differential geometry michaelmas 2006 dr. g.p. paternain∗

Differential GeometryMichaelmas 2006Dr. G.P. Paternain∗

Chris Almost†

See Dr. Paternain’s website for the room and time of the next examples class(at the beginning of Lent term).

Contents

Contents 1

1 Manifolds 21.1 Definition and First Examples . . . . . . . . . . . . . . . . . . . . . . . 21.2 Tangent Space and Differentials . . . . . . . . . . . . . . . . . . . . . . 51.3 Tangent Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Forms and Bundles 132.1 Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2 Orientability and Integration . . . . . . . . . . . . . . . . . . . . . . . . 162.3 Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.4 Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.5 Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.6 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Riemannian Metrics 333.1 Metric Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.2 Levi-Civita connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.3 Curvature revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.4 Sectional, Ricci, and Scalar curvature . . . . . . . . . . . . . . . . . . 373.5 Laplace(-Bertrami) operator . . . . . . . . . . . . . . . . . . . . . . . . 373.6 Yang-Mills Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

Index 47∗[email protected]†[email protected]

1

2 Differential Geometry

Notes: (available on the internet)

(i) Alexei Kovalev (followed most closely)

(ii) Mihalis Dafermos (these have an emphasis on calculus of variations)

Book:

(i) J. Jost, Riemannian Geometry & Geometric Analysis, Springer, Universitext(chapters 1, 2, 3)

1 Manifolds

1.1 Definition and First Examples

1.1.1 Definition. A topological space M is a set with a specified class of open sets,such that

(i) ∅ and M are open;

(ii) the intersection of two open sets is open;

(iii) an arbitrary union of open sets is open.

M is Haudorff if given p1, p2 ∈ M there are open sets Ui (i = 1, 2) such that pi ∈ Uiand U1 ∩ U2 = ∅. M is second countable if one can find a coutable collection Bof open sets of M such that any open set U ⊆ M can be written as a union ofelements inB .

1.1.2 Definition. Recall that a continuous map is a map such that the preimageof every open set is open. A homeomorphism is a continuous bijection with acontinuous inverse.

1.1.3 Definition. A homeomorphism ϕ : U → V , where U ⊆ M is open andV ⊆ Rd is open is called a chart and U the coordinate neighbourhood.

Insert standard chart picture here with change of local coordinates.Suppose you have charts ϕ : U → V and ψ : U ′ → V ′. Then ϕ ψ−1 is a map

between subsets of Rd .

1.1.4 Definition. A C∞ differentiable structure (or a smooth structure) on M is acollection of coordinate charts ϕα : Uα→ Vα ⊆ Rd (same d for all α) such that

(i)⋃

α Uα = M ;

(ii) any two charts are compatible, namely for all α,β the change of local co-ordinates ϕβ ϕ−1

α is C∞ on its domain ϕα(Uβ ∩ Uα) (this is equivalent torequiring that it has continuous partial derivatives of all orders);

Definition and First Examples 3

(iii) the collection of charts (Uα,ϕα) is maximal with respect to property (ii),namely if a chart (U ,ϕ) is compatible with (Uα,ϕα) for all α then (U ,ϕ) isincluded in the collection.

In this case d is the dimension of M , denoted dim M .

Remark.(i) The change of local coordinates is a diffeomorphism, a C∞ map with C∞

inverse.

(ii) We only really need to worry about (i) and (ii), since there is a unique wayof extending a collection of charts satisfying (i) and (ii) to a maximal one:just add all the compatible charts (proof is a Zorn’s Lemma arugment).

(iii) If we start just with bijective charts, we can mount a topology on M bydefining D ⊆ M to be open if and only if ϕ(D ∩ U) is open in Rd for everychart ϕ : U → V ⊆ Rd . This is the topology induced by the C∞ structure.

1.1.5 Examples.(i) Rd is a manifold, as is any open subset of Rd .

(ii) Sn = (x0, . . . , xn) ∈ Rn+1 |∑n

i=0 x2i = 1, with smooth structure given by

stereographic projection from the north and south poles. This is the canoni-cal smooth structure on Sn.

(iii) Real projective space RPn = (Rn+1 \ 0)/R∗, with smooth structure givenby the charts (0≤ i ≤ n)

ϕi : Ui = [x0 : · · · : xn] | x i 6= 0 → Rn

: [x0 : · · · : xn] 7→

x1

x i, . . . , i, . . . ,

xn

x i

omitting the ith component. It is clear that these charts are compatible.Check that RPn is a compact manifold. (Hint: there is a continuous projec-tion Sn→ RPn.)

In the 50’s J. Milnor showed that S7 has many different smooth structures.He won a Field’s Medal for his work. In the 80’s an even more startling fact wasdiscovered: R4 has uncountably many smooth structures! The Smooth PoincaréConjecture asks whether S4 has an exoctic (non-canonical) smooth structure.

1.1.6 Definition. Let M and N be smooth manifolds and f : M → N be continu-ous. We say that f is a smooth map if for all p ∈ M and charts (U ,ϕ) containing pand (V,ψ) containing f (p) the map ψ f ϕ−1 is C∞ on its domain of definition.

1.1.7 Definition. A continuous map f : M → N is a diffeomorphism if f is asmooth bijection with smooth inverse. In this case we say that M and N arediffeomorphic


1.1.8 Exercise. Show that if M and N are manifolds then M ×N is a manifold ina canonical way.

1.1.9 Example (Lie groups). A Lie group is a group G which is also a smoothmanifold such that the map G× G→ G : (σ,τ) 7→ στ−1 is a smooth map.

(i) GL(n,R) is a Lie group since it embeds naturally in Rn2as an open set and

matrix multiplication is given by polynomials in the coefficients.

(ii) O(n) = A∈ GL(n,R) | AAT = I is a Lie group, as we shall see below.

1.1.10 Exponential and Logarithm of Matrices. Let A be an n× n real matrix.The exponential of A is defined to be

exp(A) = I + A+1

2A2 +

1

6A3 + · · ·+

1

n!An + · · · .

It can be shown, using the Weierstrass M -test, that this series is uniformly con-vergent on compact subsets of M(n,R). In fact the map A 7→ exp(A) is C∞. Theexponential map has the following properties.

(i) exp(AT ) = (exp(A))T ;

(ii) exp(C−1AC) = C−1 exp(A)C;

(iii) Warning: it does not hold in general that exp(A+ B) = exp(A)exp(B), but itdoes hold if A and B commute. In particular, exp(A)exp(−A) = I .

Similarily, we define the logarithm of a matrix A to be

log(A) = A−1

2A2 +

1

3A3 + · · ·+

(−1)n+1

nAn + · · · .

This series also converges uniformly on compact sets, provided that |A− I | < 1,and A 7→ log(A) is C∞ for all A with |A− I |< 1. We have

(i) exp(log(A)) = A if |A− I |< 1;

(ii) log(exp(A)) = A if |A|< log 2.

Take A such that |A− I | < 1. If A ∈ O(n) then log(A) ∈ S, the set of skew-symmetric matrices. Indeed, AA= I implies that

exp(log(A))exp((log(A))T ) = I

by (i), soexp((log(A))T ) = exp(− log(A)).

But | log(A)| ≤ log |A|< 2 since |A− I |< 1, so by (ii)

(log(A))T =− log(A).

Tangent Space and Differentials 5

The dimension of the space of skew-symmetric matrices is 12n(n− 1). Consider

ϕ : U → S ∼= R12

n(n−1) : A 7→ log(A),

where U = A∈ O(n) | |A− I |< 1. Then ϕ is a chart around the identity. For anyC ∈ O(n), let ϕC(A) = log(C−1A) for A in a small open set around C . It remainsto check that change of coordinates is a smooth map, but this follows from thedefinition of Lie group (in particular that multiplication is smooth).

1.2 Tangent Space and Differentials

The tangent space TpM

For each point p ∈ M we would like to define the space of tangent vectors to Mat p, which we will denote Tp M . It should be a vector space of dimension dim M ,and whenever f : M → N is a smooth map, there should be an associated linearmap d fp : Tp M → T f (p)N that satisfies the chain rule, which will be called thedifferential of f

Intuitively, any tangent vector to a curve α : I → M , where 0 ∈ I ⊆ R isan open interval and α(0) = p, is a tangent vector to M at p, and any tangentvector should arise in this way. Let us first consider the simplest case, where themanifold under consideration is an open set U ⊆ Rn. Let p ∈ U and considerall smooth curves through p (i.e. all smooth curves α : I → U , where 0 ∈ I ⊆ Ris an open interval and α(0) = p). We say that two such curves α1 and α2 areequivalent if α′1(0) = α

′2(0), since all we really care about is the tangent vector at

p. Let TpU be the set of equivalence classes. There is a natural bijection betweenTpU → Rn given by [α] 7→ α′(0), since each equivalence class is uniquely definedby its tangent vector at p. If f : U → Rm is a smooth map then, from multivariablecalculus,

d

d t

t=0f (α(t)) = d fp(α

′(0)),

where d fp = (∂ fi

∂ x j)i j is the usual differential. It follows that d fp([α]) = [ f α].

Now let M be a manifold and p ∈ M . Fix a chart (U ,ϕ) around p. If α1 andα2 are two curves through p then we say that α1 is equivalent to α2 if

(ϕ α1)′(0) = (ϕ α2)

′(0).

We need to check that this definition does not depend on the chart chosen. Let(V,ψ) be another chart around p and let h= ψ ϕ−1 : ϕ(U ∩ V )→ ψ(U ∩ V ) bethe change of coordinates. If α1 and α2 are equivalent with respect to ϕ, then byapplying dhϕ(p) we get

dhϕ(p)((ϕ α1)′(0)) = dhϕ(p)((ϕ α2)

′(0))

(h ϕ α1)′(0) = (h ϕ α2)

′(0)(ψ α1)

′(0) = (ψ α2)′(0)

so they are equivalent with respect to ψ as well.


1.2.1 Definition. Tp M , the tangent space to M at p is the set of all equivalenceclasses of smooth curves α : I → M , where 0 ∈ I ⊆ R is an open interval andα(0) = p. Two curves α1 and α2 through p are equivalent if (ϕ α1)′(0) =(ϕ α2)′(0) for some chart (U ,ϕ) containing p.

There is a linear structure on Tp M induced by the map Φϕ : Tp M → Rn : [α]→(ϕ α)′(0) (i.e. there is a unique linear structure on Tp M such that Φϕ is a linearisomorphism). Again, this linear structure is defined in terms of a particular chart,so again we need to check that it is well-defined. As above, we apply dhϕ(p).

Tp MΦϕ

zzzz

zzzz Φψ

!!DDDD

DDDD

Rndhϕ(p)

∼= // Rn

Since h is a change of coordinates, dhϕ(p) is a linear isomorphism, so the linearstructures are isomorphic.

The differential d fp

Now we move to the definition of the differential of a smooth map. We take as thedefinition the result noted at the end of the discussion from the beginning of thesection.

1.2.2 Definition. Let f : M → N be a smooth map between manifolds. Thedifferential of f at p is

d fp : Tp M → T f (p)N : [α] 7→ [ f α].

First we check that this defines a linear map. Let (U ,ϕ) be a chart containingp and (V,ψ) be a chart containing f (p). Then we have the following commutativediagram.

Tp Md fp //

Φϕ ∼=

T f (p)N

Φψ∼=

Rnd(ψ f ϕ−1)ϕ(p)

// Rm

Notice that Φϕ = dϕp with this definition of d fp, so we will no longer use thenotation Φϕ.

As in multvariable calculus, we have a Chain Rule for compositions of smoothmaps.

1.2.3 Theorem (Chain Rule). Let Mf−→ N

g−→ P be smooth maps between mani-

folds, so that there are linear maps

Tp Md fp−→ T f (p)N

d g f (p)−−−→ Tg( f (p))P.

Tangent Space and Differentials 7

Then d(g f )p = d g f (p) d fp.

PROOF: Exercise.

Expressions in local coordinates

Let (U ,ϕ) be a chart around p ∈ M . Let e1, . . . , en be the canonical basis of Rn.Pulling back this basis through the linear isomorphism dϕp : Tp M → Rn gives a“canonical” basis for Tp M (which depends on the chart chosen).

1.2.4 Definition. The canonical basis for Tp M with respect to a fixed chart (U ,ϕ)containing p is ∂

∂ x1 , . . . , ∂

∂ xn , where ∂

∂ x i := (dϕp)−1(ei).

Notation. In class expressions of the form “x = yi” will be written, by which it ismeant “x =

∑ni=1 yi .” Namely, whenever a lonely index occurs on the righthand

side of an equality there is an implicit sum from 1 to the dimension of the space.

How does the canonical basis change when a different chart is used? Letv ∈ Tp M and suppose v = ai

∂

∂ x i . Let (V,ψ) be another chart containing p andh=ψ ϕ−1 be the change of coordinates. Suppose that h changes x i coordinatesinto x ′i coordinates via

x ′j = x ′j(x1, . . . , xn) = h(x1, . . . , xn),

and v = a′i∂

∂ x ′i.

Since ∂∂ x1 , . . . , ∂

∂ xn and ∂

∂ x ′1, . . . , ∂

∂ x ′n are both bases of Tp M , there are (bi j)

such that

(dϕp)−1(ei) =

∂

∂ x i = bi j∂

∂ x ′ j= bi j(dψp)

−1(e j).

Apply dψp to both sides and the Chain Rule to see that dhϕ(p)(ei) = bi je j , so

(bi j) = dhϕ(p) = (∂ x ′j∂ x i)i j .

