,y) =12

x2y1 +12

x2y2−12

y2x1−12

x2y2

= −12

x1y2 +12

x2y1 =−2

T ∧S(x,y),

which gives that S∧T = −T ∧S. Note that if T = e∗1, i.e. T (x) = x1, and S = e∗2,i.e. S(x) = x2, then

T ∧S(x,y) = x1y2− x2y1 = det(x,y).

8.9 Remark. Some authors use the more logical definition

S∧T = Alt (S⊗T ),

which is in accordance with the definition of the symmetric product. This definitionis usually called the alt convention for the wedge product, and our definition isusually referred to as the determinant convention. For computational purposes thedeterminant convention is more appropriate.

If e∗1, · · · ,e∗n is the standard dual basis for (Rn)∗, then for vectors a1, · · · ,an ∈R

n,det(a1, · · · ,an) = e∗1∧ · · ·∧ e∗n(a1, · · · ,an).

Using the multilinearity the more general statement reads

(7) β1∧ · · ·∧βn(a1, · · · ,an) = detβi(a j)

,

58

where βi are co-vectors.The alternating tensor det = e∗1∧ · · ·∧e∗n is called the determinant function on R

n.If f : V →W is a linear map between vector spaces then the pullback f ∗T ∈Λr(V )of any alternating tensor T ∈ Λr(W ) is given via the relation:

f ∗T (v1, · · · ,vr) = T

f (v1), · · · , f (vr), f ∗ : Λr(W )→ Λr(V ).

In particular, f ∗(T ∧S) = ( f ∗T )∧ f ∗(S). As a special case we have that if f : V →V , linear, and dimV = n, then

(8) f ∗T = det( f )T,

for any alternating tensor T ∈Λn(V ). This can be seen as follows. By multilinearitywe verify the above relation for the vectors ei. We have that

f ∗T (e1, · · · ,en) = T ( f (e1), · · · , f (en))

= T ( f1, · · · , fn) = cdet( f1, · · · , fn) = cdet( f ),

where we use the fact that Λn(V )∼= R (see below). On the other hand

det( f )T (e1, · · ·en) = det( f )c ·det(e1, · · · ,en)

= cdet( f ),

which proves (8).

Lemma 8.10. Let θ1, · · · ,θn be a basis for V ∗, then the set

BΛ =

θi1 ∧ · · ·∧θir : 1≤ i1 < · · · < ir ≤ n,

is a basis for Λr(V ), and dimΛr(V ) = n!(n−r)!r! . In particular, dimΛr(V ) = 0 for

r > n.

Proof: From Theorem 7.5 we know that any alternating tensor T ∈ Λr(V ) canbe written as

T = Tj1··· jr θ j1 ⊗ · · ·⊗θ jr .

We have that Alt T = T , and so

T = Tj1··· jr Alt (θ j1 ⊗ · · ·⊗θ jr) =1r!

Tj1··· jr θ j1 ∧ · · ·∧θ jr

In the expansion the terms with jk = j are zero since θ jk ∧θ j = 0. If we order theindices in increasing order we obtain

T = ± 1r!

Ti1···ir θi1 ∧ · · ·∧θir ,

which show that BΛ spans Λr(V ).Linear independence can be proved as follows. Let 0 = λi1···ir θi1 ∧ · · ·∧θir , and

thus λi1···ir = θi1 ∧ · · ·∧θir(vi1 , · · · ,vir) = 0, which proves linear independence.

59

It is immediately clear that BΛ consists of

nr

elements.

As we mentioned before the operation Alt is called the alternating projection.As a matter of fact Sym is also a projection.

Lemma 8.11. Some of the basic properties can be listed as follows;

(i) Sym and Alt are projections on T r(V ), i.e. Sym2 = Sym, and Alt2 = Alt;(ii) T is symmetric if and only if Sym T = T , and T is alternating if and only

if Alt T = T ;(iii) Sym(T r(V )) = Σr(V ), and Alt(T r(V )) = Λr(V );(iv) Sym Alt = Alt Sym = 0, i.e. if T ∈ Λr(V ), then Sym T = 0, and if

T ∈ Σr(V ), then Alt T = 0;(v) let f : V →W, then Sym and Alt commute with f ∗ : T r(W )→ T r(V ), i.e.

Sym f ∗ = f ∗ Sym, and Alt f ∗ = f ∗ Alt.

9. Tensor bundles and tensor fields

Generalizations of tangent spaces and cotangent spaces are given by the tensorspaces

T r(TpM), Ts(TpM), and T rs (TpM),

where T r(TpM) = Tr(T ∗p M). As before we can introduce the tensor bundles:

T rM =G

p∈MT r(TpM),

TsM =G

p∈MTs(TpM),

T rs M =

G

p∈MT r

s (TpM),

which are called the covariant r-tensor bundle, contravariant s-tensor bundle,and the mixed (r,s)-tensor bundle on M. As for the tangent and cotangent bundlethe tensor bundles are also smooth manifolds. In particular, T 1M = T ∗M, andT1M = T M. Recalling the symmetric and alternating tensors as introduced in theprevious section we also define the tensor bundles ΣrM and ΛrM.

Theorem 9.1. The tensor bundles T rM,TrM and T rs M are smooth vector bundles.

Proof: Using Section 6 the theorem is proved by choosing appropriate localtrivializations Φ. For coordinates x = ϕα(p) and x = ϕβ(p) we recall that

∂∂xi

p =

∂xj∂xi

∂∂xj

p, dx j|p =

∂xj∂xi

dxi|p.

60

For a covariant tensor T ∈ T rM this implies the following. The components aredefined in local coordinates by

T = Ti1···ir dxi1 ⊗ · · ·⊗dxir , Ti1···ir = T ∂

∂xi1, · · · , ∂

∂xir

.

The change of coordinates x→ x then gives

Ti1···ir = T∂xj1

∂xi1

∂∂xj1

, · · · ,∂xjr∂xir

∂∂xjr

= T j1··· jr

∂xj1∂xi1

· · ·∂xjr∂xir

.

DefineΦ(Ti1···ir dxi1 ⊗ · · ·⊗dxir) =

p,(Ti1···ir)

,

then

Φ Φ−1p,(T j1··· jr)

=

p,

T j1··· jr

∂xj1∂xi1

· · ·∂xjr∂xir

.

The products of the partial derivatives are smooth functions. We can now applyTheorem 6.4 as we did in Theorems 6.8 and 6.12.

On tensor bundles we also have the natural projection

π : T rs M →M,

defined by π(p,T ) = p. A smooth section in T rs M is a smooth mapping

σ : M → T rs M,

such that πσ = idM. The space of smooth sections in T rs M is denoted by F

rs(M).

For the co- and contravariant tensors these spaces are denoted by Fr(M) and Fs(M)

respectively. Smooth sections in these tensor bundles are also called smooth tensor

fields. Clearly, vector fields and 1-forms are examples of tensor fields. Sections intensor bundles can be expressed in coordinates as follows:

σ =

σi1···ir dxi1 ⊗ · · ·⊗dxir , σ ∈ Fr(M),

σ j1··· js ∂∂x j1

⊗ · · ·⊗ ∂∂x js

, σ ∈ Fs(M),

σ j1··· jsi1···ir dxi1 ⊗ · · ·⊗dxir ⊗ ∂

∂x j1⊗ · · ·⊗ ∂

∂x js, σ ∈ F

rs(M).

Tensor fields are often denoted by the component functions. The tensor and tensorfields in this course are, except for vector fields, all covariant tensors and covarianttensor fields. Smoothness of covariant tensor fields can be described in terms ofthe component functions σi1···ir .

Lemma 9.2.27 A covariant tensor field σ is smooth at p ∈U if and only if

(i) the coordinate functions σi1···ir : U → R are smooth, or equivalently if andonly if

27See Lee, Lemma 11.6.

61

(ii) for smooth vector fields X1, · · · ,Xr defined on any open set U ⊂M, then thefunction σ(X1, · · · ,Xr) : U → R, given by

σ(X1, · · · ,Xr)(p) = σp(X1(p), · · · ,Xr(p)),

is smooth.

The same equivalences hold for contravariant and mixed tensor fields.

Proof: Use the identities in the proof of Theorem 9.1 and then the proof goes asLemma 6.11.

9.3 Example. Let M = R2 and let σ = dx1⊗dx1 +x2

1dx2⊗dx2. If X = ξ1(x) ∂∂x1

+ξ2(x) ∂

∂x2and Y = η1(x) ∂

∂x1+η2(x) ∂

∂x2are arbitrary smooth vector fields on TxR

2 ∼=R

2, then

σ(X ,Y ) = ξ1(x)η1(x)+ x21ξ2(x)η2(x),

which clearly is a smooth function in x ∈ R2.

For covariant tensors we can also define the notion of pullback of a mappingf between smooth manifolds. Let f : N → M be a smooth mappings, then thepullback f ∗ : T r(Tf (p)M)→ T r(TpN) is defined as

( f ∗T )(X1, · · ·Xr) := T ( f∗X1, · · · , f∗Xr),

where T ∈ T r(Tf (p)M), and X1, · · · ,Xr ∈ TpN. We have the following properties.

Lemma 9.4.28 Let f : N → M, g : M → P be smooth mappings, and let p ∈ N,

S ∈ T r(Tf (p)M), and T ∈ T r(Tf (p)M), then:

(i) f ∗ : T r(Tf (p)M)→ T r(TpN) is linear;(ii) f ∗(S⊗T ) = f ∗S⊗ f ∗T ;

(iii) (g f )∗ = f ∗ g∗ : T r(T(g f )(p)P)→ T r(TpM);(iv) id∗MS = S;(v) f ∗ : T rM → T rN is a smooth bundle map.

Proof: Combine the proof of Lemma 5.3 with the identities in the proof ofTheorem 9.1.

9.5 Example. Let us continue with the previous example and let N = M = R2.

Consider the mapping f : N →M defined by

f (x) = (2x1,x32− x1).

28See Lee, Proposition 11.8.

62

For given tangent vectors X ,Y ∈ TxN, given by X = ξ1∂

∂x1+ξ2

∂∂x2

and Y = η1∂

∂x1+

η2∂

∂x2, we can compute the pushforward

f∗ =

2 0−1 3x2

2

, and

f∗X = 2ξ1∂

∂y1+(−ξ1 +3x2

2ξ2)∂

∂y2,

f∗Y = 2η1∂

∂y1+(−η1 +3x2

2η2)∂

∂y2.

Let σ be given by σ = dy1⊗dy1 + y21dy2⊗dy2. This then yields

σ( f∗X , f∗Y ) = 4ξ1η1 +4x21(ξ1−3x2

1ξ2)(η1−3x21η2)

= 4(1+ x21)ξ1η1−12x2

1x22ξ1η2

−12x21x2

2ξ2η1 +36x21x4

2x21ξ2η2.

We have to point out here that f∗X and f∗Y are tangent vectors in TxN and notnecessarily vector fields, although we can use this calculation to compute f ∗σ,which clearly is a smooth 2-tensor field on T 2N.

9.6 Example. A different way of computing f ∗σ is via a local representation ofσ directly. We have σ = dy1⊗dy1 + y2

1dy2⊗dy2, and

f ∗σ = d(2x1)⊗d(2x1)+4x21d(x3

2− x1)⊗d(x32− x1)

= 4dx1⊗dx1 +4x213x2

2dx2−dx1⊗3x2

2dx2−dx1

= 4(1+ x21)dx1⊗dx1−12x2

1x22dx1⊗dx2

−12x21x2

2dx2⊗dx1 +36x21x4

2x21dx2⊗dx2.

Here we used the fact that computing the differential of a mapping to R producesthe pushforward to a 1-form on N.

9.7 Example. If we perform the previous calculation for an arbitrary 2-tensor

σ = a11dy1⊗dy1 +a12dy1⊗dy2 +a21dy2⊗dy1 +a22dy2⊗dy2.

Then,

f ∗σ = (4a11−2a12−2a21 +a22)dx1⊗dx1 +(6a12−3a22)x22dx1⊗dx2

+(6a21−3a22)x22dx2⊗dx1 +9x4

2a22dx2⊗dx2,

63

which produces the following matrix if we identify T 2(TxN) and T 2(TyM) with R4:

f ∗ =

4 −2 −2 10 6x2

2 0 −3x22

0 0 6x22 −3x2

20 0 0 9x4

2

,

which clearly equal to the tensor product of the matrices (J f )∗, i.e.

f ∗ = (J f )∗ ⊗ (J f )∗ =

2 −10 3x2

2

⊗

2 −10 3x2

2

.

This example show how to interpret f ∗ : T 2(TyM)→ T 2(TxN) as a linear mapping.

