Tensor fields and differential forms

C H A P T E R 2

Tensor fields and differential forms

2.1 Multilinear algebra

LetV be a real vector space. In this section, we construct the tensor algebra T(V) and the exterior algebra Λ(V) overV. Elements ofT(V)are called tensors onV. Later we will apply these constructions to the tangent space TpM of a manifoldM and letpvary inM, similarly to the definition of the tangent bundle.

Tensor algebra

All vector spaces are real and finite-dimensional. LetV andW be vector spaces. It is less important what the tensor product of V and W is than what itdoes. Namely, a tensor product ofV andW is a vector spaceV ⊗W together with a bilinear mapι:V ×W → V ⊗W such that the following universal propertyholds: for every vector space U and every bilinear map B :V×W →U, there exists a unique linear mapB˜ :V ⊗W →U such that B˜◦ι=B.

V ⊗W

V ×W ι∧

B> U B˜ ...>

There are different ways to constructV ⊗W. It does not actually matter which one we choose, in view of the following exercise.

2.1.1 Exercise Prove that the tensor product ofV and W is uniquely de- fined by the universal property. In other words, if(V⊗1W, ι1),(V⊗2W, ι2) are two tensor products, then there exists an isomorphismℓ : V ⊗₁W → V ⊗₂W such thatℓ◦ι₁ =ι₂.

We proceed as follows. Start with the canonical isomorphismV^∗∗ ∼=V betweenV and its bidual. It says that we can view an element v inV as

41

the linear map onV^∗ given by f 7→ f(v). Well, we can extend this idea and consider the spaceBil(V, W)of bilinear forms onV ×W. Then there is a natural mapι :V ×W → Bil(V, W)^∗ given byι(v, w)(b) = b(v, w) for b∈Bil(V, W). We claim that(Bil(V, W)^∗, ι)satisfies the universal property:

given a bilinear mapB : V ×W → U, there is an associated mapU^∗ → Bil(V, W),f 7→f◦B; letB˜ : Bil(V, W)^∗ →U^∗∗=U be its transpose.

2.1.2 Exercise Check thatB˜◦ι=B.

2.1.3 Exercise Let {ei}, {fj} be bases ofV, W, respectively. Definebij ∈ Bil(V, W)to be the bilinear form whose value on(e_k, f_ℓ)is1if(k, ℓ) = (i, j) and0 otherwise. Prove that{bij} is a basis ofBil(V, W). Prove also that {i(ei, fj)}is the dual basis ofBil(V, W)^∗. Deduce that the image ofιspans Bil(V, W)^∗ and henceB˜ as in Exercise 2.1.2 is uniquely defined.

Now thatV ⊗W is constructed, we can forget about its definition and keep in mind its properties only (in the same way as when we work with real numbers and we do not need to know that they are equivalence classes of Cauchy sequences), namely, the universal property and those listed in the sequel. Henceforth, we writev⊗w=ι(v, w)forv∈V andw∈W. 2.1.4 Proposition LetV andW be vector spaces. Then:

a. (v₁+v₂)⊗w=v₁⊗w+v₂⊗w; b. v⊗(w1+w2) =v⊗w1+v⊗w2; c. av⊗w=v⊗aw =a(v⊗w);

for allv,v₁,v₂ ∈V;w,w₁,w₂∈W;a∈R.

2.1.5 Proposition LetU, V andW be vector spaces. Then there are canonical isomorphisms:

a. V ⊗W ∼=W ⊗V;

b. (V ⊗W)⊗U ∼=V ⊗(W ⊗U);

c. V^∗⊗W ∼= Hom(V, W); in particular,dimV ⊗W = (dimV)(dimW). 2.1.6 Exercise Prove Propositions 2.1.4 and 2.1.5.

2.1.7 Exercise Let {e₁, . . . , em}and{f₁, . . . , fn}be bases forV andW, re- spectively. Prove that{e_i⊗f_j : i= 1, . . . , mandj = 1, . . . , n}is a basis forV ⊗W.

2.1.8 Exercise LetA= (a_ij)be a realm×nmatrix, viewed as an element of Hom(Rⁿ,R^m). Use the canonical inner product inRⁿto identify(Rⁿ)^∗ ∼= Rⁿ. What is the element ofRⁿ⊗R^mthat corresponds toA?

TakingV =W and using Proposition 2.1.5(b), we can now inductively form the tensornth power⊗ⁿV =⊗ⁿ⁻¹V ⊗V forn ≥1, where we adopt

2.1. MULTILINEAR ALGEBRA 43 the convention that⊗⁰V =R. Thetensor algebraT(V)overV is the direct sum

T(V) = M

r,s≥0

V^r,s where

V^r,s= (⊗^rV)⊗(⊗^sV^∗)

is called thetensor space of type(r, s). The elements ofT(V)are calledtensors, and those ofV^r,sare calledhomogeneous of type(r, s). The multiplication⊗, read “tensor”, is theR-linear extension of

(u₁⊗ · · · ⊗ur1 ⊗u^∗₁⊗ · · · ⊗u^∗_s1)⊗(v₁⊗ · · · ⊗vr2⊗v^∗₁⊗ · · · ⊗v_s^∗₂)

= u1⊗ · · · ⊗ur1⊗v1⊗ · · · ⊗vr2 ⊗u^∗₁⊗ · · · ⊗u^∗_s1 ⊗v₁^∗⊗ · · · ⊗v^∗_s2. T(V) is a non-commutative, associative graded algebra, in the sense that tensor multiplication is compatible with the natural grading:

V^r1,s1 ⊗V^r2,s2 ⊂V^r1+r2,s1+s2.

Note thatV^0,0 = R, V^1,0 = V, V^0,1 = V^∗, so real numbers, vectors and linear forms are examples of tensors.

Exterior algebra

Even more important to us will be a certain quotient of the subalgebra T⁺(V) =L

k≥0V^k,0ofT(V). LetIbe the two-sided ideal ofT⁺(V)gener- ated by the set of elements of the form

(2.1.9) v⊗v

forv∈V.

2.1.10 Exercise Prove that another set of generators for Iis given by the elements of the formu⊗v+v⊗uforu,v ∈V.

Theexterior algebraoverV is the quotient Λ(V) =T⁺(V)/I.

The induced multiplication is denoted by∧, and read “wedge” or “exterior product”. In particular, the class ofv₁⊗· · ·⊗vkmoduloIis denotedv₁∧· · ·∧

v_k. This is also a graded algebra, where the space of elements of degreekis Λ^k(V) =V^k,0/I∩V^k,0.

