INTRODUCTION TODIFFERENTIABLE

MANIFOLDS

spring 2007

Course by Prof. dr. R.C.A.M. van der Vorst1

2

Introduction to differentiable manifoldsLecture notes version 2.1, May 25, 2007

This is a self contained set of lecture notes. The notes were written by Rob van derVorst. We follow the book ‘Introduction to Smooth Manifolds’ by John M. Lee asa 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 Commons

Attribution-NonCommercial-ShareAlike License.

To view a copy of this license, visit

http://creativecommons.org/licenses/by-nc-sa/2.0/

or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California

94305, USA.

CONTENTS

I. Manifolds 51. Topological manifolds 52. Differentiable manifolds and differentiable structures 113. Immersions, submersions and embeddings 184. Manifolds in Euclidean space 29

II. Tangent and cotangent spaces 325. Tangent spaces 326. Cotangent spaces 387. Vector bundles 407.1. The tangent bundle and vector fields 447.2. The cotangent bundle and differential 1-forms 468. Tangent and cotangent spaces for embedded manifolds 488.1. Tangent spaces 498.2. Cotangent spaces 54

III. Tensors and differential forms 559. Tensors and tensor products 5510. Symmetric and alternating tensors 6011. Tensor bundles and tensor fields 66

3

12. Differential forms 7013. Orientations 74

IV. Integration on manifolds 8014. Integrating m-forms onRm 8015. Partitions of unity 8116. Integration on of m-forms on m-dimensional manifolds. 8617. The exterior derivative 9218. Stokes’ Theorem 97

V. De Rham cohomology 10219. Definition of De Rham cohomology 10220. Homotopy invariance of cohomology 103

VI. Exercises 107

ManifoldsTopological Manifolds 107Differentiable manifolds 108Immersion, submersion and embeddings 109Manifolds in Euclidean space 110

Tangent and cotangent spacesTangent spaces 111Cotangent spaces 112Vector bundles 113

TensorsTensors and tensor products 114Symmetric and alternating tensors 114Tensor bundles and tensor fields 115Differential forms 116Orientations 116

Integration on manifoldsIntegrating m-forms onRm 117Partitions of unity 117Integration of m-forms on m-dimensional manifolds 117The exterior derivative 118Stokes’ Theorem 119Extra’s 120

De Rham cohomology

4

Definition of De Rham cohomology 122

5

I. Manifolds

1. Topological manifolds

Basically anm-dimensional (topological) manifold is a topological spaceM

which is locally homeomorphic toRm. A more precise definition is:

Definition 1.1. 1 A topological spaceM is called anm-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 subsetV ⊂ Rm, and(iii) M has a countable basis of open sets.

Axiom (ii) is equivalent to saying thatp∈ M has a open neighborhoodU 3 p

homeomorphic to the open discDm in Rm. We sayM is locally homeomorphictoRm. Axiom (iii) says thatM can be covered by countably many of such neighbor-hoods.

Rm

M

ϕ

Up

x

FIGURE 1. Coordinate charts(U,ϕ).

1See Lee, pg. 3.2Usually an open neighborhoodU of a pointp∈M is an open set containingp. A neighborhood

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

neighborhood is an open set.

6

M

ϕiϕj

Rm

Rm

ϕij = ϕj ϕ−1

i

UiUj

Vj

Vi

p

x y

FIGURE 2. The transition mapsϕi j .

Recall some notions from topology: A topological spaceM is calledHausdorffif for any pair p,q∈ M, there exist (open) neighborhoodsU 3 p, andU ′ 3 q suchthatU ∩U ′ = ∅. For a topological spaceM with topologyτ, a collectionβ ⊂ τ isa basisif and only if eachU ∈ τ can be written as union of sets inβ. A basis iscalled acountable basisif β is a countable set.

Figure 37 displayscoordinate charts(U,ϕ), whereU are coordinate neighbor-hoods, or charts, andϕ are (coordinate) homeomorphisms. Transitions betweendifferent choices of coordinates are calledtransitions mapsϕi j = ϕ j ϕ−1

i , whichare again homeomorphisms by definition. We usually writex = ϕ(p), ϕ : U →V ⊂Rn, ascoordinatesfor U , see Figure 2, andp= ϕ−1(x), ϕ−1 : V →U ⊂M, asa parametrizationof U . A collectionA = (ϕi ,Ui)i∈I of coordinate charts withM = ∪iUi , is called anatlasfor 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.

J 1.5Example. M= Rm; the axioms (i) and (iii) are obviously satisfied. As for (ii)we takeU = Rm, andϕ the identity map. I

J 1.6Example. M= S1 = p= (p1, p2) ∈R2 | p21+ p2

2 = 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 areϕ1(p) = p1, ϕ2(p) = p1, ϕ3(p) = p2, andϕ4(p) = p2. For instance

ϕ−11 (x) =

(x,√

1−x2),

and the domain isV1 = (−1,1). It is clear thatϕi andϕ−1i are continuous, and

therefore the mapsϕi are homeomorphisms. With these choices we have found anatlas forS1 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 fromUi to R. For example

ϕ−11 (x) =

( 4xx2 +4

,x2−4x2 +4

),

which is continuous fromR to U1.

(p1, p2)

(p′1, p

′

2)

p1

p2

xx′

Np

Sp

FIGURE 3. The stereographic projection describingϕ1.

(c): (Polar coordinates) Consider the two chartsU1 = S1\(1,0), andU2 =S1\(−1,0). The homeomorphism areϕ1(p) = θ ∈ (0,2π) (polar angle counterclockwise rotation), andϕ2(p) = θ ∈ (−π,π) (complex argument). For exampleϕ−1

1 (θ) = (cos(θ),sin(θ)). I

8

J 1.8Example. M= Sn = p = (p1, · · · pn+1) ∈ Rn+1 | |p|2 = 1. Obvious exten-sion of the choices forS1. I

J 1.9Example.(see Lee) LetU ⊂Rn be an open set andg : U →Rm a continuousfunction. Define

M = Γ(g) = (x,y) ∈ Rn×Rm : x∈U, y = g(x),

endowed with the subspace topology. This topological space is ann-dimensionalmanifold. Indeed, defineπ : Rn×Rm→ Rn asπ(x,y) = x, andϕ = π|Γ(g), whichis continuous ontoU . The inverseϕ−1(x) = (x,g(x)) is also continuous. Therefore(Γ(g),ϕ) is an appropriate atlas. The manifoldΓ(g) is homeomorphic toU . I

J 1.10 Example. M= PRn, the real projective spaces. Consider the followingequivalence relation on points onRn+1\0: For any twox,y∈ Rn+1\0 define

x∼ y if there exists aλ 6= 0, such thatx = λy.

DefinePRn = [x] : x ∈ Rn+1\0 as the set of equivalence classes. One canthink of PRn as the set of lines through the origin inRn+1. Consider the naturalmapπ : Rn+1\0 → PRn, via π(x) = [x]. A setU ⊂ PRn is open ifπ−1(U) isopen inRn+1\0. This makesPRn a compact Hausdorff space. In order to verifythat we are dealing with ann-dimensional manifold we need to describe an atlasfor PRn. For i = 1, · · ·n+1, defineVi ⊂ Rn+1\0 as the set of pointsx for whichxi 6= 0, and defineUi = 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 chartsUi coverPRn. In dimensionn = 1 we have thatPR ∼= S1, and in thedimensionn = 2 we obtain an immersed surfacePR2 as shown in Figure 4. I

The examples 1.6 and 1.8 above are special in the sense that they are subsets ofsomeRm, which, as topological spaces, are given a manifold structure.

Define

Hm = (x1, · · · ,xm) | xm≥ 0,

as the standard Euclidean half-space.

9

FIGURE 4. Identification of the curves indicated above yields animmersion ofPR2 into R3.

Definition 1.12. A topological spaceM is called anm-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 neighborhoodU of p, which is homeo-

morphic to an open subsetV ⊂Hm, and(iii) M has a countable basis of open sets.

Axiom (ii) can be rephrased as follows, any pointp∈M is contained in a neigh-borhoodU , which either homeomorphic toDm, or toDm∩Hm. The setM is locallyhomeomorphic toRm, or Hm. Theboundary∂M ⊂M is a subset ofM which con-sists of pointsp for which any neighborhood cannot be homeomorphic to an opensubset of int(Hm). In other words pointp∈ ∂M are points that lie in the inverseimage ofV ∩ ∂Hm for some chart(U,ϕ). Pointsp∈ ∂M are mapped to points on∂Hm and∂M is an(m−1)-dimensional topological manifold.J 1.14Example.Consider the bounded cone

M = C = p = (p1, p2, p3) ∈ R3 | p21 + p2

2 = p23, 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) 6= (0, p3),

U3 = p∈C | 12

< p3 ≤ 1, (p1, p2) 6= (0,−p3).

For instanceϕ2 can be constructed as follows. Define

q = ψ(p) =( p1

p3,

p2

p3, p3

), σ(q) =

( 2q1

1−q2,1−q3

),

andx = ϕ2(p) = (σ ψ)(p), ϕ2(U2) = R× [0, 12) ⊂ H2. The mapϕ3 is defined

similarly. The boundary is given by∂C= ϕ−12 (R×0)∪ϕ−1

3 (R×0), see Figure6. I

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

2 = p23, 0≤

p3 < 1, can be described by one coordinate chart, seeU1 above, and is therefore amanifold without boundary (non-compact), andC is homeomorphic toD2, or R2,see definition below. I

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

Let N andM be manifolds, and letf : N → M be a continuous mapping. Amapping f is called ahomeomorphismbetweenN andM if f is continuous andhas a continuous inversef−1 : M →N. In this case the manifoldsN andM are said

11

FIGURE 6. Coordinate maps for the coneC.

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

Recall the subspace topology. LetX be a topological space and letS⊂ X beany subset, then thesubspace, or relative topologyon S (induced by the topologyon X) is defined as follows. A subsetU ⊂ S is open if there exists an open setV ⊂ X such thatU = V ∩S. In this caseS is called a (topological)subspaceof X.A (topological) embeddingis a continuous injective mappingf : X →Y, which isa homeomorphism onto its imagef (X)⊂Y with respect to the subspace topology.Let f : N→M be an embedding, then its imagef (N)⊂M is called asubmanifoldof M. Notice that an open submanifold is the special case whenf = i : U → M isan inclusion mapping.

2. Differentiable manifolds and differentiable structures

A topological manifoldM for which the transition mapsϕi j = ϕ j ϕ−1i for all

pairsϕi ,ϕ j in the atlas are diffeomorphisms is called adifferentiable, or smoothmanifold. The transition maps are mappings between open subsets ofRm. Dif-feomorphisms between open subsets ofRm areC∞-maps, whose inverses are alsoC∞-maps. For two chartsUi andU j the transitions maps are mappings:

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

and as such are homeomorphisms between these open subsets ofRm.

12

FIGURE 7. Differentiability ofϕ′′ ϕ−1 is achieved via the map-pingsϕ′′ (ϕ′)−1, andϕ′ ϕ−1, which are diffeomorphisms sinceA∼A′, andA′ ∼A′′ by assumption. This establishes the equiva-lenceA∼A′′.

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

(i) M = ∪i∈IUi ,(ii) the transition mapsϕi j are diffeomorphisms betweenϕi(Ui ∩U j) and

ϕ j(Ui ∩U j), for all i 6= j (see Figure 2 ).

The charts in aC∞-atlas are said to beC∞-compatible. Two C∞-atlasesA andA′ are equivalent ifA∪A′ is again aC∞-atlas, which clearly defines a equivalencerelation onC∞-atlases. An equivalence class of this equivalence relation is calledadifferentiable structureD onM. The collection of all atlases associated withD,denotedAD, is called themaximal atlasfor the differentiable structure. Figure 7shows why compatibility of atlases defines an equivalence relation.

Definition 2.3. Let M be a topological manifold, and letD be a differentiablestructure onM with maximal atlasAD. Then the pair(M,AD) is called a(C∞-)differentiable manifold.

13

Basically, a manifoldM 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; exoticR4’s. One can also considerCr -differentiable structures andmanifolds. Smoothness will be used here for theC∞-case here.

Theorem 2.4. 4 Let M be a topological manifold with a C∞-atlasA. Then there

exists a unique differentiable structureD containingA, i.e. A⊂AD.

J 2.5 Remark.In our definition of ofn-dimensional differentiable manifold weuse the local model overstandardRn, i.e. on the level of coordinates we expressdifferentiability with respect to the standard differentiable structure onRn, or dif-feomorphic to the standard differentiable structure onRn. I

J 2.6Example.The coneM = C⊂R3 as in the previous section is a differentiablemanifold 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. I

J 2.7 Example. M= S1 (or more generallyM = Sn). As in Section 1 considerS1 with two charts via stereographic projection. We have the overlapU1∩U2 =S1\(0,±1), and the transition mapϕ12 = ϕ2 ϕ−1

1 given by y = ϕ12(x) = 4x ,

x 6= 0,±∞, andy 6= 0,±∞. Clearly,ϕ12 and its inverse are differentiable functionsfrom ϕ1(U1∩U2) = R\0 to ϕ2(U1∩U2) = R\0. I

pϕ12

S1

x = ϕ1(p)

y = ϕ2(p)

Sp

Np

FIGURE 8. The transition maps for the stereographic projection.

4Lee, Lemma 1.10.

14

J 2.9Example.The real projective spacesPRn, see exercises Chapter VI. I

J 2.10Example.The generalizations of projective spacesPRn, the so-called(k,n)-GrassmanniansGkRn are examples of smooth manifolds. I

Theorem 2.11. 5 Given a set M, a collectionUαα∈A of subsets, and injective

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

(i) ϕα(Uα)⊂ Rm is open for allα;

(ii) ϕα(Uα∩Uβ) andϕβ(Uα∩Uβ) are open inRm for any pairα,β ∈ A;

(iii) for Uα∩Uβ 6= ∅, the mappingsϕα ϕ−1β : ϕβ(Uα∩Uβ)→ ϕα(Uα∩Uβ) are

diffeomorphisms for any pairα,β ∈ A;

(iv) countably many sets Uα cover M.

(v) for any pair p 6= q∈ M, either p,q∈Uα, or there are disjoint sets Uα,Uβsuch that p∈Uα and q∈Uβ.

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

charts.

Let N andM be smooth manifolds (dimensionsn andm respectively). Letf :N→M be a mapping fromN to M.

Definition 2.12. A mapping f : N → M is said to beC∞, or smoothif 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 aC∞-mapping (fromRn to Rm).

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

5See Lee, Lemma 1.23.

15

The above definition also holds true for mappings defined on open subsets ofN, i.e. letW ⊂ N is an open subset, andf : W ⊂ N → M, then smoothness onWis defined as above by restricting to piontsp∈W. With this definition coordinatemapsϕ : U → ϕ(U) are smooth maps.

The definition of smooth mappings allows one to introduce the notion of differ-entiable homeomorphism, ofdiffeomorphismbetween manifolds. LetN andM besmooth manifolds. AC∞-mapping f : N → M, is called a diffeomorphism if it isa homeomorphism and alsof−1 is a smooth mapping, in which caseN andM aresaid to bediffeomorphic. The associated differentiable structures are also calleddiffeomorphic. Diffeomorphic defines an equivalence relation. In the definitionof diffeomorphism is suffices require thatf is a differentiable bijective mappingwith smooth inverse (see Theorem 2.15). A mappingf : N → M is called alocaldiffeomorphismif for every p∈N there exists a neighborhoodU , with f (U) openin M, such thatf : U → f (U) is a diffeomorphism. A mappingf : N → M is adiffeomorphism if and only if it is a bijective local diffeomorphism.J 2.14Example.ConsiderN = R with atlas(R,ϕ(p) = p), andM = R with atlas(R,ψ(q) = q3). Clearly these define different differentiable structures. BetweenN andM we consider the mappingf (p) = p1/3, which is a homeomorphism be-tweenN andM. The claim is thatf is also a diffeomorphism. TakeU = V = R,thenψ f ϕ−1(p) = (p1/3)3 = p is the identity and thusC∞ 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 to

defining the differential quotient via limh→0f 3(p+h)− f 3(p)

h . I

Theorem 2.15. 6 Let N,M be smooth manifolds with atlasesAN andAM respec-

tively.

(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 → M

such 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 mapsfαβ = ψβ f ϕ−1α ,

for charts(Uα,ϕα)∈AN, and(Vα,ψβ)∈AM, are smooth on their domains

for all α,β. Then f is smooth.

6See Lee Lemmas 2.1, 2.2 and 2.3.

16

Rm

M

ϕU

p

x

Rm

V

ψ

ϕ = ϕ ψ−1

ϕ−1

= ψ ϕ−1

q

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

With this theorem at hand we can verify differentiability locally. In order toshow thatf is a diffeomorphism we need thatf is a homeomorphism, or a bijectionthat satisfies the above local smoothness conditions, as well as for the inverse.J 2.17 Example.Let N = M = PR1 and points inPR1 are equivalence classes[x] = [(x1,x2)]. Define the mappingf : PR1 → PR1 as[(x1,x2)] 7→ [(x2

1,x22)]. Con-

sider the chartsU1 = x1 6= 0, andU2 = x2 6= 0, with for exampleϕ1(x) =tan−1(x2/x1), thenϕ−1(θ1) =

([cos(θ1),sin(θ1)]

), with θ1 ∈V1 = (−π/2,π/2). In

local coordinates onV1 we have

f (θ1) = tan−1(sin2(θ1)/cos2(θ1)),

and a similar expression forV2. Clearly, f is continuous and the local expressionsalso prove thatf is a differentiable mapping using Theorem 2.15. I

The coordinate mapsϕ in a chart(U,ϕ) are diffeomorphisms. Indeed,ϕ : U ⊂M→Rn, then if(V,ψ) is any other chart withU ∩V 6= ∅, then by the definition thetransition mapsψϕ−1 andϕψ−1 are smooth. Forϕ we then haveϕ = ϕψ−1

andϕ−1 = ψϕ−1, which proves thatϕ is a diffeomorphism, see Figure 10. Thisjustifies the terminologysmooth charts.J 2.18 Remark.For arbitrary subsetsU ⊂ Rm andV ⊂ Rn a map f : U ⊂ Rm

→ V ⊂ Rn is said to be smooth atp ∈ U , if there exists an open setU† ⊂ Rm

containingp, and a smooth mapf † : U† → Rm, such thatf and f † coincide onU ∩U†. The latter is called anextensionof f . A mapping f : U → V betweenarbitrary subsetsU ⊂Rm andV ⊂Rn is called a diffeomorphism iff is smooth (as

17

just described above) and has a smooth inversef−1. In this caseU andV are saidto be diffeomorphic. I

J 2.19Example.An important example of a class of differentiable manifolds areappropriately chosen subsets of Euclidean space that can be given a smooth mani-fold structure. LetM ⊂ R` be a subset such that everyp∈ M has a neighborhoodU 3 p in M (open in the subspace topology, see Section 3) which is diffeomorphicto an open subsetV ⊂Rm (or, equivalently an open discDm⊂Rm). In this case thesetM is a smoothm-dimensional manifold. Its topology is the subsapce topologyand the smooth structure is inherited from the standard smooth structure onR`,which can be described as follows. By definition a coordinate mapϕ is a diffeo-

Rm

M

ϕ

Up

x

M

Rm

Rm

p

x y

V VV

′

ϕ ψ

ψ ϕ−1

ϕ ψ−1

W

U

Rℓ

Rℓ

FIGURE 11. The transitions mapsψ ϕ−1 andϕ ψ−1 smoothmappings establishing the smooth structure onM.

morphisms which means thatϕ : U →V = ϕ(U) is a smooth mapping (smoothnessvia a smooth mapϕ†), and for which alsoϕ−1 is a smooth map. This then directlyimplies that the transition mapsψ 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 4 we will refer to the above class of differentiable manifoldsas manifolds in Euclidean space, which are smoothly embedded submanifolds inEuclidean space. I

J 2.21Example.Let us consider the coneM = C described in Example 1 (see alsoExample 2 in Section 1). We already established thatC is manifold homeomrphicto R2, and moreoverC 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 ofR3. Indeed, following the definition in theabove remark, we haveU = C, and coordinate homeomorphismϕ(p) = (p1, p2) =x. By the definition of smooth maps it easily follows thatϕ is smooth. The inverse

is given byϕ−1(x) =(x1,x2,

√x2

1 +x22

), which is clearlynot differentiable at the

18

cone-top(0,0,0). The coneC is not a smoothly embedded submainfold ofR3

(topological embedding). I

LetU,V andW be open subsets ofRn, Rk andRm respectively, and letf :U →V

andg : V →W be smooth maps withy = f (x), andz= g(y). Then theJacobiansare

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 themapsf , g andg f yields a commutative diagram for the Jacobians:

V

U W

AAAAAAU

g

f

-g f

Rk

Rn Rm

AAAAAAU

Jg|y

J f |x

-J(g f )|x

For diffeomorphisms between open sets inRn we have a number of importantproperties. LetU ⊂Rn andV ⊂Rm be open sets, and letf : U →V is a diffeomor-phism, thenn = m, andJ f |x is invertible for anyx∈U . Using the above describedcommutative diagrams we have thatf−1 f yields thatJ( f−1)|y= f (x) · J f |x is theidentity onRn, and f f−1 yields thatJ f |x · J( f−1)|y= f (x) is the identity onRm.Thus J fx has an inverse and consequentlyn = m. Conversely, if f : U → Rn,U ⊂ Rn, open, then we have the Inverse Function Theorem;

Theorem 2.22.If J f |x : Rn→Rn is invertible, then f is a diffeomorphism between

sufficiently small neighborhoods U′ and f(U ′) of x and y respectively.

