1 Manifolds, tangent planes, and the implicitfunction theorem

If U ⊆ Rn and V ⊆ Rm are open sets, a map f : U → V is called smooth orC∞ if all partial derivatives of all orders exist. If instead A ⊆ Rn and B sseRmare arbitrary subsets, we say that f : A → B is smooth if there is an openneighborhood Ux ⊆ Rn around every x ∈ A so that f extends to a smooth mapF : Ux → Rm, that is, F = f on Ux ∩ A. A smooth bijection with a smoothinverse is called a diffeomorphism.

A subset M ⊆ Rk is called an m–manifold if M is locally diffeomorphic toRm. (To say that a space A is “locally X” means that each point x ∈ A iscontained in an open neighborhood U ⊆ A so that U is X .) Notice that forany ε > 0, the open ball Bm(ε) ⊆ Rm is diffeomorphic to Rm via the radialmap which sends any point x ∈ Bm(ε) of distance r from the origin to 1

ε−rx.Therefore, saying that “M is locally diffeomorphic to the entire space Rm” isequivalent to saying that “M is locally diffeomorphic to open sets in Rm”. Inparticular any open set of Rn is an n–manifold, but many n–manifolds are not ofthis form. For example the sphere Sn =

∑x2i = 1 ⊆ Rn+1 is an n–manifold

(prove this), but since it is compact it is not diffeomorphic to any open set inRn.

Suppose again U ⊆ Rn V ⊆ Rm are open sets, f is a smooth map betweenthem, and x ∈ U . Then we define TUx = Rn, called the tangent plane to U atx. We define Dfx : TUx → TVf(x) by the formula

Dfx(v) = limh→0

1

h(f(x+ hv)− f(x))

where v ∈ TUx. We note that Dfx is a linear map of vector spaces. (Dependingon the text, dfx, (f∗)x, or even Tfx is used to denote the same thing. dfxis probably most common, but we avoid this to avoid confusion with exteriordifferentiation later.)

Let M ⊆ Rk be an m–manifold, let x ∈ M , and let ϕ : Rm → M be adiffeomorphism with the neighborhood U = Im(ϕ) ⊆ M containing x. (Such aϕ is called a coordinate system or local coordinates or similar, also U is oftencalled a coordinate chart.) We then define TMx = Im(Dϕx). (Here we think ofϕ as a map Rm → Rk and apply the previous definition.) TMx does not dependon ϕ, only on M and x (prove this, by choosing a different coordinate systemψ).

Let M ⊆ Rk and N ⊆ Rl be manifolds, and let f : M → N be a smoothmap. Then for each x ∈M we define Dfx : TMx → TNf(x) by choosing a localextension of f to F : U → N , and defining Dfx = DFx|TMx

. If instead wechoose coordinate systems ϕ : Rm →M and ψ : Rn → N we note that

DFx|TMx= Dψψ−1(f(x)) D(ψ−1 f ϕ)ϕ−1(x) D(ϕ−1)x

citing the chain rule. This shows that the image of Dfx is contained in TNf(x),and that it does not depend on the choice of F . Again we see that Dfx is

1

a linear map, and we also have the chain rule in the context of manifolds: iff : M → N and g : N → Q are maps of manifolds then

D(g f)x = Dgf(x) Dfx.

In particular, if f : M → N is a diffeomorphism (globally, or near a pointx ∈ M), then Dfx = (Df−1

f(x))−1. So Dfx is a linear isomorphism, and in

particular dimTMx = dimTNf(x). Thus the tangent plane to an n–manifoldat any point is an n–dimensional vector space, and two diffeomorphic manifoldsmust have the same dimension.

The inverse function theorem is the converse of this fact:

Theorem 1.1 (Inverse function theorem). Let f : M → N be a smooth map,and suppose Dfx : TMx → TNf(x) is a linear isomorphism for some x ∈ M .Then there is an open set U ⊆ M containing x, so that f |U : U → f(U) is adiffeomorphism.

Proof. Because the theorem is local, we can work in a coordinate chart. There,the theorem reduces to the standard inverse function theorem from multivariableanalysis class.

Note however that, even if Dfx is an isomorphism at every point x ∈ M ,it does not follow that f is a diffeomorphism: a smooth map which is a localdiffeomorphism everywhere does not need to be a diffeomorphism. For example,if S1 = eiθ; θ ∈ R ⊆ C = R2 is the circle and f : S1 → S1 is definedby f(z) = z2, then f is a local diffeomorphism everywhere but it is injectivenowhere.

Definition 1.2. Let f : M → N be a smooth map of manifolds, and let x ∈M .We say x is a regular point of f if Dfx is surjective, and x is a critical pointotherwise. For y ∈ N , we say that y is a regular value of f , if each point x ∈Mwith f(x) = y is a regular point. If y is not a regular value, we call y a criticalvalue.

Notice that, if dimM < dimN then Dfx cannot be surjective for any x.However if y ∈ N is not in the image of f , then vacuously y is a regular value.

Proposition 1.3. Let f : M → N and let x ∈ M be a regular point. Thenfor some coordinate choice near x and f(x), we can present f : Rm → Rn asf(x1, . . . , xm) = (x1, . . . , xn).

Proof. We can choose coordinates so that f : Rm → Rn with x = 0 ∈ Rm andf(x) = 0 ∈ Rn (here and throughout these notes we will often drop the notationfor the coordinates, writing x = 0 instead of x = ϕ(0) for coordinates ϕ : Rm →M). Df0 is assumed to be surjective so after composing by a linear coordinatechange we can assume that Df0 =

[Idn

0

]. Then let F : Rm → Rm be the

map F (x1, . . . xm) = (f(x1, . . . , xm), xm+1, . . . , xn), so DF0 = Idn. The inversefunction theorem states that F−1 exists (locally), and if we re-coordinatize Rmvia F−1 then f is these coordinates is equal to f F−1, which is the standardmap as claimed.

2

Theorem 1.4. Let f : M → N be a smooth map and let y ∈ N be a regularvalue. Then f−1(y) ⊆ M is a smooth manifold. It’s dimension is equal todimM − dimN and for x ∈ f−1(y) we have Tf−1(y)x = kerDfx.

Proof. The theorem is local so we can assume that M = Rm, N = Rn, andf(x1, . . . , xm) = (x1, . . . , xm), as proved above. But in this case the statementis obvious.

Definition 1.5. Let f : M → N be a smooth map of manifolds. If Dfx issurjective at all points x ∈M , we say that f is a submersion. If instead Dfx isinjective at each point we say that f is an immersion.

Proposition 1.6. Suppose f : M → N is an immersion. Then around anypoint x ∈M , we can find coordinate systems near x and f(x) so that

f(x1, x2, . . . , xm) = (x1, x2, . . . , xn, 0, . . . , 0).

Here dimM = m ≥ n = dimN and p corresponds to the point 0 in the givencoordinates.

Proof. Again this immediately follows from the implicit function theorem.

Proposition 1.7. Let f : M → N be an injective, proper immersion. Thenf(M) is a submanifold and f : M → f(M) is a diffeomorphism.

(A submanifold is a subset of a manifold which is itself a manifold (as asubset of Rk).

Proof. By the standard model for immersions, it follows that on small setsU ⊆ M the map f |U : U → f(U) is a diffeomorphism onto its image. Ifadditionally f(U) is an open set of f(M), then the inverse function is smooth(it is continuous since it is an open map and it is smooth by the inverse functiontheorem). Thus it remains to show that f : M → f(M) is an open map.

Let y ∈ f(U) be the limit of a sequence yk ∈ f(M), and let x = f−1(y),xk = f−1(yk). Since f is proper the set x, xk is compact, being the preimageof a compact set. Then xk has a convergent subsequence with limit x∞, andby continuity of f , f(x∞) is the subsequential limit of f(xk) = yk, which is y.Thus f(x∞) = y so x∞ = x. Since x ∈ U and U is open then xk ∈ U for largek, so yk ∈ f(U) for large k, implying that f(U) is open.

Definition 1.8. Let f : M → N be a smooth map, and let Q ⊆ N be asubmanifold. Then we say f is transverse to Q (writing f t Q) if Im(Dfx) +TQf(x) = TNf(x) for all x ∈ f−1(Q).

A couple notes about transversality. First, notice that when Q = y (asingle point is a 0–manifold), then f t Q means exactly that y is a regularvalue of f . Secondly notice that, if M ⊆ N is a submanifold and f is theinclusion map, then M t Q if and only if Q tM .

3

2 The tangent bundle, vector fields, and flows

A useful tool is to fit all the tangent planes together in a single package.

Definition 2.1. Let M ⊆ Rk be a smooth n–manifold. We define the tangentbundle of M to be the space TM = (x, v) ∈ Rk × Rk;x ∈M,v ∈ TMx.

We note a few properties:

1: TM is a smooth manifold. (This follows because the derivatives of smoothmaps are also smooth.)

2: TM is equipped with a projection π : TM →M , so that TMx = π−1(x).

3: If f : M → N is a smooth map, then we obtain a smooth map Df : TM →TN by the equation Df(x, v) = (f(x), Dfx(v)). We have πN Df =f πM .

From here, we can define a smooth vector field to be a smooth choice oftangent vector at each point. More precisely:

Definition 2.2. Let M be a manifold. A smooth vector field (often simplycalled a vector field) is a smooth map V : M → TM so that π V = id.