In particular, ∂

∂ x i =∂ x ′j∂ x i

∂

∂ x ′ j, and a′i =

∂ x ′i∂ xk

ak. Reiterating, the change of basismatrix changing from the canonical basis induced by ϕ to the canonical basisinduced by ψ is the Jacobian at ϕ(p) of the change of coordinate map ψ ϕ−1.

Remark. C∞(M , N) denotes the space of all C∞ maps from M to N . Supposef ∈ C∞(M ,R). A element v ∈ Tp M induces a linear map v : C∞(M ,R)→ R : f 7→d fp(v). But more than that, v satisfies the Leibniz rule

v( f g) = f (p)v(g) + v( f )g(p),

so v is a so-called derivation. The tangent space can be defined to be the collectionof derivations on C∞(M ,R). We may also write

v( f ) = ai∂

∂ x i f ϕ−1.


1.3 Tangent Bundles

1.3.1 Definition. T M , the tangent bundle, is the disjoint union of all the tangentspaces, qp∈M Tp M . Let π : T M → M be the canonical projection. π−1(p) is thefibre at p. If (U ,ϕ) is a chart around p, define the bundle chart

ϕT : π−1(U)→ ϕ(U)×Rn : (p, v) 7→ (ϕ(p), dϕp(v)) = (ϕ(p), (a1, . . . , an)).

Check that the collection of bundle charts induces a C∞ structure on T M . Thetangent bundle is a prime example of a vector bundle.

1.3.2 Definition. Similarily, T ∗M , the cotangent bundle, is the disjoint union ofthe duals of all the tangent spaces.

Describe the charts on T ∗M . With the proper charts, the cotangent bundle isalso a vector bundle. In classical mechanics, the tangent bundle corresponds to thespace of all positions and velocities, (q,~q), while the cotangent bundle correspondsto the space of all positions and momentums (q,~p). It is also know as the phasespace.

1.3.3 Definition. A vector field on M is a smooth map X : M → T M such thatπ X = id.

A vector field is a prime example of a section of a vector bundle. Supposethat (U ,ϕ) is a chart around p ∈ M . Then X (p) = (ai(p)

∂

∂ x i ) for some smoothfunctions ai : M → R. Vector fields have integral curves, which are curves γ : I →M such that γ(t) = X (γ(t)). (This follows from the existence and uniqueness ofODEs in Rn.) If M is compact then we may take I = R (i.e. there is a solution tothe ODE for all times t ∈ R). A flow is a one parameter family of diffeomorphismsφt : M → M (t ∈ R) such that φt+s = φt φs and φt(p) = γp(t), where γp isthe unique solution to γ(t) = X (γ(t)) around p. A flow is an action of R on M bydiffeomorphisms.

1.3.4 Definition. A smooth 1-form ω on T ∗M is a smooth map ω : M → T ∗Msuch that π ω= id. i.e. a smooth 1-form is a section of T ∗M .

1.3.5 Proposition. Let M be a smooth manifold of dimension n, and suppose thatthere are vector fields X1, . . . , Xn such that X1(p), . . . , Xn(p) is a basis of Tp M forall p ∈ M . Then T M is “isomorphic” to M ×Rn.

Here by “isomorphic” we mean diffeomorphic via a diffeomorphism with takesfibres to fibres in a linear fashion. Manifolds M satisfying 1.3.5 are called paral-lelizable in the literature of differential geometry.

PROOF: For any (p, v) ∈ T M , X1(p), . . . , Xn(p) is a basis of Tp M , so suppose vhas coordinates v = (aiX i(p)). Let

Φ : T M → M ×Rn : (p, v) 7→ (p, (a1, . . . , an)).

Tangent Bundles 9

Clearly Φ is a bijection and maps fibres to fibres in a linear fashion, so we needonly check that Φ and Φ−1 are smooth. Let (U ,ϕ) be a chart around a point p. Thisinduces a chart (π−1(U),ϕT ) on T M around (p, v) for any v ∈ Tp M . On M ×Rn

we take ϕ× id around (p, (a1, . . . , an)). Then for any ((x i), (b j)) ∈ ϕ(U)×Rn, letq = φ−1(x i), so

(ϕ× id) Φ ϕ−1T ((x i), (b j)) = (ϕ× id) Φ

q,

b j∂

∂ x j

.

Write X i =∑

X i j∂

∂ x j , where the X i j depend on q and are smooth since X is smooth.

Write b j∂

∂ x j as aiX i(q) with respect to the basis X1(q), . . . , Xn(q), we have b j =∑

aiX i j . Let (ci j) be the inverse matrix of (X i j), so that ai =∑

ci j b j . Then ci jdepends smoothly on q and

(ϕ× id) Φ

q,

b j∂

∂ x j

= (ϕ× id)(q, (ci j b j)) = ((x i), (ci j b j)).

Parallelizability is special! It can be shown that every even dimensional sphereS2n is not parallelizable. Lie groups are always parallelizable thanks to the left-regular representation.

Lie bracket of vector fields

Let V (M) denote the space of all smooth vector fields on M . For X , Y ∈ V (M),we will define the Lie bracket [X , Y ] ∈ V (M). But first, for p ∈ M consider thatX (p) ∈ Tp M , so we may regard X as an operator on C∞(M ,R), acting as

X ( f )(p) =ÕX (p)( f ) = d fp(X (p)).

We say that X is a first order linear operator, and in any fixed local coordinates(U ,ϕ) X acts as

X ( f )(p) = ai(p)∂

∂ x i ( f ϕ−1),

or X (p) = ai(p)∂

∂ x i .

1.3.6 Definition. Let X (p) = ai(p)∂

∂ x i and Y (p) = bi(p)∂

∂ x i be vector fields anddefine the Lie bracket of X and Y by its action on C∞(M ,R) as

[X , Y ]( f ) = X (Y ( f ))− Y (X ( f )).


Notice that since f is smooth the mixed partials are equal, and

[X , Y ]( f ) = X (Y ( f ))− Y (X ( f ))

= a j∂

∂ x j

bi∂

∂ x i ( f ϕ−1)

−

b j∂

∂ x j

ai∂

∂ x i ( f ϕ−1)

= a j∂ bi

∂ x j

∂

∂ x i ( f ϕ−1) + a j bi

∂ 2

∂ x j∂ x i( f ϕ−1)

−

b j∂ ai

∂ x j

∂

∂ x i ( f ϕ−1) + b jai

∂ 2

∂ x j∂ x i( f ϕ−1)

=

a j∂ bi

∂ x j− b j

∂ ai

∂ x j

∂

∂ x i ( f ϕ−1).

Therefore [X , Y ] is also a first order linear operator, and we are justified in writing

[X , Y ] = (a j∂ bi

∂ x j− b j

∂ ai

∂ x j)∂

∂ x i .

1.3.7 Properties of Lie bracket. For all vector fields X , Y, Z ∈ V (M),

(i) [·, ·] is bilinear;

(ii) [X , Y ] =−[Y, X ] (anti-commutative);

(iii) [[X , Y ], Z] + [[Y, Z], X ] + [[Z , X ], Y ] = 0 (Jacobi identity).

The vector space V (M) with [·, ·] is an example of a Lie algebra, (i.e. a vectorspace with a product satisfying (i), (ii), and (iii)).

Left invariant vector fields on Lie groups

1.3.8 Definition. Let G be a Lie group and g = TeG. For g ∈ G, the half-translation by g is the map

Lg : G→ G : h 7→ hg.

Half-translations are diffeomorphisms.

1.3.9 Proposition. Any Lie group is parallelizable.

PROOF: Notice that(d Lg)e : TeG = g→ Tg G,

so given ξ ∈ g we may define a vector field Xξ(g) = (d Lg)e(ξ). Then Xξ : G→ T Gis a smooth vector field. If ξ1, . . .ξn is a basis of g (where n = dim G) thenXξ1

, . . . , Xξn is a basis at every g ∈ G. By 1.3.5, G is parallelizable.

Submanifolds 11

Now consider

(d Lg)h(Xξ(h)) = (d Lg)h((d Lg)e(ξ)) = d(Lg Lh)e = d(Lgh)e(ξ) = Xξ(gh)

for h ∈ G, since Lg Lh(x) = Lg(hx) = ghx = Lgh(x). Therefore

(d Lg)h(Xξ(h)) = Xξ(Lg(h)).

1.3.10 Definition. A left-invariant vector field is a vector field satisfying. . . Let`(G) denote the set of all left-invariant vector fields. (they all arise as Xξ for someξ).

1.3.11 Theorem. (`(G), [·, ·]) is a Lie algebra for any Lie group G.

PROOF: We have g→ `(G) ,→ V (M) : ξ 7→ Xξ, so we will show that `(G) is a Liesubalgebra of V (M).

For ξ,η ∈ g, [Xξ, Xη] ∈ `(G). Indeed, let f ∈ C∞(G,R).

(d Lg)([Xξ, Xη])( f ) = [Xξ, Xη]( f Lg)

= Xξ(Xη( f Lg))− Xη(Xξ( f Lg))

= Xξ(d Lg(Xη)( f ))− Xη(d Lg(Xξ)( f ))

= Xξ(Xη( f ))− Xη(Xξ( f ))

= [Xξ, Xη]( f )

Recalling for F : M → N , p ∈ M , v ∈ Tp M , and f ∈ C∞(M ,R) we have

dFp(v)( f ) = d fF(p) dFp(v) = d( f F)(v) = v( f F)

1.4 Submanifolds

1.4.1 Definition. Let M and N be two manifolds. Suppose that N ⊂ M , and leti : N → M be the inclusion map. We say that N is an embedded submanifold if

(i) i is smooth;

(ii) dip : TpN → Ti(p)M is one-to-one for all p ∈ N (i.e. i is an immersion); and

(iii) i is a homoeomorphism onto its image (i.e. D ⊆ N is open in N if and onlyif D = N ∩ E for some open E ⊆ M).

1.4.2 Examples.(i) Take M = T 2, the 2-dimensional torus, and N = R, embedded as a line of

irrational slope. Then N is dense in M , so (iii) fails though (i) and (ii) hold.

(ii) Take M = R2 and N = R embedded in the shape of a “ρ”, but with an openend instead of a double point (insert a diagram). Again (iii) fails but (i) and(ii) hold.


We would like to decide when a set of equations define a submanifold.

1.4.3 Definition. Let f : M → N be a smooth map between two manifolds. Thenf is a submersion if d fp : TP M → T f (p)N is onto for all p ∈ M . It is an immersion ifd fp is one-to-one. A point q ∈ N is a regular value if d fp : Tp M → TqN is onto forevery p ∈ f −1(|[q) (i.e. f is a submersion for every p in the preimage of q).

1.4.4 Example. Let π : Rk → R` : (a1, . . . , ak) 7→ (a1, . . . , a`) be the canonicalprojection (where k ≥ `). Then π is a submersion, the canonical submersion.

1.4.5 Theorem (Inverse Function Theorem). If f : M → N be a smooth mapbetween manifolds then f is a local diffeomorphism at p if and only if d fp : Tp M →T f (p)N is a linear isomorphism.

( f is a local diffeomorphism at p if there is an open neighbourhood U of p suchthat f |U : U → f (U) is a diffeomorphism.)

PROOF: Use charts plus the inverse function theorem in Rn.

1.4.6 Theorem (Preimage Theorem). Let f : M → N be a smooth map betweenmanifolds. If q ∈ N is a regular value, then f −1(q) (when non-emtpy) is anembedded submanifold of M of dimension dim M − dim N .

PROOF: The theorem will follow from the “local form of submersions:”

Claim. Let f : M → N , where dim M = k and dim N = `. If f is a submersionat p then there are charts (U ,ϕ) around p and (V,ψ) around f (p) such thatψ f ϕ−1 = π, the canonical submersion.

We have the picture

Mf //

ϕ

N

ψ

ϕ(U)⊆ Rk

h// ψ(V )⊆ R`

where h=ψ f ϕ−1. Without loss of generality assume that ϕ(p) = 0,ψ( f (p)) =0, and h(0) = 0. Then dh0 : Rk → R` is onto since f is a submersion at p. By alinear change of coordinates we may suppose that dh0 = [I`×`|0]`×k. Define

H : ϕ(U)→ Rk : (a1, . . . , ak) 7→ (h(a1, . . . , ak), a`+1, . . . , ak).

Then by construction dH0 : Rk → Rk is the identity map. Therefore H is a localdiffeomorphism at 0, so there is a neighbourhood W ⊆ ϕ(U) around 0 such that

Forms and Bundles 13

H|W : W →W ′ ⊆ Rk is a diffeomorphism.

Mf // N

ψ

W

ϕ−1

OO

h// ψ(V )

W ′ ⊆ Rk

H−1

OO

π

::ttttttttt

where π := hH−1 truly is the canonical submersion. Notice that (ϕ−1(W ), H ϕ)is a chart around p such that

ψ f (H ϕ)−1 =ψ f ϕ−1 H−1 = h H−1 = π,

so the claim is proved.To complete the proof, note that with the new charts (namely (ϕ−1(W ), H

ϕ) and (V,ψ), making f into a canonical submersion) x`+1, . . . , xk is coordinatesystem for f ∈ (q) around p. f −1(q) is locally given by x1 = · · · = x` = 0. (Checkthat this really finishs the proof.)

1.4.7 Theorem (Whitney Embedding Theorem). Any smooth manifold of dimen-sion n can be embedded in R2n.

2 Forms and Bundles

2.1 Differential Forms

Facts from multilinear algebra

Let V be a vector space over R of dimension n.