SinceIis generated by elements of degree2, we immediately get Λ⁰(V) =R and Λ¹(V) =V.

Λ(V)is not commutative, but we have:

2.1.11 Proposition α∧β= (−1)^kℓβ∧αforα ∈Λ^k(V),β ∈Λ^ℓ(V).

Proof. Sincev⊗v ∈Ifor allv∈V, we havev∧v= 0. SinceRis not a field of characteristic two, this relation is equivalent tov₁∧v₂ = −v₂ ∧v₁ for allv₁,v₂∈V.

By linearity, we may assume that α = u₁∧ · · · ∧u_k, β = v₁ ∧ · · · ∧v_ℓ. Now

α∧β = u1∧ · · · ∧uk∧v1∧ · · · ∧vℓ

= −u₁∧ · · · ∧u_k−1∧v₁∧u_k∧v₂· · · ∧v_ℓ

= u₁∧ · · · ∧u_k−1∧v₁∧v₂∧u_k∧v₃· · · ∧v_ℓ

= · · ·

= (−1)^ℓu₁∧ · · · ∧u_k−1∧v₁∧ · · · ∧vℓ∧uk

= (−1)^2ℓu1∧ · · · ∧u_k−2∧v1∧ · · · ∧vℓ∧u_k−1∧uk

= · · ·

= (−1)^kℓβ∧α,

as we wished.

2.1.12 Lemma IfdimV =n, thendim Λⁿ(V) = 1andΛ^k(V) = 0fork > n.

Proof. Let{e1, . . . , en}be a basis ofV. Since

(2.1.13) {e_i1⊗ · · · ⊗e_ik : i₁, . . . , i_k∈ {1, . . . , n}}

is a basis ofV^k,0 (see Exercise 2.1.7), the image of this set under the pro- jectionV^k,0 → Λ^k(V)is a set of generators ofΛ^k(V). Taking into account Proposition 2.1.11 yieldsΛ^k(V) = 0fork > nand thatΛⁿ(V)is generated bye₁∧ · · · ∧en, so we need only show that this element is not zero.

Suppose, on the contrary, thate1 ⊗ · · · ⊗en ∈ I. Thene1⊗ · · · ⊗en is a linear combination of elements of the formα⊗v⊗v⊗β wherev ∈ V, α ∈ V^k,0, β ∈ V^ℓ,0 andk+ℓ+ 2 = n. Writingα (resp.β) in terms of the basis (2.1.13), we may assume that the only appearing base elements are of the forme₁⊗ · · · ⊗ek(resp.e_n−ℓ+1⊗ · · · ⊗en). It follows that we can write (2.1.14) e₁⊗ · · · ⊗e_n =

n−2X

k=0

c_ke₁⊗ · · · ⊗e_k⊗v_k⊗v_k⊗e_k+3⊗ · · · ⊗e_n whereck∈Randvk∈V for allk. Finally, writevk =Pn

i=1aikeiforak∈R. Form= 0, . . . , n−2, the coefficient of

e₁⊗ · · · ⊗em⊗e_m+2⊗e_m+1⊗e_m+3⊗ · · · ⊗en

on the right hand side of (2.1.14) is

cma_m+2,ma_m+1,m,

2.2. TENSOR BUNDLES 45 thus zero. However, the coefficient ofe₁⊗ · · · ⊗enon the right hand side is

n−2X

k=0

cka_k+1,ka_k+2,k,

hence also zero, a contradiction.

2.1.15 Proposition If{e₁, . . . , en}be a basis ofV, then {ei1 ∧ · · · ∧eik : i1 <· · ·< ik}

is a basis ofΛ^k(V)for all0≤k≤n; in particular,dim Λ^k(V) = ⁿ_k .

Proof. Fixk ∈ {0, . . . , n}. The above set is clearly a set of generators of Λ^k(V)and we need only show linear independence. Suppose

Xai1···ikei1 ∧ · · · ∧ei_k = 0, which we write as

XaIeI = 0

where the I denotes increasing k-multi-indices, and e∅ = 1. Multiply through this equation by eJ, where J is an increasing n−k-multi-index, and note that e_I ∧e_J = 0 unlessI is the multi-index J^c complementary to J, in which caseeJ^c ∧eJ = ±e₁ ∧ · · · ∧en. Since e₁ ∧ · · · ∧en 6= 0 by Lemma 2.1.12, this shows thataI = 0for allI.

2.2 Tensor bundles

Cotangent bundle

In the same way as the fibers of the tangent bundle of M are the tangent spacesTpM forp ∈M, the fibers of the cotangent bundle ofM will be the dual spacesTpM^∗. Indeed, form the disjoint union

T^∗M =[˙

p∈MT_pM^∗.

There is a natural projectionπ^∗ :T^∗M →Mgiven byπ(τ) =pifτ ∈TpM^∗. Recall that every local chart(U, ϕ)ofM induces a local chartϕ˜:π⁻¹(U)→ Rⁿ×Rⁿ=R²ⁿofT M, whereϕ(v) = (ϕ(π(v)), dϕ(v))˜ , and thus a mapϕ˜^∗: (π^∗)⁻¹(U) → Rⁿ×(Rⁿ)^∗ = R²ⁿ,ϕ˜^∗(τ) = (ϕ(π^∗(τ)),((dϕ)^∗)⁻¹(τ)), where (dϕ)^∗ denotes the transpose map ofdϕand we have identifiedRⁿ = R^n∗

using the canonical Euclidean inner product. The collection (2.2.1) {((π^∗)⁻¹(U),ϕ˜^∗)|(U, ϕ)∈ A},

for an atlasAofM, satisfies the conditions of Proposition 1.2.10 and defines a Hausdorff, second-countable topology and a smooth structure onT^∗M such thatπ^∗:T M →M is smooth.

A section of T^∗M is a map ω : M → T^∗M such that π^∗ ◦ω = idM. A smooth section ofT^∗M is also called adifferential form of degree1or dif- ferential 1-form. For instance, if f : M → R is a smooth function then dfp : TpM → R is an element ofTpM^∗ for all p ∈ M and hence defines a differential1-formdfonM.

If(U, x₁, . . . , xn)is a system of local coordinates onM, the differentials dx₁, . . . , dxnyield local smooth sections ofT^∗Mthat form the dual basis to

∂

∂x1, . . . ,_∂x^∂

n at each point (recall (1.3.7)). Therefore any sectionω of T^∗M can be locally written as ω|U = Pn

i=1aidxi, and one proves similarly to Proposition 1.6.4 thatωis smooth if and only if theaiare smooth functions onU, for every coordinate system(U, x₁, . . . , x_n).