A closely related concept is that of an isotopy. Let I ⊆ R be an interval1

containing 0, which we coordinatize with the variable t ∈ I. An isotopy is asmooth map ϕ : M × I →M so that ϕ0 = id and ϕt is a diffeomorphism for allt ∈ I (we denote the function ϕ(·, t) : M →M by ϕt). Given an isotopy ϕt anda ∈ I, we can define the time derivative of ϕt at t = a, denoted by ϕt|t=a (orsometimes simply ϕa). This time derivative is a vector field on M , defined asfollows:

ϕt|t=a = Dϕ|t=a(∂

∂t

) ϕ−1

a .

This definition deserves some explanation. The manifold I has a canonicalvector field which we denote ∂

∂t , defined by the canonical identification TI =

I × R, and ∂∂t is the constant function t 7→ (t, 1). Notice that this is not a

diffeomorphism invariant! It depends on the parametrization of I.Since T (M × I) = TM ×TI, ∂

∂t is also a vector field on M × I (which is thezero vector in the M direction). Then, at each point x ∈ M , we get a vectorDϕ(x,a)

(∂∂t

)∈ TMϕa(x). This is something like a vector field, except we get a

vector at the wrong point! That is, π Dϕ|t=a(∂∂t

)= ϕa. So instead we can

precompose with the function ϕ−1a . Thus

ϕt|t=a(x) = Dϕ(ϕ−1a (x),a)

(∂

∂t

)∈ TMx.

Often times we will write simply ϕt to denote a 1–parametric family of vectorfields on M .

1We may assume that I is an open interval, though it is easy to check that everythingworks equally well if I has endpoints.

4

Proposition 2.3. Let ft : M →M and gt : M →M both be isotopies, and letht be the isotopy ht = ft gt. Then

ht = Dft(gt) f−1t + ft.

Proof. Think of h, f and g as maps M × I → M . By definition of ht we can

write Dh =[Dht ht ht

], and similar for f and g. Since h(x, t) = f(g(x, t), t),

the chain rule says

Dh(x,t) =[Dft ft ft

](g(x,t),t)

[Dgt gt gt

0 1

]x

.

Then the second column of this matrix, which is (ht ht)(x), is equal to[Dft]g(x,t)(gt gt) + ft (ft gt), composing with h−1

t completes the result.

Corollary 2.4. Let ϕt : M → M be an isotopy, and let f : M → M be adiffeomorphism. Then

˙(ϕt f) = ϕt, and ˙(f ϕt) = Df(ϕt) f−1.

Of course, the Corollary follows from the above by letting one of the twoisotopies be constant in t.

Theorem 2.5 (Integration of flows). Let Vt be a time-dependent family of vectorfields on a manifold M (without boundary), i.e. V : M × I → TM is a smoothmap so that Vt = V (·, t) is a vector field for all t ∈ I. Then there is an openset U ∈ M × I, containing M × 0, and a smooth map ϕ : U → M satisfyingϕa(x) = Va(x) whenever (x, a) ∈ U , which is a “partially defined isotopy”. Thatis to say ϕ0 = id, and for any t ∈ [0, 1] and any open set U ∈ M so thatU × t ⊆ U , we have that ϕt : U →M is a diffeomorphism onto its image.

If additionally M is compact, then U = M × [0, 1], i.e. ϕt is an isotopy.

Proof. First we consider the problem locally. For an open set U ∈ Rn, a vectorfield is a map V : U → Rn. If ϕ : U0× (−ε, ε)→ U is a partially defined isotopyand x ∈ U is fixed, then the map t 7→ ϕt(x) defines a curve γx : (−ε, ε) → U ,and

ϕt|t=a =dγxdt

∣∣∣∣t=a

∈ Rn.

So to start, we are looking to find a curve γx : (−ε, ε)→ U so that γx(0) = x

and dγxdt

∣∣∣t=a

= Va(γx(t)). But this is just a first order ordinary differential

equation, with initial condition x! By the existence theorem for solutions tosuch equations, we know that γx exists, at least for sufficiently small ε > 0.Furthermore γx is smooth, (solutions of smooth first order ODEs are smooth),and furthermore the map ϕ defined by ϕ(x, t) = γx(t) is smooth (since solutionsdepend smoothly on initial conditions). This proves the first claim.

5

Notice that ε depends on x, so ϕ may not be able to be defined on M×(−ε, ε)for any positive ε. For example, if M = (0, 1) and V = ∂

∂x (constant with respectto t), then the largest possible domain of ϕ is the set U = (x, t) ∈ (0, 1)×R; 0 <x+ t < 1.

However, there is a well-defined maximal set Umax which is the largest possi-ble set in which ϕ can be defined. For if we have solutions ϕ1 defined on U1 andϕ2 defined on U2, the fact that solutions to first order ODEs is unique showsthat ϕ1 = ϕ2 on U1∩U2 and therefore we can define ϕ on U1∪U2 in the obviousway. Umax is an open set in M × I, because if (x, t) ∈ Umax then the existencetheorem lets us define ϕ in an open neighborhood of (x, t). Thus to prove thatUmax = M × I it suffices to prove that Umax is closed. In fact, since for each x,Umax ∩ x × I 6= ∅, it suffices to show that the sets Umax ∩ x are closed.

Now suppose that M is compact. Let Ix = Umax ∩ x, let γx : Ix → M bedefined by γx(t) = ϕt(x), and suppose that (x, t) lies in the closure of Ix. Giventhe embedding M ⊆ Rk, let ||Vt(x)|| be the Euclidean norm of Vt(x) ∈ TMx ⊆Rk. Restricting to the set M × [0, t], compactness gives a uniform upper bound

K of ||Vt(x)|| on this set, and therefore∣∣∣dγxdx ∣∣∣ < K for all t ∈ [0, t). Therefore

the image of γx|[0, t) lies in the ball centered at x whose radius is Kt. From thisit follows that limt→t γx(t) exists: a subsequence must converge since it lives ina compact set, and there can be at most one limit point since the derivative ofγx is bounded. Defining γx(t) to be this limit, it follows from the definition ofderivative that dγx

dt is well defined and continuous at t = t, and therefore equalsVt(γx(t)). This shows that (x, t) ∈ Ix which completes the proof.

Notice that, if M is not compact but Vt 6= 0 on a compact subset of M , thenwe can find a global isotopy ϕt. But even this is not necessary, for example thevector field ∂

∂x on M = R globally integrates to an isotopy.Also notice that, if V is a vector field that does not depend on time, then

ψt = ϕt+a ϕ−1a also satisfies ψt = V . It follows that ψt = ϕt, so ϕs+t = ϕs ϕt

for all s, t ∈ R. Therefore the space of vector fields on M is homeomorphic withthe space of smooth group homomorphisms from R to Diff(M).

3 Abstract manifolds and the Whitney embed-ding theorem

Theorem 3.1. Let X be a compact Hausdorff space with open covering X =⋃i Ui. Suppose we are given homeomorphisms fi : Ui → Mi where Mi are n–

manifolds. Finally suppose that, for every pair i, j, the homeomorphism fif−1j :

fj(Ui)→ fi(Uj) is smooth. Then X is an n-manifold.

Note. If X is not compact, but second countable, the theorem remains true.Usually, the hypotheses of the theorem are taken as the definition of an abstractn-manifold. Then the Whitney embedding theorem, stated in the traditionalway, is “any (abstract) manifold can be realized as a submanifold of RN forsome N”.

6

Lemma 3.2. Let M be a manifold. Then there is a locally finite open coveringVj of M , so that V j is compact, and smooth functions ϕj : M → [0,∞) sothat ϕj(x) > 0 if and only if x ∈ Vj.

Proof: Choose an arbitrary locally finite open covering of M by charts Vj, sothat each Vj is diffeomorphic to an open subset of Rn. Any open set Vj ⊆ Rnadmits a locally finite open covering by sets whose closures are diffeomorphicto the closed ball, thus we can assume Vj ∼= Bn(1). To construct the functionsϕj , simply note that the open ball Bn(1) ⊆ Rn admits such a function:

ϕ(x) =

exp( −1

1−r ) r ∈ [0, 1)

0 r > 1.

Proof of Whitney’s embedding theorem: Since X is compact the given opencovering can be chosen to be finite: i = 1, . . . ,K. For each Mi, find a coveringVij as in the lemma, and corresponding functions ϕij . We define ψij = ϕijfi :Ui → [0,∞), and we extend the domain of definition ψij : X → [0,∞) by lettingit be zero outside of Ui. This extension is continuous since X is Hausdorff.

(To show this let A ⊆ [0,∞) be a closed set, and consider ψ−1ij (A). If 0 ∈ A

then ψ−1ij ([0,∞) \A) ⊆ Ui so this set is open since fi is a homeomorphism onto

its image. If 0 /∈ A then ψ−1ij (A) is compact (since Uij is compact), and since X

is Hausdorff this set is closed.)

Define ψi : X → [0,∞) by

ψi =

∑j

ψij∑i

∑j

ψij

then∑i ψi = 1 and ψi(x) > 0 if and only if x ∈ Ui).

Suppose that Mi ⊆ RN , we let ei : Mi → RN be the inclusion maps. Wethen define e : X → RNK+K by

e(x) =(ψ1(x)(e1 f1)(x), . . . , ψK(x)(eK fK)(x), ψ1(x), . . . , ψK(x)

).