2.1.1 Definition. Ap(V ) is the set of alternating multilinear p-forms on V , i.e.ω ∈ Ap(V ) if

ω : V × · · · × V︸︷︷︸

p copies

→ R,

ω is linear in each entry, and

ω(x1, . . . , xp) = sg(σ)ω(xσ(1), . . . , xσ(p))

for all σ ∈Sn.There is a wedge product (or exterior product) Ap(V )× Aq(V ) → Ap+q(V ), de-

fined by

(ω∧η)(x1, . . . , xp+q) =∑

σ

sg(σ)ω(xσ1, . . . , xσp

)η(xσp+1, . . . , xσp+q

)

where σ runs over all permutations of 1,2, . . . , p + q such that σ1 < · · · < σpand σp+1 < · · ·< σp+q.


We have A0(V ) = R and A1(V ) = V ∗ (the dual vector space).

2.1.2 Properties of Wedge Product.(i) It is bilinear and associative;

(ii) If ω1, . . . ,ωk ∈ A1(V ) then (ω1 ∧ · · · ∧ωk)(x1, . . . , xk) = det(ωi(x j));

(iii) If ω1, . . . ,ωn is a basis of V ∗ then ωi1 ∧ · · · ∧ωip| i1 < · · ·< ip is a basis of

Ap(V );

(iv) dim Ap(V ) =n

p

(v) If ω ∈ Ap(V ) and η ∈ Aq(V ), then ω∧η= (−1)pqη∧ω;

(vi) In particular, if p is odd then ω∧ω= 0, and specifically, the wedge productis anti-commutative on 1-forms.

Remark. In the case p = n, so-called top dimensional forms, dim An(V ) = 1, sothe choice of a basis is really a choice of determinant (or volume form).

2.1.3 Definition. Let T : V →W be linear. There is a natural map T ∗ : Ap(W )→Ap(V ) defined by

(T ∗ω)(x1, . . . , xp) =ω(T x1, . . . , T xp),

the pullback of T .

The pullback interacts nicely with the wedge product. Indeed,

T ∗(ω∧η) = T ∗ω∧ T ∗η.

Remark. For T : V → V with n = dim V and ω ∈ An(V ), we have T ∗ω = λωwhere λ= det(T ).

Back to manifolds

Let M be a manifold and (U ,ϕ) be a chart giving a basis ∂∂ x1 , . . . , ∂

∂ xn of Tp M ,for p ∈ U . The dual basis is d x1, . . . , d xn, a basis of T ∗p M . The notation is notaccidental, d x i : Tp(M)→ R really is the differential of the coordinate functionsx i : U → R, and we have d x i

∂

∂ x j = δi j (check).

2.1.4 Definition. A differential p-form on M is a function x 7→ ωx , where ωx ∈Ap(Tx M), such that if (U ,ϕ) is a chart and we write

ω=∑

i1<···<ip

fi1,...,ipd x i1 ∧ · · · ∧ d x ip

,

then the functions fi1,...,ipare smooth. The space of all p-forms will be denoted by

Ωp(M) (in particular Ω0(M) = C∞(M ,R)).

Differential Forms 15

Contrast the above definition with the definition of a vector field. We couldhave instead given qx∈M Ap(Tx M) = Ap(M) the structure of a vector bundle overM and defined p-forms to be sections of this bundle.

Remark.(i) We have a wedge product of differential forms: ifω ∈ Ωp(M) and η ∈ Ωq(M)

then ω∧η ∈ Ωp+q(M).

(ii) Smooth functions are 0-forms, so they may be wedged with p-forms. Wewill not use such pompous notation as “ f ∧ω”, and simply write fω.

2.1.5 Definition. Let f : M → N be a smooth map. Then there is a map f ∗ :Ωp(N)→ Ωp(M), the pullback of f , defined by

( f ∗ω)x(v1, . . . , vp) =ω f (x)(d fx(v1), . . . , d fx(vp)).

As in the case of the pullback of a linear map, f ∗(ω∧η) = f ∗ω∧ f ∗η.

Exterior differentiation

2.1.6 Theorem. There exists a unique linear operator d : Ωp(M) → Ωp+1(M)(p ≥ 0) such that

(i) if f ∈ Ω0(M) then d f coincides with the differential of f ;

(ii) d(ω∧η) = dω∧η+ (−1)degω ∧ dη (the Leibniz rule);

(iii) d(dω) = 0 for all ω ∈ Ωp(M) (the chain condition).

PROOF (SKETCH): For smooth functions f we are forced to take d f = ∂ f∂ x i

d x i . De-fine

d( f d x i1 ∧ · · · ∧ d x ip) = d f ∧ d x i1 ∧ · · · ∧ d x ip

+ (−1)deg f f d(d x i1 ∧ · · · ∧ d x ip)

=∂ f

∂ x id x id x i1 ∧ · · · ∧ d x ip

,

and extend d linearly to all p-forms.Let’s see why d(dω) = 0.

d

∂ f

∂ x id x id x i1 ∧ · · · ∧ d x ip

=∂ 2 f

∂ x i∂ x jd x j ∧ d x id x i1 ∧ · · · ∧ d x ip

.

In the double sum over i, j there a lot of cancellation since f is smooth, d x j ∧d x i = −d x i ∧ d x j and d x i ∧ d x i = 0. In fact, nothing remains. Check that dis unique (which is clear since we were forced to define it the way we did) andwell-defined.


Consider the de Rham complex

Ω0(M)d−→ Ω1(M)

d−→ . . .d−→ Ωp(M)

d−→ Ωp+1(M)d−→ . . .

A p-form is closed if dω = 0, and exact if there is η such that ω = dη. Notice thatexact implies closed. The de Rham cohomology is

H pdR(M) =

ω ∈ Ωp(M) | dω= 0dη | η ∈ Ωp−1(M)

.

2.1.7 Example. Consider M = R3. Then a 1-form is something of the form ω =f d x + gd y + hdz, so 1-forms are in one-to-one correspondence with vector fieldsF = ( f , g, h). A 2-form is something of the form

η= f d y ∧ dz+ gdz ∧ d x + hd x ∧ d y,

so they are also in one-to-one correspondence with vector fields. Of course, a3-form is just a multiple of the determinant. Now

dω=

∂ f∂ x

d x + ∂ f∂ y

d y + ∂ f∂ z

dz

+

∂ g∂ x

d x + ∂ g∂ y

d y + ∂ g∂ z

dz

+

∂ h∂ x

d x + ∂ h∂ y

d y + ∂ h∂ z

dz

=

∂ g∂ x− ∂ f∂ y

d x ∧ d y +

∂ f∂ z− ∂ h∂ x

dz ∧ d x +

∂ h∂ y− ∂ g∂ z

d y ∧ dz

Therefore dω corresponds exactly to the curl of F . Similarly,

dη=

∂ f

∂ x+∂ f

∂ y+∂ f

∂ z

d x ∧ d y ∧ dz

so dη corresponds to the divergence of F . (I really wish that I had taken vectorcalculus.)

2.2 Orientability and Integration

Orientable manifolds

Suppose that h : U → V is a diffeomorphism between open sets in Rn. Then wecan write coordinates for the tangent space to V as y j = h j(x1, . . . , xn), wherex1, . . . , xn are coordinates for the tangent space of U . Then

h∗(d y1 ∧ · · · ∧ d yn) = det(dhx)d x1 ∧ · · · ∧ d xn.

(Recall that det(dhx) = det( ∂ yi

∂ x j) is the Jacobian.)

2.2.1 Theorem (Orientability). Let M be an n-manifold. The following are equiv-alent.

(i) There exists a nowhere vanishing smooth differential n-form.

Orientability and Integration 17

(ii) There is a family of charts that cover M such that the determinant of theJacobian of coordinate changes is positive on all overlaps.

(iii) An(T M) is isomorphic to M ×R.

2.2.2 Definition. M is orientable if it satisfies any (and hence all) of the threeequivalent conditions.

PROOF: (i) if and only if (iii): Proved exactly as 1.3.5.(i) implies (ii): Consider all charts (U ,ϕ) such that d x1 ∧ · · · ∧ d xn = f Ω,

where f > 0 and Ω is the non-vanishing n-form. We can cover M with such chartsand if (U ,ϕ) and (V,ψ) are two overlapping charts then

d x1 ∧ · · · ∧ d xn = det

∂ x i

∂ x ′j

!

d x ′1 ∧ · · · ∧ d x ′n

implies that det( ∂ x i

∂ x ′j)> 0.

(ii) implies (i): We need to introduce partitions of unity.

2.2.3 Theorem (Partitions of unity). For any open cover⋃

α∈A Uα = M there ex-ists a countable collection of functions ρi ∈ C∞(M ,R) (i = 1,2, . . . ) such that

(i) for any i, the support x | ρi(x) 6= 0 of ρi is compact and contained in someUα;

(ii) the collection is locally finite: every point x ∈ M has a neighbourhood Wxsuch that ρi 6= 0 on Wx only for finitely many i’s;

(iii) ρi ≥ 0 on M , and∑

i ρi(x) = 1 for all x ∈ M .

The collection ρi is a partition of unity, and is said to be subordinate to thecover Uα.

PROOF: Deferred.

PROOF (OF ORIENTABILITY, CONTINUED):(ii) implies (i): Cover M with compatible charts (Uα,ϕα) (i.e. the determinantof the Jacobian is positive on overlaps). Take a partition of unity ρi subordinateto Uα. Define ωi = d x i

1 ∧ · · · ∧ d x in, where the i indicates the (a?) chart that

contains the support of ρi . Then ρiωi is a smooth form defined everywhere on M .Let Ω =

∑

i ρiωi , a smooth form on M since the sum is finite around any point. Ωis nowhere vanishing because of the third property of partitions of unity and thefact that the Jacobian is positive on overlaps.


Integration of n-forms

2.2.4 Definition. Let M be an orientable n-manifold. Let ω ∈ Ωn(M) have com-pact support (i.e. x |ωx 6= 0 is compact).

(i) If M = U ⊆ Rn is an open set then ω= f d x1 ∧ · · · ∧ d xn, and define

∫

U

ω :=

∫

U

f (x1, . . . , xn)d x1 . . . d xn,

where the latter is the Riemann integral.

(ii) Suppose that the support of ω is contained in U , for some chart (U ,ϕ).Then define

∫

M

ω :=

∫

ϕ(U)

(ϕ−1)∗ω.

(iii) Consider a partition of unity ρi subordinate to a cover of M by compatiblecharts. Define

∫

M

ω :=∑

i

∫

M

ρiω.

The sum is finite since ω has compact support.

Suppose that h : V → U is a diffeomorphism (where V ⊆ Rn is open). Then

h∗ω= h∗( f d x1 ∧ · · · ∧ d xn) = ( f h)det(dh)d y1 ∧ · · · ∧ d yn.

Hence∫

V

h∗ω=

∫

V

( f h)det(dh)d y1 . . . d yn,

and if det(dh) > 0 then the change of variables formula implies that∫

Vh∗ω =

∫

Uω. In this case we say that h is orientation preserving.In the second part of the definition, we need to check that if (V,ψ) is another

chart containing the support of ω then the two possible definitions of the integralagree. Assume without loss of generality that U = V . Then if h = ψ ϕ−1 :ϕ(U)→ψ(V ) is the change of coordinates, we have

∫

ψ(V )

(ψ−1)∗ω=

∫

ϕ(U)

h∗((ψ−1)∗ω) =

∫

ϕ(U)

(ψ−1 h)∗ω=

∫

ϕ(U)

(ϕ−1)∗ω

In the third part one needs to check that the definition is independent of thepartition of unity.

Orientability and Integration 19

Stoke’s theorem

Let M be an oriented n-manifold and ω an n-form on M .

Notation. We will often write M n to remind us that M is an n-manifold. Thenotation does not mean the Cartesian product of M with itself n times.

2.2.5 Definition. N ⊆ M n is a domain with smooth boundry (or a codimensionzero submanifold with boundry) if for all p ∈ N there is a chart (U ,ϕ) around p inM such that

ϕ(U ∩ N) = ϕ(U)∩Rn−. (∗)

In this case we define

∂ N = p ∈ N | ϕ satisfies (∗) and ϕ(p) ∈ ∂Rn−,

the boundry of N , where

Rn− = x ∈ R

n | x1 ≤ 0 and ∂Rn− = x ∈ R

n | x1 = 0.

Remark. The boundry of N is an embedded submanifold of M . Indeed, we havethe map

ϕ|U∩∂ N : U ∩ ∂ N → ϕ(U)∩ ∂Rn−,

and ∂Rn− = R

n−1, so it is a chart.

The orientation of M induces an orientation on ∂ N . By flipping the sign of x2if necessary, we may cover M with charts that are positively oriented and satisfying(∗). The restricted charts from the remark above give an orientation to ∂ N .

2.2.6 Definition. For p ∈ ∂ N and x ∈ Tp M , we say that v is outward directedif dϕp(v) has positive first coordinate. A basis v2 . . . , vn of Tp(∂ N) is positivelyoriented if and only if v, . . . , vn is a postively oriented basis of Tp M , where v isany outward directed vector.

Check that this is a well-definition.

2.2.7 Theorem (Stoke’s Theorem).Let N be a domain with smooth boundry in an oriented smooth n-manifold M .Let ∂ N have the induced orientation. For every ω ∈ Ωn−1(M) with N ∩ supp(ω)compact, we have

∫

∂ N

i∗ω=

∫

N

dω,

where i : ∂ N → M is the inclusion map.

PROOF: Suppose first that we can prove the theorem when

(i) supp(ω)⊆ U; where

(ii) (U ,ϕ) is a positively oriented chart satisfying (∗).


Then cover M with positively oriented charts satisfying (∗) and take a partition ofunity ρi subordinate to this covering. We have∫

N

dω=

∫

N

d∑

i

ρiω=∑

i

∫

N

d(ρiω) =∑

i

∫

∂ N

ρiω=

∫

∂ N

∑

i

ρiω=

∫

∂ N

ω.