3. Immersions, submersions and embeddings

Let N andM be smooth manifolds of dimensionsn andm respectively, and letf : N → M be a smooth mapping. In local coordinatesf = ψ f ϕ−1 : ϕ(U)→ψ(V), with respects to charts(U,ϕ) and(V,ψ). Therank of f at p∈ N is definedas the rank off at ϕ(p), i.e. rk( f )|p = rk(J f )|ϕ(p), whereJ f |ϕ(p) is the Jacobianof f at p:

J f |x=ϕ(p) =

∂ f1∂x1

· · · ∂ f1∂xn

......

...∂ fm∂x1

· · · ∂ fm∂xn

19

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 ruleJ ˜f |x′ = J(ψ′ ψ−1)|y ·J f |x ·J(ϕ′ ϕ−1)−1|x′ . Sinceψ′ ψ−1 andϕ′ 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 allp∈N we sim-

FIGURE 12. Representations off via different coordinate charts.

ply write rk( f ). These are calledconstant rankmappings. Let us now considerthe various types of constant rank mappings between manifolds.

Definition 3.2. A mapping f : N → M is called animmersionif rk( f ) = n, and asubmersionif rk( f ) = m. An immersion that is injective, or 1-1, and is a home-omorphism onto its imagef (N) ⊂ M, with respect to the subspace topology, iscalled asmooth embedding.

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

Theorem 3.3. 7 Let f : N →M be smooth with constant rankrk( f ) = k. Then for

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

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

J 3.4 Example.Let N = (−π2 , 3π

2 ), andM = R2, and the mappingf is given byf (t) = (sin(2t),cos(t)). In Figure 13 we displayed the image off . The Jacobian is

7See Lee, Theorem 7.8 and 7.13.

20

given by

J f |t =

(2cos(2t)−sin(t)

)Clearly, rk(J f |t) = 1 for all t ∈ N, and f is an injective immersion. SinceN isan open manifold andf (N) ⊂ M is a compact set with respect to the subspacetopology, it follows f is not a homeomorphism ontof (N), and therefore is not anembedding. I

FIGURE 13. Injective parametrization of the figure eight.

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

((cos(t),sin(t)), andϕ−12 (t) = ((sin(t),cos(t)), andt ∈ (−π

2 , 3π2 ). Furthermore, let

M = R2, and the mappingf : N → M is given in local coordinates; inU1 as inExample 1. Restricted toS1 ⊂ R2 the mapf can also be described by

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

This then yields forU2 that f (t) = (sin(2t),sin(t)). As before rk( f ) = 1, whichshows thatf is an immersion ofS1. However, this immersion is not injective atthe origin inR2, indicating the subtle differences between these two examples, seeFigures 13 and 14. I

FIGURE 14. Non-injective immersion of the circle.

J 3.8Example.Let N = R, M = R2, and f : N→M defined byf (t) = (t2, t3). Wecan see in Figure 15 that the origin is a special point. Indeed, rk(J f)|t = 1 for allt 6= 0, and rk(J f)|t = 0 for t = 0, and thereforef is not an immersion. I

21

f

R R2

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

J 3.10Example.ConsiderM = PRn. We establishedPRn as smooth manifolds.For n = 1 we can construct an embeddingf : PR1 → R2 as depicted in Figure 16.For n = 2 we find an immersionf : PR2 → R3 as depicted in Figure 17. I

FIGURE 16. PR is diffeomorphic toS1.

J 3.13Example.Let N = R2 andM = R, then the projection mappingf (x,y) = x

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

J 3.14Example.Let N = M = R2 and consider the mappingf (x,y) = (x2,y)t . TheJacobian is

J f |(x,y) =

(2x 00 1

),

and rk(J f)|(x,y) = 2 for x 6= 0, and rk(J f)|(x,y) = 1 for x = 0. See Figure 18. I

J 3.16Example.We alter Example 2 slightly, i.e.N = S1 ⊂ R2, andM = R2, andagain use the atlasA. We now consider a different mapf , 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 sinceS1 is compact it follows from Lemma3.18 below thatS1 is homeomorphic to it imagef (S1), which show thatf is asmooth embedding (see also Appendix in Lee). I

22

FIGURE 17. Identifications forPR2 giving an immersed non-orientable surface inR3.

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

J 3.17 Example. Let N = R, M = R2, and consider the mappingf (t) =(2cos(t),sin(t)). As in the previous examplef is an immersion, not injective how-ever. Also f (R) = f (S1) in the previous example. The manifoldN is the universalcovering ofS1 and the immersionf descends to a smooth embedding ofS1. I

Lemma 3.18. 8 Let f : N→M be an injective immersion. If

(i) N is compact, or if

(ii) f is a proper map,9

then f is a smooth embedding.

8Lee, Prop. 7.4, and pg. 47.9A(ny) map f : N→M is called proper if for any compactK ⊂M, it holds that alsof−1(K)⊂N

is compact. In particular, whenN is compact, continuous maps are proper.

23

Proof: For (i) we argue as follows. SinceN is compact any closed subsetX ⊂N

is compact, and thereforef (X) ⊂ M is compact and thus closed;f is a closedmapping.10 We are done if we show thatf is a topological embedding. By theassumption of injectivity,f : N → f (N) is a bijection. LetX ⊂ 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 thatf 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 smoothembeddedn-dimensionalsubmanifoldin M if for every p∈N, there exists a chart(U,ϕ) for M, with p∈U,

such that

ϕ(U ∩N) = ϕ(U)∩(Rn×0

)= x∈ ϕ(U) : xn+1 = · · ·= xm = 0.

Theco-dimensionof N is defined ascodimN = dimM−dimN.

The setW = U ∩N in N is called an-dimensional slice, orn-slice of U , seeFigure 19, and(U,ϕ) aslice chart. The associated coordinatesx = (x1, · · · ,xn) arecalledslice coordinates. Embedded submanifolds can be characterized in terms ofembeddings.

FIGURE 19. Take ak-sliceW. On the left the imageϕ(W) corre-sponds to the Eucliden subspaceRk ⊂ Rm.

Theorem 3.21. 11 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).

10 A mapping f : N → M is calledclosedif f (X) is closed inM for any closed setX ⊂ N.

Similarly, f is calledopenif f (X) is open inM for every open setX ⊂ N.11See Lee, Thm’s 8.2

24

To get an idea let us show thatN is a topological manifold. Axioms (i) and(iii) are of course satisfied. Consider the projectionπ : Rm→ Rn, and an inclusionj : Rn → Rm defined by

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

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

Now setZ = (π ϕ)(W) ⊂ Rn, and ϕ = (π ϕ)|W, then ϕ−1 = (ϕ−1 j)|Z, andϕ : W → Z is a homeomorphism. Therefore, pairs(W, ϕ) are charts forN, whichform an atlas forN. The inclusioni : 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 thatϕ ϕ−1 = π ϕ′ ϕ−1 j, which arediffeomorphisms, which defines a smooth atlas forN. The inclusioni : N →M canbe expressed in local coordinates;

i = ϕ i ϕ−1, (x1, · · · ,xn) 7→ (x1, · · · ,xn,0, · · · ,0),

which is an injective immersion. Sincei 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 3.22.12 The image of an embedding is a smooth embedded submanifold.

Proof: By assumption rk( f ) = n and thus for anyp∈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 inV, sinceψ( f (U) satisfies Definition 3.19. By assump-tion f (U) is open inf (N) and thusf (U) = A∩ f (N) for some open setA⊂M. ByreplacingV byV ′ = A∩V and restrictingψ toV ′, (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 intoRm is notnot really a restriction for defining smooth manifolds.

Theorem 3.23. 13 Any smooth n-dimensional manifold M can be (smoothly) em-

bedded intoR2n+1.

A subsetN ⊂ M is called animmersed submanifoldif N is a smoothn-dimensional manifold, and the mappingi : N →M is a (smooth) immersion. Thismeans that we can endowN with an appropriate manifold topology and smooth

12See Lee, Thm’s 8.313See Lee, Ch. 10.

25

structure, such that the natural inclusion ofN into M is an immersion. Iff : N→M

is an injective immersion we can endowf (N) with a topology and unique smoothstructure; a setU ⊂ f (N) is open if and only if f−1(U) ⊂ N is open, and the(smooth) coordinate maps are taken to beϕ f−1, whereϕ’s are coordinate mapsfor N. This way f : N→ f (N) is a diffeomorphism, andi : f (N) →M an injectiveimmersion via the compositionf (N)→ N→M. This proves:

Theorem 3.24. 14 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 immersionf : N → M, theimage f (N) is called animmersed manifoldin M.

In this setting Whitney established some improvements of Theorem 3.23.Namely, for dimensionn > 0, any smoothn-dimensional manifold can be em-bedded intoR2n (e.g. the embedding of curves inR2). Also, for n > 1 any smoothn-dimensional manifold can be immersed intoR2n−1 (e.g. the Klein bottle). Inthis course we will often think of smooth manifolds as embedded submanifolds ofRm. An important tool thereby is the general version of Inverse Function Theorem,

FIGURE 20. An embedding ofR [left], and an immersion ofS2

[right] called the Klein bottle.

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

Theorem 3.26. 15 Let N,M be smooth manifolds, and f: N → M is a smooth

mapping. If, at some point p∈ N, it holds that Jf |ϕ(p)=x is an invertible matrix,

then there exist sufficiently small neighborhoods U0 3 p, and V0 ∈ f (p) such that

f |U0 : U0 →V0 is a diffeomorphism.

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

14See Lee, Theorem 8.16.15See Lee, Thm’s 7.6 and 7.10.

26

Theorem 3.27.Let f : N → M be a constant rank mapping withrk( f ) = k. Then

for each q∈ f (N), the level set S= f−1(q) is an embedded submanifold in N with

co-dimension equal to k.

Proof: Clearly, by continuityS is closed inN. By Theorem 3.3 there are coor-dinates(U,ϕ) of p∈ Sand(V,ψ) of f (p) = q, such that

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

The points inS∩U are characterized byϕ(S∩U) = x | (x1, · · · ,xk) = 0. There-fore,S∩U is a(n−k)-slice, and thusS is a smooth submanifold inM of codimen-sionk.

In particular, whenf : N → M is a submersion, then for eachq ∈ f (N), thelevel setS= 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 aregular point if rk( f )|p = m= dimM, otherwisea point is called acritical point. A level q ∈ f (N) is called aregular level if allpointsp∈ f−1(q) are regular points, otherwise a level is called acritical 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 todimM.

Proof: Let us illustrate the last result for the important caseN = Rn andM = Rm.For anyp∈ S= f−1(q) the JacobianJ f |p is surjective by assumption. Denote thekernel ofJ f |p by kerJ f |p ⊂ Rn, which has dimensionn−m. Define

g : N = Rn → Rn−m×Rm∼= Rn,

by g(ξ) = (Lξ, f (ξ)− q)t , whereL : N = Rn → Rn−m is any linear map whichis invertible on the subspace kerJ f |p ⊂ Rn. Clearly,Jg|p = L⊕ J f |p, which, byconstruction, is an invertible (linear) map onRn. Applying Theorem 2.22 (InverseFunction Theorem) tog we conclude that a sufficiently small neighborhood ofU

of p maps diffeomorphically onto a neighborhoodV of (L(p),0). Sinceg is adiffeomorphism it holds thatg−1 maps

(Rn−m×0

)∩V onto f−1(q)∩U (the 0∈

Rm corresponds toq). This exactly says that every pointp∈ Sallows an(n−m)-slice and is therefore an(n−m)-dimensional submanifold inN = Rn (codimS=m).

J 3.30Example.Let us start with an explicit illustration of the above proof. LetN = R2, M = R, and f (p1, p2) = p2

1 + p22. Consider the regular valueq = 2, then

J f |p = (2p1 2p2), and f−1(2) = p : p21 + p2

2 = 2, the circle with radius√

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

27

FIGURE 21. The mapg yields 1-slices for the setS.

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 isL = (1 −1). The mapg is a local diffeomorphism andonS∩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 form(R×0

)∩V. This procedure can be

carried out for any pointp∈ S, see Figure 21. I

J 3.31Example.Let N = R2\(0,0)×R = R3\(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 onN. This map is not a submersion, but is ofconstant rank, and therefore for any valueq∈ f (N) ⊂ M it holds thatS= f−1(q)is a embedded submanifold, see Figure 22. I

J 3.33Example.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 everyq∈ M, the setS= f−1(q) is aembedded submanifold, see Figure 23. I

J 3.35Example.Let N = R2, M = R, and f (x,y) = 14(x2−1)2+ 1

2y2. The JacobianisJ f |(x,y) = (x(x2−1) y). This is not a constant rank, nor a submersion. However,

28

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

FIGURE 23. An embedding circle of an submersion.

anyq > 0 is a regular value since then the rank is equal to 1. Figure 24 shows thelevel setf−1(0) (not an embedded manifold), andf−1(1) (an embedded circle)I

FIGURE 24. A regular and critical level set.

Theorem 3.37. 16 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

16See Lee, Prop.8.12.

29

there exists a neighborhood U3 p such that U∩S is the level set of a submersion

f : U → Rm−k.

Proof: Assume thatS⊂ M is a smooth embedded manifold. Then for eachp∈ S there exists a chart(U,ϕ), p∈U , such thatϕ(S∩U) = x : xk+1 = · · · =xm = 0. Clearly,S∩U is the sublevel set off : U → Rm−k at 0, given byf (x) =(xk+1, · · · ,xm), which is a submersion.

Conversely, ifS∩U = f−1(0), for some submersionf : U → Rm−k, then byTheorem 3.28,S∩U is an embedded submanifold ofU . This clearly shows thatSis an embedded submanifold inM.

4. Manifolds in Euclidean space

In the following sections and chapters we will mainly deal with smooth embed-ded submanifolds inR` due to the Whitney result. We will refer to these manifoldsin R`. For completeness we restate the definition again.

Definition 4.1. A subsetM ⊂ R` is called a smoothm-dimensional manifold inR`, if for every p ∈ M there exists an neighborhoodU ⊂ M of p (open in thesubspace topology), an open setU ′ ⊂ Rm and a diffeomorphismϕ : U →U ′. Pairs(U,ϕ) are smooth charts forM.

In other wordsϕ allows a smooth extensionϕ† : U† → Rm, with ϕ†|U∩U† =ϕ|U∩U†, andϕ−1 : U ′→ R` smooth. In Example 2.19 we established that these aresmooth manifolds. We want to show now that we that manifolds inR` are smoothembedded manifolds inR`.

Start with a differentiable manifoldM as in Definition 4.1 with charts(U,ϕ).By definition there exists a smooth functionϕ† on a neighborhoodU† ⊂ R`, withϕ†|U∩U† = ϕ|U∩U†. Using the Inverse Function Theorem we can construct a diffeo-morphismϕ† : U† ⊂ R` →U ′† ⊂ R`, such that

ϕ†(U†∩M) = ϕ†(U†)∩(Rm×0

).

Indeed,Jϕ−1|x is an injective linear map, and

R` = Jϕ−1|x(Rm)⊕[Jϕ−1|x(Rm)

]⊥.

Now letL : R`−m→[Jϕ−1|x(Rm)

]⊥be a surjective linear map, and define

G(x1, · · · ,xm,xm+1, · · · ,x`) = ϕ−1(x1, · · · ,xm)+L(xm+1, · · · ,x`).

By construction the JacobianJG|x is invertible, and thus by the Inverse FunctionTheorem there exists a local diffeomorphismG−1 extendingϕ. The extension isdenotedϕ†, and(U†,ϕ†) is a slice chart with a smooth coordinate mapping. This

30

shows thatM as defined in Definition 4.1 is a smooth embeddedm-dimensionalsubmanifold ofR`. In addition the coordinate maps are local diffeomorphisms.

Let us start with an embedded submanifoldM ⊂ R` with smooth coordinatemaps. Consider the projectionπ : R` → Rm, and immersionj : Rm→ R` definedby

π(x1, · · · ,xm,xm+1, · · · ,x`) = (x1, · · · ,xm),

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

Then the compositionπϕ = ϕ† is a smooth coordinate map fromU ⊂ R` to Rm.By definition it holds thatϕ†|U∩M = ϕ|U∩M = ϕ defined onW = U ∩M. If werestrict the inverseϕ−1 to Rm×0, then

ϕ−1 = (ϕ−1 j)|Z,

whereZ = ϕ(W). showing thatM is a manifold inR` as defined in Definition 4.1,with charts(W, ϕ). This can be summarized as follows.

Lemma 4.2. A subset M⊂ R` is a smooth m-dimensional manifold inR` in the

sense of Definition 4.1 if and only if for every p∈M there exists an neighborhood

U ⊂ R` of p, an open set V⊂ R` and a diffeomorphismϕ : U →V, such that

ϕ(U ∩M) = V ∩Rm = x∈V : xm+1 = · · ·= x` = 0,

i.e. if W = U ∩M is an m-slice of U.

This characterization can serve as a useful equivalent way of defining smoothmanifolds inR`.

In Section 2 we introduced the notion of a differentiable function between differ-entiable manifolds. In the case of manifolds embedded in Euclidean space there isan equivalent notion of differentiability that is convenient in many practical cases.

Lemma 4.3. Let N⊂ Rk and M⊂ R` be manifolds. A mapping f: N → M is a

smooth mapping (in the sense of Definition 2.12) if and only if for every p∈ N

there exists a neighborhood U⊂ Rk and smooth mapping f† : U → R` such that

f †|U∩M = f |U∩M. The map f† is called anextensionof f .

Proof: To prove this lemma we need to show that the existence of a smoothmapping f in the sense of Definition 2.12 yields the existence of a smooth exten-sion f †. Via the charts(U,ϕ) and(V,ψ) (as in Definition 4.1) we have a smoothrepresentationf = ψ f ϕ−1 : U ′∩Rk→V ′∩R`. Define the smooth map

˜f (x1, · · ·xn,xn+1, · · · ,xk) = f (x1, · · ·xn),

mapping fromU ′ toV ′. This trivial extension now allows us to definef † : U →R`

by:

f † = ψ−1 ˜f ϕ.