2.2.2 Exercise Prove that the differential of a smooth function onM indeed gives a a smooth section ofT^∗M by using the atlas (2.2.1).

Tensor bundles

We now generalize the construction of the tangent and cotangent bundles using the notion of tensor algebra. LetM be a smooth manifold. Set:

T^r,s(M) = S

p∈M(TpM)^r,s tensor bundle of type(r, s)overM; Λ^k(M) = S

p∈MΛ^k(TpM^∗) exteriork-bundle overM;

Λ(M) = S

p∈MΛ(TpM^∗) exterior algebra bundle overM. ThenT^r,s(M),Λ^k(M)andΛ(M)admit natural structures of smooth man- ifolds such that the projections ontoM are smooth. If (U, x₁, . . . , x_n) is a coordinate system onM, then the bases {_∂x^∂

i|p}ⁿ_i=1 ofTpM and{dxi|p}ⁿ_i=1 ofTpM^∗, forp ∈ U, define bases of(TpM)^r,s,Λ^k(TpM^∗)andΛ(TpM). For instance, a sectionωofΛ^k(M)can be locally written as

(2.2.3) ω|U = X

i1<···<ik

ai1···ikdxi1 ∧ · · · ∧dxik,

where theai1,...,ikare functions onU.

2.2.4 Exercise Check that T^1,0(M) = T M, T^0,1(M) = T^∗M = Λ¹(M) andΛ⁰(M) =M×R.

The smooth sections ofT^r,s(M),Λ^k(M),Λ^∗(M)are respectively called tensor fields of type(r, s), differentialk-forms,differential formsonM. For in- stance, a sectionωofΛ^k(M)is a differentialk-form if and only if the func- tionsaiin all its local representations (2.2.3) are smooth.

2.2. TENSOR BUNDLES 47 We will denote the space of differential k-forms on M by Ω^k(M) and the space of all differential forms onM byΩ^∗(M). Note that Ω^∗(M) is a graded algebra overRwith wedge multiplication and a module over the ringC^∞(M).

It follows from Problems 4 and 7(a) that a differential k-formω onM is an object that, at each pointp ∈ M, yields a mapωp that can be evalu- ated onktangent vectorsv₁, . . . , vk atpto yield a real number, with some smoothness assumption. The meaning of the next proposition is that we can equivalently think of ω as being an object that, evaluated at k vector fieldsX₁, . . . , Xkyields the smooth function

ω(X₁, . . . , Xk) :p7→ωp(X₁(p), . . . , Xk(p)).

We first prove a lemma.

Hereafter, it shall be convenient to denote theC^∞(M)-module of smooth vector fields onM byX(M).

2.2.5 Lemma Let

ω:X(M)× · · · ×X(M)

| {z }

kfactors

→C^∞(M)

be a C^∞(M)-multilinear map. Then the value of ω(X₁, . . . , X_k) at any given pointpdepends only on the values ofX₁, . . . , X_katp.

Proof. For simplicity of notation, let us do the proof fork = 1; the case k > 1is similar. We first show that ifX|U =X^′|U for some open subsetU ofM, thenω(X)|U =ω(X^′)|U. Indeed letp∈ U be arbitrary, take an open neighborhoodV ofpsuch thatV¯ ⊂U and a smooth functionλ∈C^∞(M) withλ|V¯ = 1andsuppλ⊂U (Exercise 1.5.1). Then

ω(X)(p) = λ(p)ω(X)(p)

= (λ(ω(X)))(p)

= ω(λX)(p)

= ω(λX^′)(p)

= λ(ω(X^′)))(p)

= λ(p)ω(X^′)(p)

= ω(X^′)(p),

where in the third and fifth equalities we have usedC^∞(M)-linearity ofω, and in the fourth equality we have used that λX = λX^′ as vector fields onM.

Finally, we prove that ω(X)(p) depends only on X(p). By linearity, it suffices to prove thatX(p) = 0implies ω(X)(p) = 0. Let(W, x₁, . . . , xn)

be a coordinate system around p and write X|W = Pn i=1ai ∂

∂xi for ai ∈ C^∞(W). By assumption,ai(p) = 0for alli. Letλbe a smooth function on M with support contained inW and such that it is equal to1on an open neighborhoodU ofpwithU¯ ⊂W. Define also

X˜i = λ_∂x^∂

i onW

0 onM\U¯ and ˜ai=

λai onW 0 onM \U¯. ThenX˜ := Pn

i=1˜aiX˜i is a globally defined smooth vector field onM such thatX|˜ U = X|U and we can apply the result in the previous paragraph to write

ω(X)(p) = ω( ˜X)(p)

=

Xn

i=1

˜ aiω( ˜Xi)

! (p)

= Xn

i=1

˜

ai(p)ω( ˜Xi)(p)

= 0

because˜ai(p) =ai(p) = 0for alli.

2.2.6 Proposition Ω^∗(M)is canonically isomorphic as aC^∞(M)-module to the C^∞(M)-module of alternatingC^∞(M)-multilinear maps

(2.2.7) X(M)× · · · ×X(M)

| {z }

kfactors

→C^∞(M)

Proof. Letω ∈ Ω^k(M). Then ωp ∈Λ^k(TpM^∗) ∼= Λ^k(TpM)^∗ ∼=Ak(TpM) for everyp ∈ M, owing to Problems 4 and 7(a), namely, ω_p can be consid- ered to be an alternatingk-multilinear form onTpM. Therefore, for vector fieldsX₁, . . . , X_konM,

˜

ω(X₁, . . . , Xk)(p) :=ωp(X₁(p), . . . , Xk(p))

defines a smooth function onM, ω(X˜ ₁, . . . , Xk) isC^∞(M)-linear in each argumentX_i, thusω˜is an alternatingC^∞(M)-multilinear map as in (2.2.7).

Conversely, let ω˜ be a C^∞(M)-multilinear map as in (2.2.7). Due to Lemma 2.2.5, we haveω˜p ∈ Ak(TpM) ∼= Λ^k(TpM^∗), namely, ω˜ defines a sectionω of Λ^k(M): given v₁, . . . , v_k ∈ T_pM, chooseX₁, . . . , X_k ∈ X(M) such thatXi(p) =vifor alliand put

ωp(v1, . . . , vk) := ˜ω(X1, . . . , Xk)(p).