Now let η ∈ Ωn−1c (Rn) be the (n− 1)-form that is (ϕ−1)∗ω in ϕ(U) and zero

outside of ϕ(U). Then∫

∂ N

ω=

∫

ϕ(U)∩∂Rn−

(ϕ−1)∗ω=

∫

∂Rn−

η,

while∫

N

dω=

∫

ϕ(U)∩Rn−

dω=

∫

Rn−

dη.

Therefore it suffices to prove the theorem when M = Rn, N = Rn−, and ω ∈

Ωn−1c (Rn).

Suppose

ω=n∑

i=1

fi(x)d x1 ∧ · · · ∧dd x i ∧ · · · ∧ dn.

Choose b > 0 such that supp( fi)⊆ [−b, b]n for i = 1, . . . , n. Then

i∗ω=ω|∂Rn−= f1(0, x2, . . . , xn)d x2 ∧ · · · ∧ d xn,

so∫

∂Rn−

ω=

∫

f1(0, x2, . . . , xn)d x2 . . . d xn.

On the other hand,

dω=n∑

i=1

(−1)i−1 ∂ fi

∂ x id x1 ∧ · · · ∧ d xn,

so∫

Rn

dω=n∑

i=1

(−1)i−1

∫

Rn−

∂ fi

∂ x id x1 ∧ · · · ∧ d xn.

When 2≤ i ≤ n, we have∫ ∞

−∞

∂ fi

∂ x i(x1, . . . , x i−1, t, x i+1, . . . , xn)d t = 0,

so∫

Rn−

∂ fi

∂ x id x1 ∧ · · · ∧ d xn = 0.

Metrics 21

When i = 1,∫ 0

−∞

∂ f1

∂ x1(t, x2, . . . , xn)d t = f1(0, x2, . . . , xn),

so∫

Rn−

∂ f1

∂ x1d x1 ∧ · · · ∧ d xn =

∫

f1(0, x2, . . . , xn)d x2 ∧ · · · ∧ d xn,

and the theorem is proved.

2.3 Metrics

Riemannian metrics

2.3.1 Definition. Let M be an n-manifold. A Riemannian metric on M is a func-tion g : x ∈ M 7→ gx , where gx is a positive definite inner product on Tx Mand such that, for any chart (U ,ϕ), the functions gi j(x) = gx(

∂

∂ x i ,∂

∂ x j ) | i, j =1, . . . , n are smooth. In this case the pair (M ,ω) is a Riemannian manifold.

Remark.(i) There are variations on this definition e.g. a semi-Riemannian metric re-

quires only that the metric be non-degenerate.

(ii) We could also say that g is a smooth section of the bundle of symmetricbilinear forms with the positivity property.

(iii) In the case when n = 2 and M is a surface, in classiscal notation we writeg =

E FF G

or as gi jd x id x j .

A metric induces natural isomorphisms (though they depend on the metricchosen),

Lg : T M → T ∗M and L −1g : T ∗M → T M ,

the Legendre transform, defined by Lg : (x , v) 7→ (x , w 7→ gx(v, w)).

Remark. Sometimes we write Lg = [ and L −1g = ] for these isomorphisms and

call them the musical isomorphisms. The reason for this notation is from physics,as in abstract index notation [ lowers indices and ] raises indices. (If v = v i ∂

∂ x i

and p = pid x i then pi = gi j vj , and v j = g i j pi , where g i j is the inverse of gi j .)

Notation. For v ∈ Tx M , by convention we write gx(v, v) = |v|2x = |v|2.

Symplectic forms

2.3.2 Definition. Let M be an n-manifold. A symplectic form is a non-degeneratesmooth closed 2-form, i.e.ω ∈ A2(T M), dω= 0, such thatωx is a non-degeneratebilinear form for all x ∈ M . The pair (M ,ω) is a symplectic manifold.


For an n× n matrix A, At =−A implies that

det A= det(−A) = (−1)n det A,

so if A is invertible then n is even. Hence if (M ,ω) is a symplectic manifold thenM is even dimensional.

2.3.3 Example. Let M = R2n = Rn × Rn (coordinates (qi , p j)). The canonicalsymplectic form on M isω=

∑ni=1 dqi∧dpi . (Check this, take the derivative, etc.)

If ω is a symplectic form on M2n, then ω∧ · · · ∧ω = ωn is a top-dimensionalform. Hence if ω is non-degenerate then ωn is never zero (prove this). Thisimplies in particular that M is orientable. Any orientable surface is automaticallya symplectic manifold (use the form that shows orientability).

Suppose now that (M2n,ω) is a compact symplectic manifold. Then ω is notexact. Indeed, if ω= dη for some smooth 1-form η, then ωn is a volume form, so∫

Mωn 6= 0 (in fact it will be greater than zero). But

d(η∧wn−1) = dη∧ωn−1 ±η∧ d(ωn−1) =ωn

since ω is closed (indeed, exact), so ωn is exact. Applying Stoke’s theorem,

∫

M

ωn =

∫

M

d(η∧wn−1) = 0

since M has no boundry in itself. Recall that H2dR(M) is the quotient of the closed

2-forms by the exact 2-forms, so in the case where (M ,ω) is a compact symplecticmanifold 0 6= [ω] ∈ H2

dR(M).

2.3.4 Theorem (de Rham). HkdR(M)

∼= Hk(M ,R) (topological cohomology).

In the case of S4, H2(S4,R) = 0, so there is no symplectic form on S4 and itcannot be made into a symplectic manifold.

2.3.5 Example. The cotangent bundle is a very important example of a symplecticmanifold. Let N be any manifold and M = T ∗N . A canonical 1-form λ is definedby

λ(x ,p)(ξ) = p(dπ(x .p)(ξ))

(recall that π : T ∗N → N is the foot-point projection, so

dπ(x ,p) : T(x ,p)(T∗N)→ Tx N).

With local coordinates (q1, . . . , qn, p1, . . . , pn) on T ∗N , we have λ = pidqi . Locallytake ω = −dλ = dqi ∧ dpi . Then (T ∗N ,−dλ) is a (non-compact) symplecticmanifold.

Bundles 23

2.3.6 Example (Classical mechanics). Let (M ,ω) be a a symplectic manifold.For any smooth function H : M → R, dH is an exact 1-form. There exists a uniquevector field XH such that dHx(·) =ωx(XH(x), ·). This XH is called the Hamiltonianvector field of H. If ϕt is the flow of XH then this flow is the Hamiltonian flow ofH.

Notice that

d

d tH(ϕt x) = dHϕt x(XH(ϕt x)) =ωϕt x(XH(ϕt x), XH(ϕt x)) = 0.

Therefore “energy is preserved along the gradients of the Hamiltonian flows.”Let (M , g) be a Riemannian manifold. For Lg(x , v) = (x , p), write |p|x =de f

|v|x . Say v = v i ∂∂ x i , p = pid x i , so |p|2 = g i j pi p j and |v|2 = gi j v

i v j . Then define

H(x , p) = 12|p|2x (the Hamiltonian) and let XH be the vector field in T ∗M , so that its

integral curves are γ : t 7→ (x(t), p(t)) ∈ T ∗M , where γ(t) = XH(γ(t)). t 7→ x(t)is a geodesic curve in M .

Define a symplectic form ω=Lg(−dλ) on T M . There is a canonical functionL : T M → R : (x , v) 7→ 1

2|v|2x . There is an associated vector field on T M , with

integral curves γ : t 7→ (x(t), v(t)), with˙γ(t) = X L(γ(t)). Then tλx(t) is also ageodesic. (In fact, v(t) = x(t).)

(insert diagram)Finally, potential forces are represented by smooth functions V : M → R and

incorportated by taking H(x , p) = 12|p|2x + V (x) (F =−∇V ).

2.4 Bundles

Vector bundles

2.4.1 Definition. Let B be a smooth manifold. A manifold E together with asmooth submersion π : E → B is called a vector bundle of rank k over B if thefollowing hold.

(i) There exists a k-dimensional vector space V (called the typical fibre) suchthat each fibre Ep = π−1(p) is a vector space isomorphic to V .

(ii) Any point p ∈ B has a neighbourhood U such that there exists a diffeomor-phism ϕU : π−1(U)→ U × V which makes the diagram below commute.

π−1(U)ϕU //

π

U × V

pr1yyttttttttttt

U

(iii) ϕU |Ep: Ep → V is a linear isomorphism.

B is called the base space, E is called the total space, ϕU is called a trivialization,and U is a trivializing neighbourhood.


2.4.2 Examples.(i) The trivial bundle is E = B× V with π(b, r) = b.

(ii) Tangent bundles, cotangent bundles, exterior bundles Ap(T M), . . .

V may be an R-vector space or a C-vector space.Suppose that Uα is a complete family of trivializing neighbourhoods, and

ϕα : π−1(Uα)→ Uα × V are the corresponding trivializations. Notice that

ϕβ ϕ−1α : (Uα ∩ Uβ)× V → (Uα ∩ Uβ)× V

has the form ϕβ ϕ−1α (b, v) = (b,ψβα(b)v), where ψβα : Uα∩Uβ → GL(V ). Such

ψβα is a transition function. Transition functions satisfy the cocyle conditions

(i) ψαα = idUα ;

(ii) ψαβψβα = idUα∩Uβ ;

(iii) ψαβψβγψγα = idUα∩Uβ∩Uγ .

In fact, given a collection of functions satisfying all the properties of transitionfunctions, there is a unique bundle which has those functions as its transitionfunctions.

2.4.3 Steenrod construction. Let E = qα(Uα ×Rn)/ ∼, where (x , v) ∼ (y, w) ifand only if x = y and w =ψβα(x)v, where x ∈ Uα and y ∈ Uβ .

Very often it will happen that ψβα takes values in subgroup G < GL(V ). Whenthis happens we say that the bundle has structure group G.

2.4.4 Examples.(i) For E = B×Rn, the transition function is ψα(x) = In.

(ii) M is orientable if and only if the tangent bundle T M has structure group inGL+(k,R), linear maps with positive determinant.

(iii) The structure group of the Möbius band over S1 is ±1 ∼= Z2.

(iv) O(k), SO(k), . . .

Principal bundles

2.4.5 Definition. Let G be a Lie group and P be a smooth manifold. A smoothaction (in this case a smooth right action) is just an action P×G→ P : (p, g) 7→ pgwhich is also a smooth map. The action is a free action if pg = p implies g = 1.

2.4.6 Definition. Let B be a smooth manifold. A principal bundle (or principalG-bundle) is a manifold P together with a smooth submersion π : P → B togetherwith a smooth right free action satisfying the following conditions.

(i) π(pg) = π(p) for all p ∈ P and g ∈ G (i.e. the fibres of π are the orbits ofG);

Bundles 25

(ii) for any b ∈ B there is a neighbourhood U of b and a diffeomorphism ϕU :π−1(U)→ U × G such that pr1 ϕU = π|π−1(U);

(iii) the actions are “intertwined” by ϕU , i.e. for all h ∈ G, ϕU(ph) = (b, gh),where (b, g) = ϕU(p) and π(p) = b ∈ U .

As in the case of vector bundles, we have trivializing neighbourhoods andtransition functions. Let Uα be a complete family of trivializing neighbourhoods,and consider the transition function defined by

ϕβ ϕ−1α (b, g) = (b,ψβα(b, g)).

As before, for each b ∈ Uα ∩ Uβ , ϕβα(b, ·) is a map from G to G. Since the actionis intertwined,

ψβα(b, g)h=ψβα(b, gh).

Taking g = 1, we have ψβα(b, 1)h = ϕβα(b, h), so ψβ ,α(b, ·) (as a map from G toG) is left-translation on G by the element ψβα(b, 1). As is usual in geometry, wenow simplify the notation by renaming

ϕβα : Uα ∩ Uβ → G : b 7→ψβα(b, 1),

the transition function associated with the principal bundle. Transition functionsfor principal bundles satisfy the cocycle conditions (namely,ψαβ ψβγψβγ = id).As in the case of vector bundles, if we know the transition functions then we canrecover the bundle by the Steenrod construction. Take P = qα(Uα×G)/∼, where(b, h)∼ (b′, h′) if and only if b = b′ and h′ =ψβα(b)h.

Suppose that G ⊆ GL(k) (or, what amounts to the same thing, a representationof G in GL(k)) and we have a collection ψβα : Uα ∩ Uβ → G of transistionfunctions which satisfy the cocyle conditions. Then there is an associated vectorbundle E and an associated principal bundle P. In this case E and P are said to beassociated bundles

2.4.7 Definition. A section of a vector bundle E (resp. principal bundle P) is asmooth map s : B→ E (resp. P) such that π s = id.

2.4.8 Example (Hopf bundle). Let B = CP1 = (C2 \ (0,0))/C∗, the collectionof (complex) lines through the origin in C2. We have charts CP1 = U1∪U2, whereUi = [z1 : z2] ∈ CP1 | zi 6= 0, with coordinates z : U1 → C : [z1 : z2] 7→

z2

z1and

η : U2→ C : [z1 : z2] 7→z1

z2. On U1 ∩ U2 we have η= 1

z.

Let E be the disjoint union of the (complex) lines through the origin in C2, abundle over CP1 with trivial submersion π : E→ CP1 : ` 7→ `. E is the tautologicalbundle of CP1. A point in E will be written as (ωz1,ωz2), with |z1|2 + |z2|2 6= 0.Trivializations are

ϕ1 : π−1(U1)→ U1 ×C : (ω,ωz) 7→ ([1 : z],ωp

1+ |z|2)

andϕ2 : π−1(U2)→ U2 ×C : (ωη,ω) 7→ ([η : 1],ω

p

1+ |η|2).