31

This extension clearly satisfies the requirements in the lemma.

The notion of differentiable maps for manifolds in Euclidean space is thereforethe usual notion of multi-variable differentiability. Due to Lemma 4.3 it makessense to talk about the Jacobian off at a pointp∈ N, i.e. givenp∈ N define

J f |p = J f†|p,

for any extensionf †. The commuting diagram below shows thatJ f |p is indepen-dent of the chosen extensionf †:

U ⊂ Nf−−−−→ V ⊂M

ϕy yψ

U ′˜f−−−−→ V ′

In Section 1 we also introduced manifolds with boundary. The theory in thissection immediately goes through for manifolds with boundary. For such objectthe same holds as for manifolds; manifolds with boundary can always be embed-ded into someR`. For this reason we also give the definition of a manifold withboundary inR`.

Definition 4.4. A subsetM⊂R` is called a smoothm-dimensional manifold inR`

with boundary∂M ⊂ M, if for every p∈ M there exists an neighborhoodU ⊂ M

of p (open in the subspace topology), an open setU ′ ⊂Hm and a diffeomorphismϕ : U →U ′. The boundary∂M are the points inM which corresponds to points on∂Hm under the diffeomorphismsϕ. Notation:(M,∂M).

As in the case of manifolds we have an equivalent characterization for manifoldswith boundary as images of smooth embeddings.

Lemma 4.5. A subset M⊂ R` is a smooth m-dimensional manifold inR` withboundary∂M ⊂M in the sense of Definition 4.4 if and only if for every p∈M\∂M

there exists an neighborhood U⊂ R` of p, an open set V⊂ R` and a diffeomor-

phismϕ : U →V, such that

ϕ(U ∩M) = V ∩Rm = x∈V : xm+1 = · · ·= x` = 0,

and for every p∈ ∂M there exists an neighborhood U⊂R` of p, an open set V⊂R`

and a diffeomorphismϕ : U →V, such that

ϕ(U ∩M) = V ∩Hm = x∈V : xm≥ 0, xm+1 = · · ·= x` = 0,

andϕm(p) = 0.

This idea can be further extended by considering manifolds with corner, see forinstance Lee, pg. 363.

32

II. Tangent and cotangentspaces

5. Tangent spaces

For a(n) (embedded) manifoldM ⊂ R` the tangent spaceTpM at a pointp ∈M can be pictured as a hyperplane tangent toM. In Figure 25 we consider theparametrizationsx+tei in Rm. These parametrizations yield curvesγi(t) = ϕ−1(x+tei) onM whose velocity vectors are given by

γ′(0) =ddt

ϕ−1(x+ tei)∣∣∣t=0

= Jϕ−1|x(ei).

The vectorsp+γ′(0) are tangent toM at p and span anm-dimensional affine linearsubspaceMp of R`. Since the vectorsJϕ−1|x(ei) spanTpM the affine subspace isgiven by

Mp := p+TpM ⊂ R`,

which is tangent toM at p.

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

33

The considerations are rather intuitive in the sense that we consider only em-bedded manifolds, and so the tangent spaces are tangentm-dimensional affine sub-spaces ofR`. 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 mappingγ : I = (a,b)⊂R→M, such thatγ(0) = p. This mapping is called a(smooth) curveon M, and is parametrized byt ∈ I . If the mappingγ (between the manifoldsN = I , andM) is an immersion,thenγ is called an immersed curve. For such curves the ‘velocity vector’Jγ|t =(ϕ γ)′(t) in Rm is nowhere zero. We build the concept of tangent spaces in orderto define the notion velocity vector to a curveγ.

Let (U,ϕ) be a chart atp. Then, two curvesγ andγ are equivalent,γ∼ γ, if

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

The equivalence class of a curveγ throughp∈M is denoted by[γ].

Definition 5.2. 17 At a p ∈ M define the tangent spaceTpM as the space of allequivalence classes[γ] of curvesγ throughp. A tangent vectorXp, 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 ofTpM.

p

0

Rm

M

N

x

γ

ϕϕ γ

(ϕ γ)′(0)

FIGURE 26. Immersed curves and velocity vectors inRm.

17 Lee, Ch. 3.

34

The above definition does not depend on the choice of charts atp ∈ M. Let(U ′,ϕ′) be another chart atp ∈ 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 atp∈M.

One can prove thatTpM ∼= Rm. Indeed,TpM can be given a linear structure asfollows; 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 atp∈M, and the operations yield non-empty equivalenceclasses. The mapping

τϕ : TpM → Rm, τϕ([γ])

= (ϕ γ)′(0),

is a linear isomorphism andτϕ′ = J(ϕ′ ϕ−1)|xτϕ. Indeed, by considering curvesγi(x) = ϕ−1(x+ tei), i = 1, ...,m, it follows that[γi ] 6= [γ j ], i 6= j, since

(ϕ γi)′(0) = ei 6= ej = (ϕ γ j)′(0).

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

Given a smooth mappingf : N →M we can define how tangent vectors inTpN

are mapped to tangent vectors inTqM, with q = f (p). Choose charts(U,ϕ) forp∈ N, and(V,ψ) for q∈ M. We define thetangent mapor pushforward of f asfollows, see Figure 27. For a given tangent vectorXp = [γ] ∈ TpN,

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

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

TpNf∗−−−−→ TqM

τϕ

y yτψ

Rn J(ψ fϕ−1)|x−−−−−−−→ Rm

35

N M

TpN TqMf

f

ϕψ

x y

p qf∗Xp

Xp

hh f

R

Rm

Rn

FIGURE 27. Tangent vectors inXp ∈ TpN yield tangent vectorsf∗Xp ∈ TqM under the pushforward off .

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

(ψ f (γ))′(0) = (ψ f ϕ−1ϕ γ)′(0) = J f∣∣x·(ϕ γ)′(0).

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

thenτϕ can be identified withϕ∗, andJ(ψ f ϕ−1)|x with (ψ f ϕ−1)∗. Indeed,τϕ([γ]) = (ϕ γ)′(0) andϕ∗([γ]) = [ϕ γ], and inRn 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 identifyTxRn with Rn by identifying ϕ∗ andτϕ. This justifiesthe notation

ϕ∗([γ]) = [ϕ γ] := (ϕ γ)′(0).

Properties of the pushforward can be summarized as follows:

36

Lemma 5.5. 18 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 thatf∗([γ]) = [ f γ], andg∗([ 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 parametrizationϕ−1 : Rm → M coming from a chart(U,ϕ) is a local dif-feomorphism, and can be used to find a canonical basis forTpM. Choosing localcoordinatesx = (x1, · · · ,xn) = ϕ(p), and the standard basis vectorsei for Rm, wedefine

∂∂xi

∣∣∣p:= ϕ−1

∗ (ei).

By definition ∂∂xi

∣∣p∈ TpM, and since the vectorsei form a basis forRm, the vectors

∂∂xi

∣∣p form a basis forTpM. An arbitrary tangent vectorXp ∈ TpM can now be

written with respect to the basis ∂∂xi

∣∣p:

Xp = ϕ−1∗ (Xiei) = Xi

∂∂xi

∣∣∣p.

where the notationXi∂

∂xi

∣∣p= ∑i Xi

∂∂xi

∣∣p denotes the Einstein summation convention,

and(Xi) is a vector inRm!We now define the directional derivative of a smooth functionh : M → R in the

direction ofXp ∈ TpM by

Xph := h∗Xp = [h γ].

In fact we have thatXph = (h γ)′(0), with Xp = [γ]. For the basis vectors ofTpM

this yields the following. Letγi(t) = ϕ−1(x+ tei), andXp = ∂∂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 vectorsXp, Xph = Xi

∂h∂xi

.

18Lee, Lemma 3.5.

37

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

ddt

∣∣∣t=0

:= id−1∗ (1).

We have the following commuting diagrams:

Nγ−−−−→ M

id

y yϕ

R γ=ϕγ−−−−→ Rm

TtNγ∗−−−−→ TpM

id∗

y yϕ∗

R γ∗−−−−→ Rm

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 34 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 coordinatesx = ϕ(p)andx′ = ϕ′(p), andp∈U ∩U ′. This yields two different basis forTpM, namely ∂

∂xi

∣∣p

, and

∂∂x′i

∣∣p

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

forward yields the identity onTpM. If we use the different choices of coordinatesas described above we obtain(

id∗∂

∂xi

∣∣∣p

)h = (ϕ′ ϕ−1)∗

∂h∂x′j

=∂x′j∂xi

∂h∂x′j

.

In terms of the different basis forTpM this gives

∂∂xi

∣∣∣p=

∂x′j∂xi

∂∂x′j

∣∣∣p.

Let us prove this formula by looking a slightly more general situation. LetN,M

be smooth manifolds andf : N → M a smooth mapping. Letq = f (p) and(V,ψ)is a chart forM containingq. The vectors ∂

∂y j

∣∣q form a basis forTqM. 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

38

which implies thatTpN is mapped toTqM under the mapf∗. In general a tangentvectorXi

∂∂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 takingN = M and f = id we obtain the abovechange of variables formula.

6. Cotangent spaces

In linear algebra it is often useful to study the space of linear functions on agiven vector spaceV. This space is denoted byV∗ and called thedual vector spaceto V — again a linear vector space. So the elements ofV∗ are linear functionsθ : V → R. As opposed to vectorsv∈V, the elements, or vectors inV∗ are calledcovectors.

Lemma 6.1. Let V be a n-dimensional vector space with basisv1, · · · ,vn, then

the there exist covectorsθ1, · · · ,θn such that

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

and the covectorsθ1, · · · ,θn form a basis for V∗.

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

Definition 6.2. Let M be a smoothm-dimensional manifold, and letTpM be thetangent space at somep∈M. The thecotangent spaceT∗

p M is defined as the dualvector space ofTpM, i.e.

T∗p M := (TpM)∗.

By definition the cotangent spaceT∗p M is alsom-dimensional and it has a canon-

ical basis as described in Lemma 6.1. As before have the canonical basis vectors∂

∂xi|p for TpM, the associated basis vectors forT∗

p M are denoteddxi |p. Let us nowdescribe this dual basis forT∗

p M and explain the notation.The covectorsdxi |p are calleddifferentials and we show now that these are

indeed relateddhp. Let h : M → R be a smooth function, thenh∗ : TpM → Rand h∗ ∈ T∗

p M. Since the differentialsdxi |p form a basis ofT∗p M we have that

h∗ = λidxi |p, and therefore

h∗∂

∂xi

∣∣∣p= λ jdxj |p ·

∂∂xi

∣∣∣p= λ jδ j

i = λi =∂h∂xi

,

39

and thus

(4) dhp = h∗ =∂h∂xi

dxi∣∣p.

Chooseh such thath∗ satisfies the identity in Lemma 6.1, i.e. leth = xi ( h = xi ϕ= 〈ϕ,ei〉). These linear functionsh of course spanT∗

p M, and

h∗ = (xi ϕ)∗ = d(xi ϕ)p = dxi |p.

Cotangent vectors are of the form

θp = θidxi∣∣p.

The pairing between a tangent vectorXp and a cotangent vectorθp is expressedcomponent wise as follows:

θp ·Xp = θiXjδij = θiXi .

In the case of tangent spaces a mappingf : N → M pushes forward to a linearf∗;TpN → TqM for eachp ∈ N. For cotangent spaces one expects a similar con-struction. Letq = f (p), then for a given cotangent vectorθq ∈ T∗

q M define

( f ∗θq) ·Xp = θq · ( f∗Xp) ∈ T∗p N,

for any tangent vectorXp∈TpN. The homomorphismf ∗ : T∗q M→ T∗

p N, defined byθq 7→ f ∗θq is called thepullbackof f at p. It is a straightforward consequence fromlinear algebra thatf ∗ defined above is indeed the dual homomorphism off∗, 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 ∗dyj∣∣q

)·Xi

∂∂xi

∣∣∣p

= dyj∣∣q· f∗(

Xi∂

∂xi

∣∣∣p

)= dyj

∣∣q

[∂ f j

∂xiXi

]∂

∂y j

∣∣∣q=

∂ f j

∂xiXi

Using this relation we obtain that(f ∗σ jdyj

∣∣q

)·Xi

∂∂xi

∣∣∣p= σ j ∂ f j

∂xiXi = σ j ∂ f j

∂xidxi∣∣pXi

∂∂xi

∣∣∣p,

which produces the local formula

(5) f ∗σ jdyj∣∣q=

[σ j ∂ f j

∂xi

]dxi∣∣p=

[σ j f

∂ f j

∂xi

]x

dxi∣∣p.

Lemma 6.3. Let f : N → M, and g: M → P be smooth mappings, and let p∈ M,

then

40

(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 off andg we have

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

For the compositiong f it holds that(g f )∗ω(g f )(p) ·Xp = ω(g f )(p) ·(g f )∗(Xp).Using Lemma 5.5 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 mappingϕ : U ⊂ M → Rm, which is local diffeo-morphism. Using Lemma 6.3 we then obtain an isomorphismϕ∗ : Rm → T∗

p M,which justifies the notation

dxi |p = ϕ∗(ei).

7. Vector bundles

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

Definition 7.1. A triple (E,M,π) is called areal vector bundleof rankk overM if

(i) for eachp∈ M, the setEp = π−1(p) is a realk-dimensional linear vectorspace, called thefiber overp, such that

(ii) for every p∈M there exists a open neighborhoodU 3 p, and a homeomor-phismΦ : π−1(U)→U×Rk;(a)(πΦ−1

)(p,ξ) = p, for all ξ ∈ Rk;

(b) ξ 7→Φ−1(p,ξ) is a vector space isomorphism betweenRk andEp.

The homeomorphismΦ is called alocal trivialization of the bundle.

41

FIGURE 28. Charts in a vector bundleE overM.

If there is no ambiguity about the base spaceM we often denote a vector bundleby E for short. Another way to denote a vector bundle that is common in theliterature isπ : E→M. It is clear from the above definition that ifM is a topologicalmanifold then so isE. Indeed, via the homeomorphismsΦ it follows that E

is Hausdorff and has a countable basis of open set. Defineϕ =(ϕ× IdRk

)Φ,

andϕ : π−1(U)→V×Rk is a homeomorphism. Figure 28 explains the choice ofbundle charts forE related to charts forM.

For two trivializationsΦ : π−1(U)→U×Rk andΨ : π−1(V)→V×Rk, we havethat the transition mapΨΦ−1 : (U ∩V)×Rk → (U ∩V)×Rk has the followingform

ΨΦ−1(p,ξ) = (p,τ(p)ξ)),

whereτ : 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 thatξ 7→ Φ−1(p,ξ =A(p)ξ, andξ 7→Ψ−1(p,ξ) = B(p)ξ. Now,

ΨΦ−1(p,ξ) = Ψ(A(p)ξ) = (p,σ(p,ξ)),

andΨ−1(p,σ) = A(p)ξ = B(p)σ. Thereforeσ(p,ξ) = B−1Aξ =: τ(p)ξ. Continuityis clear from the assumptions in Definition 7.1.

If both E andM are smooth manifolds andπ 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 mapsτ are smooth. The following resultallows us to construct smooth vector bundles and is important for the special vectorbundles used in this course.

42

FIGURE 29. Transition mappings for local trivializations.

Theorem 7.4. 19 Let M be a smooth manifold, and letEpp∈M be a family of k-

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

p∈M Ep, and

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

(i) an open coveringUαα∈A for M;

(ii) for each α ∈ A, a bijectionΦα : π−1(Uα) → Uα ×Rk, such thatξ 7→Φ−1

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

(iii) for eachα,β ∈ I, with Uα ∩Uβ 6= ∅, smooth mappingsταβ : 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 smooth

vector bundle of rank k over M.

Proof: The proof of this theorem goes by verifying the hypotheses in Theorem2.11. Let us propose charts forE. Let (Vp,ϕp), Vp ⊂Uα, be a smooth chart forM containingp. As before defineϕp : π−1(Vp) → Vp×Rk. We show now, usingTheorem 2.11, that(π−1(Vp), ϕp), p∈M are smooth charts, which gives a uniquedifferentiable manifold structure.

19See Lee, Lemma 5.5.

43

J 7.5Example.Consider the setE 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 overS1 of rank 1, embedded inR4. Forexample ifU = S1\(1,0) (θ ∈ (0,2π)), thenΦ(π−1(U)) = (p1, p2,ξ1) is a localtrivialization. This bundle is called the Mobius strip.

To show thatE is a smooth vector bundle we need to verify the conditions inTheorem 7.4. Consider a second chartU ′ = S1\(−1,0) (θ ∈ (−π,π)), and thetrivialization Ψ(π−1(U ′)) = (p1, p2,ξ2). For theΦ 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 thatτ(p) is represented byτ =−cos(θ/2)sin(θ/2) , for θ ∈ (0,π), which invert-

ible and smooth inp. I

Mappings between smooth vector bundles are calledbundle maps, and are de-fined as follows. Given vector bundlesπ : E → M andπ′ : E′ → M′ and smoothmappingsF : E → E′ and f : M →M′, such that the following diagram commutes

EF−−−−→ E′

πy yπ′

Mf−−−−→ M′

andF |Ep : Ep → E′f (p) is linear, then the pair(F, f ) is a called a smooth bundle

map.A section, or cross sectionof a bundleE is a continuous mappingσ : M → E,

such thatπσ = IdM. A section is smooth ifσ is a smooth mapping. The space of

FIGURE 30. A cross sectionσ in a bundleE.

44

smooth section inE is denoted byE(M). Thezero sectionis a mappingσ : M → E

such thatσ(p) = 0∈ Ep for all p∈M.J 7.7Remark.By identifyingM with the trivial bundleE0 = M×0, a section isa bundle map fromE0 = M×0 to E. Indeed, defineσ′ : E0 → E by σ′(p,0) =σ(p), and let f = IdM. Thenπσ′ = πσ = IdM. I

7.1. The tangent bundle and vector fields

The disjoint union of tangent spaces

TM :=G

p∈M

TpM

is called thetangent bundleof M. We show now thatTM is a fact a smooth vectorbundle overM.

Theorem 7.8. The tangent bundle TM is smooth vector bundle over M of rank m,

and as such TM is a smooth2m-dimensional manifold.

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

Φ(

Xi∂

∂xi

∣∣∣p

)=(p,(Xi)

),

which clearly is a bijective map betweenπ−1(U) andU ×Rm. Moreover,Xp =

(Xi) 7→ Φ−1(p,Xp) = Xi∂

∂xi

∣∣∣p

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

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

Φβ Φ−1α (p,Xp) = (p,ταβ(p)Xp),

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

α = (ϕβ × Id) Φβ Φ−1α (ϕ−1

α × Id). Using the change of coordinatesformula derived in Section 5 we have that

ϕβ ϕ−1α(x,(Xi)

)=

(x′1(x), · · · ,x′m(x),

(∂x′j∂xi

Xj

)),

which proves the smoothness ofταβ. Theorem 7.4 can be applied now showing thatTM is asmooth vector bundle overM. From the charts(π−1(U), ϕ) we concludethatTM is a smooth 2m-dimensional manifold.

45

Definition 7.9. A smooth (tangent) vector fieldis a smooth mapping

X : M → TM,

with the property thatπX = idM. In other wordsX is a smooth (cross) section inthe vector bundleTM, see Figure 31. The space of smooth vector fields onM isdenoted byF(M).

M

M

TMTpM

p

XpXp

p

X

FIGURE 31. A smooth vector fieldX on a manifoldM [right],and as a ‘curve’, or section in the vector bundleTM [left].

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

X = Xi∂

∂xi

∣∣∣p,

whereXi : U → R. Smoothness of vector fields can be described in terms of thecomponent functionsXi .

Lemma 7.11. A mapping X: M → TM is a smooth vector field at p∈U if and

only if the coordinate functions Xi : U → R are smooth.

Proof: In standard coordinatesX is given by

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

which immediately shows that smoothness ofX is equivalent to the smoothness ofthe coordinate functionsXi .