As for smooth 1-forms this operation extends to smooth covariant tensor fields:( f ∗σ)p = f ∗(σ f (p)), σ ∈ F

r(N), which in coordinates reads

( f ∗σ)p(X1, · · ·Xr) := σ f (p)( f∗X1, · · · , f∗Xr),

for tangent vectors X1, · · · ,Xr ∈ TpN.

Lemma 9.8.29 Let f : N →M, g : M→ P be smooth mappings, and let h∈C∞(M),

σ ∈ Fr(M), and τ ∈ F

r(N), then:

(i) f ∗ : Fr(M)→ F

r(N) is linear;(ii) f ∗(hσ) = (h f ) f ∗σ;

(iii) f ∗(σ⊗ τ) = f ∗σ⊗ f ∗τ;(iv) f ∗σ is a smooth covariant tensor field;(v) (g f )∗ = f ∗ g∗;

(vi) id∗Mσ = σ;

Proof: Combine the Lemmas 9.4 and 9.2.

10. Differential forms

A special class of covariant tensor bundles and associated bundle sections arethe so-called alternating tensor bundles. Let Λr(TpM) ⊂ T r(TpM) be the spacealternating tensors on TpM. We know from Section 8 that a basis for Λr(TpM) isgiven by

dxi1 ∧ · · ·∧dxir : 1≤ i1, · · · , ir ≤ m

,

and dimΛr(TpM) = m!r!(m−r)! . The associated tensor bundles of atlernating covariant

tensors is denoted by ΛrM. Smooth sections in ΛrM are called differential r-forms,and the space of smooth sections is denoted by Γr(M) ⊂ F

r(M). In particular

29See Lee, Proposition 11.9.

64

Γ0(M) = C∞(M), and Γ1(M) = F∗(M). In terms of components a differential r-

form, or r-form for short, is given by

σi1···ir dxi1 ∧ · · ·∧dxir ,

and the components σi1···ir are smooth functions. An r-form σ acts on vector fieldsX1, · · · ,Xr as follows:

σ(X1, · · · ,Xr) = ∑a∈Sr

(−1)aσi1···ir dxi1(Xa(1)) · · ·dxir(Xa(r))

= ∑a∈Sr

(−1)aσi1···ir Xi1a(1) · · ·X

ira(r).

10.1 Example. Let M = R3, and σ = dx∧dz. Then for vector fields

X1 = X11

∂∂x

+X21

∂∂y

+X31

∂∂z

,

and

X2 = X12

∂∂x

+X22

∂∂y

+X32

∂∂z

,

we have thatσ(X1,X2) = X1

1 X32 −X3

1 X12 .

An important notion that comes up in studying differential forms is the notionof contracting an r-form. Given an r-form σ ∈ Γr(M) and a vector field X ∈ F(M),then

iX σ := σ(X , ·, · · · , ·),

is called the contraction with X , and is a differential (r−1)-form on M. Anothernotation for this is iX σ = Xσ. Contraction is a linear mapping

iX : Γr(M)→ Γr−1(M).

Contraction is also linear in X , i.e. for vector fields X ,Y it holds that

iX+Y σ = iX σ+ iY σ, iλX σ = λ · iX σ.

Lemma 10.2.30 Let σ ∈ Γr(M) and X ∈ F(M) a smooth vector field, then

(i) iX σ ∈ Γr−1(M) (smooth (r−1)-form);(ii) iX iX = 0;

(iii) iX is an anti-derivation, i.e. for σ ∈ Γr(M) and ω ∈ Γs(M),

iX(σ∧ω) = (iX σ)∧ω+(−1)rσ∧ (iX ω).

30See Lee, Lemma 13.11.

65

A direct consequence of (iii) is that if σ = σ1 ∧ · · ·∧σr, where σi ∈ Γ1(M) =F∗(M), then

(10) iX σ = (−1)i−1σi(X)σ1∧ · · ·σi∧ · · ·∧σr,

where the hat indicates that σi is to be omitted, and we use the summation conven-tion. 10.3 Example. Let σ = x2dx1∧dx3 be a 2-form on R

3, and X = x21

∂∂x1

+ x3∂

∂x2+

(x1 +x2) ∂∂x3

a given vector field on R3. If Y = Y 1 ∂

∂y1+Y 2 ∂

∂y2+Y 3 ∂

∂y3is an arbitrary

vector fields then

(iX σ)(Y ) = σ(X ,Y ) = dx1(X)dx3(Y )−dx1(Y )dx3(X)

= x21Y 3− (x1 + x2)Y 1,

which gives thatiX σ = x2

1dx3− (x1 + x2)dx1.

Since σ is a 2-form the calculation using X ,Y is still doable. For higher forms

this becomes to involved. If we use the multilinearity of forms we can give a simpleprocedure for computing iX σ using Formula (10). 10.4 Example. Let σ = dx1∧dx2∧dx3 be a 3-form on R

3, and X the vector fieldas given in the previous example. By linearity

iX σ = iX1σ+ iX2σ+ iX3σ,

where X1 = x21

∂∂x1

, X2 = x3∂

∂x2, and X3 = (x1 + x2) ∂

∂x3. This composition is chosen

so that X is decomposed in vector fields in the basis directions. Now

iX1σ = dx1(X1)dx2∧dx3 = x21dx2∧dx3,

iX2σ = −dx2(X2)dx1∧dx3 =−x3dx1∧dx3,

iX3σ = dx2(X3)dx1∧dx2 = (x1 + x2)dx1∧dx2,

which gives

iX σ = x21dx2∧dx3− x3dx1∧dx3 +(x1 + x2)dx1∧dx2,

a 2-form on R3. One should now verify that the same answer is obtained by com-

puting σ(X ,Y,Z). For completeness we recall that for a smooth mapping f : N →M, the pullback

of a r-form σ is given by

( f ∗σ)p(X1, · · · ,Xr) = f ∗σ f (p)(X1, · · · ,Xr) = σ f (p)( f∗X1, · · · , f∗Xr).

We recall that for a mapping h : M → R, then pushforward, or differential of hdhp = h∗ ∈ T ∗p M. In coordinates dhp = ∂h

∂xidxi|p, and thus the mapping p → dhp

66

is a smooth section in Λ1(M), and therefore a differential 1-form, with componentσi = ∂h

∂xi(in local coordinates).

If f : N →M is a mapping between m-dimensional manifolds with charts (U,ϕ),and (V,ψ) respectively, and f (U)⊂V . Set x = ϕ(p), and y = ψ(q), then

(11) f ∗σdy1∧ · · ·∧dym

= (σ f )detJ f |x

dx1∧ · · ·∧dxm.

This can be proved as follows. From the definition of the wedge product andLemma 9.8 it follows that f ∗(dy1∧ · · ·∧dym) = f ∗dy1∧ · · ·∧ f ∗dym, and f ∗dy j =∂ f j∂xi

dxi = dF j, where F = ψ f and f = ψ f ϕ−1. Now

f ∗(dy1∧ · · ·∧dym) = f ∗dy1∧ · · ·∧ f ∗dym

= dF1∧ · · ·∧dFm,

and furthermore, using (7),

dF1∧ · · ·∧dFm ∂

∂x1, · · · , ∂

∂xm

= det

dFi

∂∂x j

= det

∂ f j

∂x j

,

which proves the above claim.As a consequence of this a change of coordinates f = ψϕ−1 yields

(12) dy1∧ · · ·∧dym = detJ f |x

dx1∧ · · ·∧dxm.

10.5 Example. Consider σ = dx∧dy on R2, and mapping f given by x = r cos(θ)

and y = r sin(θ). The map f the identity mapping that maps R2 in Cartesian coor-

dinates to R2 in polar coordinates (consider the chart U = (r,θ) : r > 0, 0 < θ <

2π). As before we can compute the pullback of σ to R2 with polar coordinates:

σ = dx∧dy = d(r cos(θ))∧d(r sin(θ))

= (cos(θ)dr− r sin(θ)dθ)∧ (sin(θ)dr + r cos(θ)dθ)

= r cos2(θ)dr∧dθ− r sin2(θ)dθ∧dr

= rdr∧dθ.

Of course the same can be obtained using (12).

10.6 Remark. If we define

Γ(M) =∞M

r=0Γr(M),

which is an associative, anti-commutative graded algebra, then f ∗ : Γ(N)→ Γ(M)is a algebra homomorphism.

67

11. Orientations

In order to explain orientations on manifolds we first start with orientations offinite-dimensional vector spaces. Let V be a real m-dimensional vector space. Twoordered basis v1, · · · ,vm and v1, · · · ,vm are said to be consistently oriented ifthe transition matrix A = (ai j), defined by the relation

vi = ai jvj,

has positive determinant. This notion defines an equivalence relation on orderedbases, and there are exactly two equivalence classes. An orientation for V is achoice of an equivalence class of order bases. Given an ordered basis v1, · · · ,vmthe orientation is determined by the class O = [v1, · · · ,vm]. The pair (V,O) is calledan oriented vector space. For a given orientation any ordered basis that has thesame orientation is called positively oriented, and otherwise negatively oriented.

Lemma 11.1. Let 0 = θ ∈ Λm(V ), then the set of all ordered bases v1, · · · ,vmfor which θ(v1, · · · ,vm) > 0 is an orientation for V .

Proof: Let vi be any ordered basis and vi = ai jvj. Then θ(v1, · · · ,vm) =det(A)θ(v1, · · · ,vm) > 0, which proves that the set of bases vi for which it holdsthat θ(v1, · · · ,vm) > 0, characterizes an orientation for V .

Let us now describe orientations for smooth manifolds M. We will assume thatdimM = m ≥ 1 here. For each point p ∈ M we can choose an orientation Op forthe tangent space TpM, making (TpM,Op) oriented vector spaces. The collectionOp∈M is called a pointwise orientation. Without any relation between thesechoices this concept is not very useful. The following definition relates the choicesof orientations of TpM, which leads to the concept of orientation of a smooth man-ifold.

Definition 11.2. A smooth m-dimensional manifold M with a pointwise orienta-tion Opp∈M, Op =

X1, · · · ,Xm

, is oriented if for each point in M there exists a

neighborhood U and a diffeomorphism ϕ, mapping from U to an open subset ofR

m, such that

(13)ϕ∗(X1), · · · ,ϕ∗(Xm)

p= [e1, · · · ,em], ∀p ∈U,

where [e1, · · · ,em] is the standard orientation of Rm. In this case the pointwise

orientation Opp∈M is said to be consistently oriented. The choice of Opp∈M iscalled an orientation on M, denoted by O = Opp∈M.

68

If a consistent choice of orientations Op does not exist we say that a manifold isnon-orientable. 11.3 Remark. If we look at a single chart (U,ϕ) we can choose orientations Op

that are consistently oriented for p ∈ Op. The choices Xi = ϕ−1∗ (ei) = ∂

∂xi|p are

ordered basis for TpM. By definition ϕ∗(Xi) = ei, and thus by this choice we obtaina consistent orientation for all p∈U . This procedure can be repeated for each chartin an atlas for M. In order to get a globally consistent ordering we need to worryabout the overlaps between charts.

11.4 Example. Let M = S1 be the circle in R2, i.e. M = p = (p1, p2) : p2

1 + p22 =

1. The circle is an orientable manifold and we can find a orientation as follows.Consider the stereographic charts U1 = S1\N p and U2 = S1\Sp, and the associatedmappings

ϕ1(p) =2p1

1− p2, ϕ2(p) =

2p1

1+ p2

ϕ−11 (x) =

4xx2 +4

,x2−4x2 +4

, ϕ−1

2 (x) = 4x

x2 +4,4− x2

x2 +4

.

Let us start with a choice of orientation and verify its validity. Choose X(p) =(−p2, p1), then

(ϕ1)∗(X) = Jϕ1|p(X) = 2

1− p2

2p1

(1− p2)2

−p2

p1

=

21− p2

> 0,

on U1 (standard orientation). If we carry out the same calculation for U2 we obtain(ϕ2)∗(X) = Jϕ2|p(X) = −2

1+p2< 0 on U2, with corresponds to the opposite orienta-

tion. Instead by choosing ϕ2(p) = ϕ2(−p1, p2) we obtain

(ϕ2)∗(X) = Jϕ2|p(X) =2

1+ p2> 0,

which chows that X(p) is defines an orientation O on S1. The choice of vectorsX(p) does not have to depend continuously on p in order to satisfy the definitionof orientation, as we will explain now.

For p ∈U1 choose the canonical vectors X(p) = ∂∂x |p, i.e.

∂∂x

p= Jϕ−1

1 |p(1) =

16−4x2

(x2+4)2

16x(x2+4)2

.