It can be shown that, for ([1 : z],ω) ∈ (U1 ∩ U2)×C,

ϕ2 ϕ−11 ([1 : z],ω) = ϕ2

ωp

1+ |z|2,

ωzp

1+ |z|2

= ϕ2

|η|ωp

1+ |η|2,

|η|ω

ηp

1+ |η|2

=

[η : 1],|η|ηω

=

[1 : z],z

|z|ω

Therefore ψ21([1 : z]) = z|z| ∈ U(1) ⊆ C∗. This transition function gives a vector

bundle or a principal bundle over CP1, both of which are called the Hopf bundle.

2.4.9 Example. S3 embeds in C2 as S3 = (z1, z2) ∈ C2 | |z1|2 + |z2|2 = 1. Foreiθ ∈ U(1),

S3 × U(1)→ S3 : ((z1, z2), eiθ ) 7→ (eiθ z1, eiθ z2)

is often called the Hopf map. The orbits are circles in S3, and we have S3 π−→ CP1 =S2. This is an example of a non-trivial principal bundle. (I don’t understand. . . )

Pullback bundles, morphisms, and automorphisms

For this section we will phrase everything in terms of vector bundles, but thedefinitions work, with appropriate modification, for principal bundles as well.

2.4.10 Definition. Let Eπ−→ B be a vector bundle and f : M → B be a smooth

map. Let f ∗E = (x , e) ∈ M × E | π(e) = f (x), F(x , e) = e, and π′(x , e) = x .Then the following diagram commutes.

f ∗E F //

π′

E

π

M

f// B

f ∗E is called the pullback bundle of E through f , a vector bundle over M .

From the point of view of transition functions, if

ψβα : Uα ∩ Uβ → G ⊆ GL(V ) then f ∗ψβα =ψβα f .

2.4.11 Definition. Suppose that Eπ−→ B and E′

π′−→ B′ are two vector bundles overthe same vector space V and f : B → B′ is smooth. A smooth map F : E → E′

if a bundle morphism covering f if for any b ∈ B, F restricts to a linear map

Connections 27

F |Eb: Eb → E′f (b) and such that the following diagram commutes.

EF //

π

E′

π′

B

f// B′

If F is a diffeomorphism and F |Eb: Eb → E′f (b) is a linear isomorphism for all b ∈ B

then we say that F is a bundle isomorphism. An isomorphism E → E covering theidentity map is a bundle automorphism. The groups of all autmorphisms is denotedAut(E).

From a local point of view, take trivializing neighbourhoods U and U ′ suchthat f (U) ⊆ U ′, ϕU : π−1(U) → U × V , and ϕ′U ′ : (π′)−1(U ′) → U ′ × V . LetFU = ϕ′U ′ F ϕ−1

U , so that FU(b, v) = ( f (b), h(b)v), where h : U →L (V, V ).

2.4.12 Example. Take E = B × V , the trivial bundle. Then one chart suffices, sothe local behavior determines the map, and we have Aut(E) = C∞(B, GL(V )).

Remark.(i) If the bundle has structure group G ⊆ GL(V ) then we often only care about

AutG(E)⊆ Aut(E), where AutG(E) are those automorphisms for which whenwritten in trivializations giving the G-struture, h above takes values in G.

(ii) Using bundle automorphisms one can change transition functions.

ψ′βα = hαψβαh−1α and ϕ′α(e) = hα(π(e))ϕα(e).

This change of trivializations is analogous to the change of basis in GL(V ).

2.5 Connections

First definition

We would like to generlize the fact that the derivative of a vector field (vi(x)) inRm is another vector field in Rm, namely (v′i (x)). If E

π−→ B is a vector bundle ands : B → E is a section then we would like a way of differentiating s in such a waythat the derivative is also a section.

With the notation above, suppose dim B = n and V = Rm. Let U ⊆ B bea trivializing neighbourhood (so there is ϕU : π−1(U) → U × Rm), and let x k,k = 1, . . . , n be local coordinates in U . Let a j , j = 1, . . . , m be canonical coordinatesin Rm. Using the trivialization, Tp E (for p ∈ π−1(U)) has a basis ∂

∂ xk , ∂

∂ a j . Forπ : E → B, dπp : Tp E → Tπ(p)B is a submersion and we call its kernel the verticalsubspace. The vertical subspace is spanned by ∂

∂ a j .

2.5.1 Definition. A subspace Sp ⊆ Tp E is called a horizontal subspace if

Sp ∩ ker(dπp) = 0 and Sp ⊕ ker(dπp) = Tp E.


We will not simply take Sp = ∂

∂ x j , since this depends heavily on the choice oflocal coordinates; notice that the vertical subspace does not depend on the localcoordinates.

2.5.2 Lemma. If θ 1, . . . ,θm ∈ (Rm+n)∗ are linear functionals then

dim m⋂

i=1

kerθ i

= n

if and only if θ 1, . . . ,θm is linearly independent.

PROOF: Exercise (or possibly omitted).

Let θ 1p , . . . ,θm

p be m linearly independent covectors (i.e. elements of (Tp E)∗).

Take Sp =⋂m

i=1 ker(θ ip). (Tp E)∗ has basis d x k, da j, so we can write θ i

p = f ik d x k+

g ijda j . If Sp is a horizontal subspace then θ i

p(v) = 0 for all i implies v = 0, for

all vertical vectors v. Restating, if Sp is a horizontal subspace and v = c j ∂∂ a j then

0 = θ ip(v) = g i

jcj for all i implies c j = 0. It follows that Sp is horizontal if and

only if (g ij)i j is an invertible m×m matrix. Say (g i

j)−1 = (d i

j). Replace θ i by θ i by

multiplying by (d ij), i.e. θ i = d i

jθj = dai + ei

kd x k. This change does not alter Sp.Now let p vary, so the ei

k become functions of p. If eik(p) are smooth functions of

p then we say that Sp varies smoothly with p.The following proposition summarizes the above discussion.

2.5.3 Proposition. Let S = Sp (p ∈ E) be any smooth field of horizontal subspacesin T E. Let x k, a j be local coordinates on π−1(U) arising from a local trivialization(as above). Then Sp =

⋂mi=1 ker(θ i

p), where θ ip = da j + e j

k(x , a)d x k for smooth

functions e jk(x , a) which are uniquely determined by the trivialization.

2.5.4 Definition. A smooth field of horizontal subspaces Sp is a connection if inevery local trivialization, the functions ei

k(x , a) are linear in the fibre variables,i.e. ei

k(x , a) = Γijk(x)a

j , where Γijk : U ⊆ B → R are smooth functions called the

coefficients of the connection Sp.

Notation. We write θ i = dai+Aija

j , where Aij = Γ

ijkd x k. (Ai

j)i j is an m×m matrixof 1-forms on U .

Coming up, an “index festival”. . .

Transformation law for connections and the second definition

Let U ′ be another trivializing neighbourhood with trivialization ϕ′ : π−1(U ′) →Rm, and x k′, ai′ coordinates in U ′ ×Rm.

Notation. A prime “′” refers to U ′ e.g. the transition matrix on U ∩ U ′ is ψi′i and

its inverse is ψii′.

Connections 29

Let x k′ = x k′(x). Then ai′ =ψi′i ai , so

dai′ = dψi′i ai +ψi′

i dai =∂ψi′

i

d x kai +ψi′

i dai ,

and d x k′ = ∂ xk′

∂ xk d x k. Therefore

θ i′ = dai′ +Γi′j′k′a

j′x k′ =ψi′i dai +

∂ψi′j

d x k+Γi′

j′k′∂ x k′

∂ x kψ

j′j

a jd x k,

so

ψii′θ

i′ = dai +

ψii′

∂ψi′j

d x k+ψi

i′Γi′j′k′ψ

j′j

∂ x k′

∂ x k

a jd x k.

From this is follows that

Γijk = Γ

i′j′k′ψ

ii′ψ

j′j

∂ x k′

∂ x k+ψi

i′

∂ψi′j

d x k(1)

soAi

j = Γijkd x k and Ai′

j′ =ψi′i Ai

jψjj′ +ψ

i′j dψi

j′. (2)

Rewritting, if Aϕ is the matrix of 1-forms in U and Aϕ′ is the matrix of 1-forms inU ′, then

Aϕ′ =ψAϕψ−1 +ψ(dψ−1) =ψAϕψ−1 − (dψ)ψ−1. (3)

2.5.5 Theorem. Any system of functions Γijk (1 ≤ k ≤ n, 1 ≤ i, j ≤ m) attached

to local trivializations satisfying the transformation law (1) defines a connectionA on E whose coefficients are Γi

jk.

Third definition

Notation. For any vector bundle Eπ−→ B, Γ(E) denotes the space of sections s :

B→ E (so s(x) ∈ Ex).

2.5.6 Definition. A connection is an R-linear map ∇ : Γ(E) → Γ(T ∗B ⊗ E) suchthat

∇( f s) = d f ⊗ s+ f∇s Leibniz rule

for f ∈ C∞(B,R) and s ∈ Γ(E).

Or better,

2.5.7 Definition. A connection is a map∇ : Γ(T B)×Γ(E)→ Γ(E) which we writeas ∇(X , s) =∇X s and such that

(i) ∇ f X+gY s = f∇X s+ g∇Y s for all f , g ∈ C∞(B,R) and X , Y ∈ Γ(T B);

(ii) ∇X (s1 + s2) =∇X s1 +∇X s2 and ∇X (cs) = c∇X s for all si ∈ Γ(E) and c ∈ R;


(iii) ∇X ( f s) = X ( f )s+ f∇X s (= d f (X )s+ f∇X s).

Recall that if V and W are vector spaces then there is a canonical linear iso-morphism

V ∗ ⊗W → Hom(V, W ) : f ⊗w 7→ “v 7→ f (v)w”.

Whence T ∗B ⊗ E ∼= Hom(T B, E), so a section s : B → T ∗B ⊗ E gives linear mapssx : Tx B→ Ex .

We can think of Γ(T ∗B ⊗ E) as the E-valued 1-forms, and we will denote itΩ1

B(E) by analogy with the notation for forms. (Our old notation becomes Ω1(B) =Ω1

B(B×R).) Sections of E are thought of as E-valued 0-forms, denoted Ω0B(E)

2.5.8 Definition. Let ΩrB(E) be the collection of E-valued r-forms, alternating

multilinear mapsTx B× · · · × Tx B︸︷︷︸

r

→ Ex .

With these definitions, ∇ : Ω0B(E)→ Ω

1B(E). This will of course be generalized

later.The definitions 2.5.6 and 2.5.7 at the beginning of this section are see to be

the same via the identification ∇X (s) = (∇s)(X ).

Bringing it all together

(Insert several puns on the word “connection” here.)Let A be a connection (i.e. Ai

j = Γijkd x k with the correct transformation law).

Work in a local trivialization, and define ∇ := dA by dAs = ds+As. Locally we maywrite s(x) = (s1(x), . . . , sm(x)) (a vector in Rm), so

dA(s1, . . . , sm) =

∂ s1

∂ x k+Γ1

jks j

d x k, . . . ,

∂ sm

∂ x k+Γm

jks j

d x k

.

Now dA is well-defined since if we take another trivialization U ′ with transitionmatrix ψ from U ′ to U , then from above we have A = ψA′ψ−1 − (dψ)ψ−1. Wealso have s =ψs′. Checking,

dAs = ds+ As = d(ψs′) + A(ψs′) = (dψ)s′ +ψds′ + (ψA′ψ−1 − (dψ)ψ−1)(ψs′)= (dψ)s′ +ψds′ +ψA′s′ − (dψ)s′ =ψds′ +ψA′s′ =ψ(dA′s

′)

Remark. From the definition of ∇, it follows that ∇ is a local operator, i.e. ifs ∈ Γ(E) vanishes on an open set U then ∇s vanishes on U as well. Indeed, ifx ∈ U then let V be a neighbourhood of x such that V ⊆ U . Let α be a C∞ cut-off function such that 0 ≤ α ≤ 1, α|V ≡ 1, and supp(α) ⊆ U . Then αs = 0, so0=∇(αs) = dα⊗s+α∇s by linearity and the Leibniz rule. Evaluating at x shows(∇s)(x) = 0.

This implies that a connection ∇ on E also defines a connection on E|U , whereU is any open set of B.

Connections 31

2.5.9 Theorem. Any connection ∇ arises as dA for some A.

PROOF: Take a trivialization over U ⊆ B. Local sections

Γ(E|U) = s : U → E | s(x) ∈ Ex

are just smooth functions U → Rm. We have a frame of sections e j, the constantsections which map everything to (0, . . . , 0, 1, 0, . . . , 0) ∈ Rm. Then we can writean arbitrary section as s = s je j . Therefore

∇s =∇(s je j) = ds j ⊗ e j + s j∇e j .

But for some Γ, we can write ∇e j = (Γijkd x k)ei . Let Ai

j = Γijkd x k. We now check

that ∇s = dAs.

∇s = ds j ⊗ e j + s jΓijkd x kei = (dsi + s jΓi

jkd x k)ei = dAs

To finish the proof one has to check that the Γijk (in A) transform in the correct

way.

2.5.10 Example. Let M n ⊆ Rn+k, and let s ∈ Γ(T M) (so s is a vector field).Then s : M → T M such that s(x) ∈ Tx M ⊆ TxRn+k = Rn+k. For v ∈ Tx M , itmakes sense to write dsx , and dsx(v) ∈ Rn+k. But this may have jumped outof Tx M , so orthogonal projection onto Tx M brings it back. Define (∇vs)(x) tobe the orthogonal projection of dx s(v) onto Tx(v). As the notations suggests,∇ : Γ(T M) × Γ(T M) → Γ(T M) is a connection on T M . This is the Levi-Civitaconnection of the metric induced on M by Rn+k.