In Section 5 we introduced the notion of push forward of a mappingf : N→M.Under submersions and immersion a vector fieldX : N→ TN does not necessarilypush forward to a vector field onM. If f is a diffeomorphism, thenf∗X = Y is avector field onM. We remark that the definition off∗ also allows use to restatedefinitions about the rank of a map in terms of the differential, or pushforwardf∗at a pointp∈ N.

46

7.2. The cotangent bundle and differential 1-forms

The disjoint union

T∗M :=G

p∈M

T∗p M

is calledcotangent bundleof M.

Theorem 7.12.20 The cotangent bundle T∗M is a smooth vector bundle over M of

rank m, and as such T∗M is a smooth2m-dimensional manifold.

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

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

Definition 7.13. A smooth covector fieldis a smooth mapping

θ : M → T∗M,

with the property thatπ θ = idM. In order wordsθ is a smooth section inT∗M.The space of smooth covector fields onM is denoted byF∗(M). Usually the spaceF∗(M) is denoted byΓ1(M), and covector fields are referred to a(smooth) differ-ential 1-formonM.

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

θ = θidxi∣∣p,

whereθi : U → R. Smoothness of a covector fields can be described in terms ofthe component functionsθi .

Lemma 7.14. A covector fieldθ is smooth at p∈U if and only if the coordinate

functionsθi : U → R are smooth.

Proof: See proof of Lemma 7.11.

We saw in the previous section that for arbitrary mappingsf : N →M, a vectorfield X ∈ F(N) does not necessarily push forward to a vector onM under f∗. Thereason is that surjectivity and injectivity are both needed to guarantee this, whichrequiresf 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-formθ ∈ Λ1(M) we define a 1-formf ∗θ ∈ Λ1(N) as follows

( f ∗θ)p = f ∗θ f (p).

20See Lee, Proposition 6.5.

47

Theorem 7.15. 21 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 = θidyi∣∣q. By Formula (5) we

then obtain

f ∗θidyi∣∣

f (p)=

[θi∣∣y

∂ fi∂x j

]dxj∣∣p=

[θi f

∂ fi∂x j

]x

dxj∣∣p,

which proves thatf ∗θ ∈ Λ1(N).

Given a mappingg : M → R, the differential, or push forwardg∗ = dg definesan element inT∗

p M. In local coordinates we havedg∣∣p=

∂g∂xi dxi

∣∣p, and thus defines

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

f ∗dg= f ∗∂g∂xi

dxi∣∣

f (p)=∂g∂xi f

∂ fi∂x j

dyj∣∣p.

By applying the same to the 1-formd(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,ej〉= y j , andω = dg= dyj |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 7.16. A differential 1-formθ ∈Λ1(N) is called anexact1-form if thereexists a smooth functiong : N→ R, such thatθ = 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. LetN = R, andθ a 1-form onN given in local coordinates byθt = h(t)dt, which can be identified with a functionh. The notation makes sensebecauseθ can be integrated over any interval[a,b]⊂ R:Z

[a,b]θ :=

Z b

ah(t)dt.

21Lee, Proposition 6.13.

48

Let M = R, and consider a mappingf : M = R→N = R, which satisfiesf ′(t) > 0.Thent = f (s) is an appropriate change of variables. Let[c,d] = f ([a,b]), thenZ

[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 curveγ on a manifoldN.

Definition 7.17. Let γ : [a,b] ⊂ R → N, and letθ be a 1-form onN andγ∗θ thepullback ofθ, which is a 1-form onR. Denote the image ofγ in N also byγ, thenZ

γθ :=

Z[a,b]

γ∗θ =Z

[a,b]θi(γ(t))γ′i(t)dt,22

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))γ′i(t)dt, andZγθ =

Z[a,b]

γ∗θ =Z

[a,b]θγ(t) · γ′(t)dt =

Z[a,b]

θi(γ(t))γ′i(t)dt.

If γ′ is nowhere zero then the mapγ : [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 = dgbe an exact 1-form, thenZ

γdg= g

∣∣∂γ= g(γ(b))−g(γ(a)).

Indeed,Zγdg=

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 theFundamental Theorem for Line Integralsand is a spe-cial case of the Stokes Theorem (see Section 18).

8. Tangent and cotangent spaces for embedded manifolds

In this section we discuss tangent and cotangent spaces for manifolds embeddedin Euclidean space. We will give a number of examples of tangent, cotangentspaces, pull backs and push forwards.

22The expressionsγi(t) ∈ R are the components ofγ′(t) ∈ Tγ(t)M.

49

p

MM

p

FIGURE 32. Tangent spaces to embedded manifolds.

Rm

U

x

p

V

M

ϕ−1

Rℓ

FIGURE 33. Parametrization of an embedded manifold.

8.1. Tangent spaces

Intuitively the tangent space at a point on a smooth (embedded) curve is theline tangent to the curve at that point, and similarly, for an embedded surface it isthe tangent plane to the surface at the given point, see Figure 32. For embeddedmanifoldsM we can define the notion of tangent space at a pointp∈M as follows.Consider anm-dimensional manifoldM ⊂ R`. Given a chart(ϕ,U), consider the(smooth) parametrization

ϕ−1 : V = ϕ(U)⊂ Rm→U ⊂M ⊂ R`,

whereV is a neighborhood of a pointx∈ Rm, such thatϕ−1(x) = p. Sinceϕ−1 issmooth the JacobianJϕ−1|x : Rm→ R` is well-defined. For embedded manifoldswe now use the following definition of tangent space at a pointp∈M:

TpM := Jϕ−1|x(Rm) = rangeJϕ−1|x.

50

It is straightforward to see that this definition does not depend on the chosenparametrization. Let(ϕ′,U ′) be another chart such thatp∈U ′ and letV ′ = ϕ′(U),andϕ′−1(x′) = p. Figure 34 shows thatJ(ϕ′ϕ−1)|x is an isomorphism and thus

FIGURE 34. The tangent space is independent of the chosenparametrization.

Jϕ−1|x(Rm) = Jϕ′−1|x(Rm), which proves our assertion. For anm-dimensionalmanifoldM, the tangent spaceTpM at a pointp∈M is anm-dimensional subspaceof R`. In order to prove this we use an essential property of smooth maps betweenembedded manifolds. The commuting diagrams below show that the commutingdiagrams yield that the rank ofJϕ−1|x is equalm. It holds thatϕ : U →V = ϕ(U),has an extensionϕ† : U† → Rm, andϕ andϕ† coincide onU ∩U†, which provesthatTpM is well-defined.

U†

V Rm

AAAAAAU

ϕ†

ϕ−1

-i

R`

Rm Rm

AAAAAAU

Jϕ†|p

Jϕ−1|x

-id

J 8.4Example.Consider the standard circleS1 = (p1, p2) ∈ R2 : p21 + p2

2 = 1,and the parametrizationϕ−1 : (0,2π)→S1⊂R2 given byϕ−1(x)= (cos(x),sin(x))t .We computeTpS1 at p = (1

2

√2, 1

2

√2). The pointp corresponds tox = π

4 . We havethat Jϕ−1|x = (−sin(x),cos(x))t : R → R2, and thusJϕ−1| π

4= (−1

2

√2, 1

2

√2)t ,

which yields thatTpS1 =[(−1,1)t

]∼= R. I

51

By identifyingTpM with a pairp×TpM, tangent spaces are given the propertythatTpM∩TqM = ∅, wheneverp 6= q. The union of these tangent spacesTM :=∪p∈M

(p×TpM

)⊂ R` ×R` gives the tangent bundle ofM. In this definition

TM is smooth embedded manifold ofR2` (see Exercises). A tangent vector atp∈M will be denoted byXp ∈ TpM. Smooth functionsX : M → TM, defined byX(p) = (p,Xp), for everyp∈ M (i.e. Xp ∈ TpM), are vector fields. Of course onemay regardX as a mappingX : M → R`.

J 8.5Example.In the above exampleM = S1 the mapping

S1 3 (p1, p2) 7→ (p1, p2)× (−p2, p1),

is a smooth vector field onS1. Indeed, for a givenp described by the chartU = S1\(1,0) we have thatp1 = cos(x) and p2 = sin(x), and thereforeXp =(−sin(x),cos(x))t = Jϕ−1|x(1) ∈ TpM, which proves that the mappingX takesvalues inTM. I

A parametrizationϕ−1 which comes from with a chart(U,ϕ) can be used tofind a canonical basis forTpM. Recall that we Choosing local coordinatesx =(x1, · · · ,xn) = ϕ(p), and the standard basis vectorsei , we define

∂∂xi

∣∣∣p:= Jϕ−1|x(ei).

By definition ∂∂xi

∣∣p∈ TpM, and since the vectorsei form a basis forRm, the vectors

∂∂xi

∣∣p form a basis forTpM. An arbitrary tangent vectorXp ∈ TpM can now be

written with respect to the basis ∂∂xi

∣∣p:

Xp = Jϕ−1|x(Xiei) = Xi∂

∂xi

∣∣∣p.

This notation can easily be justified via the following elementary calculus exercise.Consider a smooth functionh : M → R, then the JacobianJh|p = ( ∂h

∂pi)ni=1 is well-

defined by virtue of Lemma 4.3. Given a tangent vectorXp ∈ TpM we can definethe directional derivative ofh in the direction ofXp

Xph := 〈Jh|p,Xp〉,

where〈·, ·〉 denotes the standard inner product inR` . Now, by settingh = hϕ−1,

Xph = 〈Jh|p,Xp〉= Jh|p ·Jϕ−1|x(Xiei) = Xi∂

∂xi(hϕ−1) = Xi

∂h∂xi

,

justifies the notation of tangent vector as a representation in local coordinates. Inparticular we have

∂∂xi

∣∣∣ph =

∂h∂xi

∣∣∣ϕ(p)=x

.

52

FIGURE 35. The functionh andh† coincide onW∩U .

J 8.7 Example.Let us go back to the above example ofM = S1, and consider apoint p∈ S1 ⊂ R2. SinceS1 is 1-dimensional,e1 = (1), andx = x1. We have that∂∂x

∣∣p= Jϕ−1|x(1) = (−sin(x),cos(x))t . Let h : S1 → R be given byf (p1, p2) = p2

1,andXp = (−p2, p1)t . Then

Xph = (2p1,0) · (−p2, p1)t =−2p1p2.

In local coordinates this reads:Xph = −2cos(x)sin(x). If we expressf in lo-cal coordinates we obtainh(x) = (h ϕ−1)(x) = cos2(x), and ∂

∂x(h ϕ−1)(x) =−2cos(x)sin(x) = ((Xph)ϕ−1)(x). I

J 8.8 Example.Consider the embedded torusT2 = (p1, · · · p4) ∈ R4 : p21 +

p22 = 1, p2

3 + p24 = 1. Consider the chart(ϕ,U), with U = (0,2π)× (0,2π),

and parametrizationϕ−1(x1,x2) = (cos(x1),sin(x1),cos(x2),sin(x2)). Consider thefunctionh(p) = p2

1 + p23, which is a mapping fromT2 ⊂ R4 → R. It follows that

Jh|p = (2p1,0,2p3,0) : R4 → R. Let us compute the directional derivative ofh inthe directionXp = (−p2, p1,−p4, p3)t . Then

Xph = (2p1,0,2p3,0) · (−p2, p1,−p4, p3)t =−2p1p2−2p3p4.

Let us now use the local coordinatesx = (x1,x2) to compute ∂∂xi

∣∣p, i = 1,2. It

follows that

∂∂x1

∣∣∣p

= Jϕ−1|x(e1) = (−sin(x1),cos(x1),0,0)t ,

∂∂x2

∣∣∣p

= Jϕ−1|x(e2) = (0,0,−sin(x2),cos(x2)t .

Therefore the tangent vectorXp is given byXp = ∂∂x1

∣∣p+

∂∂x2

∣∣p. In local coordi-

natesh reads:h(x1,x2) = ( f ϕ−1)(x) = cos2(x1)+ cos2(x2). We now apply the

53

differential operatorXpϕ−1 = ∂∂x1

∣∣p+

∂∂x2

∣∣p to the functionh = hϕ−1:( ∂

∂x1

∣∣∣p+

∂∂x2

∣∣∣p

)h =−2cos(x1)sin(x1)−2cos(x2)sin(x2),

which agrees with the direct calculation inp. I

FIGURE 36. Commutative diagrams.

Consider two embedded manifoldsN andM in Rk andR` respectively, and letf : N → M be a differentiable mapping. As before we introduce the notion of thedifferential of f at a pointp∈ N. We define for anyXp ∈ TpN

f∗Xp = d fp ·Xp := J f |p(Xp) ∈ R`.

The commutative diagrams in Figure 36 show thatJ f |p ·Jϕ−1|x = Jψ−1|y ·J(ψ f ϕ−1)|x, which implies thatJ f |p mapsTpN = rangeJϕ−1|x into rangeJψ−1|y = TqM.This does not depend on the choice the extensionf †, since by the ‘bottum part’ inthe commutative diagram in Figure 36, i.e.Jψ−1|y ·J(ψ f ϕ−1)|x · (Jϕ−1|x)−1 =J f |p, we obtain the same linear mapping.

Another way of definingf∗ is via

( f∗Xp)h := Xp(h f ),

whereh : M→R arbitrary smooth function. Figure 27 in Section 5 gives a schematicaccount of this construction, and shows thatf∗ is given byJ f†|p. In local co-ordinates this is exactly the Jacobian off . These considerations show thatf∗ iswell-defined (independent of the extensionf †), and

f∗ = d fp : TpN→ TqM.

54

8.2. Cotangent spaces

For a functionh : M → R we have23

∂∂xi

∣∣∣ph = h∗

∂∂xi

∣∣∣p= 〈Jh|p,

∂∂xi

∣∣∣p〉=

∂h∂xi

,

whereh = hϕ−1, h∗ : TpM →R is an element ofT∗p M. We now seek a functionh

such thath∗ satisfies the identity in Lemma 6.1. In order to do so we chooseh= xi ,i.e. h = xi ϕ = 〈ϕ,ei〉. This then yields

dxi∣∣p:= J〈ϕ,ei〉

∣∣p= (ei)t ·Jϕ

∣∣p.

This definition gives the following relation

dxi∣∣p·

∂∂x j

∣∣∣p= δi

j ,

which shows thatdxi |p is a basis forT∗p M. Since f∗ is also given byJ f†|p we

have thatJ f†|p = J( f ϕ)|x = J f |x · Jϕ|p = J f |x(ei) · (ei)tJϕ|p again justifies ourdefinitions. An alternative direct calculation gives

f∗∂

∂xi

∣∣∣p=

∂ f∂xi

=∂ f ϕ−1

∂xi= J f†|p ·Jϕ−1|x(ei) = J f†|p

∂∂xi

∣∣p,

and thusJ f†|p = ∂ f∂xi

dxi∣∣p. In a general a cotangent vector atp ∈ M is denoted

by θp ∈ T∗p M. As in the case of tangent space one can identify the spacesT∗

p M

with p×T∗p M, so that they are all mutually disjoint whenp 6= q. The union

∪p∈M(p×T∗

p M)

= T∗M is can thecotangent bundleof M, As before alsoT∗M

is a smooth embedded manifold inR2`. The differentiald f : M → T∗M, definedby d f(p) = (p,d fp) is an example of a smooth function. The above considerationgive the coordinate wise expression ford f .

23 In the relation(h∗Xp)g = Xp(gh) we can takeh,g : M → R, with g = id, which thenh∗Xp =Xph.

55

III. Tensors and differentialforms

9. 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 9.1. Let V1, · · · ,Vr , and W be real vector spaces. A mapping T: V1×·· ·×Vr →W is called amultilinear mapping if

T(v1, · · · ,λvi +µv′i , · · · ,vr) = λT(v1, · · · ,vi , · · ·vr)+µT(v1, · · · ,v′i , · · ·vr), ∀i,

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

Now consider the special case thatW = R, thenT becomes amultilinear func-tion, or form, and a generalization of linear functions. If in additionV1 = · · · =Vr = V, then

T : V×·· ·×V → R,

is a multilinear function onV, and is called acovariantr-tensoronV. The numberof copiesr is called the rank ofT. The space of covariantr-tensors onV is denotedby Tr(V), which clearly is a real vector space using the multilinearity property inDefinition 9.1. In particular we have thatT0(V) ∼= R, T1(V) = V∗, andT2(V) isthe space of bilinear forms onV. If we consider the caseV1 = · · ·= Vr = V∗, then

T : V∗×·· ·×V∗→ R,

is a multilinear function onV∗, and is called acontravariant r-tensoronV. Thespace of contravariantr-tensors onV is denoted byTr(V). Here we have thatT0(V)∼= R, andT1(V) = (V∗)∗ ∼= V.J 9.2 Example.The cross product onR3 is an example of a multilinear (bilinear)function mapping not toR to R3. Let x,y∈ R3, then

T(x,y) = x×y∈ R3,

which clearly is a bilinear function onR3. I

56

Since multilinear functions onV can be multiplied, i.e. given vector spacesV,W

and tensorsT ∈ Tr(V), andS∈ Ts(W), the multilinear function

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

is well defined and is a multilnear function onVr ×Ws. This brings us to thefollowing definition. LetT ∈ Tr(V), andS∈ Ts(W), then

T⊗S: Vr ×Ws→ R,

is given by

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

This product is called thetensor product. By takingV = W, T⊗S is a covariant(r + s)-tensor onV, which is a element of the spaceTr+s(V) and⊗ : Tr(V)×Ts(V)→ Tr+s(V).

Lemma 9.3. Let T∈ Tr(V), S,S′ ∈ Ts(V), and R∈ Tt(V), then

(i) (T⊗S)⊗R= T⊗ (S⊗R) (associative),

(ii) T⊗ (S+S′) = T⊗S+T⊗S′ (distributive),

(iii) T⊗S 6= 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.J 9.4Example.The last property can easily be seen by the following example. LetV = R2, andT,S∈ T1(R2), given byT(v) = v1 +v2, andS(w) = w1−w2, then

T⊗S(1,1,1,0) = 2 6= 0 = S⊗T(1,1,1,0),

which shows that⊗ is not commutative in general. I

The following theorem shows that the tensor product can be used to build thetensor spaceTr(V) from elementary building blocks.

Theorem 9.5. 24 Letv1, · · · ,vn be a basis for V , and letθ1, · · · ,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 Tr(V).

Proof: Compute

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

i1(v j1) · · ·θir (v jr )

= Ti1···ir δj1j1 · · ·δ

jrjr = Tj1··· jr

= T(v j1, · · · ,v jr ),

24See Lee, Prop. 11.2.

57

which shows by using the multilinearity of tensors thatT can be expanded in thebasisB as follows;

T = Ti1···ir θi1⊗·· ·⊗θir ,

whereTi1···ir = T(vi1, · · · ,vir ), the components of the tensorT. Linear independencefollows form the same calculation.

J 9.6Example.Consider the the 2-tensorsT(x,y) = x1y1 +x2y2, T ′(x,y) = x1y2 +x2y2 andT ′′ = x1y1+x2y2+x1y2 onR2. With respect to the standard basesθ1(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 ofT are given byT11 = 1, T12 = 0, T21 = 0, andT22 = 1.Also notice thatT ′ = S⊗S′, whereS(x) = x1 + x2, andS′(y) = y2. Observe thatnot every tensorT ∈ T2(R2) is of the formT = S⊗S′. For exampleT ′′ 6= S⊗S′,for anyS,S′ ∈ T1(R2). I

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 andW be two (finite dimensional) real vector spaces, with basesv1, · · ·vnandw1, · · ·wm respectively, and for their dual spacesV∗ andW∗ we have thedual basesθ1, · · · ,θn andσ1, · · · ,σm respectively. If we use the identificationV∗∗ ∼= V, andW∗∗ ∼= W we can defineV⊗W as follows:

Definition 9.7. The tensor product ofV andW is the real vector space of (finite)linear combinations

V⊗W :=

λi j vi ⊗w j : λi j ∈ R

=[

vi ⊗w j

i, j

],

wherevi⊗w j(v∗,w∗) := v∗(vi)w∗(w j), using the identificationvi(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 spacesV∗ andW∗. We obtain the real vectorspace of (finite) linear combinations

V∗⊗W∗ :=

λi j θi ⊗σ j : λi j ∈ R

=[

θi ⊗σ ji, j

],

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

58

we have thatξiη j = θi(v)σ j(w) = θi ⊗σ j(v,w), which shows the isomorphism bysettingλi j = b(vi ,w j).