In terms of p this gives X(p) = 12(1− p2)

−p2

p1

. By definition (ϕ1)∗(X) = 1

for p∈U1. For p = N p we choose X(p) = (−1 0)t , so X(p) is defined for p∈ S1

69

and is not a continuous function of p! It remains to verify that (ϕ2)∗(X) > 0 forsome neighborhood of N p. First, at p = N p,

Jϕ2|p

−10

=

21+ p2

= 1 > 0.

The choice of X(p) at p = N p comes from the canonical choice ∂∂x

p

with respect

to ϕ2. Secondly,

Jϕ2|p(X(p)) =12(1− p2)

−21+ p2

2p1

(1+ p2)2

−p2

p1

=

1− p2

1+ p2,

for all p ∈U2\N p. Note that ϕ2 ϕ−11 (x) = −4

x , and

Jϕ2|p(X(p)) = Jϕ2|p(Jϕ−11 |p(1)) = J(ϕ2 ϕ−1

1 )|p(1) =4x2 ,

which shows that consistency at overlapping charts can be achieved if the transitionmappings have positive determinant. Basically, the above formula describes thechange of basis as was carried out in the case of vector spaces.

What the above example shows us is that if we choose X(p) = ∂∂xi

|p for allcharts, it remains to be verified if we have the proper collection of charts, i.e. doesJϕβJϕ−1

α (ei)= [ei] hold? This is equivalent to having det(JϕβJϕ−1

α ) > 0.These observations lead to the following theorem.

Theorem 11.5. A smooth manifold M is orientable if and only if there exists anatlas A = (Uα,ϕα such that for any p ∈M

det(Jϕαβ|x) > 0, ϕαβ = ϕβ ϕ−1α ,

for any pair α,β for which x = ϕα(p), and p ∈Uα∩Uβ.

Proof: Obvious from the previous example and Lemma 11.1.

As we have seen for vector spaces m-forms can be used to define an orientationon a vector space. This concept can also be used for orientations on manifolds.

Theorem 11.6.31 Let M be a smooth m-dimensional manifold. A nowhere van-

ishing differential m-form θ ∈ Γm(M) determines a unique orientation O on M forwhich θp is positively orientated at each TpM. Conversely, given an orientation O

on M, then there exists a nowhere vanishing m-form θ ∈ Γm(M) that is positivelyoriented at each TpM.

31See Lee, Prop. 13.4.

70

Proof: Let θ∈ Γm(M) be a nowhere vanishing m-form, then in local coordinates(U,ϕ), θ = f dx1∧ · · ·∧dxm. For the canonical bases for TpM we have

θ ∂

∂x1, · · · , ∂

∂xm

= f = 0.

Without loss of generality can be assumed to be positive, otherwise change ϕ bymeans of x1 →−x1. Now let ∂

∂yi be canonical bases for TpM with respect to a

chart (V,ψ). As before θ = gdy1∧ · · ·∧dym, g > 0 on V . Assume that U ∩V = ∅,then at the overlap we have

0 < g = θ ∂

∂y1, · · · , ∂

∂ym

= det

J(ϕψ−1)

θ ∂

∂x1, · · · , ∂

∂xm

= f det

J(ϕψ−1),

which shows that det

J(ϕψ−1)

> 0, and thus M is orientable.For the converse we construct non-vanishing m-forms on each chart (U,ϕ). Via

a partition of unity we can construct a smooth m-form on M, see Lee.

This theorem implies in particular that non-orientable m-dimensional manifoldsdo not admit a nowhere vanishing m-form, or volume form. 11.7 Example. Consider the Mobius strip M. The Mobius strip is an exampleof a non-orientable manifold. Let us parametrize the Mobius strip as an embeddedmanifold in R

3:

g : R× (−1,1)→ R3, g(θ,r) =

r sin(θ/2)cos(θ)+ cos(θ)r sin(θ/2)sin(θ)+ sin(θ)

r cos(θ/2)

,

where g is a smooth embedding when regarded as a mapping from R/2πZ×(−1,1) → R

3. Let us assume that the Mobius strip M is oriented, then by theabove theorem there exists a nonwhere vanishing 2-form σ which can be given asfollows

σ = a(x,y,z)dy∧dz+b(x,y,z)dz∧dx+ c(x,y,z)dx∧dy,

where (x,y,z) = g(θ,r). Since g is a smooth embedding (parametrization) thepullback form g∗σ = ρ(θ,r)dθ∧ dr is a nowhere vanishing 2-form on R/2πZ×(−1,1). In particular, this means that ρ is 2π-periodic in θ. Notice that σ =iX dx∧dy∧dz, where

X = a(x,y,z)∂∂x

+b(x,y,z)∂∂y

+ c(x,y,z)∂∂z

.

For vector fields ξ = ξ1∂∂x +ξ2

∂∂y +ξ3

∂∂z and η = η1

∂∂x +η2

∂∂y +η3

∂∂z we then have

thatσ(ξ,η) = iX dx∧dy∧dz(ξ,η) = X · (ξ×η),

71

where ξ× η is the cross product of ξ and η. The condition that θ is nowherevanishing can be translated to R

2 as follows:

g∗σ(e1,e2) = σ(g∗(e1),g∗(e2)).

Since g∗(e1)×g∗(e2)|θ=0 =−g∗(e1)×g∗(e2)|θ=2π, the pullback g∗σ form cannotbe nowhere vanishing on R/2πZ×R, which is a contradiction. This proves thatthe Mobius band is a non-orientable manifold.

Let N,M be oriented manifolds of the same dimension, and let f : N → M bea smooth mapping. A mapping is called orientation preserving at a point p ∈ Nif f∗ maps positively oriented bases of TpN to positively oriented bases in Tf (p)M.A mapping is called orientation reversing at p ∈ N if f∗ maps positively orientedbases of TpN to negatively oriented bases in Tf (p)M.

FIGURE 32. Out(in)ward vectors and the induced orientation to∂M.

Let us now look at manifolds with boundary; (M,∂M). For a point p ∈ ∂M wedistinguish three types of tangent vectors:

(i) tangent boundary vectors X ∈ Tp(∂M) ⊂ TpM, which form an (m− 1)-dimensional subspace of TpM;

(ii) outward vectors; let ϕ−1 : W ⊂ Hm → M, then X ∈ TpM is called an out-

ward vector if ϕ−1∗ (Y ) = X , for some Y = (y1, · · · ,ym) with y1 < 0;

(iii) inward vectors; let ϕ−1 : W ⊂Hm →M, then X ∈ TpM is called an inward

vector if ϕ−1∗ (Y ) = X , for some Y = (y1, · · · ,ym) with y1 > 0.

72

Using this concept we can now introduce the notion of induced orientation on ∂M.Let p ∈ ∂M and choose a basis X1, · · · ,Xm for TpM such that [X1, · · · ,Xm] = Op,X2, · · · ,Xm are tangent boundary vectors, and X1 is an outward vector. In this case[X2, · · · ,Xm] = (∂O)p determines an orientation for Tp(∂M), which is consistent,and therefore ∂O = (∂O)pp∈∂M is an orientation on ∂M induced by O. Thus foran oriented manifold M, with orientation O, ∂M has an orientation ∂O, called theinduced orientation on ∂M. 11.8 Example. Any open set M ⊂ R

m (or Hm) is an orientable manifold.

11.9 Example. Consider a smooth embedded co-dimension 1 manifold

M = p ∈ Rm+1 : f (p) = 0, f : Rm+1 → R, rk( f )|p, p ∈M.

Then M is an orientable manifold. Indeed, M = ∂N, where N = p∈Rm+1 : f (p)>

0, which is an open set in Rm+1 and thus an oriented manifold. Since M = ∂N the

manifold M inherits an orientation from M and hence it is orientable.

73

IV. Integration on manifolds

12. Integrating m-forms on Rm

We start off integration of m-forms by considering m-forms on Rm.

Definition 12.1. A subset D⊂ Rm is called a domain of integration if

(i) D is bounded, and(ii) ∂D has m-dimensional Lebesgue measure dµ = dx1 · · ·dxm equal to zero.

In particular any finite union or intersection of open or closed rectangles is adomain of integration. Any bounded32 continuous function f on D is integrable,i.e.

−∞ <Z

Df dx1 · · ·dxm < ∞.

Since Λm(Rm)∼= R, a smooth m-form on Rm is given by

ω = f (x1, · · · ,xm)dx1∧ · · ·∧dxm,

where f : Rm → R is a smooth function. For a given (bounded) domain of integra-

tion D we defineZ

Dω :=

Z

Df (x1, · · · ,xm)dx1 · · ·dxm =

Z

Df dµ

=Z

Dωx(e1, · · · ,em)dµ.

An m-form ω is compactly supported if supp(ω) = clx ∈ Rm : ω(x) = 0

is a compact set. The set of compactly supported m-forms on Rm is denoted by

Γmc (Rm), and is a linear subspace of Γm(Rm). Similarly, for any open set U ⊂ R

m

we can define ω ∈ Γmc (U). Clearly, Γm

c (U) ⊂ Γmc (Rm), and can be viewed as a

linear subspace via zero extension to Rm. For any open set U ⊂ R

m there exists adomain of integration D such that U ⊃ D⊃ supp(ω) (see Exercises).

32Boundedness is needed because the rectangles are allowed to be open.

74

Definition 12.2. Let U ⊂ Rm be open and ω ∈ Γm

c (U), and let D be a domain ofintegration D such that U ⊃ D⊃ supp(ω). We define the integral

Z

Uω :=

Z

Dω.

If U ⊂Hm open, then

Z

Uω :=

Z

D∩Hmω.

The next theorem is the first step towards defining integrals on m-dimensionalmanifolds M.

Theorem 12.3. Let U,V ⊂ Rm be open sets, f : U →V an orientation preserving

diffeomorphism, and let ω ∈ Γmc (V ). Then,

Z

Vω =

Z

Uf ∗ω.

If f is orientation reversing, thenR

U ω =−R

V f ∗ω.

Proof: Assume that f is an orientation preserving diffeomorphism from U toV . Let E be a domain a domain of integration for ω, then D = f−1(E) is a do-main of integration for f ∗ω. We now prove the theorem for the domains D andE. We use coordinates xi and yi on D and E respectively. We start withω = g(y1, · · · ,ym)dy1∧ · · ·∧ dym. Using the change of variables formula for inte-grals and the pullback formula in (11) we obtain

Z

Eω =

Z

Eg(y)dy1 · · ·dym (Definition)

=Z

D(g f )(x)det(J f |x)dx1 · · ·dxm

=Z

D(g f )(x)det(J f |x)dx1∧ · · ·∧dxm

=Z

Df ∗ω (Definition).

One has to introduce a − sign in the orientation reversing case.

13. Partitions of unity

We start with introduce the notion partition of unity for smooth manifolds. Weshould point out that this definition can be used for arbitrary topological spaces.

75

Definition 13.1. Let M be smooth m-manifold with atlas A = (ϕi,Ui)i∈I . Apartition of unity subordinate to A is a collection for smooth functions λi : M →Ri∈I satisfying the following properties:

(i) 0≤ λi(p)≤ 1 for all p ∈M and for all i ∈ I;(ii) supp(λi)⊂Ui;

(iii) the set of supports supp(λi)i∈I is locally finite;(iv) ∑i∈I λi(p) = 1 for all p ∈M.

Condition (iii) says that every p ∈ M has a neighborhood U p such that onlyfinitely many λi’s are nonzero in U . As a consequence of this the sum in Condition(iv) is always finite.

Theorem 13.2. For any smooth m-dimensional manifold M with atlas A thereexists a partition of unity λi subordinate to A.

In order to prove this theorem we start with a series of auxiliary results andnotions.

Lemma 13.3. There exists a smooth function h : Rm → R such that 0 ≤ h ≤ 1 on

Rm, and h|B1(0) ≡ 1 and supp(h)⊂ B2(0).

Proof: Define the function f1 : R→ R as follows

f1(t) =

e−1/t , t > 0,

0, t ≤ 0.

One can easily prove that f1 is a C∞-function on R. If we set

f2(t) =f1(2− t)

f1(2− t)+ f1(t−1).

This function has the property that f2(t) ≡ 1 for t ≤ 1, 0 < f2(t) < 1 for 1 < t <

2, and f2(t) ≡ 0 for t ≥ 2. Moreover, f2 is smooth. To construct f we simplywrite f (x) = f2(|x|) for x ∈ R

m\0. Clearly, f is smooth on Rm\0, and since

f |B1(0) ≡ 1 it follows that f is smooth on Rm.