The smoothness of the sectionωfollows from the fact that, in a coordinate system(U, x₁, . . . , xn), we can writeω|U =P

i1<···<ikai1···ikdxi1 ∧ · · · ∧dxi_k

2.3. THE EXTERIOR DERIVATIVE 49 whereai1···ik(q) =ωq(_∂x^∂

i1

_q, . . . ,_∂x^∂

ik

_q) = ˜ω(_∂x^∂

i1, . . . ,_∂x^∂

ik)(q)for allq∈U, and thusai1···ik ∈C^∞(U). It follows thatωis a differentialk-form onM. Henceforth we will not distinguish between differential k-forms and alternating multilinear maps (2.2.7). Similarly to Proposition 2.2.6:

2.2.8 Proposition The C^∞(M)-module of tensor fields of type (r, s) on M is canonically isomorphic to theC^∞(M)-module ofC^∞(M)-multilinear maps

Ω¹(M)× · · · ×Ω¹(M)

| {z }

rfactors

×X(M)× · · · ×X(M)

| {z }

sfactors

→C^∞(M).

2.3 The exterior derivative

Recall thatΛ⁰(M) = M×R, so a smooth section of this bundle is a map M → M ×Rof the form p 7→ (p, f(p)) wheref ∈ C^∞(M). This shows that Ω⁰(M) ∼= C^∞(M). Furthermore, we have seen that the differential off ∈ C^∞(M) can be viewed as a differential1-formdf ∈ Ω¹(M), so we have an operatorC^∞(M) → Ω¹(M), f 7→ df. In this section, we extend this operator to an operatord: Ω^∗(M) → Ω^∗(M), calledexterior derivative, mappingΩ^k(M) to Ω^k+1(M) for all k ≥ 0. It so happens that dplays an extremelyimportant rôle in the theory of smooth manifolds.

2.3.1 Theorem There exists a uniqueR-linear operator d : Ω^∗(M) → Ω^∗(M) with the following properties:

a. d Ω^k(M)

⊂Ω^k+1(M) for allk≥0 (dhasdegree+1);

b. d(ω∧η) =dω∧η+ (−1)^kω∧dη for everyω∈Ω^k(M),η∈Ω^ℓ(M) (dis ananti-derivation);

c. d² = 0;

d. dfis the differential off for everyf ∈C^∞(M)∼= Ω⁰(M).

Proof. We start with uniqueness, so letdbe as in the statement. The first case is when M is a coordinate neighborhood (U, x₁, . . . , xn). Then any ω ∈ Ω^k(U) can be written asω = P

IaIdxI, whereI runs over increasing multi-indices(i1, . . . , ik)andaI ∈C^∞(U), and we get

dω = X

I

d(aIdxi1 ∧ · · · ∧dxik) (byR-linearity)

= X

I

d(aI)∧dxi1 ∧ · · · ∧dxi_k

+ Xk

r=1

(−1)^r−1aIdxi1 ∧ · · · ∧d(dxir)∧ · · · ∧dxi_k (by (b)) (2.3.2)

= X

I

Xn

r=1

∂a_I

∂xr

dx_r∧dx_i1 ∧ · · · ∧dx_ik (by (c) and (d).)

Next we go to the case of a general manifold M and show thatdis a localoperator, in the sense that(dω)|U = 0wheneverω|U = 0andU is an open subset ofM. So assumeω|_U = 0, take an arbitrary pointp ∈ U, and chooseλ∈C^∞(M) such that0 ≤λ≤1,λis flat equal to1onM \U and has support disjoint fromV¯, whereV is a neighborhood ofpwithV¯ ⊂U. Thenω=λωon the entireMso that, using (b) we get

(dω)p =d(λω)p =dλp∧ ωp

|{z}=0

+λ(p)

|{z}

=0

dωp = 0,

as wished.

To continue, we verify that d induces an operatordU on Ω^∗(U) satis- fying (a)-(d) for every open subset U of M. So given ω ∈ Ω^k(U) and p ∈ U, constructω˜ ∈ Ω^k(M) which coincides withω on a neighborhood V of p with V¯ ⊂ U, as usual by means of a bump function, and define (dUω)p := (d˜ω)p. The definition is independent of the chosen extension, asdis a local operator. It is easy to check thatd_U indeed satisfies (a)-(d);

for instance, for (b), note that ω˜ ∧ η˜ is an extension of ω ∧η and hence dU(ω∧η)p = (d(˜ω ∧η))˜ p = (d˜ω)p ∧η˜p + (−1)^{deg ˜ω}ω˜p ∧(d˜η)p = (dUω)p ∧ η_p+ (−1)^degωω_p∧(d_Uη)_p. Note also that the collection{d_U}isnatural with respect to restrictions, in the sense that ifU ⊂V are open subsets ofM then dV|U =dU.

Finally, forω ∈ Ω^∗(M) and a coordinate neighborhood(U, x₁, . . . , xn), on one handdU(ω|U)is uniquely defined by formula (2.3.2). On the other hand, ω itself is an extension of ω|U, and hence (dω)p = (dU(ω|U))p for everyp∈U. This proves thatdωis uniquely defined.

To prove existence, we first use formula (2.3.2) to define an R-linear operatordUonΩ^k(U)for every coordinate neighborhoodUofM. It is clear thatd_U satisfies (a) and (d); let us prove that it also satisfies (b) and (c). So letω=P

IaIdxI ∈Ω^k(U). ThendUω=P

IdaI ∧dxI and

d²_Uω = X

I,r

dU

∂aI

∂xr

dxr∧dxI

= X

I,r,s

∂²a_I

∂xs∂xr

dx_s∧dx_r∧dx_I

= 0,

2.3. THE EXTERIOR DERIVATIVE 51 since ∂²aI

∂x_s∂x_r is symmetric anddxs∧dxris skew-symmetric inr,s. Let also η=P

JbJdxJ. Thenω∧η=P

I,JaIbJdxI∧dxJ and dU(ω∧η) = X

I,J

dU(aIbJdxI ∧dxJ)

= X

I,J,r

∂a_I

∂xr

bJdxr∧dxI ∧dxJ +X

I,J,s

aI

∂b_J

∂xs

dxs∧dxI∧dxJ

=



X

I,r

∂aI

∂xr

dxr∧dxI



∧ X

J

bJdxJ

!

+(−1)^|I| X

I

aIdxI

!

∧



X

J,s

∂bJ

∂xs

dxs∧dxJ





= dUω∧η+ (−1)^degωω∧dUη,

where we have used Proposition 2.1.11 in the third equality to writedx_s∧ dxI = (−1)^|I|dxI ∧dxs.