Let Eπ−→ B and α : I → B be a smooth curve, and s : I → E be a section along

α, i.e. s(α(t)) ∈ Eα(t) for all t ∈ I . One can introduce an operator Ddt

which takess into another section along α and such that

(i) Ddt(s1 + s2) =

Ds1

d t+ Ds2

d t;

(ii) Ddt( f s) = f s+ f Ds

d t, where f is a smooth function of t; and

(iii) If s(t) =ω(α(t)), where ω ∈ Γ(E), then Dsd t=∇αω.

Let α : I → U ⊆ B, where U is a trivializing neighbourhood, with trivializationϕU : π−1(U)→ U ×Rm, with s(t) = (s1(t), . . . , sm(t)) (so s = s je j). We must have

D

dt(s je j) = s je j + s j∇αe j

and if α = (α1, . . . ,αn) thenDe j

d t= ∇αe j = Γi

jkd x k(α)ei = Γijkα

kei . Bringing it alltogether,

D

dt(s je j) = (s

i + s jαkΓi

jk)ei .

It can be shown that this is the unique local definition of Ddt

.


2.5.11 Definition.(i) s ∈ Γ(E) is parallel (or covariant-constant) if ∇s = 0.

(ii) If s is a section along a curve α : I → B is parallel if Dsd t= 0.

Notice that Dsd t= 0 means that si + s jα

kΓijk = 0 for every i. This is a linear

system of ODE’s. Take s0 ∈ Eα(t0), so there is a unique parallel section s alongα such that s(t0) = s0. This defines P : Eα(t0) → Eα(t1), P(s0) = s(t1), a linearisomorphism. Such a thing is a parallel transform. The velocity vectors s of theparallel sections t 7→ s(t) ∈ E are live in the horizontal subspaces at s(t) (exercise,use the forms θ i = dai +Γi

jka jd x k to show that θ i(s) = 0 if s is parallel).

Remark. If ∇1 and ∇2 are connections on Eπ−→ B, then ∇1( f s) − ∇2( f s) =

f (∇1s −∇2s), so s 7→ ∇1s −∇2s is a C∞-linear map from sections to E-valued1-forms. We may think of having ∇1 − ∇2 ∈ Ω1

B(End E). Whence the space ofconnection is an affine space.

In a similar way we have ΩrB(End E), the End E-valued r-forms. In priniple,

they act on E-valued sections (or `-forms), getting ΩrB(End E)×Ω`B(E)→ Ω

r+`B (E).

2.6 Curvature

“I’m not going to tell you what it is, I’m going to tell you where it lives. . . ”Recall the covariant derivative dA : Ω0

B(E)→ Ω1B(E). We would like to extend

it dA : ΩrB(E)→ Ω

r+1B (E) analogously to how we extended d. For σ ∈ Ωr

B(E) andω ∈ Ωq(B) (and ordinary q-form), we require

dA(σ∧ω) = dAσ∧ω+ (−1)rσ∧ dω.

Locally in a trivialization we get dAσ = dσ+ A∧σ. We have

Ω0B(E)

dA−→ Ω1B(E)

dA−→ . . .dA−→ Ωi

B(E)dA−→ . . .

dA−→ 0

But what is d2A?. For σ ∈ Ωr

B(E), we have

dA(dAσ) = dA(dσ+ A∧σ)= d(dσ+ A∧σ) + A∧ (dσ+ A∧σ)= 0+ dA∧σ+ (−1)A∧ dσ+ A∧ dσ+ A∧ A∧σ= (dA+ A∧ A)∧σ

Let F = dA+ A∧ A, so that dA(dAσ) = F ∧σ. Notice that dAdA( f σ) = f dAdAσ, soF ∈ Ω2

B(End E). F is the curvature, the deviation of our chain from being a chaincomplex.

2.6.1 Definition. A flat connection is a connection with F = 0.

Riemannian Metrics 33

2.6.2 Example. Let E = B × R (or B × Rm if you wish). Then ΩrB(E) = Ω

r(B)we may take dA to simply be d, i.e. dAω = dω, the trivial connection or productconnection. Since d2 = 0, the curvature is zero and the connection is flat.

Suppose that Aij = Γ

ijkd x k, and A = Akd x k, where Ak is an m× m matrix of

functions. Then

F = dA+ A∧ A

= d(Akd x k) + (Aid x i)∧ (Akd x k)

=∂ Ak

∂ x i d x i ∧ d x k + AiAkd x i ∧ d x k

=1

2

∂ Ak

∂ x i −∂ Ai

∂ x k+ [Ai , Ak]

d x i ∧ d x k (C1)

Finally, dA also extends naturally to ΩrB(End E), but how? Let µ ∈ Ωr

B(End E),σ ∈ Ω`B(E), so that µ∧σ ∈ Ωr+`

B (E). We would like to define dA so that

dA(µ∧σ) = (dAµ)∧σ+ (−1)rµ∧ dAσ.

But we may take that as the definition. Locally we have

(dAµ)∧σ = dA(µ∧σ)− (−1)rµ∧ dAσ

= d(µ∧σ) + A∧µ∧σ− (−1)rµ∧ (dσ+ A∧σ)= (dµ)∧σ) + (−1)rµ∧ dσ+ A∧µ∧σ− (−1)rµ∧ dσ− (−1)rµ∧ A∧σ)

= (dµ+ A∧µ− (−1)rµ∧ A)∧σ

Therefore dAµ= dµ+ A∧µ− (−1)rµ∧ A.

2.6.3 Theorem (Bianchi Identity). dAF = 0

PROOF: Since dAdAσ = F ∧σ,

(dAF)∧σ = dA(F ∧σ)− F ∧ dAσ = dA(dAdAσ)− dAdA(dAσ) = 0

3 Riemannian Metrics

3.1 Metric Connections

Let Eπ−→ B, A be a connection, and ⟨·, ·⟩ a Riemannian metric on E over R (or a

Hermitian metric on E over C).

3.1.1 Definition. A metric connection (or orthogonal connection or unitary connec-tion) is a connection such that for any X ∈ V (B) and s1, s2 ∈ Γ(E),

X ⟨s1, s2⟩= ⟨∇X s1, s2⟩+ ⟨s1,∇X s2⟩.


Remark. If E is endowed with a Riemannian metric then we may choose trivializa-tions ϕ : π−1(U)→ U ×Rm such that the vectors ei(p) such that they are mappedby ϕ to the canonical basis in Rm are orthogonal (exercise: check this (use Gram-Schmidt method)). Such a trivialization is called an orthogonal trivialization.

3.1.2 Proposition. A metric connection has skew-symmetric matrix of coefficientsin any orthogonal trivialization.

PROOF: Take ei , an orthonormal set of sections in the trivialization. Then for anyvector field X ,

0= X ⟨ei , e j⟩= ⟨∇X ei , e j⟩+ ⟨ei ,∇X e j⟩

= ⟨Aki (X )ei , e j⟩+ ⟨ei , Ak

j (X )e j⟩

= Aji(X ) + Ai

j(X ).

3.1.3 Corollary. Let A be a metric curvature. Then in an orthogonal local trivial-ization, the matrix of F is also skew-symmetric.

(In the form (C1) for curvature one can always swap i and k and get thenegative of what one started with, but the corollary refers to taking the transpose,so we see a different kind of skew-symmetry.)

3.2 Levi-Civita connection

Let M be a manifold with a Riemannian metric g (also written ⟨·, ·⟩). A connectionon M is a connection on T M , ∇ : Γ(T M) → Γ(T ∗ ⊗ T M). We have Γ(T M) =V (M), the vector fields on M . Notice that we may consider ⟨X , Y ⟩ a function onM , taking value, for p ∈ M , ⟨X (p), Y (p)⟩p.

Recall the compatibility conditions for a connection to be a metric connection.They imply (?) that

Z⟨X , Y ⟩= ⟨∇Z X , Y ⟩+ ⟨X ,∇Z Y ⟩.

Notice that

∇X ( f Y )−∇ f Y (X ) = X ( f )Y + f (∇X Y −∇Y X ),

while[X , f Y ] = X ( f )Y + f [X , Y ],

so T (X , Y ) := ∇X Y −∇Y X − [X , Y ] is a tensor, called the torsion of ∇. Locally,if T ( ∂

∂ x i ,∂

∂ x j ) = T ki j∂

∂ xk then we may write T (X , Y ) = X iY j T ki j∂

∂ xk by C∞(M ,R)-linearity. If the torsion is zero then the connection is said to be symmetric.

3.2.1 Theorem. For any Riemannian manifold, there is a unique connection ∇such that

Curvature revisited 35

(i) ∇ is compatible with g; and

(ii) ∇ is symmetric, i.e. ∇X Y −∇Y X = [X , Y ].

This connection is the Levi-Civita connection of M .

PROOF: Suppose that ∇ exists. Then

(i) X ⟨Y, Z⟩= ⟨∇X Y, Z⟩+ ⟨Y,∇X Z⟩;

(ii) Y ⟨X , Z⟩= ⟨∇Y X , Z⟩+ ⟨X ,∇Y Z⟩;

(iii) Z⟨X , Y ⟩= ⟨∇Z X , Y ⟩+ ⟨X ,∇Z Y ⟩.

Then (i) + (ii) - (iii) gives

X ⟨Y, Z⟩+ Y ⟨X , Z⟩ − Z⟨X , Y ⟩= ⟨[X , Z], Y ⟩+ ⟨[Y, Z], X ⟩+ ⟨[X , Y ], Z⟩+ 2⟨X ,∇Y X ⟩ (4)

If ∇′ is another connection then ⟨Z ,∇Y X ⟩ = ⟨Z ,∇′Y X ⟩ for all Z , so ∇Y X = ∇′Y Xfor all X , Y . To show existence, define ∇ by (4) and check that ∇ is compatibleand symmetric.

In local coordinates, if ∇ ∂

∂ x i

∂

∂ x j = Γki j∂

∂ xk , then

Γki j g`k =

1

2

∂ g jk

∂ x i +∂ gki

∂ x j −∂ gi j

∂ x k

.

3.3 Curvature revisited

From now on we deal with a Levi-Civita connection on a Riemannian manifold.Denote the curvature of ∇ by R ∈ Ω2

M (End T M). For X , Y, Z ∈ V (M), R(X , Y )Z ∈V (M). This is C∞(M ,R)-linear in all three variables. Further, R(X , Y ) =−R(Y, X ).

Remark. If M = Rn and g is the usual inner product then Γki j ≡ 0, whence R ≡ 0.

(So Rn is flat, surprise, surprise. . . )

Locally, write R( ∂∂ xk , ∂

∂ x l )∂

∂ x j = Rijkl

∂

∂ x i . We have seen that Rijkl = −Ri

jlk. Wehave the expression

R=1

2

∂ Al

∂ x k−∂ Ak

∂ x l+ [Ak, Al]

d x k ∧ d x l ,

where A= Akd x k, so

R

∂

∂ x k,∂

∂ x l

=∂ Al

∂ x k−∂ Ak

∂ x l+ [Ak, Al].

There is a direct formula for Rijkl in terms of the gi j and their first and second

derivatives.


3.3.1 Proposition. R(X , Y )Z =∇X∇Y Z −∇Y∇X Z −∇[X ,Y ]Z .

The conclusion of this proposition may be taken as the definition of the curva-ture, given a Levi-Civita connection.

PROOF: It suffices to check this equation locally. Check first for ∂∂ x i , and then

use C∞(M ,R)-linearity. Use the fact that ∇ ∂

∂ x i= ∂

∂ x i + Ai (exercise).

Warning: some authors/books will use the convention that

R(X , Y )Z =∇Y∇X Z −∇X∇Y Z +∇[X ,Y ]Z

i.e. with the opposite sign.Recall that R(X , Y ) = −R(Y, X ), or Ri

jkl = Rijlk. Many times we will express

the symmetry conditions in terms of the 4-tensor ⟨R(X , Y )Z , T ⟩, locally given byRi jkl = giqRq

jkl .

3.3.2 Proposition (First Bianchi Identity).

R(X , Y )Z + R(Y, Z)X + R(Z , X )Y = 0

PROOF: Since the left hand side is linear, it suffices to check the equation locallyon a basis ∂

∂ x i with [ ∂∂ x i ,

∂

∂ x j ] = 0.

Locally this identity is Rijkl + Ri

l jk + Rikl j = 0.

3.3.3 Proposition.(i) ⟨R(X , Y )Z , T ⟩=−⟨R(Y, X )Z , T ⟩=−⟨R(X , Y )T, Z⟩

(ii) ⟨R(X , Y )Z , T ⟩=−⟨R(Z , T )X , Y ⟩

Locally, Ri jkl =−Ri jlk =−R jikl and Ri jkl = Rkli j .

PROOF:(i) The first equality we have done before. The second equality is a consequence

of 3.1.3 since the Levi-Civita connection is a metric connection, so the mapZ 7→ R(X , Y )Z is anti-symmetric.

(ii) From the first Bianchi identity,

⟨R(X , Y )Z , T ⟩+ ⟨R(Y, Z)X , T ⟩+ ⟨R(Z , X )Y, T ⟩= 0,

⟨R(Y, Z)T, X ⟩+ ⟨R(Z , T )Y, X ⟩+ ⟨R(T, Y )Z , X ⟩= 0,

⟨R(Z , T )Y, Y ⟩+ ⟨R(T, X )Z , Y ⟩+ ⟨R(X , Z)T, Y ⟩= 0,

⟨R(T, X )Z , Z⟩+ ⟨R(X , Y )T, Z⟩+ ⟨R(Y, T )X , Z⟩= 0.