Notice that e is well defined since whenever x /∈ Ui) then ψi(x) = 0. e is

obviously continuous, and it is injective because if e(x) = e(y) then ψi(x) =

ψi(y) for all i. Choosing such an i so that ψi(x) 6= 0, we get (ei fi)(x) =(ei fi)(y), and the map ei fi is injective. Since X is compact it follows thate is a homeomorphism onto its image.

It remains to show that e(X) is smooth, it suffices to show that e(Ui) issmooth for all i. But e(Ui) is equal to the image of the smooth map Mi →RNK+K given by

z 7→(

(ψ1 f−1i )(z)(e1 f1 f−1

i )(z), . . . , (ψK f−1i )(z)(eK fK f−1

i )(z),

7

(ψ1 f−1i )(z), . . . , (ψK f−1

i )(z)).

4 Vector fields as operators

4.1 Covectors and 1–forms

Let V be a finite dimensional vector space over R. Recall that the dual of V ,V ∗ = L : V → R is defined to be the set of all linear maps to R. V ∗ isisomorphic to V , but there is no canonical isomorphism.

In the particular case of tangent planes TMx, we call the dual T ∗Mx =(TMx)∗ the cotangent plane to M at x. Just as with tangent planes, we canbundle all of the cotangent planes together to build a smooth vector bundle.

Definition 4.1. Let M be an n–manifold. Then the cotangent bundle of Mis the space of pairs T ∗M = (x, α);x ∈ M and α ∈ T ∗Mx. T ∗M is giventhe structure of a smooth 2n–manifold by choosing charts ϕ : U → M on Mwith U ⊆ Rn open, and letting the map (u, α) 7→ (ϕ(x), α Dϕ−1

ϕ(u)) be a chart

T ∗U = U × (Rn)∗ → T ∗M . (Here we use Whitney embedding.)By definition, T ∗M comes equipped with a submersion π : T ∗M →M whose

fiber is π−1(x) = T ∗Mx.

Maps α : M → T ∗M satisfying π α are called either covector fields or1-forms. We denote the space of all 1-forms by either Γ(T ∗M) or Ω1(M).If we denote the space of all smooth vector fields by Γ(TM), we can writeΩ1(M) ⊆ Hom(Γ(TM), C∞(M)). Here we can think of Hom as being simplyhomomorphisms of real vector spaces, but if instead we restrict our homomor-phisms we get equality.

Proposition 4.2. Let Hom denote the set of homomorphisms which are linearover C∞M and continuous. Then Ω1(M) = Hom(Γ(TM), C∞(M)).

If f : M → N is a smooth map, and α ∈ Ω1(N), we can define f∗α ∈ Ω1(M)by the formula (f∗α)(v) = α(Df(v)) for any v ∈ TM . We call f∗α the pull-backof α under f .

Notice that this contrasts with vectors and vector fields. Though there isalways a diffeomorphism TM → T ∗M which is linear on each fiber, this map isnot canonical. Making this choice, we have a bijective correspondence betweenvector field and 1-forms. Nevertheless, you should not think of vector fields and1-forms as being essentially the same thing! Since this correspondence dependson arbitrary choices, the properties of vector fields up to diffeomorphism arequite different than the properties of 1-forms.

To sum up all the structure we’ve defined thus far: a smooth f : M → Ninduces a map Df : TM → TN , but there is no map Γ(TM) → Γ(TN). Wehave a map f∗ : Ω1(N) → Ω1(M), but there is no map T ∗N → T ∗M . Bothsatisfy a chain rule: if g : N → P then D(gf) = DgDf and (gf)∗ = f∗g∗.

8

Vector fields arise naturally as the time derivative of isotopies, and nothing likethis is true for covector fields.

4.2 Directional derivative

Recall that at each point y ∈ R, the tangent space TRy is canonically isomorphicto R. This allows us to define a canonical 1–form on R denoted by dy: itsdefinition is dy( ∂∂y = 1. If h : M → R is a smooth function on a manifold,

we define dh = h∗dy ∈ Ω1M . Thus by definition at each point x ∈ M wehave dhx = Dhx.2 We call this operation d : C∞M → Ω1(M) the exteriordifferential. Notice that it is Leibnitzian: if h1, h2 : M → R, then d(h1h2) =h1dh2 + h2dh1. Also exterior differentiation commutes with pullbacks: it f :N → M is a smooth map and h : M → R then the chain rule immediatelyimplies that d(h f) = f∗(dh).

This concept gives another important distinction between vector fields andcovector fields: non-vanishing vector fields are locally standard but non-vanishingcovector fields are not! Since a vector field generates an isotopy by taking itsflow, any non-vanishing vector field can be given as ∂

∂x1for some choice of local

coordinate system (prove this!). On the other hand, some non-vanishing cov-ector fields can be locally described as the differential of a function, and somecannot. If it can, we say that it is closed or sometimes conservative.

To make things quite clear, on R2 the 1-form dx is not locally diffeomorphicto the 1-form (y2 + 1)dx, because the latter cannot be written as df for anyfunction f ∈ C∞(R2), even locally (why?). However, using the standard metric,these 1-forms are respectively dual to the vector fields ∂

∂x and (y2+1) ∂∂x , and the

diffeomorphism (x, y) 7→ ((y2 + 1)x, y) maps the former to the latter (globallyeven).

Suppose V is a vector field on M , and f ∈ C∞(M). Then df ∈ Ω1M , and wecan evaluate df(V ) which is itself a smooth function on M . Thus given any Vwe get a map V : C∞M → C∞M : we will often write V (f) = df(V ) if we wishto emphasize V as an operator on smooth functions. This is simply the familiardirectional derivative from vector calculus. If x ∈M , notice that V (f)(x) onlydepends on the vector Vx ∈ TMx, but it depends on f in a neighborhood of x.Phrased differently, this means that V (f) is tensorial in V but not in f .

Notice that the function V : C∞(M) → C∞(M) is linear and also Leib-nitzian, that is, it satisfies the equality V (fg) = fV (g) + gV (f). This followsimmediately from the corresponding property of exterior differentiation. In fact,these properties characterize vector fields.

Proposition 4.3. Let F : C∞(M)→ C∞(M) be any function which is linear,Leibnitzian, and continuous. Then F is induced by a unique vector field V .

Proof. First, notice that if F and G are linear and Leibnitzian, and h ∈ C∞(M),then F + G and hF are also linear and Leibnitzian. Therefore we can work

2We use slightly different notation here to distinguish between Dh : TM → TR anddh : TM → R. Thus while Dh and dh are not exactly the same thing, they contain essentiallythe same data, Dh = (h, dh).

9

locally: if every function F : C∞(Rn) → C∞(Rn) is induced by a vector fieldon Rn, then for an arbitrary manifold M we can use a partition of unity todecompose F into functions which are zero outside of single charts.

Let fi : Rn → R be defined by F (xi) (where x1, . . . , xn are coordinatefunctions on Rn), and let V = f1V1 + . . . + fnVn, where V1, . . . , Vn are thestandard basis vectors on Rn. By definition F and V are the same map onthe functions xi, and since they are both linear and Leibnitzian it follows thatthey are equal on all polynomials. But every function can be approximatedarbitrarily closely by polynomials (in the C∞ topology on compact subsets),therefore since F and V are both continuous functions on C∞(Rn) it followsthat they are equal everywhere.

For uniqueness, suppose that V and W are two vector fields inducing thesame operation C∞(M) → C∞(M). Then the vector field V −W induces thezero operation. Suppose that Vx −Wx 6= 0 at some point x ∈M . Then we canchoose coordinates so that Vx −Wx = ∂

∂x1at the origin, but then the function

x1 (cut off to be zero outside of this chart) cannot be sent to zero.

The next proposition makes more precise what it means to say that V (f) isthe directional derivative of f in the direction V .

Proposition 4.4. Let ϕt : M → M be an isotopy with ϕt|t=0 = V and letf ∈ C∞M . Then

limh→0

1

h((f ϕh)(x)− f(x)) = V (f)(x).

The left hand side is often denoted by LV f and in this form it is called theLie derivative of f in the direction of V . The reason for this extra vocabulary isbecause we can take a Lie derivative of many objects, we explore another caseof this below. We leave the proof of the proposition to the reader.

4.3 Lie bracket, flows, and Frobenius integrability theo-rem

Definition 4.5. Let X and Y be vector fields on a manifold M , and let f :M → R be a smooth function. We define [X,Y ](f) = X(Y (f)) − Y (X(f)).This defines [X,Y ] as a linear Leibnitzian operator on C∞(M), therefore [X,Y ]is a vector field on M , called the Lie bracket of X and Y .

Note that [X,Y ] is Leibnitzian by simply calculation:

[X,Y ](fg) = X(Y (fg))− Y (X(fg))

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

= fX(Y (g)) +X(f)Y (g) + gX(Y (f)) +X(g)Y (f)

−fY (X(g))− Y (f)X(g)− gY (X(f))− Y (g)X(f))

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

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= f [X,Y ](g) + g[X,Y ](f).