We finish by noting that the operatorsdU for each coordinate systemU ofM can be pieced together to define a global operatord. Indeed for two coordinate systemsU andV, the operatorsd_Uandd_V induce two operators onΩ^∗(U ∩V)satisfying (a)-(d) by the remarks above which must coincide by the uniqueness part. Note also that the resultingdsatisfies (a)-(d) since

it locally coincides with somedU.

2.3.3 Remark We have constructed the exterior derivativedas an operator between sections of vector bundles which, locally, is such that the local coordinates ofdωare linear combinations of partial derivatives of the local coordinates ofω(cf. 2.3.2). For this reason,dis called adifferential operator.

Pull-back

A nice feature of differential forms is that they can always be pulled-back under a smooth map. In contrast, the push-forward of a vector field under a smooth map need not exist if the map is not a diffeomorphism.

Letf : M → N be a smooth map. The differentialdfp : TpM → T_f(p) at a pointpinM has a transpose map(df_p)^∗ :T_f(p)N^∗ → T_pM^∗ and there is an induced algebra homomorphism δfp := Λ((dfp)^∗) : Λ(T_f(p)N^∗) → Λ(TpM^∗)(cf. Problem 6). For varyingp∈M, this yields mapδf : Λ^∗(N)→ Λ^∗(M). Recall that a differential formω onN is a section ofΛ^∗(N). The pull-back ofωunderf is the section ofΛ^∗(M)given byf^∗ω =δf ◦ω◦f, so

that the following diagram is commutative:

Λ^∗(M) <δf

Λ^∗(N)

M f^∗ω∧

f > N

∧...ω ...

(We prove below thatf^∗ω is smooth, so that it is in fact a differential form onM. This fact would also follow from the formulaf^∗ω =δf ◦ω◦f if we checked thatδf is a smooth.) In more detail, we have

(f^∗ω)_p =δf(ω_f(p))

for allp ∈ M. In particular, ifω is ak-form, then(f^∗ω)p ∈ Λ^k(TpM^∗) = Λ^k(TpM)^∗=A_k(TpM)and

(2.3.4) (f^∗ω)p(v₁, . . . , vk) =ω_f(p)(dfp(v₁), . . . , dfp(vk)) for allv₁, . . . , v_k∈T_pM.

2.3.5 Exercise Letf :M →N be a smooth map.

a. In the case of 0-forms, that is smooth functions, check that f^∗(g) = g◦f for allg∈Ω⁰(N) =C^∞(N).

b. In the caseω=dg∈Ω¹(N)for someg∈C^∞(N), check thatf^∗(dg) = d(g◦f).

2.3.6 Proposition Letf :M →N be a smooth map. Then:

a. f^∗ : Ω^∗(N)→Ω^∗(M)is a homomorphism of algebras;

b. d◦f^∗ =f^∗◦d;

c. (f^∗ω)(X₁, . . . , Xk)(p) = ω_f(p)(df(X₁(p)), . . . , df(Xk(p))) for all ω ∈ Ω^∗(N)and allX₁, . . . , X_k∈X(M).

Proof. Result (c) follows from (2.3.4). The fact that f^∗ is compatible with the wedge product is a consequence of Problem 6(b) applied to local expressions of the form (2.2.3). For (a), it only remains to prove thatf^∗ωis actually asmoothsection of Λ^∗(M) for a differential formω ∈ Ω^∗(M). So letp∈M, choose a coordinate system(V, y₁, . . . , y_n)ofN aroundf(p)and a neighborhoodU of p inM with f(U) ⊂ V. Sincef^∗ is linear, we may assume thatωis ak-form. Asωis smooth, we can write

ω|V =X

I

aIdyi1 ∧ · · · ∧dyik. It follows from Exercise 2.3.5 that

(2.3.7) f^∗ω|_U =X

I

(a_I ◦f)d(y_i1 ◦f)∧ · · · ∧d(y_ik ◦f),

2.4. THE LIE DERIVATIVE OF TENSORS 53 which indeed is a smooth form onU. Finally, (b) is proved using (2.3.7):

d(f^∗ω)_p = d X

I

(a_I ◦f)d(y_i1◦f)∧ · · · ∧d(y_ik◦f)

!

p

= X

I

(d(aI ◦f)∧d(yi1◦f)∧ · · · ∧d(yi_k◦f))|p

= f^∗ X

I

da_I ∧dy_i1∧ · · · ∧dy_ik

! _p

= f^∗(dω)p,

as desired.

2.4 The Lie derivative of tensors

In section 1.6, we defined the Lie derivative of a smooth vector fieldY on M with respect to another smooth vector fieldX by using the flow{ϕ_t}of X to identify different tangent spaces ofM along an integral curve ofX.

The same idea can be used to define the Lie derivative of a differential form ωor tensor fieldSwith respect toX. The main point is to understand the action of{ϕt}on the space of differential forms or tensor fields.

So let {ϕt} denote the flow of a vector field X on M, and let ω be a differential form onM. Then the pull-backϕ^∗_tω is a differential form and t7→ (ϕ^∗_tω)p is a smooth curve inΛ(TpM^∗), for allp ∈M. TheLie derivative ofωwith respect toXis the sectionL_XωofΛ(M)given by

(2.4.1) (LXω)p= d

dt

t=0(ϕ^∗_tω)p.

We prove below thatLXωis smooth, so it indeed yields a differential form onM. In view of (2.3.4), it is clear that the Lie derivative preserves the degree of a differential form.

We extend the definition of Lie derivative to an arbitrary tensor fieldS of type(r, s)as follows. Suppose

S_ϕt(p)=v₁⊗ · · · ⊗v_r⊗v^∗₁⊗ · · · ⊗v^∗_s. Then we define(ϕ^∗_tS)p ∈(TpM)^r,sto be

dϕ_−t(v₁)⊗ · · · ⊗dϕ_−t(vr)⊗δϕt(v^∗₁)⊗ · · · ⊗δϕt(v_s^∗) and put

(2.4.2) (LXS)p= d

dt

t=0(ϕ^∗_tS)p.

One can view definition 2.4.1 as the operator in the quotient obtained from definition 2.4.2 in the sense that the exterior algebra is a subquotient of the tensor algebra.

Before stating properties of the Lie derivative, it is convenient to intro- duce two more operators. ForX∈X(M)andω∈Ω^k+1(M)withk≥0, the interior multiplicationι_Xω∈Ω^k(M)is thek-differential form given by

ι_Xω(X₁, . . . , X_k) =ω(X, X₁, . . . , X_k) forX₁, . . . , X_k∈X(M), andι_X is zero on0-forms.