In the caseV∗⊗W the tensors represent linear maps fromV to W. Indeed,from the previous we know that elements inV∗⊗W represent bilinear maps fromV×W∗ to R. For an elementb∈V∗⊗W this means thatb(v, ·) : W∗→R, and thusb(v, ·) ∈ (W∗)∗ ∼= W.J 9.8 Example.Consider vectorsa∈V andb∗ ∈W, thena∗⊗ (b∗)∗ can be iden-tified with a matrix, i.ea∗⊗ (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 thatA = (a·b∗)∗ = b·a∗. I

Lemma 9.9. 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 spacesTr(V) andVr(V) are given as follows;

Tr(V) = V∗⊗·· ·⊗V∗︸ ︷︷ ︸r times

, Tr(V) = V⊗·· ·⊗V︸ ︷︷ ︸r times

.

By considering tensor products ofV ’s andV∗’s we obtain the tensor space ofmixed tensors;

Trs (V) := V∗⊗·· ·⊗V∗︸ ︷︷ ︸

r times

⊗V⊗·· ·⊗V︸ ︷︷ ︸s times

.

Elements in this space are called(r,s)-mixed tensorsonV — r copies ofV∗, ands

copies ofV. Of course the tensor product described above is defined in general fortensorsT ∈ Tr

s (V), andS∈ Tr ′s′ (V):

⊗ : Trs (V)×Tr ′

s′ (V)→ Tr+r ′s+s′ (V).

59

The analogue of Theorem 9.5 can also be established for mixed tensors. In the nextsections we will see various special classes of covariant, contravariant and mixedtensors.J 9.10Example.The inner product on a vector spaceV is an example of a covariant2-tensor. This is also an example of a symmetric tensor. I

J 9.11Example.The determinant ofn vectors inRn is an example of covariantn-tensor onRn. The determinant is skew-symmetric, and an example of an alternatingtensor. I

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

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

and f ∗ : Tr(W)→ Tr(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 representf by a matrixA withrespect to basesvi andw j for V andW respectively, then the matrix for thelinear f ∗ is given by

A∗⊗·· ·⊗A∗︸ ︷︷ ︸r times

,

with respect to the basesθi1⊗·· ·⊗θir andσ j1⊗·· ·⊗σ jr for Tr(W) andTr(V)respectively.J 9.12Remark.The direct sum

T∗(V) =∞M

r=0

Tr(V),

consisting of finite sums of covariant tensors is called thecovariant tensor algebraof V with multiplication given by the tensor product⊗. Similarly, one defines thecontravariant tensor algebra

T∗(V) =∞M

r=0

Tr(V).

For mixed tensors we have

T(V) =∞M

r,s=0

Trs (V),

which is called thetensor algebra of mixed tensorof V. Clearly,T∗(V) andT∗(V)subalgebras ofT(V). I

60

10. 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 10.1. A covariantr-tensorT on a vector spaceV is calledsymmetricif

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

for any pair of indicesi ≤ j. The set of symmetric covariantr-tensors onV isdenoted byΣr(V)⊂ Tr(V), which is a (vector) subspace ofTr(V).

If a∈ Sr is a permutation, then define

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

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

SymT =1r! ∑

a∈Sr

aT.

It is straightforward to see that for any tensorT ∈ Tr(V), SymT is a symmetric.Moreover, a tensorT is symmetric if and only if SymT = T. For that reasonSymT is called the(tensor) symmetrization.J 10.2 Example.Let T,T ′ ∈ T2(R2) be defined as follows:T(x,y) = x1y2, andT ′(x,y) = x1y1. Clearly,T is not symmetric andT ′ is. We have that

SymT(x,y) =12

T(x,y)+12

T(y,x)

=12

x1y2 +12

y1x2,

which clearly is symmetric. If we do the same thing forT ′ we obtain:

SymT ′(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. I

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

S·T = Sym(S⊗T).

61

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)).

J 10.3Example.Consider the 2-tensorsS(x) = x1 + x2, andT(y) = y2. Now S⊗T(x,y) = x1y2 + x2y2, andT⊗S(x,y) = x2y1 + x2y2, which clearly gives thatS⊗T 6= 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 thatS·T = T ·S. I

Lemma 10.4. Let v1, · · · ,vn be a basis for V , and letθ1, · · · ,θn be the dual

basis for V∗. Then the set

BΣ =

θi1 · · ·θir : 1≤ i1 ≤ ·· · ≤ ir ≤ n,

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

r

)= (n+r−1)!

r!(n−1)! .

Proof: Proving thatBΣ is a basis follows from Theorem 9.5, see also Lemma10.10. It remains to establish the dimension ofΣr(V). Note that the elements inthe basis are given by multi-indices(i1, · · · , ir), satisfying the property that 1≤i1 ≤ ·· · ≤ ir ≤ n. This means choosingr integers satisfying this restriction. To dothis redefinejk := ik +k−1. The integersj range from 1 throughn+ r−1. Now

chooser integers out ofn+ 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.

62

Definition 10.5. A covariantr-tensorT on a vector spaceV is calledalternatingif

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

for any pair of indicesi ≤ j. The set of alternating covariantr-tensors onV isdenoted byΛr(V)⊂ Tr(V), which is a (vector) subspace ofTr(V).

As before we define

Alt 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 projectionof a tensorT, and AltT is of course aalternating tensor.J 10.6 Example.Let T,T ′ ∈ T2(R2) be defined as follows:T(x,y) = x1y2, andT ′(x,y) = x1y2− x2y1. Clearly, T is not alternating andT ′(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 forT ′ 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 thatT ′(x,y) = det(x,y). I

This brings us to the fundamental product of alternating tensors called thewedgeproduct. Let S∈ Λr(V) andT ∈ 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 covectorsθ,ω ∈V∗ gives

θ∧ω = θ⊗ω−ω⊗θ.

In particular we have that

63

(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) andS∈ 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 10.7. Let T∈ Tr(V) and S∈ Ts(V), then

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

Proof: Let G∼= Sr be the subgroup ofSr+s consisting of permutations that onlypermute the element1, · · · , r. For a ∈ G, we havea′ ∈ Sr . Now a(T ⊗S) =a′T⊗S, and thus

1r! ∑

a∈G

(−1)a a(T⊗S) = (Alt T)⊗S.

For the right cosetsba : 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 factor1(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 thatr! 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 10.7 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).

64

The same formula holds forT∧ (S∧R), which prove associativity. More generallyit holds that forTi ∈ Λr i (V)

T1∧·· ·∧Tk =(r1 + · · ·+ rk)!

r1! · · · rk!Alt (T1⊗·· ·⊗Tk).

Property (iii) can be seen as follows. Each term inT ∧Scan be found inS∧T.This can be done by linking the permutationsa anda′. 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 requiresrs permutations of two elements, which shows Property (iii).J 10.8Example.Consider the 2-tensorsS(x) = x1 +x2, andT(y) = y2. As beforeS⊗T(x,y) = x1y2 +x2y2, andT⊗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 thatS∧T = −T ∧S. Note that ifT = e∗1, i.e. T(x) = x1, andS= e∗2,i.e. S(x) = x2, then

T ∧S(x,y) = x1y2−x2y1 = det(x,y).

I

J 10.9Remark.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. I

If e∗1, · · · ,e∗n is the standard dual basis for(Rn)∗, then for vectorsa1, · · · ,an ∈Rn,

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

Using the multilinearity the more general statement reads

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

),

65

whereβi are co-vectors.The alternating tensor det= e∗1∧·· ·∧e∗n is called the determinant function onRn.

If f : V →W is a linear map between vector spaces then the pullbackf ∗T ∈Λr(V)of any alternating tensorT ∈ 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 iff : V →V, linear, and dimV = n, then

(8) f ∗T = det( f )T,

for any alternating tensorT ∈Λn(V). This can be seen as follows. By multilinearitywe verify the above relation for the vectorsei. 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 thatΛn(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 10.10.Letθ1, · · · ,θn be a basis for V∗, then the set

BΛ =

θi1∧·· ·∧θir : 1≤ i1 < · · ·< ir ≤ n,

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

r > n.

Proof: From Theorem 9.5 we know that any alternating tensorT ∈ Λr(V) canbe written as

T = Tj1··· jr θj1⊗·· ·⊗θ jr .

We have that AltT = T, and so

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

Tj1··· jr θj1∧·· ·∧θ jr

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

T =± 1r!

Ti1···ir θi1∧·· ·∧θir ,

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

thusλi1···ir = θi1∧·· ·∧θir (vi1, · · · ,vir ) = 0, which proves linear independence.

66

It is immediately clear thatBΛ consists of

(n

r

)elements.

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

Lemma 10.11.Some of the basic properties can be listed as follows;

(i) Sym andAlt are projections on Tr(V), i.e. Sym2 = Sym, andAlt2 = Alt ;(ii) T is symmetric if and only ifSymT = T, and T is alternating if and only

if Alt T = T;

(iii) Sym(Tr(V)) = Σr(V), andAlt(Tr(V)) = Λr(V);(iv) SymAlt = Alt Sym= 0, i.e. if T ∈ Λr(V), thenSymT = 0, and if

T ∈ Σr(V), thenAlt T = 0;

(v) let f : V →W, thenSymandAlt commute with f∗ : Tr(W)→ Tr(V), i.e.

Sym f ∗ = f ∗ Sym, andAlt f ∗ = f ∗ Alt .

11. Tensor bundles and tensor fields

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

Tr(TpM), Ts(TpM), and Trs (TpM),

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

TrM =G

p∈M

Tr(TpM),

TsM =G

p∈M

Ts(TpM),

Trs M =

Gp∈M

Trs (TpM),

which are called thecovariant r-tensor bundle, contravariants-tensor bundle,and themixed(r,s)-tensor bundleon M. As for the tangent and cotangent bundlethe tensor bundles are also smooth manifolds. In particular,T1M = T∗M, andT1M = TM. Recalling the symmetric and alternating tensors as introduced in theprevious section we also define the tensor bundlesΣrM andΛrM.

Theorem 11.1.The tensor bundles TrM,TrM and Trs M are smooth vector bundles.

Proof: Using Section 7 the theorem is proved by choosing appropriate localtrivializationsΦ. For coordinatesx = ϕα(p) andx′ = ϕβ(p) we recall that

∂∂xi

∣∣p =

∂x′j∂xi

∂∂x′j

∣∣p, dx′ j |p =

∂x′j∂xi

dxi |p.

67

For a covariant tensorT ∈ TrM this implies the following. The components aredefined in local coordinates by

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

∂xi1, · · · , ∂

∂xir

).

The change of coordinatesx→ x′ then gives

Ti1···ir = T(∂x′j1

∂xi1

∂∂x′j1

, · · · ,∂x′jr∂xir

∂∂x′jr

)= T ′

j1··· jr∂x′j1∂xi1

· · ·∂x′jr∂xir

.

Define

Φ(Ti1···ir dxi1⊗·· ·⊗dxir ) =(p,(Ti1···ir )

),

then

Φ′ Φ−1(p,(T ′j1··· jr )

)=

(p,(

T ′j1··· jr

∂x′j1∂xi1

· · ·∂x′jr∂xir

)).

The products of the partial derivatives are smooth functions. We can now applyTheorem 7.4 as we did in Theorems 7.8 and 7.12.

On tensor bundles we also have the natural projection

π : Trs M →M,

defined byπ(p,T) = p. A smooth sectionin Trs M is a smooth mapping

σ : M → Trs M,

such thatπσ = idM. The space of smooth sections inTrs M is denoted byFr

s(M).For the co- and contravariant tensors these spaces are denoted byFr(M) andFs(M)respectively. Smooth sections in these tensor bundles are also calledsmooth tensorfields. 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 ∈ Fr

s(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 functionsσi1···ir .

Lemma 11.2. 25 A covariant tensor fieldσ is smooth at p∈U if and only if

(i) the coordinate functionsσi1···ir : U → R are smooth, or equivalently if and

only if

25See Lee, Lemma 11.6.

68

(ii) for smooth vector fields X1, · · · ,Xr defined on any open set U⊂M, then the

functionσ(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 11.1 and then the proof goesas Lemma 7.11.

J 11.3Example.LetM = R2 and letσ = dx1⊗dx1+x21dx2⊗dx2. If X = ξ1(x) ∂

∂x1+

ξ2(x) ∂∂x2

andY = η1(x) ∂∂x1

+η2(x) ∂∂x2

are arbitrary smooth vector fields onTxR2∼=R2, then

σ(X,Y) = ξ1(x)η1(x)+x21ξ2(x)η2(x),

which clearly is a smooth function inx∈ R2. I

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

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

whereT ∈ Tr(Tf (p)M), andX1, · · · ,Xr ∈ TpN. We have the following properties.

Lemma 11.4. 26 Let f : N → M, g : M → P be smooth mappings, and let p∈ N,

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

(i) f ∗ : Tr(Tf (p)M)→ Tr(TpN) is linear;

(ii) f ∗(S⊗T) = f ∗S⊗ f ∗T;

(iii) (g f )∗ = f ∗ g∗ : Tr(T(g f )(p)P)→ Tr(TpM);(iv) id∗MS= S;

(v) f ∗ : TrM → TrN is a smooth bundle map.

Proof: Combine the proof of Lemma 6.3 with the identities in the proof ofTheorem 11.1.

J 11.5Example.Let us continue with the previous example and letN = M = R2.Consider the mappingf : N→M defined by

f (x) = (2x1,x32−x1).

26See Lee, Proposition 11.8.

69

For given tangent vectorsX,Y ∈ TxN, given byX = ξ1∂

∂x1+ξ2

∂∂x2

andY = 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 byσ = 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 thatf∗X and f∗Y are tangent vectors inTxN and notnecessarily vector fields, although we can use this calculation to computef ∗σ,which clearly is a smooth 2-tensor field onT2N. I

J 11.6Example.A different way of computingf ∗σ is via a local representation ofσ directly. We haveσ = dy1⊗dy1 +y2

1dy2⊗dy2, and

f ∗σ = d(2x1)⊗d(2x1)+4x21d(x3

2−x1)⊗d(x32−x1)

= 4dx1⊗dx1 +4x21

(3x2

2dx2−dx1)⊗ (3x22dx2−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 toR producesthe pushforward to a 1-form onN. I

J 11.7Example.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,

70

which produces the following matrix if we identifyT2(TxN) andT2(TyM) with R4:

f ∗ =

4 −2 −2 10 6x2

2 0 −3x22

0 0 6x22 −3x2

2

0 0 0 9x42

,

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 interpretf ∗ : T2(TyM)→ T2(TxN) as a linear mapping.I

As for smooth 1-forms this operation extends to smooth covariant tensor fields:( f ∗σ)p = f ∗(σ f (p)), σ ∈ Fr(N), which in coordinates reads

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

for tangent vectorsX1, · · · ,Xr ∈ TpN.

Lemma 11.8. 27 Let f : N → M, g : M → P be smooth mappings, and let h∈C∞(M), σ ∈ Fr(M), andτ ∈ Fr(N), then:

(i) f ∗ : Fr(M)→ Fr(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 11.4 and 11.2.

12. Differential forms

A special class of covariant tensor bundles and associated bundle sections arethe so-called alternating tensor bundles. LetΛr(TpM) ⊂ Tr(TpM) be the spacealternating tensors onTpM. We know from Section 10 that a basis forΛr(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 byΛrM. Smooth sections inΛrM are calleddifferential r-forms,and the space of smooth sections is denoted byΓr(M) ⊂ Fr(M). In particular

27See Lee, Proposition 11.9.

71

Γ0(M) = C∞(M), andΓ1(M) = F∗(M). In terms of components a differentialr-form, or r-form for short, is given by

σi1···ir dxi1∧·· ·∧dxir ,

and the componentsσi1···ir are smooth functions. Anr-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).

J 12.1Example.Let M = R3, andσ = dx∧dz. Then for vector fields

X1 = X11

∂∂x

+X21

∂∂y

+X31

∂∂z

,

and

X2 = X12

∂∂x

+X22

∂∂y

+X32

∂∂z

,

we have that

σ(X1,X2) = X11 X3

2 −X31 X1

2 .

I

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

iXσ := σ(X, ·, · · · , ·),

is called thecontraction withX, and is a differential(r−1)-form onM. Anothernotation for this isiXσ = Xyσ. Contraction is a linear mapping

iX : Γr(M)→ Γr−1(M).

Contraction is also linear inX, i.e. for vector fieldsX,Y it holds that

iX+Yσ = iXσ+ iYσ, iλXσ = λ · iXσ.

Lemma 12.2. 28 Let σ ∈ Γr(M) and X∈ F(M) a smooth vector field, then

(i) iXσ ∈ Γr−1(M) (smooth(r−1)-form);

(ii) iX iX = 0;

(iii) iX is ananti-derivation, i.e. forσ ∈ Γr(M) andω ∈ Γs(M),

iX(σ∧ω) = (iXσ)∧ω+(−1)rσ∧ (iXω).

28See Lee, Lemma 13.11.

72

A direct consequence of (iii) is that ifσ = σ1∧ ·· · ∧σr , whereσi ∈ Γ1(M) =F∗(M), then

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

where the hat indicates thatσi is to be omitted, and we use the summation conven-tion.J 12.3Example.Let σ = x2dx1∧dx3 be a 2-form onR3, andX = x2

1∂

∂x1+x3

∂∂x2

+(x1+x2) ∂

∂x3a given vector field onR3. If Y =Y1 ∂

∂y1+Y2 ∂

∂y2+Y3 ∂

∂y3is an arbitrary

vector fields then

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

= x21Y

3− (x1 +x2)Y1,

which gives thatiXσ = x2

1dx3− (x1 +x2)dx1.

I

Sinceσ is a 2-form the calculation usingX,Y is still doable. For higher formsthis becomes to involved. If we use the multilinearity of forms we can give a simpleprocedure for computingiXσ using Formula (10).J 12.4Example.Let σ = dx1∧dx2∧dx3 be a 3-form onR3, andX the vector fieldas given in the previous example. By linearity

iXσ = iX1σ+ iX2σ+ iX3σ,

whereX1 = x21

∂∂x1

, X2 = x3∂

∂x2, andX3 = (x1 +x2) ∂

∂x3. This composition is chosen

so thatX 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

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

a 2-form onR3. One should now verify that the same answer is obtained by com-putingσ(X,Y,Z). I

For completeness we recall that for a smooth mappingf : N→M, the pullbackof 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 mappingh : M → R, then pushforward, or differential ofhdhp = h∗ ∈ T∗

p M. In coordinatesdhp = ∂h∂xi

dxi |p, and thus the mappingp 7→ dhp

73

is a smooth section inΛ1(M), and therefore a differential 1-form, with componentσi = ∂h

∂xi(in local coordinates).

If f : N→M is a mapping betweenm-dimensional manifolds with charts(U,ϕ),and(V,ψ) respectively, andf (U)⊂V. Setx = ϕ(p), andy = ψ(q), then

(11) f ∗(σdy1∧·· ·∧dym)= (σ f )det

(J f |x

)dx1∧·· ·∧dxm.