An atlas A gives an open covering for M. The set Ui in the atlas need not becompact, nor locally finite. We say that a covering U = Ui of M is locally finite

if every point p ∈ M has a neighborhood that intersects only finitely Ui ∈ U. Ifthere exists another covering V = Vj such that every Vj ∈ V is contained in someVj ⊂Ui ∈ U, then V is called a refinement of U. A topological space X for whicheach open covering admits a locally finite refinement is called paracompact.

Lemma 13.6. Any topological manifold M allows a countable, locally finite cov-ering by precompact open sets.

76

FIGURE 33. The functions f1 and f2.

FIGURE 34. A locally finite covering [left], and a refinement[right].

Proof: We start with a countable covering of open balls B = Bi. We nowconstruct a covering U that satisfies

(i) all sets Ui ∈ U are precompact open sets in M;(ii) Ui−1 ⊂Ui, i > 1;

(iii) Bi ⊂Ui.

We build the covering U from B using an inductive process. Let U1 = B1, andassume U1, · · · ,Uk have been constructed satisfying (i)-(iii). Then

Uk ⊂ Bi1 ∪ · · ·∪Bik ,

where Bi1 = B1. Now setUk+1 = Bi1 ∪ · · ·∪Bik .

Choose ik large enough so that ik ≥ k +1 and thus Bk+1 ⊂Uk+1.From the covering U we can now construct a locally finite covering V by setting

Vi = Ui\Ui−2, i > 1.

77

FIGURE 35. Constructing a nested covering U from a coveringwith balls B [left], and a locally finite covering obtained from theprevious covering [right].

Next we seek a special locally finite refinement Wj = W of V which has spe-cial properties with respect to coordinate charts of M;

(i) W is a countable and locally finite;(ii) each Wj ⊂W is in the domain of some smooth coordinate map ϕ j : j→R

m

for M;(iii) the collection Z = Z j, Z j = ϕ−1

j (B1(0)) covers M.

The covering W is called regular.

Lemma 13.8. For any open covering U for a smooth manifold M there exists aregular refinement. In particular M is paracompact.

Proof: Let V be a countable, locally finite covering as described in the previouslemma. Since V is locally finite we can find a neighborhood Wp for each p ∈ Mthat intersects only finitely many set Vj ∈ V. We want to choose Wp in a smartway. First we replace Wp by Wp∩Vj : p ∈Vj. Since p ∈Ui for some i we thenreplace Wp by Wp ∩Vi. Finally, we replace Wp by a small coordinate ball Br(p)so that Wp is in the domain of some coordinate map ϕp. This provides coordinatecharts (Wp,ϕp). Now define Zp = ϕ−1

p (B1(0)).For every k, Zp : p ∈Vk is an open covering of Vk, which has a finite subcov-

ering, say Z1k , · · · ,Z

mkk . The sets Zi

k are sets of the form Zpi for some pi ∈ Vk. Theassociated coordinate charts are (W 1

k ,ϕ1k), · · · ,(W

mkk ,ϕmk

k ). Each W ik is obtained via

the construction above for pi ∈ Vk. Clearly, W ik is a countable open covering of

M that refines U and which, by construction, satisfies (ii) and (iii) in the definitionof regular refinement. It is clear from compactness that W i

k is locally finite. Letus relabel the sets W i

k and denote this covering by W = Wi. As a consequenceM is paracompact.

78

FIGURE 36. The different stages of constructing the sets Wp col-ored with different shades of grey.

Proof of Theorem 13.2: From Lemma 13.8 there exists a regular refinement W

for A. By construction we have Zi = ϕ−1i (B1(0)), and we define

Zi := ϕ−1i (B2(0)).

Using Lemma 13.3 we now define the functions µ:M → R;

µi =

f2 ϕi on Wi

0 on M\Vi.

These functions are smooth and supp(µi)⊂Wi. The quotients

λ =µ j(p)

∑i µi(p),

are, due to the local finiteness of Wi and the fact that the denominator is positivefor each p ∈M, smooth functions on M. Moreover, 0≤λ j ≤ 1, and ∑ j

λ j ≡ 1.Since W is a refinement of A we can choose an index k j such that Wj ⊂Uk j .

Therefore we can group together some of the function λ j:

λi = ∑j : k j=i

λ j,

which give us the desired partition functions with 0 ≤ λi ≤ 1, ∑i λi ≡ 1, andsupp(λi)⊂Ui.

Some interesting byproducts of the above theorem using partitions of unity are.In all these case M is assumed to be an smooth m-dimensional manifold.

Theorem 13.10.33 For any close subset A ⊂ M and any open set U ⊃ A, there

exists a smooth function f : M → R such that

33See Lee, Prop. 2.26.

79

(i) 0≤ f ≤ 1 on M;(ii) f−1(1) = A;

(iii) supp( f )⊂U.

Such a function f is called a bump function for A supported in U.

By considering functions g = c(1− f )≥ 0 we obtain smooth functions for whichg−1(0) can be an arbitrary closed subset of M.

Theorem 13.11.34 Let A ⊂ M be a closed subset, and f : A → R

k be a smoothmapping.35 Then for any open subset U ⊂ M containing A there exists a smoothmapping f † : M → R

k such that f †|A = f , and supp( f †)⊂U.

14. Integration on of m-forms on m-dimensional manifolds.

In order to introduce the integral of an m-form on M we start with the case offorms supported in a single chart. In what follows we assume that M is an orientedmanifold with an oriented atlas A, i.e. a consistent choice of smooth charts suchthat the transitions mappings are orientation preserving.

FIGURE 37. An m-form supported in a single chart.

Let ω ∈ Γmc (M), with supp(ω) ⊂ U for some chart U in A. Now define the

integral of ω over M as follows:Z

Mω :=

Z

ϕ(U)(ϕ−1)∗ω,

where the pullback form (ϕ−1)∗ω is compactly supported in V = ϕ(U) (ϕ is adiffeomorphism). The integral over V of the pullback form (ϕ−1)∗ω is defined in

34See Lee, Prop. 2.26.35 A mapping from a closed subset A⊂M is said to be smooth if for every point p∈ A there exists

an open neighborhood W ⊂M of p, and a smooth mapping f † : W → Rk such that f †|W∩A = f .

80

Definition 12.2. It remains to show that the integralR

M ω does not depend on theparticular chart. Consider a different chart U , possibly in a different oriented atlas

FIGURE 38. Different charts U and U containing supp(ω), i.e.supp(ω)⊂U ∩U .

A for M (same orientation), then

Z

V (ϕ−1)∗ω =

Z

ϕ(U∩U )(ϕ−1)∗ω =

Z

ϕ(U∩U )(ϕ ϕ−1)∗(ϕ−1)∗ω

=Z

ϕ(U∩U )(ϕ−1)∗(ϕ)∗(ϕ−1)∗ω =

Z

ϕ(U∩U )(ϕ−1)∗ω

=Z

V(ϕ−1)∗ω,

which show that the definition is independent of the chosen chart. We crucially douse the fact that A∪A

is an oriented atlas for M. To be more precise the mappingsϕ ϕ−1 are orientation preserving.

By choosing a partition of unity as described in the previous section we can nowdefine the integral over M of arbitrary m-forms ω ∈ Γm

c (M).

Definition 14.3. Let ω ∈ Γmc (M) and let AI = (Ui,ϕii∈I ⊂A be a finite subcov-

ering of supp(ω) coming from an oriented atlas A for M. Let λi be a partitionof unity subordinate to AI . Then the integral of ω over M is defined as

Z

Mω := ∑

i

Z

Mλiω,

where the integralsR

M λiω are integrals of form that have support in single chartsas defined above.

81

We need to show now that the integral is indeed well-defined, i.e. the sum isfinite and independent of the atlas and partition of unity. Since supp(ω) is compactAI exists by default, and thus the sum is finite.

Lemma 14.4. The above definition is independent of the chosen partition of unityand covering AI .

Proof: Let AJ ⊂ A

be another finite covering of supp(ω), where A is a com-

FIGURE 39. Using a partition of unity we can construct m-formswhich are all supported in one, but possibly different charts Ui.

patible oriented atlas for M, i.e. A∪A is a oriented atlas. Let λj be a partition

of unity subordinate to AJ . We have

Z

Mλiω =

Z

M

∑

jλj

λiω = ∑

j

Z

Mλjλiω.

By summing over i we obtain ∑iR

M λiω = ∑i, jR

M λjλiω. Each term λjλiω is sup-ported in some Ui and by previous independent of the coordinate mappings. Sim-ilarly, if we interchange the i’s and j’s, we obtain that ∑ j

RM λjω = ∑i, j

RM λjλiω,

which proves the lemma.

14.6 Remark. If M is a compact manifold that for any ω ∈ Γm(M) it holds thatsupp(ω)⊂M is a compact set, and therefore Γm

c (M) = Γm(M) in the case of com-pact manifolds.

So far we considered integral of m-forms over M. One can of course integraten-forms on M over n-dimensional immersed or embedded submanifolds N ⊂ M.Given ω ∈ Γn(M) for which the restriction to N, ω|N has compact support in N,then Z

Nω :=

Z

Nω|N .

82

As a matter of fact we have i : N → M, and ω|N = i∗ω, so thatR

N ω :=R

N ω|N =R

N i∗ω. If M is compact with boundary ∂M, thenR

∂M ω is well-defined for any(m−1)-form ω ∈ Γm−1(M).

Theorem 14.7. Let ω ∈ Γmc (M), and f : N →M is a diffeomorphism. Then

Z

Mω =

Z

Nf ∗ω.

Proof: By the definition of integral above we need to prove the above statementfor the terms Z

Mλiω =

Z

Nf ∗λiω.

Therefore it suffices to prove the theorem for forms ω whose support is in a singlechart U .

FIGURE 40. The pullback of an m-form.

Z

Uf ∗ω =

Z

ϕ(U)(ϕ−1)∗ f ∗ω =

Z

ϕ(U)(ϕ−1)∗ f ∗ψ∗(ψ−1)∗ω

=Z

ψ(U )(ψ−1)∗ω =

Z

U ω.

By applying this to λiω and summing we obtain the desired result.

For sake of completeness we summarize the most important properties of theintegral.

Theorem 14.9.36 Let N,M be oriented manifolds (with or without boundary) of

dimension m , and ω,η ∈ Γmc (M), are m-forms. Then

(i)R

M aω+bη = aR

M ω+bR

M η;

36See Lee Prop. 14.6.

83

(ii) if −O is the opposite orientation to O, thenZ

M,−O

ω =−Z

M,Oω;

(iii) if ω is an orientation form, thenR

M ω > 0;(iv) if f : N →M is a diffeomorphism, then

Z

Mω =

Z

Nf ∗ω.

For practical situations the definition of the integral above is not convenient sinceconstructing partitions of unity is hard in practice. If the support of a differentialform ω can be parametrized be appropriate parametrizations, then the integral canbe easily computed from this. Let Di ⊂ R

m — finite set of indices i — be compactdomains of integration, and gi : Di →M are smooth mappings satisfying

(i) Ei = gi(Di), and gi : int (Di) → int (Ei) are orientation preserving diffeo-morphisms;

(ii) Ei∩E j intersect only along their boundaries, for all i = j.

Theorem 14.10.37 Let (gi,Di) be a finite set of parametrizations as defined

above. Then for any ω⊂ Γmc (M) such that supp(ω)⊂ ∪iEi it holds that

Z

Mω = ∑

i

Z

Di

g∗i ω.

Proof: As before it suffices the prove the above theorem for a single chart U ,

FIGURE 41. Carving up supp(ω) via domains of integration forparametrizations gi for M.

i.e. supp(ω) ⊂U . One can choose U to have a boundary ∂U so that ϕ(∂U) has

37See Lee Prop. 14.7.

84

measure zero, and ϕ maps cl(U) to a compact domain of integration K ⊂Hm. Now

set

Ai = cl(U)∩Ei, Bi = g−1i (Ai), Ci = ϕi(Ai).

We have Z

Ci

(ϕ−1)∗ω =Z

Bi

(ϕgi)∗(ϕ−1)∗ω =Z

Bi

g∗i ω.

Since the interiors of the sets Ci (and thus Ai) are disjoint it holds thatZ

Mω =

Z

K(ϕ−1)∗ω = ∑

i

Z

Ci

(ϕ−1)∗ω

= ∑i

Z

Bi

g∗i ω = ∑i

Z

Di

g∗i ω,

which proves the theorem.