The Lie bracket should be thought of as an obstruction for a vector field tocome from a coordinate system. If M ⊇ U ∼= Rn is a chart, the coordinates ofthe diffeomorphism can be thought of as n functions xi : U → R, i = 1, . . . , n.This gives a smooth basis of covectors dxi, which are simply the composition ofthe differential of the diffeomorphism U → Rn composed with the differentialwith the coordinate projections. This as defines basis vectors ∂

∂xias the dual

basis to dxi (which are the standard basis vectors in Rn, pushed forward to U

by the inverse of the diffeomorphism). Then[∂∂xi

, ∂∂xj

]= 0 for any i and j,

because partial derivatives commute on Rn.Notice that the Lie bracket is Liebnitzian with respect to scalar multiplica-

tion:[X, fY ] = X(f)Y + f [X,Y ].

In particular using skew symmetry this implies that, if [X,Y ] = 0, then [fX, gY ] =fX(g) · Y − gY (f) ·X. Since coordinate derivatives commute this allows us tocompute easily in charts. For example, if X = x ∂

∂x + y2 ∂∂z and Y = ∂

∂x + x ∂∂y

on R3, then [X,Y ] = x ∂∂y −

∂∂x − 2xy ∂

∂z .

Proposition 4.6. Let ϕt : M → M be an isotopy with with ϕt|t=0 = V , andlet W be a vector field on M . For each x ∈M define the vector

(LVW )x = limh→0

1

h

(Wx − (Dϕh)ϕ−1

h (x)(Wϕ−1h (x))

).

Then LVW = [V,W ].

Proof. Let f ∈ C∞M , and evaluate df(LVW ). We get

dfx(LVW ) = limh→0

1

h

(dfx(Wx)− (D(f ϕh))ϕ−1

h (x)(Wϕ−1h (x))

)= limh→0

1

h

(dfx(Wx)− dfϕ−1

h (x)(Wϕ−1h (x)) + dfϕ−1

h (x)(Wϕ−1h (x))− (D(f ϕh))ϕ−1

h (x)(Wϕ−1h (x))

).

But, by Proposition 4.4, we have

limh→0

1

h

(dfx(Wx)− dfϕ−1

h (x)(Wϕ−1h (x))

)= V (W (f))x

(make sure you understand why!). Also we have

dfϕ−1h (x)(Wϕ−1

h (x))− (D(f ϕh))ϕ−1h (x)(Wϕ−1

h (x)) = W (f − f ϕh)ϕ−1h (x)

which therefore implies, again using Proposition 4.4,

limh→0

1

h

(dfϕ−1

h (x)(Wϕ−1h (x))− (D(f ϕh))ϕ−1

h (x)(Wϕ−1h (x))

)= W (−V (f))x.

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This proposition is quite useful: it is not at all obvious that LVW = −LWV ,or that LXLY Z + LY LZX + LZLXY = 0, or that LVW is Liebnitzian withrespect to scalar multiplication in both factors, but all these properties areessentially immediate for [V,W ]. And of course, these are the properties thatmake [V,W ] computable in examples. On the other hand, the definition of LVWis much more geometric, and helps us relate [V,W ] to flows.

Theorem 4.7. Let V and W be two vector fields on a smooth manifold Msatisfying [V,W ] = 0 everywhere. Let ϕt : M → M be the flow of V , and letψt : M →M be the flow of W . Then for all s, t ∈ R, ϕt ψs = ψs ϕt.

Proof. First, we claim that Dϕt(W ) = W for all t. Indeed, the time derivativeof the vector field3 Dϕt(W ) is LVW = [V,W ] = 0, therefore it is constant intime, and of course Dϕ0(W ) = W .

Now, fix t ∈ R, and consider the flow ϕt ψs ϕ−1t as a flow in s. The time

derivative (i.e. ∂∂s ) of this flow is Dϕt(ψs). Since this is equal to W , uniqueness

of flows implies that ϕt ψs ϕ−1t = ψs.

This theorem gives us a useful geometric interpretation of the Lie bracket:it is measuring the failure of flows to commute. This becomes especially rel-evant when thinking of coordinate systems: a coordinate system is the sameas a commuting set of flows. This is the content of the Frobenius integrabilitytheorem.

Theorem 4.8 (Frobenius integrability theorem). Let M be an n–manifold, andlet V1, . . . Vk be a set of vector fields on M , which are linearly independent ateach point and [Vi, Vj ] = 0 for all i, j. Then in a neighborhood of each point wecan find a coordinate system f : U → M (U ⊆ Rn open) so that Df( ∂

∂xi) = Vi

for all i = 1, . . . , k.

Proof. Fix x ∈M , and choose a coordinate system g : U →M so that g(0) = xand Dg0( ∂

∂xi) = Vi(x) for all i = 1, . . . , k (since we only ask for this at one point

we can accomplish this by linear change in coordinates). Let ϕit : U → U be the(partially defined) flow obtained from integrating the vector field (Dg)−1(Vi).

Then define ψ : U → U by

ψ(x1, . . . , xn) =(ϕ1x1 . . . ϕkxk

)(0, . . . , 0, xk+1, . . . , xn),

where here U ⊆ U is an open set containing 0 chosen small enough so that all theflows appearing in the expression for ψ are defined. Notice that Dψ0 = Id (makesure you understand why!), therefore ψ is a diffeomorphism in a neighborhood

of 0, by choosing U smaller if necessary we can assume ψ is a diffeomorphismonto ψ(U) ⊆ U .

We also see that Dψ( ∂∂x1

) = ϕ1t = (Dg)−1(V1) everywhere on U . Now, using

the hypothesis that [Vi, Vj ] = 0, we see that ϕit ϕjs = ϕjs ϕit for all i, j and t, s.

3Meaning: if Xt is a time-dependent family of vector fields, then at each point Xt(x) isa path in the vector space TMx, and we take the derivative of this curve in the usual sense.Always keep track of what you mean by “derivative”!

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It follows therefore that Dψ( ∂∂xi

) = ϕit = (Dg)−1(Vi) for all i = 1, . . . , k. Andf = g ψ is the desired coordinate system.

5 Vector bundles

5.1 Basics

TM is a space with rich structure: besides being a smooth manifold, it comesequipped with a submersion π : TM →M , and the fibers π−1(x) are all vectorspaces with canonically defined vector operations. On the level of maps, asmooth map f : M → N gives a map Df : TM → TN , which is not only asmooth map, but it preserves fibers: Df(TMx) ⊆ TNf(x), equivalently πN Df = f πM . We would like to generalize this hybrid class of manifolds/vectorspaces.

Let Em+k and Mm be manifolds, and let π : E → M be a smooth map sothat Dπ is surjective at every point. We say that (E,M, π) is a (k–dimensional,smooth) vector bundle if we have an open covering of M =

⋃i Ui satisfying the

following properties. For every i we have a diffeomorphism ϕi : Ui × Rk →π−1(ψi(Ui)), so that π ϕi : Ui × Rk → Ui is the coordinate projection map.Furthermore, for all i, j, the transition map ϕ−1

i ϕj : (Ui ∩ Uj)× Rk → (Ui ∩Uj)× Rk should be of the form

(ϕ−1i ϕj)(x, v) = (x,Ax(v)),

where x 7→ Ax is a smooth map Ui∩Uj → GL(Rk) (the space of invertible linearmaps on Rk). Notice that the hypothesis on π ϕi being the projection mapit is already implies that (ϕ−1

i ϕj)(x, v) = (x, fx(v)) for some smooth familyfx ∈ Diff Rk, thus the main hypothesis is about linearity of the transition maps:that is the vector operations do not depend on coordinates.4

If σ : M → E is a smooth map satisfying π σ = id, we say that σ is asection of E. The space of all smooth sections is denoted by Γ(E). We notesome important properties of vector bundles.

Let π : E →M be a vector bundle. Then for any x ∈M , π−1(x) is a vectorspace. Thus if v1, v2 ∈ E satisfy π(v1) = π(v2), v1 + v2 ∈ E is a well definedpoint, so that π(v1 + v2) = π(v1) = π(v2). Also for any v ∈ E and c ∈ R cv ∈ Eis a point satisfying π(cv) = π(v), and these satisfy the usual vector spaceaxioms of commutativity, associativity, and distribution. Similarly if σ1 and σ2

are smooth sections and λ ∈ C∞M we can define a smooth sections σ1 + σ2 by(σ1 +σ2)(x) = σ1(x)+σ2(x), and λσ by (λσ)(x) = λ(x)σ(x). Therefore Γ(E) isalso a (infinite dimensional) vector space (and also it is a module over the ringC∞M).

4Technically speaking, the collection Ui, ϕi is the vector bundle structure which makes(E,M, π) into a vector bundle; we do not claim here that the triple (E,M, π) determine thestructure canonically (though it is true). But we will never really work with ψi, etc. by hand,and so we suppress this from the notation.

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We have some familiar examples already. E = TM is the most basic one,in the definition above ϕi = Dψi where ψi : U → M are charts. Also T ∗Mis familiar, where we take ϕi(x) = (ψi(x), D(ψ∗i )−1

x ). Another obvious exampleis E = M × Rk, called the trivial k–dimensional vector bundle. Notice thatΓ(M × R) = C∞M . Not all vector bundles are trivial! TS2 is not the samevector bundle as S2 × R2, since the trivial vector bundle has a non-vanishingsection but TS2 does not (this follows from the theorem of Poincare and Hopf).