This can be proved as follows. From the definition of the wedge product andLemma 11.8 it follows thatf ∗(dy1∧·· ·∧dym) = f ∗dy1∧·· ·∧ f ∗dym, and f ∗dyj =∂ f j

∂xidxi = dF j , whereF = 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 coordinatesf = ψϕ−1 yields

(12) dy1∧·· ·∧dym = det(J f |x

)dx1∧·· ·∧dxm.

J 12.5Example.Considerσ = dx∧dyonR2, and mappingf given byx= r cos(θ)andy = r sin(θ). The mapf the identity mapping that mapsR2 in Cartesian coor-dinates toR2 in polar coordinates (consider the chartU = (r,θ) : r > 0, 0 < θ <

2π). As before we can compute the pullback ofσ 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). I

J 12.6Remark.If we define

Γ(M) =∞M

r=0

Γr(M),

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

74

13. Orientations

In order to explain orientations on manifolds we first start with orientations offinite-dimensional vector spaces. LetV be a realm-dimensional vector space. Twoordered basisv1, · · · ,vm andv′1, · · · ,v′m are said to beconsistently orientedifthe transition matrixA = (ai j ), defined by the relation

vi = ai j v′j ,

has positive determinant. This notion defines an equivalence relation on orderedbases, and there are exactly two equivalence classes. Anorientation for V is achoice of an equivalence class of order bases. Given an ordered basisv1, · · · ,vmthe orientation is determined by the classO = [v1, · · · ,vm]. The pair(V,O) is calledan oriented vector space. For a given orientation any ordered basis that has thesame orientation is calledpositively oriented, and otherwisenegatively oriented.

Lemma 13.1. Let 0 6= θ ∈ Λm(V), then the set of all ordered basesv1, · · · ,vmfor whichθ(v1, · · · ,vm) > 0 is an orientation for V .

Proof: Let v′i be any ordered basis andvi = ai j v′j . Then θ(v1, · · · ,vm) =det(A)θ(v1, · · · ,vm) > 0, which proves that the set of basesvi for which it holdsthatθ(v1, · · · ,vm) > 0, characterizes an orientation forV.

Let us now describe orientations for smooth manifoldsM. We will assume thatdimM = m≥ 1 here. For each pointp∈ M we can choose an orientationOp forthe tangent spaceTpM, 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 ofTpM, which leads to the concept of orientation of a smooth man-ifold.

Definition 13.2. A smoothm-dimensional manifoldM with a pointwise orienta-tion Opp∈M, Op =

[X1, · · · ,Xm

], is orientedif for each point inM there exists a

neighborhoodU and a diffeomorphismϕ, mapping fromU to an open subset ofRm, such that

(13)[ϕ∗(X1), · · · ,ϕ∗(Xm)

]∣∣∣p= [e1, · · · ,em], ∀p∈U,

where [e1, · · · ,em] is the standard orientation ofRm. In this case the pointwiseorientationOpp∈M is said to be consistently oriented. The choice ofOpp∈M iscalled anorientationonM, denoted byO = Opp∈M.

75

If a consistent choice of orientationsOp does not exist we say that a manifold isnon-orientable.J 13.3Remark.If we look at a single chart(U,ϕ) we can choose orientationsOp

that are consistently oriented forp ∈ Op. The choicesXi = ϕ−1∗ (ei) = ∂

∂xi|p are

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

J 13.4Example.Let M = S1 be the circle inR2, i.e. M = p= (p1, p2) : p21+ p2

2 =1. The circle is an orientable manifold and we can find a orientation as follows.Consider the stereographic chartsU1 = S1\NpandU2 = 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. ChooseX(p) =(−p2, p1), then

(ϕ1)∗(X) = Jϕ1|p(X) =( 2

1− p2

2p1

(1− p2)2

)( −p2

p1

)=

21− p2

> 0,

onU1 (standard orientation). If we carry out the same calculation forU2 we obtain(ϕ2)∗(X) = Jϕ2|p(X) = −2

1+p2< 0 onU2, with corresponds to the opposite orienta-

tion. Instead by choosingϕ2(p) = ϕ2(−p1, p2) we obtain

(ϕ2)∗(X) = Jϕ2|p(X) =2

1+ p2> 0,

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

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

∂∂x

∣∣∣p= Jϕ−1

1 |p(1) =

(16−4x2

(x2+4)2

16x(x2+4)2

).

In terms ofp this givesX(p) = 12(1− p2)

(−p2

p1

). By definition(ϕ1)∗(X) = 1

for p∈U1. For p= Npwe chooseX(p) = (−1 0)t , soX(p) is defined forp∈S1

76

and is not a continuous function ofp! It remains to verify that(ϕ2)∗(X) > 0 forsome neighborhood ofNp. First, atp = Np,

Jϕ2|p

(−10

)=

21+ p2

= 1 > 0.

The choice ofX(p) at p = Np 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\Np. Note thatϕ2ϕ−11 (x) = −4

x , and

Jϕ2|p(X(p)) = Jϕ2|p(Jϕ−11 |p(1)) = J(ϕ2ϕ−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. I

What the above example shows us is that if we chooseX(p) = ∂∂xi|p for all

charts, it remains to be verified if we have the proper collection of charts, i.e. does[JϕβJϕ−1

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

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

Theorem 13.5. A smooth manifold M is orientable if and only if there exists an

atlasA = (Uα,ϕα such that for any p∈M

det(Jϕαβ|x) > 0, ϕαβ = ϕβ ϕ−1α ,

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

Proof: Obvious from the previous example and Lemma 13.1.

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

Theorem 13.6. 29 Let M be a smooth m-dimensional manifold. A nowhere van-

ishing differential m-formθ ∈ Γm(M) determines a unique orientationO on M for

whichθp is positively orientated at each TpM. Conversely, given an orientationO

on M, then there exists a nowhere vanishing m-formθ ∈ Γm(M) that is positively

oriented at each TpM.

29See Lee, Prop. 13.4.

77

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

θ( ∂

∂x1, · · · , ∂

∂xm

)= f 6= 0.

Without loss of generality can be assumed to be positive, otherwise changeϕ bymeans ofx1 →−x1. Now let ∂

∂yi be canonical bases forTpM with respect to a

chart(V,ψ). As beforeθ = gdy1∧·· ·∧dym, g > 0 onV. Assume thatU ∩V 6= ∅,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 thusM is orientable.

For the converse we construct non-vanishingm-forms on each chart(U,ϕ). Viaa partition of unity we can construct a smoothm-form onM, see Lee.

This theorem implies in particular that non-orientablem-dimensional manifoldsdo not admit a nowhere vanishingm-form, or volume form.J 13.7Example.Consider the Mobius stripM. The Mobius strip is an exampleof a non-orientable manifold. Let us parametrize the Mobius strip as an embeddedmanifold inR3:

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 fromR/2πZ×(−1,1) → R3. Let us assume that the Mobius stripM is oriented, then by theabove theorem there exists a nonwhere vanishing 2-formσ 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). Sinceg is a smooth embedding (parametrization) thepullback formg∗σ = ρ(θ, r)dθ∧dr is a nowhere vanishing 2-form onR/2πZ×(−1,1). In particular, this means thatρ is 2π-periodic in θ. Notice thatσ =iXdx∧dy∧dz, where

X = a(x,y,z)∂∂x

+b(x,y,z)∂∂y

+c(x,y,z)∂∂z

.

For vector fieldsξ = ξ1∂∂x +ξ2

∂∂y +ξ3

∂∂z andη = η1

∂∂x +η2

∂∂y +η3

∂∂z we then have

that

σ(ξ,η) = iXdx∧dy∧dz(ξ,η) = X · (ξ×η),

78

whereξ× η is the cross product ofξ and η. The condition thatθ is nowherevanishing can be translated toR2 as follows:

g∗σ(e1,e2) = σ(g∗(e1),g∗(e2)).

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

Let N,M be oriented manifolds of the same dimension, and letf : N → M bea smooth mapping. A mapping is calledorientation preservingat a pointp∈ N

if f∗ maps positively oriented bases ofTpN to positively oriented bases inTf (p)M.A mapping is calledorientation reversingat p∈ N if f∗ maps positively orientedbases ofTpN to negatively oriented bases inTf (p)M.

FIGURE 37. Out(in)ward vectors and the induced orientation to∂M.

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

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

(ii) outward vectors; letϕ−1 : W ⊂ Hm→ M, thenX ∈ TpM is called an out-ward vector ifϕ−1

∗ (Y) = X, for someY = (y1, · · · ,ym) with y1 < 0;(iii) inward vectors; letϕ−1 : W ⊂Hm→M, thenX ∈ TpM is called an inward

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

79

Using this concept we can now introduce the notion of induced orientation on∂M.Let p∈ ∂M and choose a basisX1, · · · ,Xm for TpM such that[X1, · · · ,Xm] = Op,X2, · · · ,Xm are tangent boundary vectors, andX1 is an outward vector. In this case[X2, · · · ,Xm] = (∂O)p determines an orientation forTp(∂M), which is consistent,and therefore∂O = (∂O)pp∈∂M is an orientation on∂M induced byO. Thus foran oriented manifoldM, with orientationO, ∂M has an orientation∂O, called theinduced orientation on∂M.J 13.8Example.Any open setM ⊂ Rm (or Hm) is an orientable manifold. I

J 13.9Example.Consider a smooth embedded co-dimension 1 manifold

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

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

0, which is an open set inRm+1 and thus an oriented manifold. SinceM = ∂N themanifoldM inherits an orientation fromM and hence it is orientable. I

80

IV. Integration on manifolds

14. Integrating m-forms on Rm

We start off integration ofm-forms by consideringm-forms onRm.

Definition 14.1. A subsetD⊂ Rm is called a domain of integration if

(i) D is bounded, and(ii) ∂D hasm-dimensional Lebesgue measuredµ= dx1 · · ·dxm equal to zero.

In particular any finite union or intersection of open or closed rectangles is adomain of integration. Any bounded30 continuous functionf on D is integrable,i.e.

−∞ <Z

Df dx1 · · ·dxm < ∞.

SinceΛm(Rm)∼= R, a smoothm-form onRm 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 :=

ZD

f (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) 6= 0is a compact set. The set of compactly supportedm-forms onRm is denoted byΓm

c (Rm), and is a linear subspace ofΓm(Rm). Similarly, for any open setU ⊂ Rm

we can defineω ∈ Γmc (U). Clearly, Γm

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

linear subspace via zero extension toRm. For any open setU ⊂ Rm there exists adomain of integrationD such thatU ⊃ D⊃ supp(ω) (see Exercises).

30Boundedness is needed because the rectangles are allowed to be open.

81

Definition 14.2. Let U ⊂ Rm be open andω ∈ Γmc (U), and letD be a domain of

integrationD such thatU ⊃ D⊃ supp(ω). We define the integralZU

ω :=Z

Dω.

If U ⊂Hm open, then ZU

ω :=Z

D∩Hmω.

The next theorem is the first step towards defining integrals onm-dimensionalmanifoldsM.

Theorem 14.3.Let U,V ⊂ Rm be open sets, f: U →V an orientation preserving

diffeomorphism, and letω ∈ Γmc (V). Then,Z

Vω =

ZU

f ∗ω.

If f is orientation reversing, thenRU ω =−

RV f ∗ω.

Proof: Assume thatf is an orientation preserving diffeomorphism fromU toV. Let E be a domain a domain of integration forω, thenD = f−1(E) is a do-main of integration forf ∗ω. We now prove the theorem for the domainsD andE. We use coordinatesxi 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 obtainZ

Eω =

ZE

g(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.

15. 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.

82

Definition 15.1. Let M be smoothm-manifold with atlasA = (ϕi ,Ui)i∈I . Apartition of unity subordinate toA is a collection for smooth functionsλi : M →Ri∈I satisfying the following properties:

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

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

Condition (iii) says that everyp∈ M has a neighborhoodU 3 p such that onlyfinitely manyλi ’s are nonzero inU . As a consequence of this the sum in Condition(iv) is always finite.

Theorem 15.2. For any smooth m-dimensional manifold M with atlasA there

exists a partition of unityλi subordinate toA.

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

Lemma 15.3. There exists a smooth function h: Rm→ R such that0≤ h≤ 1 on

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

Proof: Define the functionf1 : R→ R as follows

f1(t) =

e−1/t , t > 0,

0, t ≤ 0.

One can easily prove thatf1 is aC∞-function onR. If we set

f2(t) =f1(2− t)

f1(2− t)+ f1(t−1).

This function has the property thatf2(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 constructf we simplywrite f (x) = f2(|x|) for x∈ Rm\0. Clearly, f is smooth onRm\0, and sincef |B1(0) ≡ 1 it follows that f is smooth onRm.

An atlasA gives an open covering forM. The setUi in the atlas need not becompact, nor locally finite. We say that a coveringU = Ui of M is locally finiteif every point p ∈ M has a neighborhood that intersects only finitelyUi ∈ U. Ifthere exists another coveringV = Vj such that everyVj ∈ V is contained in someVj ⊂Ui ∈ U, thenV is called arefinement ofU. A topological spaceX for whicheach open covering admits a locally finite refinement is calledparacompact.

Lemma 15.6. Any topological manifold M allows a countable, locally finite cov-

ering by precompact open sets.

83

FIGURE 38. The functionsf1 and f2.

FIGURE 39. A locally finite covering [left], and a refinement[right].

Proof: We start with a countable covering of open ballsB = Bi. We nowconstruct a coveringU that satisfies

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

(iii) Bi ⊂Ui .

We build the coveringU from B using an inductive process. LetU1 = B1, andassumeU1, · · · ,Uk have been constructed satisfying (i)-(iii). Then

Uk ⊂ Bi1∪·· ·∪Bik,

whereBi1 = B1. Now set

Uk+1 = Bi1∪·· ·∪Bik.

Chooseik large enough so thatik ≥ k+1 and thusBk+1 ⊂Uk+1.From the coveringU we can now construct a locally finite coveringV by setting

Vi = Ui\Ui−2, i > 1.

84

FIGURE 40. Constructing a nested coveringU from a coveringwith ballsB [left], and a locally finite covering obtained from theprevious covering [right].

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

(i) W is a countable and locally finite;(ii) eachWj ⊂W is in the domain of some smooth coordinate mapϕ j : j→Rm

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

j (B1(0)) coversM.

The coveringW is calledregular.

Lemma 15.8. For any open coveringU for a smooth manifold M there exists a

regular refinement. In particular M is paracompact.

Proof: Let V be a countable, locally finite covering as described in the previouslemma. SinceV is locally finite we can find a neighborhoodWp for eachp ∈ M

that intersects only finitely many setVj ∈ V. We want to chooseWp in a smartway. First we replaceWp by Wp∩Vj : p∈Vj. Sincep∈Ui for somei we thenreplaceWp by Wp∩Vi . Finally, we replaceWp by a small coordinate ballBr(p)so thatWp is in the domain of some coordinate mapϕp. This provides coordinatecharts(Wp,ϕp). Now defineZp = ϕ−1

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

ering, sayZ1k , · · · ,Z

mkk . The setsZi

k are sets of the formZpi for somepi ∈Vk. Theassociated coordinate charts are(W1

k ,ϕ1k), · · · ,(W

mkk ,ϕmk

k ). EachWik is obtained via

the construction above forpi ∈Vk. Clearly,Wik is a countable open covering of

M that refinesU and which, by construction, satisfies (ii) and (iii) in the definitionof regular refinement. It is clear from compactness thatWi

k is locally finite. Letus relabel the setsWi

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

85

FIGURE 41. The different stages of constructing the setsWp col-ored with different shades of grey.

Proof of Theorem 15.2:From Lemma 15.8 there exists a regular refinementW

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

Zi := ϕ−1i (B2(0)).

Using Lemma 15.3 we now define the functionsµ:M → R;

µi =

f2ϕi onWi

0 onM\Vi .

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

λ =µj(p)

∑i µi(p),

are, due to the local finiteness ofWi and the fact that the denominator is positivefor eachp∈M, smooth functions onM. Moreover, 0≤ λ j ≤ 1, and∑ j λ j ≡ 1.

SinceW is a refinement ofA we can choose an indexk j such thatWj ⊂Uk j .

Therefore we can group together some of the functionλ 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 caseM is assumed to be an smoothm-dimensional manifold.

Theorem 15.10.31 For any close subset A⊂ M and any open set U⊃ A, there

exists a smooth function f: M → R such that

31See Lee, Prop. 2.26.

86

(i) 0≤ f ≤ 1 on M;

(ii) f−1(1) = A;

(iii) supp( f )⊂U.

Such a function f is called abump function forA supported inU.

By considering functionsg= c(1− f )≥ 0 we obtain smooth functions for whichg−1(0) can be an arbitrary closed subset ofM.

Theorem 15.11.32 Let A⊂ M be a closed subset, and f: A→ Rk be a smooth

mapping.33 Then for any open subset U⊂ M containing A there exists a smooth

mapping f† : M → Rk such that f†|A = f , andsupp( f †)⊂U.

16. Integration on of m-forms on m-dimensional manifolds.

In order to introduce the integral of anm-form on M we start with the case offorms supported in a single chart. In what follows we assume thatM is an orientedmanifold with an oriented atlasA, i.e. a consistent choice of smooth charts suchthat the transitions mappings are orientation preserving.

FIGURE 42. Anm-form supported in a single chart.

Let ω ∈ Γmc (M), with supp(ω) ⊂ U for some chartU in A. Now define the

integral ofω overM as follows:ZM

ω :=Z

ϕ(U)(ϕ−1)∗ω,

where the pullback form(ϕ−1)∗ω is compactly supported inV = ϕ(U) (ϕ is adiffeomorphism). The integral overV of the pullback form(ϕ−1)∗ω is defined in

32See Lee, Prop. 2.26.33A mapping from a closed subsetA⊂M is said to be smooth if for every pointp∈A there exists

an open neighborhoodW ⊂M of p, and a smooth mappingf † : W → Rk such thatf †|W∩A = f .

87

Definition 14.2. It remains to show that the integralR

M ω does not depend on theparticular chart. Consider a different chartU ′, possibly in a different oriented atlas

FIGURE 43. Different chartsU andU ′ containing supp(ω), i.e.supp(ω)⊂U ∩U ′.

A′ for M (same orientation), thenZV ′

(ϕ′−1)∗ω =Z

ϕ′(U∩U ′)(ϕ′−1)∗ω =

Zϕ(U∩U ′)

(ϕ′ ϕ−1)∗(ϕ′−1)∗ω

=Z

ϕ(U∩U ′)(ϕ−1)∗(ϕ′)∗(ϕ′−1)∗ω =

Zϕ(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 thatA∪A′ is an oriented atlas forM. 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 overM of arbitrarym-formsω ∈ Γm

c (M).

Definition 16.3. Let ω ∈ Γmc (M) and letAI = (Ui ,ϕii∈I ⊂A be a finite subcov-

ering ofsupp(ω) coming from an oriented atlasA for M. Letλi be a partition

of unity subordinate toAI . Then theintegral of ω overM is defined asZM

ω := ∑i

ZM

λiω,

where the integralsR

M λiω are integrals of form that have support in single charts

as defined above.

88

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 16.4. The above definition is independent of the chosen partition of unity

and coveringAI .

Proof: Let A′J ⊂ A′ be another finite covering of supp(ω), whereA′ is a com-

FIGURE 44. Using a partition of unity we can constructm-formswhich are all supported in one, but possibly different chartsUi .

patible oriented atlas forM, i.e. A∪A′ is a oriented atlas. Letλ′j be a partitionof unity subordinate toA′

J. We haveZM

λiω =Z

M

(∑

j

λ′j)

λiω = ∑j

ZM

λ′jλiω.

By summing overi we obtain∑iR

M λiω = ∑i, jR

M λ′jλiω. Each termλ′jλiω is sup-ported in someUi and by previous independent of the coordinate mappings. Sim-ilarly, if we interchange thei’s and j ’s, we obtain that∑ j

RM λ′jω = ∑i, j

RM λ′jλiω,

which proves the lemma.