Add and use part (i).

Sectional, Ricci, and Scalar curvature 37

3.4 Sectional, Ricci, and Scalar curvature

Sectional Curvature

For u, v ∈ Tx M (linearly independent), let |u ∧ v| =p

|u|2|v|2 − ⟨u, v⟩2, and letσ ⊆ Tx M be the 2-dimensional subspace spanned by u and v. Define

Kx(σ) :=⟨R(u, v)v, u⟩|u∧ v|2

,

the sectional curvature. Check that Kx(σ) depends only on σ and not the basischosen. In fact Kx(σ) determines R, i.e. if R′ is another multilinear map with thesame symmetry properties as R and K = K ′ then R= R′.

Remark. If dim M = 2 then there is only one plane to chose, spanned by ∂∂ x1 , ∂

∂ x2 ,so we get

Kx =⟨R( ∂

∂ x1 , ∂

∂ x2 )∂

∂ x2 , ∂

∂ x1 ⟩

EG− F2 =R1212

EG− F2

where E = | ∂∂ x1 |2, G = | ∂

∂ x2 |2, and F = ⟨ ∂∂ x1 , ∂

∂ x2 ⟩. But there is a formula for R1212

in terms of Γijk, so we get a formula for K in terms of the Christofel (sp?) symbols.

It is the same expression for K that arises in the Teoreme Egregium of Gauss.

Ricci Curvature

For u, v ∈ Tx M , define

Ricg(u, v) := Tr(w 7→ R(w, u)v),

the Ricci curvature. It is a symmetric bilinear form and Rici j = Rqiq j = g lqRil jq

(symmetry follows from the symmetry of R).

Scalar Curvature

The scalar curvature at a point x is the trace of the symmetric linear form associ-ated with Ricg , i.e. write Ricg(u, v) = ⟨Q x(u), v⟩, where Q x(u) is a symmetric map,and define sg(x) = Tr(Q x). Then sg(x) = g ik Ricik.

An Einstein manifold is a Riemannian manifold (M , g) for which Ricg = λg forsome λ ∈ R. An Einstein manifold is Ricci flat if λ= 0. A Ricci flow is a flow of theform ∂ gt

∂ t= −2Ricgt

(− λgt , here we think of the manifold being fixed, and theRiemannian metric varying with time, and λ is a “cosmological constant”). Seethe papers by M. Anderson in Notices of the AMS, 04.

3.5 Laplace(-Bertrami) operator

A bit of multilinear algebra

Let V be a real vector space with a positive definite inner product. Recall thatAp(V ) is the space of alternating p-forms, and if ω1, . . . ,ωn is a basis of V ∗,


then ωi1 ∧ · · · ∧ωip| i1 < · · · < ip is a basis of Ap(V ). We may also think of

Ap(V ) = ∧p(V ∗), the pth exterior power of V ∗.Let e1, . . . , en be an orthonormal basis of V with corresponding dual basis

ω1, . . . ,ωn of V ∗. This induces an inner product on V ∗ with respect to whichω1, . . . ,ωn is orthonormal. This induces also an inner product in Ap(V ) bydeclaring the basis ωi1 ∧ · · · ∧ ωip

| i1 < · · · < ip to be orthonormal. (Checkthat ⟨t1 ∧ · · · ∧ tp, s1 ∧ · · · ∧ sp⟩ = det⟨t i , s j⟩.) If V is oriented then we have madea choice of a non-zero top form in An(V ). But this space is one dimensional, sothere is a unique ωg that has norm 1 and defined the orientation of V .

3.5.1 Definition. The Hodge ∗ operator is a linear operator

∗ : Ap(V )→ An−p(V ) :ωi1 ∧ · · · ∧ωip7→ω j1 ∧ · · · ∧ω jn−p

such that ωi1 , . . . ,ωip,ω j1 , . . . ,ω jn−p

is a positively oriented basis of An(V ), e.g.∗(ω1 ∧ · · · ∧ωp) =ωp+1 ∧ · · · ∧ωn.

It’s easy to see (check) that for α,β ∈ Ap(V ), α ∧ ∗β = ⟨α,β⟩ωg . Also checkthat ∗∗= (−1)p(n−p).

Back to manifolds

Let (M n, g) be an oriented Riemannian manifold. We can do the constructionsabove over each Tx M and T ∗x M and we get a volume form ωg(x) for each x ∈ M ,called the Riemannian volume form. We also get a Hodge ∗ operator ∗ : Ωp(M)→Ωn−p(M) such that α∧ ∗β = ⟨α,β⟩xωg(x). Recall that d : Ωp(M)→ Ωp+1(M).

3.5.2 Definition. Define δ : Ωp(M)→ Ωp−1(M) by δ := (−1)n(p+1)+1 ∗ d∗ (thinkof divergence). The Laplace operator on Ωp(M) is ∆ : Ωp(M) → Ωp(M), definedby ∆ := dδ+δd.

(elliptic PDE)

3.5.3 Proposition. Suppose that M is compact. Then∫

M

⟨dα,β⟩ωg =

∫

M

⟨α,δβ⟩ωg

for all α ∈ Ωp−1(M) and β ∈ Ωp(M).

PROOF: We have

d(α∧ ∗β) = dα∧ ∗β + (−1)p−1α∧ d(∗β)

= dα∧ ∗β + (−1)p−1α∧ (−1)(p−1)(n−p+1) ∗ ∗d(∗β)= dα∧ ∗β −α∧ ∗(δβ)

∫

M

d(α∧ ∗β) =∫

M

dα∧ ∗β −∫

M

α∧ ∗(δβ)

0=

∫

M

⟨dα,β⟩ωg −∫

M

⟨α,δβ⟩ωg

Laplace(-Bertrami) operator 39

by Stoke’s Theorem.

3.5.3 means that for α,β ∈ Ωp(M), with respect to the inner product

(α,β)L2 :=

∫

M

⟨α,β⟩ωg ,

δ is the adjoint of d, i.e. (dα,β)L2 = (α,δβ)L2 . Notice that ∆ is self-adjoint withrespect to this inner product.

3.5.4 Definition. A form ω ∈ Ωp(M) is harmonic if ∆ω= 0.

3.5.5 Corollary. ω is harmonic if and only if dω = 0 and δω = 0 (the lattercondition is referred to as co-closed).

PROOF: One direction is obvious. Suppose that ∆ω= 0, so

0= (∆ω,ω)L2 = ((dδ+δd)ω,ω)L2 = (dδω,ω)+(δdω,ω) = (δω,δω)+(dω, dω).

Therefore both terms on the right hand side are zero since they are non-negative,andω is both closed and co-closed since the L2 inner product is positive definite.

Recall that H pdR(M ,R) is the space of closed p-forms modulo the exact forms,

so a class a ∈ H pdR(M ,R) is of the form a = α+dβ | dα= 0,β ∈ Ωp−1. LetH p =

ker∆, the space of harmonic p-forms. There is a natural map H p → H pdR(M ,R),

which turns out to be an isomorphism.Locally, let (U ,ϕ) to be a positively oriented chart inducing local coordinates

∂∂ x1 , . . . , ∂

∂ xn and d x1, . . . , d xn. Then

|d x1 ∧ · · · ∧ d xn|2 = det(⟨d x id x j⟩) = det(g i j),

so

d x1 ∧ · · · ∧ d xn =p

det(g i j)ωg and ωg =p

det(gi j)d x1 ∧ · · · ∧ d xn.

(In classical terms, ωg =p

EG− F2d x1 ∧ d x2.) We writep

g :=p

det(gi j).In the special case ∆ : Ω0 → Ω0, ∆ f = dδ f + δd f = δd f . Suppose that ψ is

a function with support contained in U . Then

−∫

1p

g

∂

∂ x j

pg g i j ∂ f

∂ x i

ψp

g d x1 · · · d xn =

∫

g i j ∂ f

∂ x i

∂ψ

∂ x j

pg d x1 · · · d xn

=

∫

⟨d f , dψ⟩ωg

= (δd f ,ψ)= (∆ f ,ψ)

=

∫

∆ fψωg

=

∫

∆ fψp

g d x1 ∧ · · · ∧ d xn,


so ∆ f = − 1p

g∂

∂ x j

pg g i j ∂ f

∂ x i

. With M = Rn and gi j = δi j , we recover the usual

Laplacian∆ f =−∑ ∂ 2 f(∂ x j)2

(sometimes the minus sign is omitted in the literature).

3.5.6 Theorem (Hodge Decomposition Theorem, 1935).Let M be a compact oriented manifold. Then dimH p <∞ for every 0 ≤ p ≤ n,and

Ωp(M) = ∆Ωp(M)⊕H p

= dδΩp(M)⊕δdΩp(M)⊕H p

= dΩp−1(M)⊕δΩp+1(M)⊕H p

and the direct sums are L2-orthogonal.

Remark. The first equality implies the other two. For orthogonality,

(dδβ ,δdβ) = (d2δβ , dβ) = 0

since d2 ≡ 0.

3.5.7 Corollary. Every de Rham cohomology class can be uniquely representedby a harmonic form.

PROOF: For a class a ∈ H pdR(M ,R), a = [α] for some closed p-form α. By the

Hodge Decomposition Theorem, we may write α= dβ +δν +ω, where ω ∈H p,β ∈ Ωp−1, and ν ∈ Ωp+1. Then 0 = dα = dδν , so 0 = (dδν ,ν) = (δν ,δν),implying δν = 0. Therefore α=ω+ dβ , so [ω] = [α] = a.

Ifω1 andω2 were two harmonic p-forms giving the same class thenω1−ω2 =dβ for some β ∈ Ωp−1, and δdβ = δ(ω1 −ω2) = 0, so (dβ , dβ) = (δdβ ,β) = 0,implying dβ = 0 and ω1 =ω2.

ThereforeH p → H pdR(M ,R) :ω 7→ [ω] is a linear isomorphism. Consequently

dim H p(M ,R) <∞ for all p when M is compact (and this is not an a priori trivialfact). These dimensions are called the Betti numbers.Aside: Write

e(α) :=1

2|α|2L2 =

1

2

∫

⟨α,α⟩ωg =1

2

∫

α∧α∗,

the energy of α. If α is closed then the cohomology class of α is represented by allforms of the form α+ dβ . Write e(t) = 1

2|α+ tdβ |2L2 , so

e(t) =1

2

(α,α) + 2t(α, dβ) + t2(dβ , dβ)

ande′(0) = (α, dβ) = (δα,β).

Then e′(0) = 0 for all β if and only if δα = 0, so the harmonic p-forms are thecritical points of e on the cohomology class.

Laplace(-Bertrami) operator 41

Poincaré Duality

Define a pairing Ωp(M)n−p(M)→ R : (α,β) 7→∫

Mα∧β . We claim that the pairing

descends to cohomology ([α], [β]) 7→∫

Mα∧ β . But

∫

M

(α+ dν)∧ β =∫

M

α∧ β +∫

M

(dν)∧ β =∫

M

d(ν ∧ β) = 0

by Stoke’s Theorem, since d(ν∧β) = (dν)∧β±ν∧dβ and β is closed. We furtherclaim that the pairing H p(M ,R)× Hn−p(M ,R)→ R is non-degenerate. Indeed, ifα is a harmonic p-form then β = ∗α is a harmonic n− p form (check), so

([α], [β]) 7→∫

α∧ ∗α= |α|2L2 6= 0

when α 6= 0. Since each class has a harmonic representative, this shows the formsis non-degenerate. It follows that when M is compact and connected, Hn(M ,R) =H0(M ,R) = R.

Outline of the proof of the Hodge decomposition theorem

“A foray into the dungeons of the analysts. They are one who make the world tick.If you spend enough time down there you might like it. Let’s hope they don’t turnout the lights. . . ”

We would like to show that Ωp = ∆Ωp ⊕ H p, and this direct sum is L2-orthogonal. In some sense we need to solve the equation ∆ω= α. Notice that forany ϕ, ⟨∆ω,ϕ⟩ = ⟨α,ϕ⟩, so (∆ω,ϕ)L2 = (α,ϕ)L2 , or (ω,∆ϕ)L2 = (α,ϕ)L2 since∆ is self-adjoint.

3.5.8 Definition. A linear functional ` : Ωp → R is said to be a weak solution of∆ω= α if

(i) ` is bounded, i.e. |`(β)| ≤ C |β | for some C constant and all β ∈ Ωp.

(ii) `(∆ϕ) = (α,ϕ)L2 for all ϕ ∈ Ωp.

If ω is an honest solution then `ω(ϕ) = (ω,ϕ)L2 is a weak solution. Can wefind even weak solutions? If we can one, can we get an honest solution out of it?

3.5.9 Theorem (Regularity Theorem). Any weak solution of ∆ω = α is of theform `ω for some ω ∈ Ωp.

3.5.10 Theorem (Compactness Theorem). Let αn ∈ Ωp, n ≥ 1. If there is aconstant C such that |αn|, |∆αn|< C for all n then αn has a Cauchy subsequence.

First note that the Compactness Theorem implies that H p is finite dimen-sional. Indeed, if not then there would exist an infinite orthonormal sequenceof orthonormal harmonic forms, which would contradict Compactness. Therefore


we may write Ωp = H p ⊕ (H p)⊥. Note that ∆Ωp ⊆ (H p)⊥, since (∆ϕ,ω) =(ϕ,∆ω) = 0 when ω ∈ H p. It remains to prove that given α ∈ (H p)⊥ we canfind ω ∈ Ωp such that ∆ω = α. To find a weak solution, proceed as follows.Define `|∆Ωp as `(∆ϕ) = (ϕ,α)L2 . This is well-defined since for ϕ1,ϕ2 ∈ Ωp with∆ϕ1 =∆ϕ2, ϕ1−ϕ2 ∈H p, so (ϕ1,α) = (ϕ2,α) for all α ∈ ϕ1−ϕ2 ∈ (H p)⊥. Wemust also check that `|∆Ωp is bounded (one can prove this again using the Com-pactness Theorem). The Hahn-Banach theorem allows us to extend ` to all of Ωp,and the Regularity Theorem gives us an honest solution.∆ is an elliptic operator. Elliptic operators have finite dimensional spaces of

solutions and the Regularity Theorem holds for them.