14.12 Remark. From the previous considerations computingR

M ω boils downto computing (ϕ−1)∗ω, or g∗ω for appropriate parametrizations, and summing thevarious contrubutions. Recall from Section 12 that in order to integrate one needsto evaluate (ϕ−1)∗ωx(e1, · · · ,em), which is given by the formula

(ϕ−1)∗ω

x(e1, · · · ,em) = ωϕ−1(x)(ϕ−1

∗ (e1), · · · ,ϕ−1∗ (em)).

For a single chart contribution this then yields the formulaZ

Uω =

Z

ϕ(U)ωϕ−1(x)(ϕ−1

∗ (e1), · · · ,ϕ−1∗ (em))dµ

=Z

ϕ(U)ωϕ−1(x)

∂∂x1

, · · · , ∂∂x1

dµ

or in the case of a parametrization g : D→M:Z

g(D)ω =

Z

Dωg(x)(g∗(e1), · · · ,g∗(em))dµ.

These expression are useful for computing integrals.

14.13 Example. Consider the 2-sphere parametrized by the mapping g : R2 →

S2 ⊂ R3 given as

g(ϕ,θ) =

xyz

=

sin(ϕ)cos(θ)sin(ϕ)sin(θ)

cos(ϕ)

.

This mapping can be viewed as a covering map. From this expression we derivevarious charts for S2. Given the 2-form ω = zdx∧ dz let us compute the pullback

85

form g∗ω on R2. We have that

g∗(e1) = cos(ϕ)cos(θ)∂∂x

+ cos(ϕ)sin(θ)∂∂y− sin(ϕ)

∂∂z

,

g∗(e2) = −sin(ϕ)sin(θ)∂∂x

+ sin(ϕ)cos(θ)∂∂y

,

and therefore

g∗ω(e1,e2) = ωg(x)(g∗(e1),g∗(e2)) =−cos(ϕ)sin2(ϕ)sin(θ),

and thus

g∗ω =−cos(ϕ)sin2(ϕ)sin(θ)dϕ∧dθ.

The latter gives thatR

S2 ω =−R 2π

0R π

0 cos(ϕ)sin2(ϕ)sin(θ)dϕdθ = 0, which showsthat ω is not a volume form on S2.

If we perform the same calculations for ω = xdy∧dz+ydz∧dx+zdx∧dy, thenZ

S2ω =

Z 2π

0

Z π

0sin(ϕ)dϕdθ = 4π.

The pullback form g∗ω = sin(ϕ)dϕ∧dθ, which shows that ω is a volume form onS2.

15. The exterior derivative

The exterior derivative d of a differential form is an operation that maps ank-form to a (k +1)-form. Write a k-form on R

m in the following notation

ω = ωi1···ik dxi1 ∧ · · ·∧dxik = ωIdxI, I = (i1 · · · ik),

then we define

(14) dω = dωI ∧dxI.

Written out in all its differentials this reads

dω = d

ωi1···ik dxi1 ∧ · · ·∧dxik

=∂ωI

∂xidxi∧dxi1 ∧ · · ·∧dxik .

Of course the Einstein summation convention is again applied here. This is thedefinition that holds for all practical purposes, but it is a local definition. We needto prove that d extends to differential forms on manifolds. 15.1 Example. Consider ω0 = f (x,y), a 0-form on R

2, then

dω0 =∂ f∂x

dx+∂ f∂y

dy.

86

For a 1-form ω1 = f1(x,y)dx+ f2(x,y)dy we obtain

dω1 =∂ f1

∂xdx∧dx+

∂ f1

∂ydy∧dx+

∂ f2

∂xdx∧dy+

∂ f2

∂ydy∧dy

=∂ f1

∂ydy∧dx+

∂ f2

∂xdx∧dy =

∂ f2

∂x− ∂ f1

∂y

dx∧dy.

Finally, for a 2-form ω2 = g(x,y)dx∧dy the d-operation gives

dω2 =∂g∂x

dx∧dx∧dy+∂g∂y

dy∧dx∧dy = 0.

The latter shows that d applied to a top-form always gives 0.

15.2 Example. In the previous example dω0 is a 1-form, and dω1 is a 2-form.Let us now apply the d operation to these forms:

d(dω0) = ∂2 f

∂x∂y− ∂2 f

∂y∂x

dx∧dy = 0,

andd(dω1) = 0,

since d acting on a 2-form always gives 0. These calculations seem to suggest thatin general d d = 0.

As our examples indicate d2ω = 0. One can also have forms ω for which dω = 0,but ω = dσ. We say that ω is a closed form, and when ω = dσ, then ω is calledan exact form. Clearly, closed forms form a possibly larger class than exact forms.In the next chapter on De Rham cohomology we are going to come back to thisphenomenon in detail. On R

m one will not be able to find closed forms that arenot exact, i.e. on R

m all closed forms are exact. However, if we consider differentmanifolds we can have examples where this is not the case. 15.3 Example. Let M = R

2\(0,0), and consider the 1-form

ω =−y

x2 + y2 dx+x

x2 + y2 dy,

which clearly is a smooth 1-form on M. Notice that ω does not extend to a 1-formon R

2! It holds that dω = 0, and thus ω is a closed form on M. Let γ : [0,2π)→M,t → (cos(t),sin(t)) be an embedding of S1 into M, then

Z

γω =

Z 2π

0γ∗ω =

Z 2π

0dt = 2π.

Assume that ω is an exact 1-form on M, then ω = d f , for some smooth functionf : M → R. In Section 5 we have showed that

Z

γω =

Z

γd f =

Z 2π

0γ∗d f = ( f γ)(t)dt = f (1,0)− f (1,0) = 0,

87

which contradicts the fact thatR

γ ω =R

γ d f = 2π. This proves that ω is not exact.

For the exterior derivative in general we have the following theorem.

Theorem 15.4.38 Let M be a smooth m-dimensional manifold. Then for all k ≥ 0

there exist unique linear operations d : Γk(M)→ Γk+1(M) such that;

(i) for any 0-form ω = f , f ∈C∞(M) it holds that

dω(X) = d f (X) = X f , X ∈ F(M);

(ii) if ω ∈ Γk(M) and η ∈ Γ(M), then

d(ω∧η) = dω∧η+(−1)kω∧dη;

(iii) d d = 0;(iv) if ω ∈ Γm(M), then dω = 0.

This operation d is called the exterior derivative on differential forms, and is aunique anti-derivation (of degree 1) on Γ(M) with d2 = 0.

Proof: Let us start by showing the existence of d on a chart U ⊂ M. We havelocal coordinates x = ϕ(p), p ∈U , and we define

dU : Γk(U)→ Γk+1(U),

via (14). Let us write d instead of dU for notational convenience. We have thatd( f dxI) = d f ∧dxI . Due to the linearity of d it holds that

d( f dxI ∧gdxJ) = d( f gdxI ∧dxJ) = d( f g)dxK

= gd f dxK + f dgdxK

= (d f ∧dxI)∧ (gdxJ)+ f dg∧dxI ∧dxJ

= (d f ∧dxI)∧ (gdxJ)+(−1)k f dxI ∧dg∧dxJ

= d( f dxI)∧ (gdxJ)+(−1)k( f dxI)∧d(gdxJ),

which proves (ii). As for (iii) we argue as follows. In the case of a 0-form we havethat

d(d f ) = d ∂ f

∂xidxi =

∂2 f∂x j∂xi

dx j ∧dxi

= ∑i< j

∂2 f∂xi∂x j

− ∂2 f∂x j∂xi

dxi∧dx j = 0.

38See Lee Thm. 12.14

88

Given a k-form ω = ωIdxI , then, using (ii), we have

d(dω) = d

dωI ∧dxI

= d(dωI)∧dxI +(−1)k+1dωI ∧d(dxI) = 0,

since ωI is a 0-form, and d(dxI) = (−1) jdxi1 ∧ · · ·∧ d(dxi j)∧ · · ·∧ dxik = 0. Thelatter follows from d(dxi j) = 0. The latter also implies (iv), finishing the existenceproof in the one chart case. The operation d = dU is well-defined, satisfying (i)-(iv), for any chart (U,ϕ).

The operation dU is unique, for if there exists yet another exterior derivative dU ,which satisfies (i)-(iv), then for ω = ωIdxI ,

dω = dωI ∧dxI +ωI d(dxI),

where we used (ii). From (ii) it also follows that d(dxI)= (−1) jdxi1∧· · ·∧ d(dxi j)∧· · ·∧dxik = 0. By (i) d(ϕ(p)i j) = dxi j , and thus by (iv) d(dxi j) = d d(ϕ(p)i j) = 0,which proves the latter. From (i) it also follows that dωI = dωI , and therefore

dω = dωI ∧dxI +ωI d(dxI) = dωI ∧dxI = dω,

which proves the uniqueness of dU .Before giving the defining d for ω ∈ Γk(M) we should point out that dU trivially

satisfies (i)-(iii) of Theorem 15.5 (use (14). Since we have a unique operation dU

for every chart U , we define for p ∈U

(dω)p = (dU ω|U)p,

as the exterior derivative for forms on M. If there is a second chart U p, then byuniqueness it holds that on

dU(ω|U∩U ) = dU∩U (ω|U∩U ) = dU (ω|U∩U ),

which shows that the above definition is independent of the chosen chart.The last step is to show that d is also uniquely defined on Γk(M). Let p ∈U , a

coordinate chart, and consider the restriction

ω|U = ωIdxI,

where ωI ∈ C∞(U). Let W ⊂U be an open set containing p, with the additionalproperty that cl(W ) ⊂ U is compact. By Theorem 13.10 we can find a bump-function g ∈C∞(M) such that g|W = 1, and supp(g)⊂U . Now define

ω = gωId(gxi1)∧ · · ·∧d(gxik).

Using (i) we have that d(gxi)|W = dxi, and therefore ω|W = ω|W . Set η = ω−ω,then η|W = 0. Let p ∈W and h ∈C∞(M) satisfying h(p) = 1, and supp(h) ⊂W .

89

Thus, hω≡ 0 on M, and

0 = d(hω) = dh∧ω+hdω.

This implies that (dω)p =−(d f ∧ω)p = 0, which proves that (dω)|W = (dω)|W .If we use (ii)-(iii) then

dω = dgωId(gxi1)∧ · · ·∧d(gxik)

= d(gωI)∧d(gxi1)∧ · · ·∧d(gxik)+gωIdd(gxi1)∧ · · ·∧d(gxik)

= d(gωI)∧d(gxi1)∧ · · ·∧d(gxik).

It now follows that since g|W = 1, and since (dω)|W = (dω)|W , that

(dω)|W =∂ωI

∂xidxi∧dxI,

which is exactly (14). We have only used properties (i)-(iii) the derive this ex-pression, and since p is arbitrary it follows that d : Γk(M)→ Γk+1(M) is uniquelydefined.

The exterior derivative has other important properties with respect to restrictionsand pullbacks that we now list here.

Theorem 15.5. Let M be a smooth m-dimensional manifold, and let ω ∈ Γk(M),k ≥ 0. Then

(i) in each chart (U,ϕ) for M, dω in local coordinates is given by (14);(ii) if ω = ω on some open set U ⊂M, then also dω = dω on U;

(iii) if U ⊂M is open, then d(ω|U) = (dω)|U ;(iv) if f : N →M is a smooth mapping, then

f ∗(dω) = d( f ∗ω),

i.e. f ∗ : Γk(M)→ Γk(N), and d commute as operations.

Proof: Let us restrict ourselves here to proof of (iv). It suffices to prove (iv) in achart U , and ω = ωIdxI . Let us first compute f ∗(dω):

f ∗(dω) = f ∗dωI ∧dxi1 ∧ · · ·∧dxik

= d(ωI f )∧d(ϕ f )i1 ∧ · · ·∧d(ϕ f )ik .

Similarly,

d( f ∗ω) = d(ωI f )∧d(ϕ f )i1 ∧ · · ·∧d(ϕ f )ik

= d(ωI f )∧d(ϕ f )i1 ∧ · · ·∧d(ϕ f )ik ,

which proves the theorem.

90

16. Stokes’ Theorem

The last section of this chapter deals with a generalization of the FundamentalTheorem of Integration: Stokes’ Theorem. The Stokes’ Theorem allows us tocompute

RM dω in terms of a boundary integral for ω. In a way the (m− 1)-form

act as the ‘primitive’, ‘anti-derivative’ of the dω. Therefore if we are interested inR

M σ using Stokes’ Theorem we need to first ‘integrate’ σ, i.e. write σ = dω. Thisis not always possible as we saw in the previous section.

Stokes’ Theorem can be now be phrased as follows.

Theorem 16.1. Let M be a smooth m-dimensional manifold with or without bound-ary ∂M, and ω ∈ Γm−1

c (M). Then,

(15)Z

Mdω =

Z

∂Mj∗ω,

where j : ∂M →M is the natural embedding of the boundary ∂M into M.