J 16.6Remark.If M is a compact manifold that for anyω ∈ Γm(M) it holds thatsupp(ω)⊂M is a compact set, and thereforeΓm

c (M) = Γm(M) in the case of com-pact manifolds. I

So far we considered integral ofm-forms overM. One can of course integraten-forms onM over n-dimensional immersed or embedded submanifoldsN ⊂ M.Given ω ∈ Γn(M) for which the restriction toN, ω|N has compact support inN,then Z

Nω :=

ZN

ω|N.

89

As a matter of fact we havei : N → M, andω|N = i∗ω, so thatR

N ω :=R

N ω|N =RN i∗ω. If M is compact with boundary∂M, then

R∂M ω is well-defined for any

(m−1)-form ω ∈ Γm−1(M).

Theorem 16.7.Let ω ∈ Γmc (M), and f : N→M is a diffeomorphism. ThenZ

Mω =

ZN

f ∗ω.

Proof: By the definition of integral above we need to prove the above statementfor the terms Z

Mλiω =

ZN

f ∗λiω.

Therefore it suffices to prove the theorem for formsω whose support is in a singlechartU .

FIGURE 45. The pullback of anm-form.

ZU

f ∗ω =Z

ϕ(U)(ϕ−1)∗ f ∗ω =

Zϕ(U)

(ϕ−1)∗ f ∗ψ∗(ψ−1)∗ω

=Z

ψ(U ′)(ψ−1)∗ω =

ZU ′

ω.

By applying this toλiω and summing we obtain the desired result.

For sake of completeness we summarize the most important properties of theintegral.

Theorem 16.9. 34 Let N,M be oriented manifolds (with or without boundary) of

dimension m , andω,η ∈ Γmc (M), are m-forms. Then

(i)R

M aω+bη = aR

M ω+bR

M η;

34See Lee Prop. 14.6.

90

(ii) if −O is the opposite orientation toO, thenZM,−O

ω =−Z

M,Oω;

(iii) if ω is an orientation form, thenR

M ω > 0;

(iv) if f : N→M is a diffeomorphism, thenZM

ω =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. LetDi ⊂ Rm — finite set of indicesi — be compactdomains of integration, andgi : Di →M are smooth mappings satisfying

(i) Ei = gi(Di), andgi : int (Di) → int (Ei) are orientation preserving diffeo-morphisms;

(ii) Ei ∩E j intersect only along their boundaries, for alli 6= j.

Theorem 16.10. 35 Let (gi ,Di) be a finite set of parametrizations as defined

above. Then for anyω⊂ Γmc (M) such thatsupp(ω)⊂ ∪iEi it holds thatZ

Mω = ∑

i

ZDi

g∗i ω.

Proof: As before it suffices the prove the above theorem for a single chartU ,

FIGURE 46. Carving up supp(ω) via domains of integration forparametrizationsgi for M.

i.e. supp(ω) ⊂U . One can chooseU to have a boundary∂U so thatϕ(∂U) has

35See Lee Prop. 14.7.

91

measure zero, andϕ maps cl(U) to a compact domain of integrationK ⊂Hm. Nowset

Ai = cl(U)∩Ei , Bi = g−1i (Ai), Ci = ϕi(Ai).

We have ZCi

(ϕ−1)∗ω =Z

Bi

(ϕgi)∗(ϕ−1)∗ω =Z

Bi

g∗i ω.

Since the interiors of the setsCi (and thusAi) are disjoint it holds thatZM

ω =Z

K(ϕ−1)∗ω = ∑

i

ZCi

(ϕ−1)∗ω

= ∑i

ZBi

g∗i ω = ∑i

ZDi

g∗i ω,

which proves the theorem.

J 16.12Remark.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 14 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 formulaZU

ω =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 parametrizationg : D→M:Zg(D)

ω =Z

Dωg(x)(g∗(e1), · · · ,g∗(em))dµ.

These expression are useful for computing integrals. I

J 16.13Example.Consider the 2-sphere parametrized by the mappingg : R2 →S2 ⊂ R3 given as

g(φ,θ) =

x

y

z

=

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 forS2. Given the 2-formω = zdx∧dz let us compute the pullback

92

form g∗ω onR2. 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

0

R π0 cos(φ)sin2(φ)sin(θ)dφdθ = 0, which shows

thatω is not a volume form onS2.If we perform the same calculations forω = xdy∧dz+ydz∧dx+zdx∧dy, thenZ

S2ω =

Z 2π

0

Z π

0sin(φ)dφdθ = 4π.

The pullback formg∗ω = sin(φ)dφ∧dθ, which shows thatω is a volume form onS2. I

17. The exterior derivative

The exterior derivatived of a differential form is an operation that maps ank-form to a(k+1)-form. Write ak-form onRm in the following notation

ω = ωi1···ikdxi1∧·· ·∧dxik = ωI dxI , I = (i1 · · · ik),

then we define

(14) dω = dωI ∧dxI .

Written out in all its differentials this reads

dω = d(

ωi1···ikdxi1∧·· ·∧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 thatd extends to differential forms on manifolds.J 17.1Example.Considerω0 = f (x,y), a 0-form onR2, then

dω0 =∂ f∂x

dx+∂ f∂y

dy.

93

For a 1-formω1 = f1(x,y)dx+ f2(x,y)dywe obtain

dω1 =∂ f1∂x

dx∧dx+∂ f1∂y

dy∧dx+∂ f2∂x

dx∧dy+∂ f2∂y

dy∧dy

=∂ f1∂y

dy∧dx+∂ f2∂x

dx∧dy=(∂ f2

∂x− ∂ f1

∂y

)dx∧dy.

Finally, for a 2-formω2 = g(x,y)dx∧dy thed-operation gives

dω2 =∂g∂x

dx∧dx∧dy+∂g∂y

dy∧dx∧dy= 0.

The latter shows thatd applied to a top-form always gives 0. I

J 17.2Example.In the previous exampledω0 is a 1-form, anddω1 is a 2-form.Let us now apply thed operation to these forms:

d(dω0) =( ∂2 f

∂x∂y− ∂2 f

∂y∂x

)dx∧dy= 0,

and

d(dω1) = 0,

sinced acting on a 2-form always gives 0. These calculations seem to suggest thatin generaldd = 0. I

As our examples indicated2ω = 0. One can also have formsω for whichdω = 0,but ω 6= dσ. We say thatω is aclosed form, and whenω = dσ, thenω is calledanexact 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. OnRm one will not be able to find closed forms that arenot exact, i.e. onRm all closed forms are exact. However, if we consider differentmanifolds we can have examples where this is not the case.J 17.3Example.Let M = R2\(0,0), and consider the 1-form

ω =−y

x2 +y2dx+x

x2 +y2dy,

which clearly is a smooth 1-form onM. Notice thatω does not extend to a 1-formonR2! It holds thatdω = 0, and thusω is a closed form onM. Let γ : [0,2π)→M,t 7→ (cos(t),sin(t)) be an embedding ofS1 into M, thenZ

γω =

Z 2π

0γ∗ω =

Z 2π

0dt = 2π.

Assume thatω is an exact 1-form onM, thenω = d f , for some smooth functionf : M → R. In Section 6 we have showed thatZ

γω =

Zγd f =

Z 2π

0γ∗d f = ( f γ)′(t)dt = f (1,0)− f (1,0) = 0,

94

which contradicts the fact thatR

γ ω =R

γ d f = 2π. This proves thatω is not exact.I

For the exterior derivative in general we have the following theorem.

Theorem 17.4. 36 Let M be a smooth m-dimensional manifold. Then for all k≥ 0there exist unique linear operations d: Γk(M)→ Γk+1(M) such that;

(i) for any 0-formω = f , f ∈C∞(M) it holds that

dω(X) = d f(X) = X f, X ∈ F(M);

(ii) if ω ∈ Γk(M) andη ∈ Γ`(M), then

d(ω∧η) = dω∧η+(−1)kω∧dη;

(iii) dd = 0;

(iv) if ω ∈ Γm(M), then dω = 0.

This operation d is called theexterior derivativeon differential forms, and is a

unique anti-derivation (of degree 1) onΓ(M) with d2 = 0.

Proof: Let us start by showing the existence ofd on a chartU ⊂ M. We havelocal coordinatesx = ϕ(p), p∈U , and we define

dU : Γk(U)→ Γk+1(U),

via (14). Let us writed instead ofdU for notational convenience. We have thatd( f dxI ) = d f ∧dxI . Due to the linearity ofd 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

)dxj ∧dxi

= ∑i< j

( ∂2 f∂xi∂x j

− ∂2 f∂x j∂xi

)dxi ∧dxj = 0.

36See Lee Thm. 12.14

95

Given ak-form ω = ωI dxI , then, using (ii), we have

d(dω) = d(

dωI ∧dxI)

= d(dωI )∧dxI +(−1)k+1dωI ∧d(dxI ) = 0,

sinceωI is a 0-form, andd(dxI ) = (−1) jdxi1 ∧ ·· · ∧d(dxi j )∧ ·· · ∧dxik = 0. Thelatter follows fromd(dxi j ) = 0. The latter also implies (iv), finishing the existenceproof in the one chart case. The operationd = dU is well-defined, satisfying (i)-(iv), for any chart(U,ϕ).

The operationdU is unique, for if there exists yet another exterior derivativedU ,which satisfies (i)-(iv), then forω = ωI dxI ,

dω = dωI ∧dxI +ωI d(dxI ),

where we used (ii). From (ii) it also follows thatd(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 thatdωI = dωI , and therefore

dω = dωI ∧dxI +ωI d(dxI ) = dωI ∧dxI = dω,

which proves the uniqueness ofdU .Before giving the definingd for ω ∈ Γk(M) we should point out thatdU trivially

satisfies (i)-(iii) of Theorem 17.5 (use (14). Since we have a unique operationdU

for every chartU , we define forp∈U

(dω)p = (dU ω|U)p,

as the exterior derivative for forms onM. If there is a second chartU ′ 3 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 thatd is also uniquely defined onΓk(M). Let p∈U , a

coordinate chart, and consider the restriction

ω|U = ωI dxI ,

whereωI ∈C∞(U). Let W ⊂U be an open set containingp, with the additionalproperty that cl(W) ⊂ U is compact. By Theorem 15.10 we can find a bump-functiong∈C∞(M) such thatg|W = 1, and supp(g)⊂U . Now define

ω = gωI d(gxi1)∧·· ·∧d(gxik).

Using (i) we have thatd(gxi)|W = dxi , and thereforeω|W = ω|W. Setη = ω−ω,thenη|W = 0. Let p∈W andh∈C∞(M) satisfyingh(p) = 1, and supp(h) ⊂W.

96

Thus,hω≡ 0 onM, 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 = d(gωI d(gxi1)∧·· ·∧d(gxik)

)= d(gωI )∧d(gxi1)∧·· ·∧d(gxik)+gωI d

(d(gxi1)∧·· ·∧d(gxik)

)= d(gωI )∧d(gxi1)∧·· ·∧d(gxik).

It now follows that sinceg|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 sincep is arbitrary it follows thatd : 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 17.5. Let M be a smooth m-dimensional manifold, and letω ∈ 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 achartU , andω = ωI dxI . Let us first computef ∗(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.

97

18. 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 forω. In a way the(m−1)-form

act as the ‘primitive’, ‘anti-derivative’ of thedω. Therefore if we are interested inRM σ using Stokes’ Theorem we need to first ‘integrate’σ, i.e. writeσ = dω. This

is not always possible as we saw in the previous section.Stokes’ Theorem can be now be phrased as follows.

Theorem 18.1.Let M be a smooth m-dimensional manifold with or without bound-

ary ∂M, andω ∈ Γm−1c (M). Then,

(15)Z

Mdω =

Z∂M

j∗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 18.2. 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)∩∂Hm 6= ∅,

where j: ∂Hm →Hm is the canonical inclusion.

Proof: We may assume without loss of generality that supp(ω) ⊂ [0,R]×·· ·×[0,R] = Im

R . Now,

ZHm

dω = (−1)i+1Z

ImR

∂ωi

∂xidx1 · · ·dxm

= (−1)i+1Z

Im−1R

(ωi |xi=R−ωi |xi=0

)dx1 · · · dxi · · ·dxm.

If supp(ω)⊂ int ImR , thenωi |xi=R−ωi |xi=0 = 0 for all i, and

ZHm

dω = 0.

98

If supp(ω)∩∂Hm 6= ∅, thenωi |xi=R−ωi |xi=0 = 0 for all i≤m−1, andωm|xm=R = 0.Therefore,Z

Hmdω = (−1)i+1

ZImR

∂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 mappingj : ∂Hm→Hm, which, in coordinates, is given byj = j ϕ : (x1, · · · ,xm−1) 7→(x1, · · · ,xm−1,0), whereϕ(x1, · · · ,xm−1,0) = (x1, · · · ,xm−1). The latter mapping isorientation preserving ifm is even and orientation reversing ifm is odd. Under themappingj we have that

j∗(ei) = ei , i = 1, · · · ,m−1,

where theei ’s on the left hand side are the unit vectors inRm−1, and the bold faceek are the unit vectors inRm. We have that the induced orientation for∂Hm isobtained by the rotatione1 →−em, em→ e1, and therefore

∂O = [e2, · · · ,em−1,e1].

Underϕ this corresponds with the orientation[e2, · · · ,em−1,e1] of Rm−1, which isindeed the standard orientation form even, and the opposite orientation form odd.The pullback form on∂Hm, 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 overHm we finally obtainZHm

dω =Z

∂Hmj∗ω,

which completes the proof.

99

Proof of Theorem 18.1:Let us start with the case that supp(ω) ⊂ U , where(U,ϕ) is an oriented chart forM. Then by the definition of the integral we havethat Z

Mdω =

Zϕ(U)

(ϕ−1)∗(dω) =Z

Hm(ϕ−1)∗(dω) =

ZHm

d((ϕ−1)∗ω

),

where the latter equality follows from Theorem 17.5. It follows from Lemma 18.2that if supp(ω)⊂ int (Hm), then the latter integral is zero and thus

RM ω. = 0. Also,

using Lemma 18.2, it follows that if supp(ω)∩∂Hm 6= ∅, thenZM

dω =Z

Hmd((ϕ−1)∗ω

)=

Z∂Hm

j ′∗(ϕ−1)∗ω

=Z

∂Hm(ϕ−1 j ′)∗ω =

Z∂M

j∗ω,

where j ′ : ∂Hm → Hm is the canonical inclusion as used Lemma 18.2, andj =ϕ−1 j ′ : ∂M →M is the inclusion of the boundary ofM into M.

Now consider the general case. As before we choose a finite coveringAI ⊂ A

of supp(ω) and an associated partition of unityλi subordinate toAI . Considerthe formsλiω, and using the first part we obtainZ

∂Mj∗ω = ∑

i

Z∂M

j∗λiω = ∑i

ZM

d(λiω)

= ∑i

ZM

(dλi ∧ω+λidω

)=

ZM

d(∑

i

λi

)∧ω+

ZM

(∑

i

λi

)dω

=Z

Mdω,

which proves the theorem.

J 18.3Remark.If M is a closed (compact, no boundary), oriented manifold mani-fold, then by Stokes’ Theorem for anyω ∈ Γm−1(M),Z

Mdω = 0,

since∂M = ∅. As a consequence volume formσ cannot be exact. Indeed,R

M σ > 0,and thus ifσ were exact, thenσ = dω, which implies that

RM σ =

RM dω = 0, a

contradiction. I

Let us start with the obvious examples of Stokes’ Theorem inR, R2, andR3.J 18.4Example.Let M = [a,b] ⊂ R, and consider the 0-formω = f , then, sinceM is compact,ω is compactly supported 0-form. We have thatdω = f ′(x)dx, andZ b

af ′(x)dx=

Za,b

f = f (b)− f (a),

100

which is the fundamental theorem of integration. Since onM any 1-form is exactthe theorem holds for 1-forms. I

J 18.5Example.Let M = γ⊂R2 be a curve parametrized given byγ : [a,b]→R2.Consider a 0-formω = f , thenZ

γfx(x,y)dx+ fy(x,y)dy=

Zγ(a),γ(b)

f = f (γ(b))− f (γ(a)).

Now let M = Ω ⊂ R2, a closed subset ofR2 with a smooth boundary∂Ω, andconsider a 1-formω = f (x,y)dx+ g(x,y)dy. Then,dω =

(gx− fy)dx∧ dy, and

Stokes’ Theorem yieldsZΩ

(∂g∂x− ∂ f

∂y

)dxdy=

Z∂Ω

f dx+gdy,

also known as Green’s Theorem inR2. I

J 18.6Example.In the case of a curveM = γ⊂ R3 we obtain the line-integral for0-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 asZγ∇ f ·ds= f |∂γ.

Next, let M = S⊂ R3 be an embedded or immersed hypersurface, and letω =f dx+gdy+hdzbe 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=

dx

dy

dz

, dS=

dydz

dzdx

dxdy

,

101

then from Stokes’ Theorem we can write the following surface and line integrals:ZS

∇×F ·dS=Z

∂SF ·ds,

which is usually referred to as the classical Stokes’ Theorem inR3. The version inTheorem 18.1 is the general Stokes’ Theorem. Finally letM = Ω a closed subsetof R3 with a smooth boundary∂Ω, and consider a a 2-formω = f dy∧dz+gdz∧dx+hdx∧dy. Then

dω =(∂ f

∂x+

∂g∂y

+∂h∂z

)dx∧dy∧dz.

Write

div F = ∇ ·F =∂ f∂x

+∂g∂y

+∂h∂z

, dV = dxdydz,

then from Stokes’ Theorem we obtainZΩ

∇ ·FdV =Z

∂ΩF ·dS,

which is referred to as the Gauss Divergence Theorem. I

102

V. De Rham cohomology

19. Definition of De Rham cohomology

In the previous chapters we introduced and integratedm-forms over manifoldsM. We recall thatk-form ω ∈ Γk(M) is closed ifdω = 0, and ak-form ω ∈ Γk(M)is exact if there exists a(k−1)-form σ ∈ Γk−1(M) such thatω = dσ. Sinced2 = 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) 3 ω = dσ

= Im(d),

and in particular

Bk(M)⊂ Zk(M).

The setsZk andBk are real vector spaces, withBk a vector subspace ofZk. Thisleads to the following definition.

Definition 19.1. Let M be a smoothm-dimensional manifold then thede Rhamcohomology groupsare defined as

(16) HkdR(M) := Zk(M)/Bk(M), k = 0, · · ·m,

whereB0(M) := 0.

It is immediate from this definition thatZ0(M) are smooth functions onM thatare constant on each connected component ofM. Therefore, whenM is con-nected, thenH0

dR(M) ∼= R. SinceΓk(M) = 0, for k > m = dim M, we havethat Hk

dR(M) = 0 for all k > m. Fork < 0, we setHkdR(M) = 0.

J 19.2Remark.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. I

An equivalence class[ω] ∈HkdR(M) is called acohomology class, and two form

ω,ω′ ∈ Zk(M) arecohomologousif [ω] = [ω′]. This means in particular thatω andω′ differ by an exact form, i.e.

ω′ = ω+dσ.

Now let us consider a smooth mappingf : N → M, then we have that the pull-back f ∗ acts as follows:f ∗ : Γk(M)→ Γk(N). From Theorem 17.5 it follows that

103

d f ∗ = f ∗ d and thereforef ∗ 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 formsZk(M) get mapped toZk(N), and the exact formBk(M) get mapped toBk(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 inHk

dR(M) to classes inHkdR(N):

f ∗ : HkdR(M)→ Hk

dR(N),

Theorem 19.3.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: Sinceg∗ f ∗ = ( f g)∗ the proof follows immediately.

As a direct consequence of this theorem we obtain the invariance of de Rhamcohomology under diffeomorphisms.

Theorem 19.4. If f : N→M is a diffeomorphism, then HkdR(M)∼= HkdR(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 thusf ∗ is an isomorphism.