If π1 : E1 → M1 and π2 : E2 → M2 are vector bundles, we say that amap F : E1 → E2 is a bundle map if, in addition to being smooth, it satisfiesπ2 F = f π1 for some map f : M1 → M2, and on each fiber the restrictedmap F : π−1

1 (x) → π−12 (f(x)) is linear. Notice that f is uniquely defined if it

exists. We also say that F is a bundle map covering f .The obvious example is, given any f : M → N , the map Df : TM → TN

is a bundle map covering f . If ω ∈ Γ(T ∗M), then evaluation is a bundle mapω : TM →M × R, covering the identity map.

5.2 Tensoriality

Let E1 and E2 be two vector bundles over M , and let F : E1 → E2 be abundle map which covers the identity. Then F defines a map of section spacesF : Γ(E1)→ Γ(E2) just by acting pointwise: (F (σ))(x) = F (σ(x)).

Definition 5.1. Let E1 and E2 be vector bundles over a manifold M , and letA : Γ(E1)→ Γ(E2) be a linear map. We say that A is tensorial (or that A is a

tensor) if A = F for some bundle map F : E1 → E2 covering the identity.

Thus a tensor is a map of section spaces A : Γ(E1) → Γ(E2), so that(A(σ))(x) only depends on σ(x). We have many examples of tensors, and alsoof maps which are not tensors.

If ω ∈ Ω1M we can think of ω : Γ(TM) → C∞M . This is induced byω : TM →M × R, and so is a tensor.

As a special case of this, for any function f , we can think of df : Γ(TM)→C∞M , and df is a tensor. However, the map d : C∞M → Γ(T ∗M) is not atensor: the derivative of a function at a point does not only depend on the valueof the function at that point.

Said differently, for a vector field X ∈ Γ(TM) and f ∈ C∞M , the associationX(f) ∈ C∞M is tensorial in X but not tensorial in f .

If A : Γ(E1) → Γ(E2) is a tensor, then is is necessarily linear over C∞M ,

not only R: for f ∈ C∞M and σ ∈ Γ(E1) F (fσ) = fF (σ) since F (f(x)σ(x)) =f(x)F (σ(x)) for all x ∈M . In fact the converse holds as well.

Proposition 5.2. Let A : Γ(E1)→ Γ(E2) be a map of section spaces of vectorbundles. If A(fσ) = fA(σ) for all f ∈ C∞M and σ ∈ Γ(E1), then A is atensor.

Proof. If (A(σ1))(x)−(A(σ2))(x) = (A(σ1−σ2))(x). Thus to show A is a tensorit suffices to show that whenever σ(x) = 0 it follows that (A(σ))(x) = 0.

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Choose a coordinate chart U around x and let χ ∈ C∞M be a function whichis equal to 1 at x and whose support is contained in U . Then (A(χσ))(x) =χ(x)(A(σ))(x) = (A(σ))(x). On this coordinate chart U we have π−1

1 (U) ∼=U × Rk ⊆ E1; we let v1, . . . vk be a basis for Rk. At all points y in U we canwrite σ(y) = f1(y)v1 + . . .+ fk(y)vk, then the functions fj ∈ C∞U are smoothand fj(x) = 0 for all j. Notice that χv1 ∈ Γ(E1) even though v1 /∈ Γ(E1): v1 isonly defined at points in U but χv1 is defined on all of M by extending outsideof U by the zero section5. Therefore we can write

(A(χσ))(y) = f1(y)(A(χv1))(y) + . . .+ fk(y)(A(χv1))(y)

and evaluating at y = x we get 0.

We note that the property of being Leibnitzian is mutually exclusive withbeing tensorial. While being tensorial is equivalent to C∞ linearity, the Leibnitzrule gives an additional term which depends on the derivative of the function.For instance since [X, fY ] = f [X,Y ]+(X(f))Y , the Lie bracket is not tensorialin the second slot (or the first slot since it is skew).

5.3 Linear algebra

In order to build a richer theory of vector bundles, we will need methods forconstructing new vector bundles from ones we already have. Before doing thiswe cover the linear algebra we will be using.

Let V and W be (finite dimensional) vector spaces. We define V ⊕W =(v, w) ∈ V ×W, where the linear operations are the standard ones: (v1, w1)+(v2, w2) = (v1 + v2, w1 + w2) and λ · (v, w) = (λv, λw).

We define V ⊗W to be the set of linear combinations of the symbols v⊗w,where v ∈ V and w ∈W , subject to the relations6 of bilinearity: (v1 +v2)⊗w =v1⊗w+v2⊗w, v⊗(w1 +w2) = v⊗w1 +v⊗w2, (λv)⊗w = λ(v⊗w) = v⊗(λw).Notice that if v1, . . . vk and w1, . . . , wl are bases for V and W respectively,then vi ⊗wj is a basis for V ⊗W , proving that dimV ⊗W = dimV · dimW .As a basis-free expression of the same fact, if V,W,X are vector spaces thenV ⊗ (W ⊕X) is canonically isomorphic to (V ⊗W ) ⊕ (V ⊗X). We also notethat the space of linear maps A : V ⊗W → X is in bijective correspondencewith the space of bilinear maps A : V ×W → X.

Hom(V,W ) is the vector space consisting of all linear maps from V to W .As a particular example, we define V ∗ = Hom(V,R), the dual of V . Noticethat Hom(V,W ) is canonically isomorphic to W ⊗ V ∗, where the identificationis defined by (w ⊗ f)(v) = f(v) · w. Here w ∈ W , f ∈ V ∗, and we definew ⊗ f ∈ Hom(V,W ) by describing its action on v ∈ V . The reader easily

5It’s important here that the closure of the points where χ is non-vanishing is containedin U .

6Whenever we would like to impose linear relations on a vector space, what we are actuallydoing is taking a vector space quotient. To impose the relation a = b, we quotient by thevector space spanned by a− b.

15

checks this is a linear isomorphism (relying on the hypothesis that V is finitedimensional!).

For n ∈ Z+, we define Symn(V ) to be the vector space

n︷ ︸︸ ︷V ⊗ . . .⊗ V , quo-

tiented by the additional relation v1 ⊗ . . . ⊗ vi ⊗ vi+1 ⊗ . . . ⊗ vn = v1 ⊗ . . . ⊗vi+1⊗ vi⊗ . . .⊗ vn. So for example, while Hom(V ⊗V,X) is the space of all bi-linear maps from V ×V to X, Hom(Sym2 V,X) is the space of all commutative,bilinear maps.

ΛnV is similar, but it is the anti–commutative tensor product. That is ΛnV

is the vector space

n︷ ︸︸ ︷V ⊗ . . .⊗ V , quotiented by the relation v1⊗ . . .⊗ vi⊗ vi+1⊗

. . . ⊗ vn = −v1 ⊗ . . . ⊗ vi+1 ⊗ vi ⊗ . . . ⊗ vn. We typically denote the elementv1 ⊗ . . . ⊗ vn ∈ ΛnV by v1 ∧ . . . ∧ vn to distinguish it from the correspondingelement in V ⊗n (for whatever reason, there doesn’t seem to be a correspondingnotation for Symn(V )). We note an important proposition:

Proposition 5.3. v1 ∧ . . . ∧ vn = 0 ∈ ΛnV if and only if the set v1, . . . vn islinearly dependent in V .

Proof. Since v ∧ v = 0, whenever one vi can be written as a linear combinationof the others we can distribute and cancel all terms. To prove the converse,notice that for any linear map A : V → Rk, the function ΛkV → R which sendsv1∧ . . .∧vk to the signed area of the parallelopiped spanned by A(v1), . . . , A(vk)is a well defined, linear map. Indeed it is obviously a well defined linear map onV ⊗k, so to prove it is well defined on ΛkV we only need to check the relationsare sent to 0.

In particular, this shows that ΛnV = 0 whenever n > dimV , and moregenerally dim ΛnV =

(dimVn

). For example, a basis of Λ2R4 is

∂x ∧ ∂y, ∂x ∧ ∂z, ∂x ∧ ∂w, ∂y ∧ ∂z, ∂y ∧ ∂w, ∂z ∧ ∂w.

We have other natural structures on these spaces. Given a ∈ ΛkV b ∈ ΛlV ,then we can define the wedge product a∧b ∈ Λk+lV as follows. If a = v1∧. . .∧vkand b = u1 ∧ . . . ul, define a ∧ b = v1 ∧ . . . vk ∧ u1 ∧ . . . ∧ ul. In general, a and bare linear combinations of such elements, so we define a ∧ b by the distributivelaw. Think about why this is well defined! For example ∂x∧∂y = ∂x∧(∂x+∂y),so if we present a in two different ways, why are the resulting products equal?Once we see that the wedge product is well defined, the following propertiesare obvious: ∧ is associative, distributes over addition, and skew commutative.That is, a ∧ b = (−1)klb ∧ a.

Here’s an important example. Consider Λ2R3, equipped with the isomor-phism A : Λ2R3 → R3 which is defined on a basis by A(∂x ∧ ∂y) = ∂z,A(∂y ∧ ∂z) = ∂x, A(∂z ∧ ∂x) = ∂y. Then, by composing the wedge productwith A, we get a bilinear, skew–symmetric product on R3. Unsurprisingly, itis easy to check that this is the cross product. Notice however, that the wedgeproduct is associative while the cross product is not, how does this happen?