In order to prove this theorem we start with the following lemma.

Lemma 16.2. Let ω ∈ Γm−1c (Hm) be given by

ω = ωidx1∧ · · ·∧ dxi∧ · · ·∧dxm.

Then,

(i)RHM dω = 0, if supp(ω)⊂ int (Hm);

(ii)RHm dω =

R∂Hm j∗ω, if supp(ω)∩∂H

m = ∅,

where j : ∂Hm →H

m is the canonical inclusion.

Proof: We may assume without loss of generality that supp(ω) ⊂ [0,R]× · · ·×[0,R] = Im

R . Now,

Z

Hmdω = (−1)i+1

Z

ImR

∂ωi

∂xidx1 · · ·dxm

= (−1)i+1Z

Im−1R

ωi|xi=R−ωi|xi=0

dx1 · · · dxi · · ·dxm.

If supp(ω)⊂ int ImR , then ωi|xi=R−ωi|xi=0 = 0 for all i, and

Z

Hmdω = 0.

91

If supp(ω)∩∂Hm = ∅, then ωi|xi=R−ωi|xi=0 = 0 for all i≤m−1, and ωm|xm=R = 0.

Therefore,Z

Hmdω = (−1)i+1

Z

ImR

∂ωi

∂xidx1 · · ·dxm

= (−1)i+1Z

Im−1R

ωi|xi=R−ωi|xi=0

dx1 · · · dxi · · ·dxm

= (−1)m+1Z

Im−1R

−ωi|xm=0

dx1 · · ·dxm−1

= (−1)mZ

Im−1R

ωi|xm=0

dx1 · · ·dxm−1.

The mapping j : ∂Hm→H

m, which, in coordinates, is given by j = jϕ : (x1, · · · ,xm−1) →(x1, · · · ,xm−1,0), where ϕ(x1, · · · ,xm−1,0) = (x1, · · · ,xm−1). The latter mapping isorientation preserving if m is even and orientation reversing if m is odd. Under themapping j we have that

j∗(ei) = ei, i = 1, · · · ,m−1,

where the ei’s on the left hand side are the unit vectors in Rm−1, and the bold face

ek are the unit vectors in Rm. We have that the induced orientation for ∂H

m isobtained by the rotation e1 →−em, em → e1, and therefore

∂O = [e2, · · · ,em−1,e1].

Under ϕ this corresponds with the orientation [e2, · · · ,em−1,e1] of Rm−1, which is

indeed the standard orientation for m even, and the opposite orientation for m odd.The pullback form on ∂H

m, using the induced orientation on ∂Hm, is given by

( j∗ω)(x1,··· ,xm−1)(e2, · · · ,em−1,e1)

= ω(x1,··· ,xm−1,0)

j∗(e2), · · · , j∗(em−2), j∗(e1)

= (−1)mωm(x1, · · · ,xm−1,0),

where we used the fact that

dx1∧ · · ·∧dxm−1

j∗(e2), · · · , j∗(em−1), j∗(e1))

= dx1∧ · · ·∧dxm−1(e2, · · · ,em−1,e1)

= (−1)mdx2∧ · · ·∧dxm−1∧dx1(e2, · · · ,em−1,e1) = (−1)m.

Combining this with the integral over Hm we finally obtain

Z

Hmdω =

Z

∂Hmj∗ω,

which completes the proof.

92

Proof of Theorem 16.1: Let us start with the case that supp(ω) ⊂ U , where(U,ϕ) is an oriented chart for M. Then by the definition of the integral we havethat

Z

Mdω =

Z

ϕ(U)(ϕ−1)∗(dω) =

Z

Hm(ϕ−1)∗(dω) =

Z

Hmd(ϕ−1)∗ω

,

where the latter equality follows from Theorem 15.5. It follows from Lemma 16.2that if supp(ω)⊂ int (Hm), then the latter integral is zero and thus

RM ω. = 0. Also,

using Lemma 16.2, it follows that if supp(ω)∩∂Hm = ∅, then

Z

Mdω =

Z

Hmd(ϕ−1)∗ω

=

Z

∂Hmj∗(ϕ−1)∗ω

=Z

∂Hm(ϕ−1 j)∗ω =

Z

∂Mj∗ω,

where j : ∂Hm → H

m is the canonical inclusion as used Lemma 16.2, and j =ϕ−1 j : ∂M →M is the inclusion of the boundary of M into M.

Now consider the general case. As before we choose a finite covering AI ⊂ A

of supp(ω) and an associated partition of unity λi subordinate to AI . Considerthe forms λiω, and using the first part we obtain

Z

∂Mj∗ω = ∑

i

Z

∂Mj∗λiω = ∑

i

Z

Md(λiω)

= ∑i

Z

M

dλi∧ω+λidω

=Z

Md∑

iλi

∧ω+

Z

M

∑

iλi

dω

=Z

Mdω,

which proves the theorem.

16.3 Remark. If M is a closed (compact, no boundary), oriented manifold mani-fold, then by Stokes’ Theorem for any ω ∈ Γm−1(M),

Z

Mdω = 0,

since ∂M = ∅. As a consequence volume form σ cannot be exact. Indeed,R

M σ > 0,and thus if σ were exact, then σ = dω, which implies that

RM σ =

RM dω = 0, a

contradiction. Let us start with the obvious examples of Stokes’ Theorem in R, R

2, and R3.

16.4 Example. Let M = [a,b] ⊂ R, and consider the 0-form ω = f , then, sinceM is compact, ω is compactly supported 0-form. We have that dω = f (x)dx, and

Z b

af (x)dx =

Z

a,bf = f (b)− f (a),

93

which is the fundamental theorem of integration. Since on M any 1-form is exactthe theorem holds for 1-forms.

16.5 Example. Let M = γ⊂R2 be a curve parametrized given by γ : [a,b]→R

2.Consider a 0-form ω = f , then

Z

γfx(x,y)dx+ fy(x,y)dy =

Z

γ(a),γ(b)f = f (γ(b))− f (γ(a)).

Now let M = Ω ⊂ R2, a closed subset of R

2 with a smooth boundary ∂Ω, andconsider a 1-form ω = f (x,y)dx + g(x,y)dy. Then, dω =

gx − fy)dx∧ dy, and

Stokes’ Theorem yieldsZ

Ω

∂g∂x− ∂ f

∂y

dxdy =

Z

∂Ωf dx+gdy,

also known as Green’s Theorem in R2.

16.6 Example. In the case of a curve M = γ⊂ R3 we obtain the line-integral for

0-forms as before:Z

γfx(x,y,z)dx+ fy(x,y,z)dy+ fz(x,y,z) = f (γ(b))− f (γ(a)).

If we write the vector field

F = f∂∂x

+g∂∂y

+h∂∂z

,

then

grad f = ∇ f =∂ f∂x

∂∂x

+∂ f∂y

∂∂y

+∂ f∂z

∂∂z

,

and the above expression can be rewritten asZ

γ∇ f ·ds = f |∂γ.

Next, let M = S ⊂ R3 be an embedded or immersed hypersurface, and let ω =

f dx+gdy+hdz be a 1-form. Then,

dω =∂h

∂y− ∂g

∂z

dy∧dz+

∂ f∂z− ∂h

∂x

dz∧dx+

∂g∂x− ∂ f

∂y

dx∧dy.

Write the vector fields

curl F = ∇×F =∂h

∂y− ∂g

∂z

∂∂x

+∂ f

∂z− ∂h

∂x

∂∂y

+∂g

∂x− ∂ f

∂y

∂∂z

.

Furthermore set

ds =

dxdydz

, dS =

dydzdzdxdxdy

,

94

then from Stokes’ Theorem we can write the following surface and line integrals:Z

S∇×F ·dS =

Z

∂SF ·ds,

which is usually referred to as the classical Stokes’ Theorem in R3. The version in

Theorem 16.1 is the general Stokes’ Theorem. Finally let M = Ω a closed subsetof R

3 with a smooth boundary ∂Ω, and consider a a 2-form ω = f dy∧dz +gdz∧dx+hdx∧dy. Then

dω =∂ f

∂x+

∂g∂y

+∂h∂z

dx∧dy∧dz.

Writediv F = ∇ ·F =

∂ f∂x

+∂g∂y

+∂h∂z

, dV = dxdydz,

then from Stokes’ Theorem we obtainZ

Ω∇ ·FdV =

Z

∂ΩF ·dS,

which is referred to as the Gauss Divergence Theorem.

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V. De Rham cohomology

17. Definition of De Rham cohomology

In the previous chapters we introduced and integrated m-forms over manifoldsM. We recall that k-form ω ∈ Γk(M) is closed if dω = 0, and a k-form ω ∈ Γk(M)is exact if there exists a (k−1)-form σ ∈ Γk−1(M) such that ω = dσ. Since d2 = 0,exact forms are closed. We define

Zk(M) =

ω ∈ Γk(M) : dω = 0

= Ker(d),

Bk(M) =

ω ∈ Γk(M) : ∃ σ ∈ Γk−1(M) ω = dσ

= Im(d),

and in particularBk(M)⊂ Zk(M).

The sets Zk and Bk are real vector spaces, with Bk a vector subspace of Zk. Thisleads to the following definition.

Definition 17.1. Let M be a smooth m-dimensional manifold then the de Rham

cohomology groups are defined as

(16) HkdR(M) := Zk(M)/Bk(M), k = 0, · · ·m,

where B0(M) := 0.

It is immediate from this definition that Z0(M) are smooth functions on M thatare constant on each connected component of M. Therefore, when M is con-nected, then H0

dR(M) ∼= R. Since Γk(M) = 0, for k > m = dim M, we havethat Hk

dR(M) = 0 for all k > m. For k < 0, we set HkdR(M) = 0.

17.2 Remark. The de Rham groups defined above are in fact real vector spaces,and thus groups under addition in particular. The reason we refer to de Rhamcohomology groups instead of de Rham vector spaces is because (co)homologytheories produce abelian groups.

An equivalence class [ω] ∈HkdR(M) is called a cohomology class, and two form

ω,ω ∈ Zk(M) are cohomologous if [ω] = [ω]. This means in particular that ω andω differ by an exact form, i.e.

ω = ω+dσ.

Now let us consider a smooth mapping f : N → M, then we have that the pull-back f ∗ acts as follows: f ∗ : Γk(M)→ Γk(N). From Theorem 15.5 it follows that

96

d f ∗ = f ∗ d and therefore f ∗ descends to homomorphism in cohomology. Thiscan be seen as follows:

d f ∗ω = f ∗dω = 0, and f ∗dσ = d( f ∗σ),

and therefore the closed forms Zk(M) get mapped to Zk(N), and the exact formBk(M) get mapped to Bk(N). Now define

f ∗[ω] = [ f ∗ω],

which is well-defined by

f ∗ω = f ∗ω+ f ∗dσ = f ∗ω+d( f ∗σ)

which proves that [ f ∗ω] = [ f ∗ω], whenever [ω] = [ω]. Summarizing, f ∗ mapscohomology classes in Hk

dR(M) to classes in HkdR(N):

f ∗ : HkdR(M)→ Hk

dR(N),

Theorem 17.3. Let f : N →M, and g : M → K, then

g∗ f ∗ = ( f g)∗ : HkdR(K)→ Hk

dR(N),

Moreover, id∗ is the identity map on cohomology.

Proof: Since g∗ f ∗ = ( f g)∗ the proof follows immediately.

As a direct consequence of this theorem we obtain the invariance of de Rhamcohomology under diffeomorphisms.

Theorem 17.4. If f : N →M is a diffeomorphism, then HkdR(M)∼= Hk

dR(N).

Proof: We have that id = f f−1 = f−1 f , and by the previous theorem

id∗ = f ∗ ( f−1)∗ = ( f−1)∗ f ∗,

and thus f ∗ is an isomorphism.

18. Homotopy invariance of cohomology

We will prove now that the de Rham cohomology of a smooth manifold M iseven invariant under homeomorphisms. As a matter of fact we prove that the deRham cohomology is invariant under homotopies of manifolds.

97

Definition 18.1. Two smooth mappings f ,g : N → M are said to be homotopic ifthere exists a continuous map H : N× [0,1]→M such that

H(p,0) = f (p)

H(p,1) = g(p),

for all p ∈ N. Such a mapping is called a homotopy from/between f to/and g. Ifin addition H is smooth then f and g are said to be smoothly homotopic, and H iscalled a smooth homotopy.