2.4.3 Exercise Prove that ι_Xω is indeed a smooth section of Λ^k−1(M) for ω∈Ω^k(M). Prove also thatιX is ananti-derivationin the sense that

ιX(ω∧η) =ιXω∧η+ (−1)^kω∧ιXη

forω ∈ Ω^k(M)andη ∈ Ω^ℓ(M). (Hint: For the last assertion, it suffices to check the identity at one point.)

Let V be a vector space. Thecontraction ci,j : V^r,s → V^r−1,s−1 is the linear map that operates on basis vectors as

v₁⊗ · · · ⊗v_r⊗v₁^∗⊗ · · · ⊗v^∗_s

7→v^∗_j(v_i)v₁⊗ · · · ⊗vˆ_i⊗ · · · ⊗v_r⊗v^∗₁⊗ · · ·vˆ_j^∗⊗ · · · ⊗v^∗_s. It is easy to see thatc_i,j extends to a mapT^r,s(M)→ T^r−1,s−1(M).

2.4.4 Exercise Let V be a vector space. Recall the canonical isomorphism V^1,1 ∼= Hom(V, V) = End(V)(Proposition 2.1.5). Check thatc1,1 :V^1,1 → V^0,0is the trace maptr : End(V)→R.

2.4.5 Proposition LetXbe a smooth vector field onM. Then:

a. LXf =X(f)for allf ∈C^∞(M).

b. LXY = [X, Y]for allX∈X(M).

c. LXis a type-preservingR-linear operator on the spaceT(M)of tensor fields onM.

d. LX :T(M)→ T(M)is a derivation, in the sense that LX(S⊗S^′) = (LXS)⊗S^′+S⊗(LXS^′) e. LX :T(M)→ T(M)commutes with contractions:

LX(c(S)) =c(LXS) for any contractionc:T^r,s(M)→ T^r−1,s−1(M).

f. L_Xis a degree-preservingR-linear operator on the space of differential forms Ω(M)which is a derivation and commutes with exterior differentiation.

2.4. THE LIE DERIVATIVE OF TENSORS 55 g. LX =ιX◦d+d◦ιX onΩ(M)(Cartan’s magical formula)

h. Forω∈Ω^k(M)andX₀, . . . , X_k ∈X(M), we have:

LX0ω(X₁, . . . , Xk) =X₀(ω(X₁, . . . , Xk))

− Xk

i=1

ω(X₁, . . . , X_i−1,[X₀, X_i], X_i+1, . . . , X_k).

i. Same assumption as in (h), we have:

dω(X₀, . . . , X_k) = Xk

i=0

(−1)ⁱX_iω(X₀, . . . ,Xˆ_i, . . . , X_k)

+X

i<j

(−1)^i+jω([Xi, Xj], X₀, . . . ,Xˆi, . . . ,Xˆj, . . . , Xk).

Proof. (a) follows from differentiation of (ϕ^∗_tf)p = f(ϕt(p)) at t = 0.

(b) was proved in section 1.6. The type-preserving part of (c) is clear from the definition. For (d), differentiate the obvious formula ϕ^∗_t(S ⊗S^′)|_p = (ϕ^∗_tS)p ⊗(ϕ^∗_tS^′)p at t = 0; the derivation property follows using the fact that tensor multiplication isR-bilinear. Smoothness ofLXSas a section of T^r,s(M)is proved noting thatL_Xis a local operator and expressingL_XSin a system of local coordinates, see below for the analogous argument in the case of differential forms. This covers (c) and (d).

(e) follows from the easily checked fact that ϕ^∗_t commutes with con- tractions. As a consequence, which we will use below, ifω ∈ Ω¹(M)and Y ∈X(M)thenω(Y) =c(Y ⊗ω)so

X(ω(Y)) = L_X(c(Y ⊗ω)) (using (a))

= c(LX(Y ⊗ω))

= c(LXY ⊗ω+Y ⊗LXω) (using (d))

= ω([X, Y]) +LXω(Y) (using (b));

in other words,

(2.4.6) LXω(Y) =X(ω(Y))−ω([X, Y]).

For (f), we first remark thatL_X is a derivation as a map fromΩ(M) to non-necessarily smooth sections of Λ(M): this is a pointwise check, and follows from the fact that (ϕt)^∗ defines an automorphism of the algebra Ω(M). Next, check thatL_X commutes withdon functions using (2.4.6):

L_X(df)(Y) = X(df(Y))−df([X, Y])

= X(Y(f))−[X, Y](f)

= Y(X(f))

= d(X(f))(Y)

= d(LXf)(Y)

for allf ∈C^∞(M)andY ∈X(M). To continue, note thatLXis a local oper- ator: formula (2.4.1) shows thatLXω|U depends only onω|U, for any open subsetU ofM, and the same applies for (2.4.1). Finally, to see thatL_Xωis smooth for anyω ∈ Ω(M), we may assume thatωhas degreekand work in a coordinate system(U, x₁, . . . , xn), whereωhas a local representation as in (2.2.3). Using the above collected facts:

LXω|U = X

i1<···<ik

X(ai1···ik)dxi1 ∧ · · · ∧dxi_k

+ Xk

j=1

ai1···ikdxi1 ∧ · · · ∧d(X(xij))∧ · · · ∧dxik

as wished. This formula can also be used to show thatLX commutes with din general.

To prove (g), letP_X =d◦ι_X+ι_X◦d. ThenP_XandL_Xare local operators, derivations of Ω(M), that coincide on functions and commmute with d.

Since any differential form is locally a sum of wedge products of functions and differentials of functions, it follows thatL_X =P_X.

The casek= 1in (h) is formula (2.4.6). The proof fork >1is completely analogous.

Finally, (i) is proved by induction onk. The initial casek= 0is imme- diate. Assuming (i) holds fork−1, one proves it forkby starting with (h)

and using (g) and the induction hypothesis.

2.4.7 Exercise Carry out the calculations to prove (h) and (i) in Proposi- tion 2.4.5.

2.5 Vector bundles

The tangent, cotangent and and all tensor bundles we have constructed so far are smooth manifolds of a special kind in that they have a fibered struc- ture over another manifold. For instance,T M fibers overM so that the fiber over any pointpinM is the tangent space TpM. Moreover, there is some control on how the fibers vary with the point. In case ofT M, this is reflected on the way a chart (π⁻¹(U),ϕ)˜ is constructed from a given chart (U, ϕ) of M. Recall that ϕ˜ : π⁻¹(U) → Rⁿ ×Rⁿ where ϕ(v) =˜ (ϕ(π(v)), dϕ(v)). Soϕ˜induces a diffeomorphism∪_p∈UTpM → ϕ(U)×Rⁿ so that each fiberTpMis mapped linearly and isomorphically onto{ϕ(p)}×

Rⁿ. We could also compose this map withϕ⁻¹×idto get a diffeomorphism T M|U :=∪_p∈UTpM →ϕ(U)×Rⁿ→U ×Rⁿ.