16

If α ∈ Λk(V ∗), and v ∈ V , we can define another type of product, the interiorproduct v yα ∈ Λk−1(V ∗). Roughly it is “plug in v in all possible ways”. Moreprecisely, if α = f1 ∧ . . . ∧ fk for fj ∈ V ∗, then

v yα = f1(v) (f2 ∧ . . . ∧ fk)−f2(v) (f1 ∧ f3 ∧ . . . ∧ fn)+. . .+(−1)kfk(v) (f1 ∧ . . . fk−1) .

Again, we extend to all of Λk(V ∗) by linearity. Again, take the time to under-stand why this is well–defined.

Proposition 5.4. Λk(V ∗) is canonically isomorphic to the space of all multilin-ear maps A : V × . . .×V → R which are alternating. That is Λk(V ∗) ∼= (ΛkV )∗

Here “alternating” means skew–symmetric in any two adjacent slots. Whatis perhaps surprising about this proposition is that, by definition, ΛkV ∗ is a quo-tient space of (V ∗)⊗k, the space of all multilinear maps, whereas the propositionidentifies it as a subspace.

Proof. Given α ∈ Λk(V ∗), we can define a multilinear map Aα : V ×. . .×V → Ras follows: given (v1, . . . vk) ∈ V ×k, defineAα(v1, . . . , vk) = vk y (vk−1 y . . . y (v1 yα).Since, for any u1, u2 ∈ V and β1, β2 ∈ ΛlV ∗ we have (u1 +u2) yβ1 = (u1 yβ1) +(u2 yβ1) and u1 y (β1 +β2) = (u1 yβ1) + (u1 yβ2) we see that Aα is multilinear,and since u1 y (u2 yβ1) = −u2 y (u1 yβ1) we see that Aα is alternating. So itremains to show that α 7→ Aα is a bijection.

Let A be a multilinear alternating map to R and let v1, . . . , vn be a basisof V . If I = (i1, . . . ik) is an increasing sequence of integers between 1 and n,define aI = A(vi1 , . . . , vik) ∈ R. The values aI over all sequences I determineA: by multilinearity it suffices to know the value of A on all collections of basiselements, and since A is alternating we can assume those values are increasing.Let f1, . . . fn ∈ V ∗ be the dual basis, so fi(vj) = 1 if i = j and 0 if i 6= j. DefinefI = fi1 ∧ . . . ∧ fik . Then define

α =∑I

aIfI .

Then Aα = A, since they are equal on all basis sequences I. This shows that theassociation α 7→ Aα is surjective, since both spaces have the same dimension itis a bijection.

We also have natural structure related to maps between spaces. If α ∈ ΛkV ∗,and A : W → V is a linear map, we can define A∗α ∈ ΛkW ∗ by the formula(A∗α)(w1, . . . , wk) = α(Aw1, . . . , Awk) (here we consider Λk as the space ofalternating multilinear maps). We call A∗α the pullback of α via A.

Here’s a cute application. Suppose A : V → V is a linear map, wheredimV = n. Then consider A∗ : ΛnV ∗ → ΛnV ∗. ΛnV ∗ is a 1–dimensional vectorspace, and since A∗ is a linear map, it follows that A∗ is just multiplication bya scalar a ∈ R. This number is detA! To see this calculate in a basis: detis multilinear and alternating (in the columns), and Id∗ is multiplication by 1,since these properties characterize the determinant they must be equal.

17

For posterity, we list the algebraic rules our operations satisfy. Let v1, v2 ∈V , let α ∈ ΛkV ∗, β ∈ ΛlV ∗; let A : W → V be a linear map, and take w ∈ W .Then:

α ∧ β = (−1)klβ ∧ α

v1 y (v2 yα) = −v2 y (v1 yα)

v1 y (α ∧ β) = (v1 yα) ∧ β + (−1)kα ∧ (v1 yβ)

w yA∗α = A∗(Aw yα)

A∗(α ∧ β) = A∗α ∧A∗β

and furthermore, all operations are linear in all slots, with respect to the vectorspace operations on V , ΛkV ∗, and Hom(W,V ).

5.4 Operations on bundles

All of the above operations we defined for vector spaces, also work for vectorbundles.

Proposition 5.5. Let π : E → M be a vector bundle over M , and let n bea positive integer. Then we have canonically defined bundles E∗, Symn(E),

and ΛnE, with the property that π−1ΛnE(x) = Λn(π−1(x)), etc. If π : E → M

is another vector bundle over M , then we can define canonically E ⊕ E, E ⊗E, and Hom(E, E), again so that i.e. π−1

Hom(E,E)(x) = Hom(π−1(x), π−1(x)).

Furthermore all the natural relationships among vector spaces holds for vectorbundles; for example Hom(E, E) = E∗ ⊗ E.

Proof. The proof follows immediately from the statement: all the above opera-tions are smoothly functorial. In other words, if A : V1 → V2 is an isomorphism,from A we have canonically defined isomorphisms between V ∗1 and V ∗2 , V1 ⊗Wand V2 ⊗W , etc. Furthermore, this association is smooth: if Ax : V1 → V2 is asmooth family of isomorphisms for x ∈M , then Ax ⊗ Id : V1 ⊗W → V2 ⊗W isalso a smooth family. Therefore, whatever the structure maps ϕi, ϕi are for Eand E, we can define E⊗ E =

⋃i(ϕi⊗ ϕi)(Ui× (Rk⊗Rl)). Then the transition

maps will be given by

((ϕi ⊗ ϕi)−1 (ϕj ⊗ ϕj))(x, v) = ((ψ−1i ψj)(x), (Ax ⊗ Ax)(v)),

which satisfies the definition. All other operations are identical.

6 The de Rham complex

In the above section, It seems that some of the most interesting structure liveson the vector spaces ΛkV ∗. For that reason, we are especially interested in thisfor manifolds.

18

Definition 6.1. Let M be a manifold, and k ∈ Z. A k–form on M is asection β of the bundle Λk(T ∗M). The space of all k–forms is denoted ΩkM =Γ(Λk(T ∗M)).

As examples, we have that Ω0M = C∞M . A 1–form is simply a covectorfield on M . If k > dimM then there are no k–forms (except 0).

The operations of wedge product, interior product, and pull–back also allgive structures on manifolds. For example, if α ∈ ΩkM and V is a vectorfield on M , then we can define V yα ∈ Ωk−1M . If β ∈ ΩlM , we can defineα∧ β ∈ Ωk+lM . If f : N →M is a smooth map, then we define f∗α ∈ ΩkN by(f∗α)(x) = Df∗x(α(x)). Because all of these are just operations that come fromlinear algebra, wedge and interior product are tensorial.

Remark 6.2. We had mentioned above that being Leibnitzian and being ten-sorial are mutually exclusive, but it’s important to be careful about what thismeans. Let α ∈ ΩkM , β ∈ ΩlM,V ∈ Γ(TM), and f ∈ C∞M . Then interiorproduct is Leibnitzian over wedge product: V y (α∧β) = (V yα)∧β±α∧(V yβ),but it is tensorial over scalar multiplication: V y (fβ) = f(V yβ) = (fV ) yβ.There is never a term than involves V (f). (What happens when k = 0?)

In addition to all of this structure, there is one more additional operation,which only exists on manifolds.

Proposition/Definition 6.3. Let M be a smooth manifold. Then there is aunique (family of) linear map d : ΩkM → Ωk+1M satisfying the following threeproperties:

• d(α ∧ β) = dα ∧ β + (−1)kα ∧ dβ for α ∈ ΩkM,β ∈ ΩlM ,

• d(dα) = 0 for any α ∈ ΩkM (briefly, d2 = 0),

• When f ∈ Ω0M = C∞M , df ∈ Ω1M is the exterior differential of f , aspreviously defined.

We call d the exterior differential on M .

Proof. First, we prove this result on M = Rn. If I = (i1, . . . , ik) is an increasingsequence, let dxI = dxi1 ∧ . . . , dxik . Then dxI is a basis for Λk(T ∗Rn)p atevery point p. So, an arbitrary α ∈ ΩkRn can be written as α =

∑I aIdxI for

some functions aI ∈ Ω0M . Then we define dα =∑I daI ∧ dxI .

Obviously d is linear and acts on C∞ as expected. d is Liebnitzian since it isLiebnitzian on C∞M , where the extra sign comes from dxI ∧daJ = (−1)kdaJ ∧dxI . To see that d2 = 0, first we check it on functions f ∈ C∞M .

d2f = d

(∑i

∂f

∂xidxi

)=∑j

∑i

∂2f

∂xj∂xidxj ∧ dxi.

In the double sum, either i = j in which case dxj ∧ dxi = 0, or else the term

appears twice with opposite sign since ∂2f∂xj∂xi

dxj ∧ dxi = − ∂2f∂xi∂xj

dxi ∧ dxj .Therefore the sum is zero.

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Since d(dxI) = 0 by definition then d2(aIdxI) = d2(aI)dxI = 0, so d2 = 0on ΩkM for all k.

Uniqueness follows easily: if d is another operator satisfying the same prop-erties, then dxi = dxi since xi ∈ C∞M . Then d(dxi) = 0 since d2 = 0. Then

ddxI = 0 by Liebnitz’s rule. So d(aIdxI) = daI ∧ dxI = daI ∧ dxI by Liebnitz’s

rule and then the fact that d = d on Ω0M . Linearity completes the claim.This completes the proof for M = Rn. Next, we show that, whenever g :

Rn → Rn is a diffeomorphism then g∗dα = dg∗α for all α ∈ ΩkRn. Let d =(g∗)−1 d g∗. Clearly d is linear, and d = d on Ω0Rn since g∗df = d(f g) for

f ∈ C∞. Obviously d2 = 0, and d is Liebnitzian since g∗(α ∧ β) = g∗α ∧ g∗β.