Using the notion of smooth homotopies we will prove the following crucialproperty of cohomology:

Theorem 18.2. Let f ,g : N →M be two smoothly homotopic maps. Then for k≥ 0it holds for f ∗,g∗ : Hk

dR(M)→ HkdR(N), that

f ∗ = g∗.

18.3 Remark. It can be proved in fact that the above results holds for two homo-topic (smooth) maps f and g. This is achieved by constructing a smooth homotopyfrom a homotopy between maps.

Proof of Theorem 18.2: A map h : Γk(M)→ Γk−1(N) is called a homotopy map

between f ∗ and g∗ if

(17) dh(ω)+h(dω) = g∗ω− f ∗ω, ω ∈ Γk(M).

Now consider the embedding it : N → N× I, and the trivial homotopy betweeni0 and i1 (just the identity map). Let ω ∈ Γk(N× I), and define the mapping

h(ω) =Z 1

0i ∂

∂tωdt,

which is a map from Γk(N× I)→ Γk−1(N). Choose coordinates so that either

ω = ωI(x, t)dxI, or ω = ωI(x, t)dt ∧dxI.

In the first case we have that i ∂∂t

ω = 0 and therefore dh(ω) = 0. On the other hand

h(dω) = h

∂ωI

∂tdt ∧dxI +

∂ωI

∂xidxi∧dxI

=Z 1

0

∂ωI

∂tdt

dxI

=ωI(x,1)−ωI(x,0)

dxI = i∗1ω− i∗0ω,

which prove (17) for i∗0 and i∗1, i.e.

dh(ω)+h(dω) = i∗1ω− i∗0ω.

98

In the second case we have

h(dω) = h

∂ωI

∂xidxi∧dt ∧dxI

=Z 1

0

∂ωI

∂dxii ∂

∂xi

dxi∧dt ∧dxIdt

= −Z 1

0

∂ωI

∂xidt

dxi∧dxI .

On the other hand

dh(ω) = dZ 1

0ωI(x, t)dt

dxI

=∂

∂xi

Z 1

0ωI(x, t)dt

dxi∧dxI

=Z 1

0

∂ωI

∂xidt

dxi∧dxI

= −h(dω).

This gives the relation that

dh(ω)+h(dω) = 0,

and since i∗1ω = i∗0ω = 0 in this case, this then completes the argument in bothcases, and h as defined above is a homotopy map between i∗0 and i∗1.

By assumption we have a smooth homotopy H : N× [0,1]→M bewteen f andg, with f = H i0, and g = H i1. Consider the composition h = hH∗. Using therelations above we obtain

h(dω)+dh(ω) = h(H∗dω)+dh(H∗ω)

= h(d(H∗ω))+dh(H∗ω)

= i∗1H∗ω− i∗0H∗ω

= (H i1)∗ω− (H i0)∗ω

= g∗ω− f ∗ω.

If we assume that ω is closed then

g∗ω− f ∗ω = dh(H∗ω),

and thus0 = [dh(H∗ω)] = [g∗ω− f ∗ω] = g∗[ω]− f ∗[ω],

which proves the theorem.

18.4 Remark. Using the same ideas as for the Whitney embedding theorem onecan prove, using approximation by smooth maps, that Theorem 18.2 holds true forcontinuous homotopies between smooth maps.

99

Definition 18.5. Two manifolds N and M are said to be homotopy equivalent, ifthere exist smooth maps f : N →M, and g : M → N such that

g f ∼= idN , f g∼= idM (homotopic maps).

We write N ∼ M., The maps f and g are homotopy equivalences are each otherhomotopy inverses . If the homotopies involved are smooth we say that N and Msmoothly homotopy equivalent.

18.6 Example. Let N = S1, the standard circle in R2, and M = R

2\(0,0). Wehave that N ∼M by considering the maps

f = i : S1 → R2\(0,0), g = id/| · |.

Clearly, (g f )(p) = p, and ( f g)(p) = p/|p|, and the latter is homotopic to theidentity via H(p, t) = t p+(1− t)p/|p|.

Theorem 18.7. Let N and M be smoothly homotopically equivalent manifold, N ∼M, then

HkdR(N)∼= Hk

dR(M),

and the homotopy equivalences f g between N and M, and M and N respectivelyare isomorphisms.

As before this theorem remains valid for continuous homotopy equivalences ofmanifolds.

100

VI. Exercises

A number of the exercises given here are taken from the Lecture notes by J. Bochnak.

Manifolds

Topological Manifolds

1 Given the function g : R→ R2, g(t) = (cos(t),sin(t)). Show that f (R) is a mani-

fold.

2 Given the set T2 = p = (p1, p2, p3) | 16(p2

1 + p22) = (p2

1 + p22 + p2

3 + 3)2 ⊂ R3,

called the 2-torus.

(i ) Consider the product manifold S1×S1 = q = (q1,q2,q3,q4) | q21 +q2

2 = 1, q23 +

q24 = 1, and the mapping f : S1×S1 → T

2, given by

f (q) =q1(2+q3),q2(2+q3),q4

.

Show that f is onto and f−1(p) = p1

r , p2r ,r−2, p3

, where r = |p|2+3

4 .

(ii ) Show that f is a homeomorphism between S1×S1 and T2.

3 (i ) Find an atlas for T2 using the mapping f in 2.

(ii ) Give a parametrization for T2.

4 Show thatA4,4 = (p1, p2) ∈ R

2 | p41 + p4

2 = 1,

is a manifold and A4,4 ∼= S1 (homeomorphic).

5 (i ) Show that an open subset U ⊂M of a manifold M is again a manifold.

(ii ) Let N and M be manifolds. Show that their cartesian production N×M is alsoa manifold.

101

6 Show that

(i ) A ∈M2,2(R) | det(A) = 1 is a manifold.

(ii ) Gl(n,R) = A ∈Mn,n(R) | det(A) = 0 is a manifold.

(iii ) Determine the dimensions of the manifolds in (a) and (b).

7 Construct a simple counterexample of a set that is not a manifold.

8 Show that in Definition 1.1 an open set U ⊂ Rn can be replaced by an open disc

Dn ⊂ Rn.

9 Show that PRn is a Hausdorff space and is compact.

10 Define the Grassmann manifold GkR

n as the set of all k-dimensional linear sub-spaces in R

n. Show that GkR

n is a manifold.

11 Consider X to be the parallel lines R×0∪R×1. Define the equivalence relation(x,0)∼ (x,1) for all x = 0. Show that M = X/∼ is a topological space that satisfies(ii) and (iii) of Definition 1.1.

12 Let M be an uncountable union of copies of R. Show that M is a topological spacethat satisfies (i) and (ii) of Definition 1.1.

13 Let M = (x,y) ∈ R2 : xy = 0. Show that M is a topological space that satisfies

(i) and (iii) of Definition 1.1.

Differentiable manifolds

14 Show that cartesian products of differentiable manifolds are again differentaiblemanifolds.

15 Show that PRn is a smooth manifold.

16 Prove that PR is diffeomorphic to the standard unit circle in R2 by constructing a

diffeomorphism.

17 Which of the atlases for S1 given in Example 2, Sect. 1, are compatible. If notcompatible, are they diffeomorphic?

18 Show that the standard torus T2 ⊂ R

3 is diffeomorphic to S1×S1, where S1 ⊂ R2

is the standard circle in R2.

19 Prove that the taking the union of C∞-atlases defines an equivalence relation (Hint:use Figure 7 in Section 2).

102

20 Show that diffeomorphic defines an equivalence relation.

21 Prove Theorem 2.15.

Immersion, submersion and embeddings

22 Show that the torus the map f defined in Exer. 2 of 1.1 yields a smooth embeddingof the torus T

2 in R3.

23 Let k,m,n be positive integers. Show that the set

Ak,m,n := (x,y,z) ∈ R3 : xk + ym + zn = 1,

is a smooth embedded submanifold in R3, and is diffeomorphic to S2 when these

numbers are even.

24 Prove Lemma 3.18.

25 Let f : Rn+1\0→ R

k+1\0 be a smooth mapping such that f (λx) = λd f (x),d ∈ N, for all λ ∈ R\0. This is called a homogeneous mapping of degree d.Show that f : PR

n → PRk, defined by f ([x]) = [ f (x)], is a smooth mapping.

26 Show that the open ball Bn ⊂ Rn is diffeomorphic to R

n.

27 (Lee) Consider the mapping f (x,y,s) = (x2 + y,x2 + y2 + s2 + y) from R3 to R

2.Show that q = (0,1) is a regular value, and f−1(q) is diffeomorphic to S1 (stan-dard).

28 Prove Theorem 3.24 (Hint: prove the steps indicated above Theorem 3.24).

29 Prove Theorem 3.26.

30 Let N, M be two smooth manifolds of the same dimension, and f : N → M is asmooth mapping. Show (using the Inverse Function Theorem) that if f is a bijec-tion then it is a diffeomorphism.

31 Prove Theorem 3.28.

32 Show that the map f : Sn → PRn defined by f (x1, · · · ,xn+1) = [(x1, · · · ,xn+1)] is

smooth and everywhere of rank n (see Boothby).

33 (Lee) Let a ∈ R, and Ma = (x,y) ∈ R2 : y2 = x(x−1)(x−a).

(i ) For which values of a is Ma an embedded submanifold of R2?

(ii ) For which values of a can Ma be given a topology and smooth structure so thatMa is an immersed submanifold?

103

34 Consider the map f : R2 →R given by f (x1,x2) = x3

1 +x1x2 +x32. Which level sets

of f are smooth embedded submanifolds of R2.

35 Let M be a smooth m-dimensional manifold with boundary ∂M (see Lee, pg. 25).Show that ∂M is an (m− 1)-dimensional manifold (no boundary), and that thereexists a unique differentiable structure on ∂M such that i : ∂M → M is a (smooth)embedding.

36 Let f : N →M be a smooth mapping. Let S = f−1(q)⊂ N is a smooth embeddedsubmanifold of codimension m. Is q necessarily a regular value of f ?

37 Use Theorem 3.27 to show that the 2-torus T2 is a smooth embedded manifold in

R4.

38 Prove Theorem 3.37.

39 Show that an m-dimensional linear subspace in R is an m-dimensional manifold.

40 Prove that the annulus A = (p1, p2)∈R2 : 1≤ p2

1 + p22 ≤ 4 is a smooth manifold

two dimensional manifold in R2 with boundary.

104

Tangent and cotangent spaces

Tangent spaces

41 Let U ⊂ Rk, y ∈ R

m, and f : U → Rm a smooth function. By Theorem 3.28 M =

f−1(y) = p∈Rk : f (p) = y is a smooth embedded manifold in R

k if rk(J f )|p =m for all p ∈M. Show that

TpM ∼= Xp ∈ Rk : Xp ·∂pi f = 0, i = 1, · · · ,m = kerJ f |p,

i.e. the tangent space is the kernel of the Jacobian. Using this identification fromnow on, prove that

Mp = p+TpM ⊂ Rk

is tangent to M at p.

42 Given the set M = (p1, p2, p3) ∈ R3 : p3

1 + p32 + p3

3−3p1 p2 p3 = 1.

(i ) Show that M is smooth embedded manifold in R3 of dimension 2.

(ii ) Compute TpM, at p = (0,0,1).

43 Given the set M = (p1, p2, p3) ∈ R3 : p2

1− p22 + p1 p3−2p2 p3 = 0, 2p1− p2 +

p3 = 3.

(i ) Show that M is smooth embedded manifold in R3 of dimension 1.

(ii ) Compute TpM, at p = (1,−1,0).

44 Prove Lemma 4.5.

45 Express the following planar vector fields in polar coordinates:

(i ) X = x ∂∂x + y ∂

∂y ;

(ii ) X =−y ∂∂x + x ∂

∂y ;

(iii ) X = (x2 + y2) ∂∂x .

46 Find a vector field on S2 that vanishes at exactly one point.

47 Let S2 ⊂ R2 be the standard unit sphere and f : S2 → S2 a smooth map defined as

a θ-degree rotation around the polar axis. Compute f∗ : TpS2 → Tf (p)S2 and give amatrix representation of f∗ in terms of a canonical basis.

105

Cotangent spaces

48 (Lee) Let f : R3 → R be given by f (p1, p2, p3) = |p|2, and g : R

2 → R3 by

g(x1,x2) = 2x1

|x|2 +1,

2x2

|x|2 +1,|x|2−1|x|2 +1

.

Compute both g∗d f , and d( f g) and compare the two answers.

49 (Lee) Compute d f in coordinates, and determine the set points where d f vanishes.

(i ) M = (p1, p2) ∈ R2 : p2 > 0, and f (p) = p1

|p|2 — standard coordinates in R2.

(ii ) As the previous, but in polar coordinates.