Of course each T_pM is abstractly isomorphic to Rⁿ, where n = dimM, but here we are saying that the part ofT M consisting of fibers lying over

2.5. VECTOR BUNDLES 57 points inU is diffeomorphic to a productU ×Rⁿin such a way thatTpM corresponds to{p} ×Rⁿ. This is the idea of a vector bundle.

2.5.1 Definition A (smooth) vector bundle of rank k over a smooth man- ifold M is a smooth manifold E, called the total space, together with a smooth projectionπ:E →Msuch that:

a. E_p :=π⁻¹(p)is a vector space of dimensionkfor allp∈M;

b. M can be covered by open setsU such that there exists a diffeomor- phismE|U =π⁻¹(U)→U×R^kmappingEplinearly and isomorphi- cally onto{p} ×R^kfor allp∈U.

Thetrivial vector bundleof rankkoverM is the direct productM×R^k with the projection onto the first factor. A vector bundle of rankk = 1is also called aline bundle.

An equivalent definition of vector bundle, more similar in spirit to the definition of smooth manifold, is as follows.

2.5.2 Definition A (smooth) vector bundle of rankk over a smooth mani- foldMis a setE, called thetotal space, together with a projectionπ :E→M with the following properties:

a. M admits a covering by open setsU such that there exists a bijection ϕU : E|U = π⁻¹(U) → U ×R^k satisfyingπ = π₁ ◦ϕU, where π₁ : U×R^k→U is the projection onto the first factor. Such aϕ_Uis called alocal trivialization.

b. Given local trivializationsϕU,ϕV withU∩V 6=∅, the change of local trivialization or transition function

ϕU ◦ϕ⁻¹_V : (U ∩V)×R^k→(U ∩V)×R^k has the form

(x, a)7→(x, gU V(x)a) where

g_{U V} :U ∩V →GL(k,R) is smooth.

2.5.3 Exercise Prove that the family of transition functions{gU V} in Defi- nition 2.5.2 satisfies thecocycle conditions:

gU U(x) = id (x∈U)

gU V(x)gV W(x)gW U(x) = id (x∈U∩V ∩W) 2.5.4 Exercise LetM be a smooth manifold.

a. Prove that for a vector bundleπ : E → M as in Definition 2.5.2, the total spaceEhas a natural structure of smooth manfifold such thatπ is smooth and the local trivializations are diffeomorphisms.

b. Prove that Definitions 2.5.1 and 2.5.2 are equivalent.

2.5.5 Example In this example, we construct a very important example of vector bundle which is not a tensor bundle, called thetautological (line) bun- dleoverRPⁿ. Recall that a pointpin real projective spaceM = RPⁿis a 1-dimensional subspace ofRⁿ⁺¹(Example 1.2.9). SetE = ˙∪_p∈MEp where Epis the subspace ofRⁿ⁺¹ corresponding top, namely,Ep consists of vec- torsv∈ Rⁿ⁺¹ such thatv∈ p. Letπ :E →M mapE_p top. We will prove that this is a smooth vector bundle by constructing local trivializations and using Definition 2.5.2. Recall the atlas{ϕi}ⁿ⁺¹_i=1 of Example 1.2.9. Set

˜

ϕ_i:π⁻¹(U_i)→U_i×R v7→(π(v), x_i(v)).

This is a bijection and the cocycle

gij(x1, . . . , xn+1) =xi/xj ∈GL(1,R) =R\ {0}

is smooth onUi∩Uj, as wished.

2.6 Problems

§ 2.1

1 LetV be a vector space and letι:Vⁿ→ ⊗ⁿV be defined asι(v₁, . . . , vn) = v1⊗ · · · ⊗vn, whereVⁿ =V × · · · ×V (nfactors on the right hand side).

Prove that⊗ⁿV satisfies the following universal property: for every vector spaceU and everyn-multilinear mapT : Vⁿ → U, there exists a unique linear mapT˜:⊗ⁿV →U such thatT˜◦ι=T.

⊗ⁿV

Vⁿ ι∧

T> U T˜ ...>

2 Prove that⊗ⁿV is canonically isomorphic to the dual space of the space n-multilinear forms onVⁿ. (Hint: Use Problem 1.)

3 Let V be a vector space. Ann-multilinear mapT : Vⁿ → U is called alternatingifT(v_σ(1), . . . , v_σ(n)) = (sgnσ)T(v₁, . . . , vn)for everyv₁, . . . , vn∈ V and every permutationσof{1, . . . , n}, wheresgndenotes the sign±1of the permutation.

Letι:Vⁿ→Λⁿ(V)be defined asι(v₁, . . . , v_n) =v₁∧· · · ∧v_n. Note thatι is alternating. Prove thatΛⁿV satisfies the following universal property: for

2.6. PROBLEMS 59 every vector spaceU and every alternatingn-multilinear mapT :Vⁿ→U, there exists a unique linear mapT˜: Λⁿ(V)→U such thatT˜◦ι=T.

ΛⁿV

Vⁿ ι

∧

T> U T˜ ...>.

4 Denote the vector space of all alternating multilinear formsVⁿ →Rby A_n(V). Prove thatΛⁿ(V)is canonically isomorphic toA_n(V)^∗.

5 Prove thatv₁, . . . , vk ∈V are linearly independent if and only ifv₁∧ · · · ∧ v_k6= 0.

6 LetV andW be vector spaces and letT :V →W be a linear map.

a. Show thatT naturally induces a linear mapΛ^k(T) : Λ^k(V)→Λ^k(W). (Hint: Use Problem 3.)

b. Show that the mapsΛ^k(V)for variouskinduce an algebra homomor- phismΛ(T) : Λ(V)→Λ(W).

c. Let nowV =W andn= dimV. The operatorΛⁿ(T)is multiplication by a scalar, asdim Λⁿ(V) = 1; define the determinantofT to be this scalar. Anyn×nmatrixA= (aij)can be viewed as the representation of a linear operator onRⁿwith respect to the canonical basis. Prove that

detA=X

σ

(sgnσ)a_i,σ(i)· · ·a_n,σ(n),

wheresgnσis the sign of the permutationσandσruns over the set of all permutations of the set{1, . . . , n}. Prove also that the determinant of the product of two matrices is the product of their determinants.

d. Using Problem 7(a) below, prove that the transpose map Λ^k(T)^∗ = Λ^k(T^∗).