Ellipticity

We will work in Rn for the moment. Consider a function u : Rn → R and ageneral linear partial differential operator (PDO) P =

∑

|α|≤k aα(x)∂ α, where α =

(α1, . . . ,αn) ∈ Nn, |α| =∑

i αi , and ∂ α = ∂ |α|u

∂ xα11 ···∂ xαn

n. The “symbol of P” is given by

replacing ∂

∂ x j with iξ j , so P(x ,ξ) =∑

|α|≤k aα(x)(iξ)α, where yα =∏

j yα j

j . The

“principal symbol” is pk(x ,ξ) =∑

|α|=k aα(x)(iξ)α. For example, if P =∑n

i=1∂ 2

∂ x2i

then P2(x ,ξ) = −|ξ|2. Associated with the “wave operator” ∂ 2u∂ x2

1= ∂ 2u∂ x2

2the symbol

is is −ξ21+ξ

22. P is elliptic at x if Pk(x ,ξ) 6= 0 for all ξ 6= 0. P is elliptic it is elliptic

at all x .Recall that

∆ f =−1p

g

∂

∂ x j

g i jpg∂ f

∂ x i

.

The higher order term is g i j ∂ 2 f∂ x i∂ x j . (g i j) is positive semi-definite, so ∆ is elliptic

(under an appropriate extension of the definition of ellipticity to vector bundlesover manifolds).

3.5.11 Theorem (Bochner, 1946). Let M be a compact, orientable, connectedRiemannian manifold. Suppose that Ric ≥ 0 (i.e. this symmetric, bilinear form ispositive semi-definite). Then every harmonic 1-form is parallel.

For ω ∈H 1, ωx(v) = ⟨X (x), v⟩, where X is the dual vector field. ω is parallelif and only if ∇X = 0. If ωx = 0 then ω≡ 0 (since ω is parallel), soH 1→ T ∗x M :ω 7→ ωx is an injective linear map. Therefore dimH 1 ≤ dim M . In particular,dim H1(M ,R)≤ dim M .

Suppose Σg is a surface of genus g. Then Σg × Sn (n ≥ 2) is an (n + 2)-dimensional manifold, so if there is a positive semi-definite Ricci curvature then2g = dim H1 ≤ n+ 2. For large enough g we get a family of neat examples.

PROOF (OF BOCHNER’S THEOREM): If ω is a harmonic 1-form then the followingidentity holds for the dual vector field X .

−∆( 12|X |2) = |∇X |2 +Ric(X , X ).

Yang-Mills Equations 43

Integrating over M yields

0= (−∆( 12|X |2), 1)L2 =

∫

−∆( 12|X |2)ωg =

∫

|∇X |2ωg +

∫

Ric(X , X )ωg .

Whence ∇X = 0 since Ric(X , X )≥ 0.

This ends the examinable material of the course.

3.6 Yang-Mills Equations

Let M n be a compact oriented manifold and E a vector bundle of rank k overM . Suppose that we have a Riemannian metric on M and a Riemannian metricon E, and a metric compatible connection D, i.e. for s1, s2 ∈ Γ(E) = Ω0

M (E) andX ∈ V (M),

X ⟨s1, s2⟩= DX s1, s2⟩+ ⟨s1, DX s2⟩,

also called an orthogonal connection. In an orthogonal trivialization D = d + A,and A(X ) is a skew-symmetric matrix. Consider the structure group O(k), andnotice that A(X ) ∈ o(k), the Lie algebra of O(k). If FD is the curvature of Dthen FD ∈ Ω2

M (End E), and FD is also skew-symmetric. Recall that ΩpM (End E) =

Γ(Ap(M)⊗ End E) is the collection of End E valued p-forms. Let ΩpM (Ad E) be the

Ad(E)-valued p-forms, where Ad E is the follection of bundles over M given byelements in End E which are skew-symmetric. E.g. FD ∈ Ω2

M (Ad E), and if D1 andD2 are two metric compatible connections then D1 − D2 ∈ Ω1

M (Ad E).We will now define an L2-inner product on Ωp

M (Ad E) = Γ(Ap(M)⊗Ad E). If Aand B are skew-symmetric matrices then A·B =− tr(AB) is a natural inner product(it is the Killing form on o(k)). For ω ∈ Ωp(M) and s ∈ Γ(Ad E), define

⟨ω1 ⊗ s1,ω2 ⊗ s2⟩x =− tr(s1s2)⟨ω1,ω2⟩x

and as before define

(ω1 ⊗ s1,ω2 ⊗ s2)L2 =

∫

M

⟨ω1 ⊗ s1,ω2 ⊗ s2⟩xωg

We also have a ∗-operator acting ΩpM (Ad E)→ Ωn−p

M (Ad E) as ω⊗ s 7→ (∗ω)⊗ s.

3.6.1 Definition. The Yang-Mills functional is

Y M : D→ (FD, FD)L2 =

∫

M

⟨FD, FD⟩xωg .


Harmonic forms pop up as critical points of the energy function. What are thecritical points of the Yang-Mills functional? Let A∈ Ω1

M (Ad E), so we have

FD+tA(σ) = (D+ tA)((D+ tA)(σ))= (D+ tA)(Dσ+ tA)(σ)

= D2σ+ tD(Aσ) + tA(Dσ) + t2A∧ A(σ)

= D2σ+ t((DA)σ− A∧ Dσ) + tA(Dσ) + t2A∧ A(σ)

= (FD + tDA+ t2(A∧ A))σ)

since we can extend D : ΩpM (Ad E) → Ωp+1

M (Ad E) to satisfy a Leibniz rule as wehave earlier in the course. Therefore

d

d t

t=0Y M(D+ tA) = 2(DA, FD)L2 .

For D : ΩpM (Ad E) → Ωp+1

M (Ad E), define D∗ : ΩpM (Ad E) → Ωp−1

M (Ad E) to be theL2-adjointof D. Then critical points of the Yang-Mills functional are exactly thoseD for which D∗FD = 0. This is the Yang-Mills equation. Recall also the BianchiIdentity which gives DFD = 0.

In the case p = 2, ∗∗ = (−1)2(n−2) = 1. One may check that when the dimen-sion of M is even then D∗ = − ∗ D∗. Whence in this case the Yang-Mills equationreduces to D ∗ FD = 0.

Remark.(i) The same construction can be done (and often is in Physics) for a Hermitian

inner product and unitary connections.

(ii) When k = 2 in the O(2) orthogonal case or k = 1 in the U(1) unitary case,we get Maxwell’s equations. We get Hodge theory for 2-forms (D ∗ FD = 0corresponds to FD being co-closed and DFD = 0 corresponds to FD beingclosed).

Locally, if gi j = δi j (say if M = R4) then if D = d + A, where A = Aid x1 andFD = Fi jd x i ∧ d x j , then

D∗FD =

−∂ Fi j

∂ x i − [Ai , Fi j]

d x j = 0

so the Yang-Mills equations are

δFi j

∂ x i + [Ai , Fi j] = 0for j = 1, . . . , n

where Fi j =δA j

∂ x i −δAi

∂ x j + [Ai , A j].Again, the set up is as follows. E → M is a vector bundle, we have Rieman-

nian metrics on E and M , and a metic/compatible/orthogonal connection D. Weextend D to Ωp

M (Ad E)→ Ωp+1M (Ad E).

Yang-Mills Equations 45

3.6.2 Definition. The Gauge group is g = AutG(E), the collection of automor-phisms h of E such that for each x ∈ M , hx : Ex → Ex is orthogonal, where G isthe structure group (O(k) in our case).

The important point is that g acts on the space of metric connections. If h ∈AutG(E) and D is a metric connection then define h∗D = h−1 D h, i.e. h∗D(s) =h−1D(hs) for all s ∈ Γ(E). Locally one may check that if D = d + A then h∗(A) =h−1dh+ h−1Ah and h∗F = h−1 F h. Further,

Y M(h∗D) = (Fh∗D, Fh∗D)L2 = (FD, FD)L2 = Y M(D)

since each element of AutG(E) is an isometry. Hence Y M is invariant under theGauge group. It follows that critcal points, i.e. Yang-Mills connections, are mappedto one another under the action of g.

3.6.3 Definition. The moduli spaceM of M is the space of Yang-Mills connectionsmodulo the action of g.

3.6.4 Example. When k = 2 and E = M ×R2 is the trival rank 2 vector bundleover a compact oriented manifold Riemannian manifold M , the structure groupis SO(2). The associated Lie algebra is skew-symmetric 2× 2 matrices and so isisomorphic to R. Then we may write D = d + A globally since E is trivial, whereA ∈ Ω1

M (Ad E). Then A(x) = 0 ax−ax 0

where ax is a 1-form, so we may think ofA as an ordinary 1-form, and Ω1

M (Ad E) = Ω1(M). For s ∈ Γ(E), s : M → R, soDs = ds+

0 a−a 0

s1s2

, and

FD = dA+ A∧ A= dA=

0 da−da 0

.

The Yang-Mills equation gives that δda = 0, so da is a harmonic 2-form. Bytriviality, δda = 0 and da harmonic imply that da = 0, so a is a closed 1-form.Hence any Yang-Mills connection is given by a closed 1-form on M , and visa versa.

What is the action of g = SO(2)? For x ∈ M , h(x) ∈ SO(2) and h∗(A) =h−1dh+ h−1Ah, and h(x) is orthogonal, so we may write h(x) = exp

0 u−u 0

. Thush∗(A) = du + A since SO(2) is Abelian. As we let u run over R, we see thatM = H1(M ,R). In particular, if H1(M ,R) = 0 thenM is just a single point.

3.6.5 Theorem (Donaldson, mid 80’s). Let M be a compact simply connected4-manifold. We have a pairing

Γ : H2(M ,R)×H2(M ,R)→ R : ([α], [β]) 7→∫

α∧ β ,

and if Γ is definite then there is a basis such that it is ±I .

Consider in particular an SU(2)-bundle over M with∫

h(F ∧ F) = −8π2. Heproved that in this case,M is compact, 5-dimensional, and is smooth a.e. exceptfor some “core singularities” and has M as a boundry. And so on. . .

Index

Ap(V ), 13C∞ differentiable structure, 2T M , 8T ∗M , 8Tp M , 6Γ(E), 29Ωp(M), 14

associated bundles, 25

base space, 23Betti numbers, 40boundry, 19bundle automorphism, 27bundle chart, 8bundle isomorphism, 27bundle morphism, 26

canonical 1-form, 22canonical basis, 7canonical submersion, 12chart, 2closed, 16co-closed, 39cocyle conditions, 24codimension zero submanifold, 19coefficients of the connection, 28compatible, 2connection, 28, 29, 34continuous map, 2coordinate neighbourhood, 2cotangent bundle, 8covariant-constant, 32curvature, 32

de Rham cohomology, 16de Rham complex, 16derivation, 7diffeomorphic, 3diffeomorphism, 3differential, 6differential p-form, 14dimension, 3domain with smooth boundry, 19dual basis, 14

Einstein manifold, 37elliptic, 42elliptic operator, 42embedded submanifold, 11energy, 40exact, 16exponential, 4exterior product, 13

fibre, 8first order linear operator, 9flat connection, 32flow, 8frame, 31free action, 24

Gauge group, 45geodesic, 23

half-translation, 10Hamiltonian flow, 23Hamiltonian vector field, 23harmonic, 39Haudorff, 2Hodge ∗ operator, 38homeomorphism, 2Hopf bundle, 26Hopf map, 26horizontal subspace, 27

immersion, 11, 12integral curves, 8

Laplace operator, 38left-invariant vector field, 11Legendre transform, 21Levi-Civita, 31Levi-Civita connection, 35Lie algebra, 10Lie bracket, 9Lie group, 4local diffeomorphism, 12local operator, 30logarithm, 4

47

48 INDEX

metric connection, 33moduli space, 45musical isomorphisms, 21

open, 2orientable, 17orientation preserving, 18orthogonal connection, 33orthogonal trivialization, 34outward directed, 19

parallel, 32parallel transform, 32parallelizable, 8partition of unity, 17phase space, 8positively oriented, 19principal bundle, 24product connection, 33pullback, 14, 15pullback bundle, 26

rank, 23regular value, 12Ricci curvature, 37Ricci flat, 37Ricci flow, 37Riemannian manifold, 21Riemannian metric, 21Riemannian volume form, 38

scalar curvature, 37second countable, 2section, 8, 25sectional curvature, 37smooth 1-form, 8smooth action, 24smooth map, 3smooth structure, 2structure group, 24submersion, 12subordinate, 17symmetric, 34symplectic form, 21symplectic manifold, 21

tangent bundle, 8

tangent space, 6tautological bundle, 25topological space, 2torsion, 34total space, 23transition function, 24, 25trivial bundle, 24trivial connection, 33trivialization, 23trivializing neighbourhood, 23typical fibre, 23

unitary connection, 33

vector bundle, 8, 23vector field, 8vertical subspace, 27volume form, 14

weak solution, 41wedge product, 13, 15

Yang-Mills equation, 44Yang-Mills functional, 43

differential geometry michaelmas 2006 dr. g.p. paternain∗

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