(iii ) M = S2 = p ∈ R3 : |p| = 1, and f (p) = p3 — stereographic coordinates.

(iv) M = Rn, and f (p) = |p|2 — standard coordinates in R

n.

50 Express in polar coordinates

(i ) θ = x2dx+(x+ y)dy;

(ii ) θ = xdx+ ydy+ zdz.

51 Prove Lemma 5.3.

52 (Lee) Let f : N → M (smooth), ω ∈ Λ1(N), and γ : [a,b]→ N is a smooth curve.Show that Z

γf ∗ω =

Z

fγω.

53 Given the following 1-forms on R3:

α =− 4z(x2 +1)2 dx+

2yy2 +1

dy+2x

x2 +1dz,

ω =− 4xz(x2 +1)2 dx+

2yy2 +1

dy+2

x2 +1dz.

(i ) Let γ(t) = (t, t, t), t ∈ [0,1], and computeR

γ α andR

γ ω.

(ii ) Let γ be a piecewise smooth curve going from (0,0,0) to (1,0,0) to (1,1,0) to(1,1,1), and compute the above integrals.

(iii ) Which of the 1-forms α and ω is exact.

(iv) Compare the answers.

106

Vector bundles

54 Let M ⊂R be a manifold in R

. Show that T M as defined in Section 6 is a smoothembedded submanifold of R

2.

55 Let S1 ⊂ R2 be the standard unit circle. Show that T S1 is diffeomorphic to S1×R.

56 Let M⊂R be a manifold in R

. Show that T ∗M as defined in Section 6 is a smoothembedded submanifold of R

2.

57 Show that the open Mobius band is a smooth rank-1 vector bundle over S1.

107

Tensors

Tensors and tensor products

58 Describe the standard inner product on Rn as a covariant 2-tensor.

59 Construct the determinant on R2 and R

3 as covariant tensors.

60 Given the vectors a =

130

, and b =

24

. Compute a∗ ⊗b and b∗ ⊗a.

61 Given finite dimensional vector spaces V and W , prove that V ⊗W and W ⊗V areisomorphic.

62 Given finite dimensional vector spaces U , V and W , prove that (U ⊗V )⊗W andU⊗ (V ⊗W ) are isomorphic.

63 Show that V ⊗RV R⊗V .

Symmetric and alternating tensors

64 Show that for T ∈ T s(V ) the tensor Sym T is symmetric.

65 Prove that a tensor T ∈ T s(V ) is symmetric if and only if T = Sym T .

66 Show that the algebra

Σ∗(V ) =∞M

k=0Σk(V ),

is a commutative algebra.

67 Prove that for any T ∈ T s(V ) the tensor Alt T is alternating.

68 Show that a tensor T ∈ T s(V ) is alternating if and only if T = Alt T .

69 Given the vectors a =

124

, b =

013

, and c =

112

. Compute a∧b∧ c,

and compare this with det(a,b,c).

70 Given vectors a1, · · · ,an show that a1∧ · · ·∧an = det(a1, · · · ,an).

108

71 Prove Lemma 8.11.

Tensor bundles and tensor fields

72 Let M ⊂R be an embedded m-dimensional manifold. Show that T rM is a smooth

manifold in R+r .

73 Similarly, show that TsM is a smooth manifold in R+s , and T r

s M is a smoothmanifold in R

+r+s .

74 Prove that the tensor bundles introduced in Section 7 are smooth manifolds.

75 One can consider symmetric tensors in TpM as defined in Section 8. Define anddescribe Σr(TpM)⊂ T r(TpM) and ΣrM ⊂ T rM.

76 Describe a smooth covariant 2-tensor field in Σ2M ⊂ T 2M. How does this relate toan (indefinite) inner product on TpM?

77 Prove Lemma 9.2.

78 Given the manifolds N = M = R2, and the smooth mapping

q = f (p) = (p21− p2, p1 +2p3

2),

acting from N to M. Consider the tensor spaces T 2(TpN) ∼= T 2(TqM) ∼= T 2(R2)and compute the matrix for the pullback f ∗.

79 In the above problem consider the 2-tensor field σ on M, given by

σ = dy1⊗dy2 +q1dy2⊗dy2.

(i ) Show that σ ∈ F2(M).

(ii ) Compute the pulback f ∗σ and show that f ∗σ ∈ F2(N).

(iii ) Compute f ∗σ(X ,Y ), where X ,Y ∈ F(N).

109

Differential forms

80 Given the differential form σ = dx1∧ dx2− dx2∧ dx3 on R3. For each of the fol-

lowing vector fields X compute iX σ:

(i ) X = ∂∂x1

+ ∂∂x2− ∂

∂x3;

(ii ) X = x1∂

∂x1− x2

2∂

∂x3;

(iii ) X = x1x2∂

∂x1− sin(x3) ∂

∂x2;

81 Given the mappingf (p) = (sin(p1 + p2), p2

1− p2),

acting from R2 to R

2 and the 2-form σ = p21dx1∧dx2. Compute the pullback form

f ∗σ.

82 Prove Lemma 10.2.

83 Derive Formula (10).

Orientations

84 Show that Sn is orientable and give an orientation.

85 Show that the standard n-torus Tn is orientable and find an orientation.

86 Prove that the Klein bottle and the projective space PR2 are non-orientable.

87 Give an orientation for the projective space PR3.

88 Prove that the projective spaces PRn are orientable for n = 2k +1, k ≥ 0.

89 Show that the projective spaces PRn are non-orientable for n = 2k, k ≥ 1.

110

Integration on manifolds

Integrating m-forms on Rm

90 Let U ⊂ Rm be open and let K ⊂U be compact. Show that there exists a domain

of integration D such that K ⊂ D⊂ Rm.

91 Show that Definition 12.2 is well-posed.

Partitions of unity

92 Show that the function f1 defined in Lemma 13.3 is smooth.

93 If U is open covering of M for which each set Ui intersects only finitely many othersets in U, prove that U is locally finite.

94 Give an example of uncountable open covering U of the interval [0,1] ⊂ R, and acountable refinement V of U.

Integration of m-forms on m-dimensional manifolds

95 Let S2 = ∂B3 ⊂ R3 oriented via the standard orientation of R

3, and consider the2-form

ω = xdy∧dz+ ydz∧dx+ zdx∧dy.

Given the parametrization

F(ϕ,θ) =

xyz

=

sin(ϕ)cos(θ)sin(ϕ)sin(θ)

cos(ϕ)

,

for S2, computeR

S2 ω.

96 Given the 2-form ω = xdy∧dz+ zdy∧dx, show thatZ

S2ω = 0.

111

97 Consider the circle S1 ⊂ R2 parametrized by

F(θ) =

xy

=

r cos(θ)r sin(θ)

.

Compute the integral over S1 of the 1-form ω = xdy− ydx.

98 If in the previous problem we consider the 1-form

ω =x

x2 + y2dy− y

x2 + y2dx.

Show thatR

S1 ω represents the induced Euclidian length of S1.

99 Consider the embedded torus

T2 = (x1,x2,x3,x4) ∈ R

4 : x21 + x2

2 = 1, x23 + x2

4 = 1.

Compute the integral over T2 of the 2-form

ω = x21dx1∧dx4 + x2dx3∧dx1.

100 Consider the following 3-manifold M parametrized by g : [0,1]3 → R4,

rst

→

rst

(2r− t)2

.

Compute Z

Mx2dx2∧dx3∧dx4 +2x1x3dx1∧dx2∧dx3.

The exterior derivative

101 Let (x,y,z) ∈ R3. Compute the exterior derivative dω, where ω is given as:

(i ) ω = exyz;

(ii ) ω = x2 + zsin(y);

(iii ) ω = xdx+ ydy;

(iv) ω = dx+ xdy+(z2− x)dz;

(v) ω = xydx∧dz+ zdx∧dy;

(vi ) ω = dx∧dz.

112

102 Which of the following forms on R3 are closed?

(i ) ω = xdx∧dy∧dz;

(ii ) ω = zdy∧dx+ xdy∧dz;

(iii ) ω = xdx+ ydy;

(iv) ω = zdx∧dz.

103 Verify which of the following forms ω on R2 are exact, and if so write ω = dσ:

(i ) ω = xdy− ydx;

(ii ) ω = xdy+ ydx;

(iii ) ω = dx∧dy;

(iv) ω = (x2 + y3)dx∧dy;

104 Verify which of the following forms ω on R3 are exact, and if so write ω = dσ:

(i ) ω = xdx+ ydy+ zdz;

(ii ) ω = x2dx∧dy+ z3dx∧dz;

(iii ) ω = x2ydx∧dy∧dz.

105 Verify that on R2 and R

3 all closed k-forms, k ≥ 1, are exact.

106 Find a 2-form on R3\(0,0,0) which is closed but not exact.

Stokes’ Theorem

107 Let Ω ⊂ R3 be a parametrized 3-manifold,i.e. a solid, or 3-dimensional domain.

Show that the standard volume of Ω is given by

Vol(M) =13

Z

∂Ωxdy∧dz− ydx∧dz+ zdx∧dy.

108 Let Ω⊂Rn be an n-dimensional domain. Prove the analogue of the previous prob-

lem.

109 Prove the ‘integration by parts’ formulaZ

Mf dω =

Z

∂Mf ω−

Z

Md f ∧ω,

where f is a smooth function and ω a k-form.

110 Compute the integralZ

S2x2ydx∧dz+ x3dy∧dz+(z−2x2)dx∧dy,

113

where S2 ⊂ R3 is the standard 2-sphere.

111 Use the standard polar coordinates for the S2 ⊂ R3 with radius r to compute

Z

S2

xdy∧dz− ydx∧dz+ zdx∧dy(x2 + y2 + z2)3/2 ,

and use the answer to compute the volume of the r-ball Br.

Extra’s

112 Use the examples in Section 16 to show that

curl grad f = 0, and div curl F = 0,

where f : R3 → R, and F : R

3 → R3.

113 Given the mapping f : R2 → R

3, ω = dx∧dz, and

f (s, t) =

cos(s)sin(t)√

s2 + t2

st

.

Compute f ∗ω.

114 Given the mapping f : R3 → R

3, ω = xydx∧dy∧dz, and

f (s, t,u) =

scos(t)ssin(t)

u

.

Compute f ∗ω.

115 Let M = R3\(0,0,0), and define the following 2-form on M:

ω =1

(x2 + y2 + z2)3/2

xdy∧dz+ ydz∧dx+ zdx∧dy

.

(i ) Show that ω is closed (dω = 0) (Hint: computeR 2

S ω).

(ii ) Prove that ω is not exact on M!

On N = R3\x = y = 0⊃M consider the 1-form:

η =−z

(x2 + y2 + z2)1/2xdy− ydx

x2 + y2 .

(iii ) Show that ω is exact as 2-form on N, and verify that dη = ω.

114

116 Let M ⊂ R4 be given by the parametrization

(s, t,u) →

st +u

ts−u

, (s, t,u) ∈ [0,1]3.

(i ) ComputeR

M dx1∧dx2∧dx4.

(ii ) ComputeR

M x1x3dx1∧dx2∧dx3 + x23dx2∧dx3∧dx4.

117 Let C = ∂∆ be the boundary of the triangle OAB in R2, where O = (0,0), A =

(π/2,0), and B = (π/2,1). Orient the traingle by traversing the boundary counterclock wise. Given the integral

Z

C(y− sin(x))dx+ cos(y)dy.

(i ) Compute the integral directly.

(ii ) Compute the integral via Green’s Theorem.

118 Compute the integralR

Σ F ·dS, where Σ = ∂[0,1]3, and

F(x,y,z) =

4xz−y2

yz

(Hint: use the Gauss Divergence Theorem).

115

De Rham cohomology

Definition of De Rham cohomology

119 Prove Theorem 17.3.

120 Let M =S

j M j be a (countable) disjoint union of smooth manifolds. Prove theisomorphism

HkdR(M)∼= ∏

jHk

dR(M j).

121 Show that the mapping ∪ : HkdR(M)×H

dR(M)→Hk+dR (M), called the cup-product,

and defined by

[ω]∪ [η] := [ω∧η],

is well-defined.

122 Let M = S1, show that

H0dR(S1)∼= R, H1

dR(S1)∼= R, HkdR(S1) = 0,

for k ≥ 2.

123 Show that

H0dR(R2\(0,0))∼= R, H1

dR(R2\(0,0))∼= R, HkdR(R2\(0,0)) = 0,

for k ≥ 2.

124 Find a generator for H1dR(R2\(0,0)).

125 Compute the de Rham cohomology of the n-torus M = Tn.

126 Compute the de Rham cohomology of M = S2.