7 LetV be vector space.

a. Prove that there is a canonical isomorphism Λ^k(V^∗)∼= Λ^k(V)^∗ given by

v^∗₁∧ · · · ∧v^∗_k7→(u₁∧ · · · ∧uk7→det(v_i^∗(uj)) ).

b. Letα,β ∈V^∗∼= Λ¹(V^∗)∼=A₁(V). Show thatα∧β ∈Λ²(V^∗), viewed as an element ofΛ²(V)^∗ ∼=A₂(V)is given by

α∧β(u, v) =α(u)β(v)−α(v)β(u) for allu,v∈V.

8 LetV be a vector space.

a. In analogy with the exterior algebra, construct thesymmetric algebra Sym(V), a commutative graded algebra, as a quotient ofT(V).

b. Determine a basis of the homogeneous subspaceSymⁿ(V).

c. State and prove thatSymⁿ(V)satisfies a certain universal property.

d. Show that theSymⁿ(V)is canonically isomorphic to the dual of the spaceSn(V)of symmetricn-multilinear formsVⁿ→R.

In view of (d),Sym(V^∗)is usually defined to be the spaceP(V)ofpolyno- mialsonV.

9 An element ofΛⁿ(V)is calleddecomposableif it lies in the subsetΛ¹(V)∧

· · · ∧Λ¹(V)(nfactors).

a. Show that in general not every element ofΛⁿ(V)is decomposable.

b. Show that, for dimV ≤ 3, every homogeneous element in Λ(V) is decomposable.

c. Letωbe a differential form. Isω∧ω= 0?

10 Let V be an oriented vector space equipped with a non-degenerate symmetric bilinear form (we do not require positive-definiteness from the outset). LetdimV =n.

a. Prove there exists an elementω ∈Λⁿ(V)such that ω=e₁∧ · · · ∧e_n

for every positively oriented orthonormal basis{e₁, . . . , e_n}ofV. b. Show that the bilinear map

Λ^k(V)×Λ^n−k(V)→Λⁿ(V), (α, β)7→α∧β together with the isomorphism

R→Λⁿ(V), a7→a ω define a canonical isomorphism

Λ^k(V)→(Λ^n−k(V))^∗.

c. Check that the bilinear form onV induces an isomorphismV ∼=V^∗, which induces an isomorphismΛ^n−k(V)∼= Λ^n−k(V^∗)via Problem 6(a).

d. Combine the isomorphims of (b) and (c) with that in Problem 7(a) to get a linear isomorphism

∗: Λ^k(V)→Λ^n−k(V) for0≤k≤n, called theHodge star.

2.6. PROBLEMS 61 e. Assume the inner product is positive definite and let{e₁, . . . , en}be a

positively oriented orthonormal basis ofV. Show that

∗1 =e1∧ · · · ∧en, ∗(e1∧ · · · ∧en) = 1, and

∗(e₁∧ · · · ∧ek) =e_k+1∧ · · · ∧en. Show also that

∗∗= (−1)^k(n−k) onΛ^k(V).

§ 2.2

11 LetM be a smooth manifold. ARiemannian metricgonMis an assign- ment of positive definite inner productgpon each tangent spaceTpMwhich is smooth in the sense thatg(X, Y)(p) = gp(X(p), Y(p))defines a smooth function for everyX,Y ∈X(M). ARiemannian manifoldis a smooth mani- fold equipped with a Riemannian metric.

a. Show that a Riemannian metricgonM is the same as a tensor fieldg˜ of type(0,2) which issymmetric, in the sense that˜g(Y, X) = ˜g(X, Y) for every X, Y ∈ X(M), with the additional property of positive- definiteness at each point.

b. Fix a local coordinate system(U, x₁, . . . , x_n)onM.

(i) Letgbe a Riemannian metric onM. Show thatg|U =P

i,jgijdxi⊗ dxj where gij = g(_∂x^∂

i,_∂x^∂

j) ∈ C^∞(U),gij = gji and the matrix (gij)is everywhere positive definite.

(ii) Conversely, given functions g_ij = g_ji ∈ C^∞(U) such that the matrix(gij)is positive definite everywhere inM, show how to define a Riemannian metric onU.

c. Use part (b)(ii) and a partition of unity to prove that every smooth manifold can be equipped with a Riemannian metric.

d. On a Riemannian manifoldM there exists a natural diffeomorphism T M ≈ T^∗M taking fibers to fibers. (Hint: There exist linear isomor- phismsv∈T_pM 7→g_p(v,·)∈T_pM^∗).

§ 2.3

12 ConsiderR³ with coordinates (x, y, z). In each case, decide whether dω= 0or there existsηsuch thatdη=ω.

a. ω=yzdx+xzdy+xydz. b. ω=xdx+x²y²dy+yzdz.

c. ω= 2xy²dx∧dy+zdy∧dz.

13 (The operatordonR³) Identify1- and2-forms onR³with vector fields onR³, and0- and3-forms onR³with smooth functions onR³, and check that:

don0-forms is the gradient;

don1-forms is the curl;

don2-forms is the divergent.

Also, interpretd²= 0is those terms.

§ 2.4

14 LetMandNbe smooth manifolds whereMis connected, and consider the projection π : M ×N → N onto the second factor. Prove that a k- formωonM ×N is of the formπ^∗ηfor somek-formηonN if and only if ι_Xω=L_Xω = 0for everyX∈X(M×N)satisfyingdπ◦X= 0.

15 LetMbe a smooth manifold.

a. Prove thatιXιX = 0for everyX∈X(M).

b. Prove that ι_[X,Y]ω = LXιYω−ιYLXω for everyX, Y ∈ X(M) and ω ∈Ω^k(M).

§ 2.5

16 TheWhitney sumE₁⊕E₂of two vector bundlesπ₁:E₁→M,π₂:E₂→ M is a vector bundleπ :E =E₁⊕E₂ → M whereE_p = (E₁)_p⊕(E₂)_p for allp∈M.

a. Show thatE₁⊕E₂ is indeed a vector bundle by expressing its local trivializations in terms of those ofE₁andE₂and checking the condi- tions of Definition 2.5.2.

b. Similarly, construct the tensor product bundle E₁⊗E₂ and the dual bundleE^∗.