20. Homotopy invariance of cohomology

We will prove now that the de Rham cohomology of a smooth manifoldM iseven invariant under homeomorphisms. As a matter of fact we prove that the deRham cohomology is invariant under homotopies of manifolds.

104

Definition 20.1. Two smooth mappingsf ,g : N → M are said to behomotopicifthere exists a continuous mapH : 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 ahomotopyfrom/betweenf to/andg. Ifin additionH is smooth thenf andg are said to besmoothly homotopic, andH iscalled asmooth homotopy.

Using the notion of smooth homotopies we will prove the following crucialproperty of cohomology:

Theorem 20.2.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∗.

J 20.3Remark.It can be proved in fact that the above results holds for two homo-topic (smooth) mapsf andg. This is achieved by constructing a smooth homotopyfrom a homotopy between maps. I

Proof of Theorem 20.2:A maph : Γk(M)→ Γk−1(N) is called ahomotopy mapbetweenf ∗ andg∗ if

(17) dh(ω)+h(dω) = g∗ω− f ∗ω, ω ∈ Γk(M).

Now consider the embeddingit : N → N× I , and the trivial homotopy betweeni0 andi1 (just the identity map). Letω ∈ Γk(N× I), and define the mapping

h(ω) =Z 1

0i ∂

∂tωdt,

which is a map fromΓk(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 thati ∂∂t

ω = 0 and thereforedh(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) fori∗0 andi∗1, i.e.

dh(ω)+h(dω) = i∗1ω− i∗0ω.

105

In the second case we have

h(dω) = h(∂ωI

∂xidxi ∧dt∧dxI ′

)=

Z 1

0

∂ωI

∂dxii ∂

∂xi

(dxi ∧dt∧dxI ′)dt

= −(Z 1

0

∂ωI

∂xidt)

dxi ∧dxI ′ .

On the other hand

dh(ω) = d((Z 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 sincei∗1ω = i∗0ω = 0 in this case, this then completes the argument in bothcases, andh as defined above is a homotopy map betweeni∗0 andi∗1.

By assumption we have a smooth homotopyH : N× [0,1]→M bewteenf andg, with f = H i0, andg = H i1. Consider the compositionh = 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 thatω is closed then

g∗ω− f ∗ω = dh(H∗ω),

and thus

0 = [dh(H∗ω)] = [g∗ω− f ∗ω] = g∗[ω]− f ∗[ω],

which proves the theorem.

J 20.4Remark.Using the same ideas as for the Whitney embedding theorem onecan prove, using approximation by smooth maps, that Theorem 20.2 holds true forcontinuous homotopies between smooth maps. I

106

Definition 20.5. Two manifoldsN andM are said to behomotopy equivalent, ifthere exist smooth mapsf : N→M, andg : M → N such that

g f ∼= idN, f g∼= idM (homotopic maps).

We write N ∼ M., The mapsf andg arehomotopy equivalencesare each otherhomotopy inverses. If the homotopies involved are smooth we say thatN andM

smoothly homotopy equivalent.

J 20.6Example.Let N = S1, the standard circle inR2, andM = R2\(0,0). Wehave thatN∼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 viaH(p, t) = t p+(1− t)p/|p|. I

Theorem 20.7.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 respectively

are isomorphisms.

As before this theorem remains valid for continuous homotopy equivalences ofmanifolds.

107

VI. Exercises

A number of the exercises given here are taken from the Lecture notes by J. Bochnak.

Manifolds

Topological Manifolds

1 Given the functiong : R→ R2, g(t) = (cos(t),sin(t)). Show thatf (R) is a mani-fold.

2 Given the setT2 = p = (p1, p2, p3) | 16(p21 + p2

2) = (p21 + p2

2 + p23 + 3)2 ⊂ R3,

called the2-torus.

(i ) Consider the product manifoldS1×S1 = q= (q1,q2,q3,q4) | q21+q2

2 = 1, q23+

q24 = 1, and the mappingf : S1×S1 → T2, given by

f (q) =(q1(2+q3),q2(2+q3),q4

).

Show thatf is onto andf−1(p) =( p1

r , p2r , r−2, p3

), wherer = |p|2+3

4 .

(ii ) Show thatf is a homeomorphism betweenS1×S1 andT2.

3 (i ) Find an atlas forT2 using the mappingf in 2.

(ii ) Give a parametrization forT2.

4 Show that

A4,4 = (p1, p2) ∈ R2 | p41 + p4

2 = 1,

is a manifold andA4,4∼= S1 (homeomorphic).

5 (i ) Show that an open subsetU ⊂M of a manifoldM is again a manifold.

(ii ) Let N andM be manifolds. Show that their cartesian productionN×M is alsoa manifold.

108

6 Show that

(i ) A∈M2,2(R) | det(A) = 1 is a manifold.

(ii ) Gl(n,R) = A∈Mn,n(R) | det(A) 6= 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 setU ′ ⊂ Rn can be replaced by an open discDn ⊂ Rn.

9 Show thatPRn is a Hausdorff space and is compact.

10 Define the Grassmann manifoldGkRn as the set of allk-dimensional linear sub-spaces inRn. Show thatGkRn is a manifold.

11 ConsiderX to be the parallel linesR×0∪R×1. Define the equivalence relation(x,0)∼ (x,1) for all x 6= 0. Show thatM = X/∼ is a topological space that satisfies(ii) and (iii) of Definition 1.1.

12 Let M be an uncountable union of copies ofR. Show thatM is a topological spacethat satisfies (i) and (ii) of Definition 1.1.

13 Let M = (x,y) ∈ R2 : xy= 0. Show thatM 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 thatPRn is a smooth manifold.

16 Prove thatPR is diffeomorphic to the standard unit circle inR2 by constructing adiffeomorphism.

17 Which of the atlases forS1 given in Example 2, Sect. 1, are compatible. If notcompatible, are they diffeomorphic?

18 Show that the standard torusT2 ⊂ R3 is diffeomorphic toS1×S1, whereS1 ⊂ R2

is the standard circle inR2.

19 Prove that the taking the union ofC∞-atlases defines an equivalence relation (Hint:use Figure 7 in Section 2).

109

20 Show that diffeomorphic defines an equivalence relation.

21 Prove Theorem 2.15.

Immersion, submersion and embeddings

22 Show that the torus the mapf defined in Exer. 2 of 1.1 yields a smooth embeddingof the torusT2 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 inR3, and is diffeomorphic toS2 when thesenumbers are even.

24 Prove Lemma 3.18.

25 Let f : Rn+1\0 → Rk+1\0 be a smooth mapping such thatf (λx) = λd f (x),d ∈ N, for all λ ∈ R\0. This is called a homogeneous mapping of degreed.Show thatf : PRn → PRk, defined byf ([x]) = [ f (x)], is a smooth mapping.

26 Show that the open ballBn ⊂ Rn is diffeomorphic toRn.

27 (Lee) Consider the mappingf (x,y,s) = (x2 + y,x2 + y2 + s2 + y) from R3 to R2.Show thatq = (0,1) is a regular value, andf−1(q) is diffeomorphic toS1 (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, andf : N → M is asmooth mapping. Show (using the Inverse Function Theorem) that iff is a bijec-tion then it is a diffeomorphism.

31 Prove Theorem 3.28.

32 Show that the mapf : Sn → PRn defined by f (x1, · · · ,xn+1) = [(x1, · · · ,xn+1)] issmooth and everywhere of rankn (see Boothby).

33 (Lee) Leta∈ R, andMa = (x,y) ∈ R2 : y2 = x(x−1)(x−a).

(i ) For which values ofa is Ma an embedded submanifold ofR2?

(ii ) For which values ofa canMa be given a topology and smooth structure so thatMa is an immersed submanifold?

110

34 Consider the mapf : R2→R given by f (x1,x2) = x31+x1x2+x3

2. Which level setsof f are smooth embedded submanifolds ofR2.

35 Let M be a smoothm-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 thati : ∂M → M is a (smooth)embedding.

36 Let f : N →M be a smooth mapping. LetS= f−1(q)⊂ N is a smooth embeddedsubmanifold of codimensionm. Is q necessarily a regular value off ?

37 Use Theorem 3.27 to show that the 2-torusT2 is a smooth embedded manifold inR4.

38 Prove Theorem 3.37.

Manifolds in Euclidean space

39 Show that anm-dimensional linear subspace inR` is anm-dimensional manifold.

40 Prove that the annulusA= (p1, p2)∈R2 : 1≤ p21+ p2

2≤ 4 is a smooth manifoldtwo dimensional manifold inR2 with boundary.

111

Tangent and cotangent spaces

Tangent spaces

41 Let U ⊂ Rk, y∈ Rm, and f : U → Rm a smooth function. By Theorem 3.28M =f−1(y) = p∈Rk : f (p) = y is a smooth embedded manifold inRk 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 toM at p.

42 Given the setM = (p1, p2, p3) ∈ R3 : p31 + p3

2 + p33−3p1p2p3 = 1.

(i ) Show thatM is smooth embedded manifold inR3 of dimension 2.

(ii ) ComputeTpM, at p = (0,0,1).

43 Given the setM = (p1, p2, p3) ∈ R3 : p21− p2

2 + p1p3−2p2p3 = 0, 2p1− p2 +p3 = 3.

(i ) Show thatM is smooth embedded manifold inR3 of dimension 1.

(ii ) ComputeTpM, at p = (1,−1,0).

44 Prove Lemma 5.5.

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 onS2 that vanishes at exactly one point.

47 Let S2 ⊂ R2 be the standard unit sphere andf : S2 → S2 a smooth map defined asa θ-degree rotation around the polar axis. Computef∗ : TpS2 → Tf (p)S

2 and give amatrix representation off∗ in terms of a canonical basis.

112

Cotangent spaces

48 (Lee) Let f : R3 → R be given byf (p1, p2, p3) = |p|2, andg : R2 → R3 by

g(x1,x2) =( 2x1

|x|2 +1,

2x2

|x|2 +1,|x|2−1|x|2 +1

).

Compute bothg∗d f , andd( f g) and compare the two answers.

49 (Lee) Computed f in coordinates, and determine the set points whered f vanishes.

(i ) M = (p1, p2) ∈ R2 : p2 > 0, and f (p) = p1|p|2 — standard coordinates inR2.

(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 inRn.

50 Express in polar coordinates

(i ) θ = x2dx+(x+y)dy;

(ii ) θ = xdx+ydy+zdz.

51 Prove Lemma 6.3.

52 (Lee) Let f : N → M (smooth),ω ∈ Λ1(N), andγ : [a,b] → N is a smooth curve.Show that Z

γf ∗ω =

Zfγ

ω.

53 Given the following 1-forms onR3:

α =− 4z(x2 +1)2dx+

2yy2 +1

dy+2x

x2 +1dz,

ω =− 4xz(x2 +1)2dx+

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-formsα andω is exact.

(iv) Compare the answers.

113

Vector bundles

54 Let M ⊂R` be a manifold inR`. Show thatTM as defined in Section 8 is a smoothembedded submanifold ofR2`.

55 Let S1 ⊂ R2 be the standard unit circle. Show thatTS1 is diffeomorphic toS1×R.

56 Let M⊂R` be a manifold inR`. Show thatT∗M as defined in Section 8 is a smoothembedded submanifold ofR2`.

57 Show that the open Mbius band is a smooth rank-1 vector bundle overS1.

114

Tensors

Tensors and tensor products

58 Describe the standard inner product onRn as a covariant 2-tensor.

59 Construct the determinant onR2 andR3 as covariant tensors.

60 Given the vectorsa =

130

, andb =

(24

). Computea∗⊗b andb∗⊗a.

61 Given finite dimensional vector spacesV andW, prove thatV⊗W andW⊗V areisomorphic.

62 Given finite dimensional vector spacesU , V andW, prove that(U ⊗V)⊗W andU⊗ (V⊗W) are isomorphic.

63 Show thatV⊗R'V ' R⊗V.

Symmetric and alternating tensors

64 Show that forT ∈ Ts(V) the tensor SymT is symmetric.

65 Prove that a tensorT ∈ Ts(V) is symmetric if and only ifT = SymT.

66 Show that the algebra

Σ∗(V) =∞M

k=0

Σk(V),

is a commutative algebra.

67 Prove that for anyT ∈ Ts(V) the tensor AltT is alternating.

68 Show that a tensorT ∈ Ts(V) is alternating if and only ifT = Alt T.

69 Given the vectorsa =

124

, b =

013

, andc =

112

. Computea∧b∧c,

and compare this with det(a,b,c).

70 Given vectorsa1, · · · ,an show thata1∧·· ·∧an = det(a1, · · · ,an).

115

71 Prove Lemma 10.11.

Tensor bundles and tensor fields

72 Let M ⊂R` be an embeddedm-dimensional manifold. Show thatTrM is a smoothmanifold inR`+`r

.

73 Similarly, show thatTsM is a smooth manifold inR`+`s, andTr

s M is a smoothmanifold inR`+`r+`s

.

74 Prove that the tensor bundles introduced in Section 9 are smooth manifolds.

75 One can consider symmetric tensors inTpM as defined in Section 10. Define anddescribeΣr(TpM)⊂ Tr(TpM) andΣrM ⊂ TrM.

76 Describe a smooth covariant 2-tensor field inΣ2M ⊂ T2M. How does this relate toan (indefinite) inner product onTpM?

77 Prove Lemma 11.2.

78 Given the manifoldsN = M = R2, and the smooth mapping

q = f (p) = (p21− p2, p1 +2p3

2),

acting fromN to M. Consider the tensor spacesT2(TpN) ∼= T2(TqM) ∼= T2(R2)and compute the matrix for the pullbackf ∗.

79 In the above problem consider the 2-tensor fieldσ onM, given by

σ = dy1⊗dy2 +q1dy2⊗dy2.

(i ) Show thatσ ∈ F2(M).

(ii ) Compute the pulbackf ∗σ and show thatf ∗σ ∈ F2(N).

(iii ) Computef ∗σ(X,Y), whereX,Y ∈ F(N).

116

Differential forms

80 Given the differential formσ = dx1∧dx2−dx2∧dx3 on R3. For each of the fol-lowing vector fieldsX computeiXσ:

(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 fromR2 to R2 and the 2-formσ = p21dx1∧dx2. Compute the pullback form

f ∗σ.

82 Prove Lemma 12.2.

83 Derive Formula (10).

Orientations

84 Show thatSn is orientable and give an orientation.

85 Show that the standardn-torusTn is orientable and find an orientation.

86 Prove that the Klein bottle and the projective spacePR2 are non-orientable.

87 Give an orientation for the projective spacePR3.

88 Prove that the projective spacesPRn are orientable forn = 2k+1, k≥ 0.

89 Show that the projective spacesPRn are non-orientable forn = 2k, k≥ 1.

117

Integration on manifolds

Integrating m-forms on Rm

90 Let U ⊂ Rm be open and letK ⊂U be compact. Show that there exists a domainof integrationD such thatK ⊂ D⊂ Rm.

91 Show that Definition 14.2 is well-posed.

Partitions of unity

92 Show that the functionf1 defined in Lemma 15.3 is smooth.

93 If U is open covering ofM for which each setUi intersects only finitely many othersets inU, prove thatU is locally finite.

94 Give an example of uncountable open coveringU of the interval[0,1] ⊂ R, and acountable refinementV of U.

Integration of m-forms on m-dimensional manifolds

95 Let S2 = ∂B3 ⊂ R3 oriented via the standard orientation ofR3, and consider the2-form

ω = xdy∧dz+ydz∧dx+zdx∧dy.

Given the parametrization

F(φ,θ) =

x

y

z

=

sin(φ)cos(θ)sin(φ)sin(θ)

cos(φ)

,

for S2, computeR

S2 ω.

96 Given the 2-formω = xdy∧dz+zdy∧dx, show thatZS2

ω = 0.

118

97 Consider the circleS1 ⊂ R2 parametrized by

F(θ) =

(x

y

)=

(r cos(θ)r sin(θ)

).

Compute the integral over S1 of the 1-formω = 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 ofS1.

99 Consider the embedded torus

T2 = (x1,x2,x3,x4) ∈ R4 : x21 +x2

2 = 1, x23 +x2

4 = 1.

Compute the integral overT2 of the 2-form

ω = x21dx1∧dx4 +x2dx3∧dx1.

100 Consider the following 3-manifoldM parametrized byg : [0,1]3 → R4, r

s

t

7→

r

s

t

(2r− t)2

.

Compute ZM

x2dx2∧dx3∧dx4 +2x1x3dx1∧dx2∧dx3.

The exterior derivative

101 Let (x,y,z) ∈ R3. Compute the exterior derivativedω, whereω 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.

119

102 Which of the following forms onR3 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 formsω onR2 are exact, and if so writeω = 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 formsω onR3 are exact, and if so writeω = dσ:

(i ) ω = xdx+ydy+zdz;

(ii ) ω = x2dx∧dy+z3dx∧dz;

(iii ) ω = x2ydx∧dy∧dz.

105 Verify that onR2 andR3 all closedk-forms,k≥ 1, are exact.

106 Find a 2-form onR3\(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 ofΩ is given by

Vol(M) =13

Z∂Ω

xdy∧dz−ydx∧dz+zdx∧dy.

108 Let Ω⊂Rn be ann-dimensional domain. Prove the analogue of the previous prob-lem.

109 Prove the ‘integration by parts’ formulaZM

f dω =Z

∂Mf ω−

ZM

d f ∧ω,

where f is a smooth function andω ak-form.

110 Compute the integralZS2

x2ydx∧dz+x3dy∧dz+(z−2x2)dx∧dy,

120

whereS2 ⊂ R3 is the standard 2-sphere.

111 Use the standard polar coordinates for theS2 ⊂ R3 with radiusr to computeZS2

xdy∧dz−ydx∧dz+zdx∧dy

(x2 +y2 +z2)3/2,

and use the answer to compute the volume of ther-ball Br .

Extra’s

112 Use the examples in Section 18 to show that

curl grad f = 0, and div curlF = 0,

where f : R3 → R, andF : R3 → R3.

113 Given the mappingf : R2 → R3, ω = dx∧dz, and

f (s, t) =

cos(s)sin(t)√s2 + t2

st

.

Computef ∗ω.

114 Given the mappingf : R3 → R3, ω = xydx∧dy∧dz, and

f (s, t,u) =

scos(t)ssin(t)

u

.

Computef ∗ω.

115 Let M = R3\(0,0,0), and define the following 2-form onM:

ω =1

(x2 +y2 +z2)3/2

(xdy∧dz+ydz∧dx+zdx∧dy

).

(i ) Show thatω is closed (dω = 0) (Hint: computeR 2

S ω).

(ii ) Prove thatω is not exact onM!

OnN = R3\x = y = 0 ⊃M consider the 1-form:

η =−z

(x2 +y2 +z2)1/2

xdy−ydxx2 +y2 .

(iii ) Show thatω is exact as 2-form onN, and verify thatdη = ω.

121

116 Let M ⊂ R4 be given by the parametrization

(s, t,u) 7→

s

t +u

t

s−u

, (s, t,u) ∈ [0,1]3.

(i ) ComputeR

M dx1∧dx2∧dx4.

(ii ) ComputeR

Mx1x3dx1∧dx2∧dx3 +x23dx2∧dx3∧dx4.

117 Let C = ∂∆ be the boundary of the triangleOAB in R2, whereO = (0,0), A =(π/2,0), andB = (π/2,1). Orient the traingle by traversing the boundary counterclock wise. Given the integralZ

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, whereΣ = ∂[0,1]3, and

F(x,y,z) =

4xz

−y2

yz

(Hint: use the Gauss Divergence Theorem).

122

De Rham cohomology

Definition of De Rham cohomology

119 Prove Theorem 19.3.

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 thecup-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 forH1dR(R2\(0,0)).

125 Compute the de Rham cohomology of then-torusM = Tn.

126 Compute the de Rham cohomology ofM = S2.