Therefore by uniqueness, d = d and so g∗ d = d g∗.Finally, for general M we let ϕi : Ui → Rn be coordinate charts for a covering

Ui ⊆M . Then we can write dα|Ui= ϕ∗i d(ϕ−1

i )∗α, as a definition. For pointsin the overlap Ui∩Uj , we know the two definitions of dα|Ui∩Uj

agree by applying

the previous claim to g = ϕi ϕ−1j . This then defines dα globally on M . This

also shows that dα does not depend on the coordinate covering, since if Vj weanother covering then we know dα is well defined for the covering Ui ∪ Vj.Since, in a neighborhood U ⊆ Rn, dα|U = d(α|U ), the fact that d satisfies thethree properties above and it is the unique operator to do so follows immediatelyfrom the local case.

Above we showed that pull-backs by diffeomorphisms commute with d, infact this is true for all smooth maps.

Proposition 6.4. Let f : M → N be any smooth map, and α ∈ ΩkN . Thend(f∗α) = f∗(dα).

Proof. If the proposition is true for α ∈ ΩkM and β ∈ ΩlM , then d(f∗(α∧β)) =d(f∗α∧f∗β) = d(f∗α)∧f∗β+(−1)kf∗α∧d(f∗β) = f∗(dα)∧f∗β+(−1)kf∗α∧f∗(dβ) = f∗(dα∧β+ (−1)kα∧ dβ) = f∗(d(α∧β)). Thus, since every k–form isa linear combination of wedge products of 1-forms, it suffices to prove the resultfor k = 1.

The proposition is local: to say d(f∗α) = f∗(dα) means they are equal atevery point. So we work in a coordinate chart. We can write α =

∑ni=1 aidxi

for some functions ai ∈ C∞Rn. Then f∗α =∑ni=1(ai f)d(xi f), and d((ai

f)d(xif)) = d(aif)∧d(xif) = f∗dai∧f∗dxi = f∗(dai∧dxi) = f∗(d(aidxi)).Thus d(f∗α) = f∗(dα).

Example 6.5. Here’s an extremely important example. On R3 with a fixedcoordinate system, we can identify Λ1(T ∗R3) ∼= TR3 via the isomorphism dx 7→∂x, dy 7→ ∂y, dz 7→ ∂z. We can also identify Λ2(T ∗R3) ∼= TR3 by dx ∧ dy 7→ ∂z,dy ∧ dz 7→ ∂x, dz ∧ dx 7→ ∂y. Finally we can identify Λ3R3 as the trivial bundleR×R3 by identifying dx∧dy∧dz 7→ 1. So under these isomorphisms Ω1R3 andΩ2R3 are both identified with the space of vector fields, and Ω0R3 and Ω3R3

are both identified with C∞R3. We have three different notions of derivatived : ΩkR3 → Ωk+1R3: for k = 0 we have a derivative sending functions to vectorfields, for k = 1 we have a derivative sending vector fields to vector fields, and

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for k = 2 we have a derivative sending vector fields to functions. These threederivatives are respectively the gradient, the curl, and the divergence. Thend2 = 0 sums up two facts: the curl of the gradient of any function is zero, andthe divergence of the curl of any vector field is zero.

7 Integration on manifolds

In the previous section, we defined the exterior differential d in two steps: firstwe defined it on Rn, and secondly we showed that it is invariant under diffeomor-phisms, and therefore we can just put it on manifolds using coordinate systems.Our approach to integration will be the same. However, integration of functionsdoes not transform correctly: in fact the correct notion will be integration offorms.

Theorem 7.1 (Change of variables). Let U, V ⊆ Rn be open sets, and letf : U → V be a diffeomorphism. Let h ∈ C∞V (or h ∈ L1V ). Then∫

V

h(x)dx1dx2 . . . dxn =

∫U

((h f)(y))|det(Dfy)|dy1dy2 . . . dyn.

The main fact worth noting is that this becomes much more natural using thelanguage of forms. Let ω = hdx1 ∧ . . .∧ dxn. Then f∗ω = (h f) det(Df)dx1 ∧. . . ∧ dxn. Indeed, as noted above in the linear case f∗(dx1 ∧ . . . dxn) =det(Df)dx1∧. . .∧dxn, and the pullback of 0–forms is simply by pre-composition.

Note also that det(Df) is never zero, so it is either positive everywhere ornegative everywhere (as long as U is connected), and the sign corresponds towhether f preserves or reverses orientation. Then the theorem above is thenequivalent to the following.

Theorem 7.2. Let U, V ⊆ Rn be connected open sets, let f : U → V be adiffeomorphism, and let ω ∈ ΩnV . Then

∫Vω = ±

∫Uf∗ω, where the sign ± is

according to whether f preserves or reverses orientation.

In other words, the integral of a top dimensional form does not depend oncoordinates. From here we can put the structure on manifolds.

Proposition/Definition 7.3. Let M be an oriented n–manifold, and let ω ∈ΩnM be compactly supported. We define

∫Mω ∈ R as follows. Let Ui be

an open covering of M , equipped with orientation preserving coordinate mapsϕi : Ui → Rn, and let ψi be a partition of unity compatible with Ui. Letωi = (ϕ−1

i )∗(ψiω), thus ωi ∈ ΩnRn is compactly supported and∑i ϕ∗iωi = ω.

Define ∫M

ω =∑i

∫ϕi(Ui)

ωi.

Then∫Mω only depends on ω and M as an oriented manifold.

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Proof. First, suppose that Ui is fixed, but we choose another partition of unity

ψi. As before let ωj = (ϕ−1j )∗(ψjω). Also let ηij = (ϕ−1

i )∗(ψiψjω). Then∑j ηij = ωi, and

∑i(ϕi ϕ

−1j )∗ηij = ωj . Then

∑i

∫ϕi(Ui)

ωi =∑i

∫ϕi(Ui)

∑j

ηij =∑i,j

∫ϕi(Ui)

ηij =∑i,j

∫ϕi(Ui∩Uj)

ηij

where the final step follows because ηij is supported on ϕi(Ui ∩ Uj) anyways.On the other hand∑j

∫ϕj(Uj)

ωj =∑i,j

∫ϕj(Uj)

(ϕ−1j ϕi)

∗ηij =∑i,j

∫ϕj(Uj)

(ϕiϕ−1j )∗ηij =

∑i,j

∫ϕi(Uj)

ηij .

This shows that the choice of partition of unity does not affect the definition.If instead we choose a different cover Vj with partition of unity ψj, notice

that both ψi and ψj are partitions of unity for the open cover Ui ∪ Vj,which completes the proof.

Also immediate from the proof is the following

Proposition 7.4. Let M and N be oriented n–manifolds, and let ω ∈ ΩnN becompactly supported. If f : M → N is a diffeomorphism, then

∫Mf∗ω =

∫Nω.

In fact, compact support of ω is not essential to defining∫Mω, as long as the

sum defining∫Mω converges for one open covering, it will converge for all, and

the integral will be invariant up to diffeomorphism. More generally it is possibleto define and work with integrable forms that are not even continuous, in thesame way as functions. We won’t have any direct application of this however,so we will restrict to compactly supported smooth forms.

If ω ∈ ΩkM for k < dimM , then there is no natural way to define∫Mω.

However, if N ⊆ M is a k–dimensional submanifold, it is natural to integrate∫Nω (here we implicitly identify ω ∈ ΩkN by pullback via the inclusion). So,

integrating ω on M can be thought of as a way to explore the space of k–dimensional submanifolds.

In some sense, however, this is insurmountable: even for a 1–form α ∈ Ω1R2,the integral

∫Nα can vary wildly, even for curves N ⊆ R2 which are isotopic

via a C0 small isotopy. In order to work with integrals from a more topologicalperspective, we will relate it with the exterior differential.

8 Stokes’ theorem

Theorem 8.1. Let M be an oriented n–manifold with boundary, and let α ∈Ωn−1M be compactly supported. Then

∫∂M

α =∫Mdα.

Stokes’ theorem is a generalization of many theorems from vector calculus.When M = [a, b] it is a restatement of the fundamental theorem of calculus, this

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is the most basic case. More generally for curves γ : [a, b]→ Rn and α ∈ Ω0Rn itrecovers the fact that a line integral of a conservative vector field over a curve isthe difference in potential function at the endpoints. When M is a compact setin R2 with smooth boundary it recovers Green’s theorem. When M is a compactset in R3 with smooth boundary it is the same as the divergence theorem, andwhen M ⊆ R3 is a surface with boundary it is equivalent to the classical Stokes’theorem.

Proof of Stokes’ theorem: Both∫∂M

α and∫Mdα are linear in α, therefore by

using a partition of unity we can assume that the support of α is contained ina coordinate chart. Furthermore both sides are invariant via diffeomorphism(since both integration and exterior differentiation are), therefore we can justassume that α is a compactly supported (n− 1)–form on either Rn or [0,∞)×Rn−1.

First we consider the case α ∈ Ωn−1Rn. Let dx∗i = dx1∧ . . .∧dxi−1∧dxi+1∧. . . ∧ dxn ∈ Ωn−1Rn. Any (n− 1)–form can be written as

∑ni=1 aidx

∗i for some

ai ∈ C∞Rn, so again by linearity if suffices to prove the result for α = aidx∗i .

Then dα = ∂ai∂xi

dxi ∧ dx∗i , and so∫Rn

dα =

∫Rn−1

(∫R

∂ai∂xi

dxi

)dx1 . . . dxi−1dxi+1 . . . dxn

=

∫Rn−1

0dx1 . . . dxi−1dxi+1 . . . dxn = 0.

Here the first equality folllows from Fubini’s theorem and the second followsfrom the fundamental theorem of calculus, together with the fact that ai iscompactly supported. Since

∫∂Rn α =

∫∅ α = 0 this proves the result for this

case.If instead we work on [0,∞) × Rn−1, let us use coordinates t ∈ [0,∞) and

(x1, . . . , xn−1). Again any (n−1)–form can be written as a0dx+∑n−1i=1 aidt∧dx∗i ,

where now dx = dx1 ∧ . . . ∧ dxn−1 ∈ Ωn−1Rn−1 and dx∗i = dx1 ∧ . . . ∧ dxi−1 ∧dxi+1 ∧ . . . ∧ dxn−1 ∈ Ωn−2Rn−1, for some ai ∈ C∞([0,∞) × Rn−1. If α =aidt ∧ dx∗i , then the same argument as before works,∫

[0,∞)×Rn−1

dα =

∫[0,∞)×Rn−2

(∫R

∂ai∂xi

dxi

)dtdx1 . . . dxi−1dxi+1 . . . dxn−1

=

∫[0,∞)×Rn−2

0dtdx1 . . . dxi−1dxi+1 . . . dxn−1 = 0,

and this agrees with∫∂[0,∞)×Rn−1 α because dt = 0 on 0×Rn−1 and so α = 0

on the boundary.Finally, if α = a0dx, then dα = ∂a0

∂t dt ∧ dx, and so∫[0,∞)×Rn−1

dα =

∫Rn−1

(∫[0,∞)

∂a0

∂tdt

)dx1 . . . dxn−1

23

=

∫Rn−1

−a0|t=0dx1 . . . dxn−1 = −∫0×Rn−1

a0dx =

∫∂([0,∞)×Rn−1

α.

The final − sign appears because ∂[0,∞) = −0, with the standard orientationon [0,∞). This completes the proof.

Corollary 8.2. Let β ∈ ΩkM for some manifold M , and let N be a closedoriented k–manifold. Suppose dβ = 0, and let f : N → M be a smooth map.Then for any map g : N → M which is homotopic to f , we have

∫Nf∗β =∫

Ng∗β. Furthermore, if W is a compact manifold satisfying ∂W = N and f

extends to F : W →M , then∫Nf∗β = 0.

Proof. The second statement implies the first, since we can apply it to W =N × [0, 1]. For the second statement, we just note that

∫Nf∗β =

∫WF ∗dβ =∫

W0.

This corollary is interesting in that it relates two questions which at first seemquite unrelated. One question that we have encountered repeatedly throughoutthe course is: given a map f : N → M , how can we show that f does notextend to F : W → M for any compact manifold W with ∂W = N? Forinstance, degree and intersection number both are approaches to answering thisquestion. This gives us a new strategy: if we can find a form β ∈ ΩkM whichsatisfies dβ = 0 but integrates over N to a non-zero number, we know thatf : N →M cannot be extended.

It also brings up another question. Suppose we are given β ∈ ΩkM , and wewould like to know whether β = dα for some α ∈ Ωk−1M . Since d2 = 0, inorder for such an α to exist we must have dβ = 0. But what other obstruction isthere? If we can find a closed oriented manifold N and a map f : N →M so that∫Nf∗β 6= 0, then α cannot exist, because

∫Nf∗dα =

∫∂N

f∗α =∫∅ f∗α = 0.

This tells us that there is some kind of relationship between k–forms satisfyingdβ = 0 and maps of manifolds which are not restrictions to boundaries. Weexplore this more systematically in the next section.

9 de Rham cohomology

Definition 9.1. Let M be a smooth manifold. Since d2 = 0, we can writeIm(d) ⊆ ker(d) ⊆ ΩkM . A k–form β is called closed if dβ = 0, i.e. if β ∈ ker(d),and β is called exact if β = dα for some α ∈ Ωk−1M , i.e. if β ∈ Im(d). Thequotient vector space, ker(d)/ Im(d), is called the de Rham cohomology in degreek, and is denoted by Hk

dR(M).

For example, we note that if M is a connected manifold then H0dR(M) ∼= R.

There are no exact 0–forms (except 0 ∈ C∞M), therefore H0dR(M) is equal to

the space of closed 0–forms. If f ∈ Ω0M , then the only way for df = 0 is if fis locally constant. Therefore H0

dR(M) is equal to the space of locally constantfunctions, which is simply the constant functions if M is connected.

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Proposition 9.2. Let M be a manifold. Then ∧ induces a product HkdRM ⊗

H ldRM → Hk+l

dR M .

Proof. This proposition entails two facts: the wedge product of two closed formsis again closed, and the wedge product of a closed form with an exact form isexact. For the first, supposing dα = dβ = 0, we see that d(α ∧ β) = dα ∧ β +(−1)kα∧dβ = 0. For the second, note that d(γ∧β) = dγ∧β as long as dβ = 0.Therefore (α + dγ) ∧ β = α ∧ β + d(γ ∧ β), so closed forms which differ by anexact form remain different only by exact forms after multiplication.

Unfortunately, we will not have the opportunity to develop the theory of deRham cohomology in depth. For reference, we state a number of nice resultsrelated to it. Assume M is a connected manifold of dimension n. (For disjointunions, clearly Hk

dR(M∐N) = Hk

dR(M)⊕HkdR(N).)

• If M is a compact manifold, then HkdR(M) is finite dimensional for all k.

• M is closed and orientable if and only ifHndR(M) = R, otherwiseHn

dR(M) =0.

• If M is closed and orientable then ∧ : HkdR(M)⊗Hn−k

dR (M)→ HndR(M) ∼=

R is non-degenerate. In particular dimHkdR(M) = dimHn−k

dR (M) (Poincareduality).

• If f, g : M → N are two homotopic maps, then f∗, g∗ : HkdR(N) →

HkdR(M) are equal for all k.

• Hk(Rn) = 0 for all k > 0.

• If M = U ∪V for open sets U and V , then dimHkdR(M) ≤ dimHk

dR(U) +dimHk

dR(V ) + dimHk−1dR (U ∩ V ).

Each of these statements can be proved directly in the theory if developedenough, for an excellent treatment of this see Madsen and Tornehave’s book“From Calculus to Cohomology”. Alternatively we can prove these statementsin any other cohomology theory because of the following.

Theorem 9.3. Let M be a manifold, and let H∗(·) be any Eilenberg-Steenrod co-homology theory, such as singular cohomology, Cech cohomology, CW cohomol-ogy, Morse cohomology, simplicial cohomology, etc. Then Hk

dR(M)congHk(M ;R).Furthermore the wedge product on H∗dR(M) is isomorphic to the cup product onany of these theories.

One method of proving this is by demonstrating the Eilenberg-Steenrodaxioms hold for de Rham cohomology. There are also direct methods of proof,depending on the cohomology theory in question.

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10 Integration and degree

Theorem 10.1. Let f : M → N be a map between closed oriented manifolds,so that dimM = dimN = n. Let ω ∈ ΩnN . Then

∫Mf∗ω = deg(f)

∫Nω.

The surprising part of this theory is the local/global behavior of the quan-tities in question. The degree of f is completely local: we just choose a regularvalue and count the points in the pre-image. But integration is inherently global:we cannot integrate an n–form without knowing its value at all points.

Proof. First, choose a regular value y ∈ N , and let U be an open neighborhoodcontaining y so that f−1(U) =

∐i Vi where f |Vi

: Vi → U is a diffeomorphism.

Lemma 10.2. If ω is supported in U then Theorem 10.1 is true.

Indeed,∫Mf∗ω =

∑i

∫Vif∗ω. But since f |Vi is a diffeomorphism onto U , we

have that∫Vif∗ω = σi

∫Nω, where σi = ±1 according to whether f |Vi

preserves

or reverses orientation. Then∫Mf∗ω =

∑i σi∫Nω. Since

∑i σi = deg(f) by

definition the result follows. Now, suppose that z ∈ N is another point, not necessarily regular for f .

Let hz : N → N be a diffeomorphism which is isotopic to the identity, so thathz(y) = z. Notice that f is homotopic to the map hz f , and that z is regularfor hz f .

Of course, dω = 0 because it lives in Ωn+1N = 0. Thus by Corollary 8.2∫M

(hz f)∗ω =∫Mf∗ω.

Lemma 10.3. If, for some z ∈ N , ω is supported in hz(U) then∫Mf∗ω =

deg(f)∫Nω.

We just apply the previous lemma to hz f , and then use Corollary 8.2.Finally the lemma implies the general result. hz(U) is an open cover of

N , therefore it admits a finite subcover. Using a partition of unity we can writeω =

∑ωj where ωj is supported in hzj (U). We then apply the previous lemma,

and sum.

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