1.8. PROBLEMS 35
4 (Real Grassmann manifolds) In this problem, we generalize Example 1.2.9.
Let Grk(Rn) denote the set of all k-dimensional subspaces of Rn, where 0 ≤ k ≤ n. For each subset I of {1, . . . , n} of cardinality k, consider the linear projectionπI : M(k×n,R) → M(k×k,R) obtained by selecting thekcolumns indexed by the elements ofI, and denote the subset comple-mentary toI in{1, . . . , n}byI′.
a. Check that the mapS : M(k×n,R) → Grk(Rn) that takes ak×n matrixxto the span of its lines inRnis surjective.
b. Let UI = {σ ∈ Grk(Rn) : detπI(x) 6= 0 for somex∈ S−1(σ)}. Check thatGrk(Rn) =∪IUI.
c. Show thatϕI :UI →M(k×(n−k),R)given byϕI(σ) =πI(x)−1πI′(x) for somex∈ S−1(σ)is well-defined and bijective.
d. For I, J ⊂ {1, . . . , n}, prove thatϕI(UI ∩UJ) is open andϕJϕ−1I : ϕI(UI ∩UJ)→ϕJ(UI ∩UJ)is smooth.
e. Deduce that this construction defines onGrk(Rn)a structure of smooth manifold of dimensionk(n−k)such that the mapSis a submersion.
This is called theGrassmann manifold of k-planes inRn. (Hint: Use Proposition 1.2.10.)
f. For σ ∈ Grk(Rn), exhibit a canonical identification TσGrk(Rn) ∼= Hom(σ, σ⊥)(the space of linear maps from σ to its orthogonal com-plement inRn). (Hint: Use theϕ−1I .)
5 Let π : ˜M → M be a topological covering of a smooth manifold M.
Check thatM˜ is necessarily Hausdorff, second-countable (here you need to know that the fundamental groupπ(M) is at most countable) and locally Euclidean. Prove also that there exists a unique smooth structure on M˜ which makesπsmooth and a local diffeomorphism (compare Appendix A).
§ 1.4
6 a. Prove that the composition and the product of immersions are im-mersions.
b. In casedimM = dimN, check that the immersionsM → N coincide with the local diffeomorphisms.
7 Show that every smooth real function on a compact manifold has at least two critical points.
8 Let M be a compact manifold of dimensionn and letf : M → Rn be smooth. Prove thatfhas at least one critical point.
9 Prove that every submersion is an open map.
10 a. Prove that ifM is compact andN is connected then every sub-mersionM →N is surjective.
1.8. PROBLEMS 37 b. Show that there are no submersions of compact manifolds into
Eu-clidean spaces.
11 Letp(z) = zm +am−1zm−1+· · ·+a0 be a polynomial with complex coefficients and consider the associated polynomial map C → C. Show that this map is a submersion out of finitely many points.
12 (Generalized inverse function theorem) Letf :M → N be a smooth map which is injective on a compact embedded submanifoldP ofM. As-sume thatdfp :TpM →Tf(p)N is an isomorphism for everyp∈P.
a. Prove thatf(P)is an embedded submanifold ofN and thatfrestricts to a diffeomorphism fromP ontof(P).
b. Prove that indeedf maps some open neighborhood of P in M dif-feomorphically onto an open neighborhood off(P) in N. (Hint: It suffices to show thatfis injective on some neighborhood ofP; if this is not the case, there exist sequences{pi},{qi}inM both converging to a pointp ∈ P, withpi 6= qi but f(pi) = f(qi) for all i, and this contradicts the non-singularity ofdfp.)
13 Letpbe a homogeneous polynomial of degreeminnvariablest1, . . . , tn. Show thatp−1(a)is a submanifold of codimension one ofRnifa6= 0. Show that the submanifolds obtained witha >0are all diffeomorphic, as well as those witha <0. (Hint: Use Euler’s identity
Xn
i=1
ti∂p
∂ti
=mp.)
14 Then×nreal matrices with determinant1form a group denotedSL(n,R).
Prove thatSL(n,R)is a submanifold ofGL(n,R). (Hint: Use Problem 13.) 15 Consider the submanifolds GL(n,R), O(n) andSL(n,R) of the vec-tor spaceM(n,R) (see Examples 1.2.7(ix) and 1.4.14(b), and Problem 14, respectively).
a. Check that the tangent space ofGL(n,R)at the identity is canonically isomorphic toM(n,R).
b. Check that the tangent space ofSL(n,R)at the identity is canonically isomorphic to the subspace ofM(n,R)consisting of matrices of trace zero.
c. Check that the tangent space ofO(n)at the identity is canonically iso-morphic to the subspace ofM(n,R)consisting of the skew-symmetric matrices.
16 Denote byM(m×n,R)the vector space of realm×nmatrices.
a. Show that the subset ofM(m×n,R)consisting of matrices of rank at leastk(0≤k≤min{m, n}) is a smooth manifold.
b. Show that the subset ofM(m×n,R) consisting of matrices of rank equal tok (0 ≤ k ≤ min{m, n}) is a smooth manifold. What is its dimension? (Hint: We may work in a neighborhood of a matrix
g=
k m−k
k A n−kB
C D
whereAis nonsingular and right multiply by I −A−1B
0 I
to check thatghas rankkif and only ifD−CA−1B = 0.)
17 Let M −→f N −→g P be a sequence of smooth maps between smooth manifolds. Assume thatg ⋔ Qfor a submanifoldQofP. Prove thatf ⋔ g−1(Q)if and only ifg◦f ⋔Q.
18 Let G ⊂ R2 be the graph ofg : R → R, g(x) = |x|1/3. Show that G admits a smooth structure so that the inclusionG→R2 is smooth. Is it an immersion? (Hint: consider the mapf :R→Rgiven by
f(t) =
te−1/t ift >0, 0 ift= 0, te1/t ift <0.)
19 Define immersed submanifolds(M1, f1),(M2, f2)ofN to beequivalent if there exists a diffeomorphismg:M1→M2such thatf2◦g=f1.
a. Show that this is indeed an equivalence relation.
b. Show that each equivalence class of submanifolds of N contains a unique representative of the form(M, ι), whereM is a subset of N with a manifold structure such that the inclusion ι : M → N is a smooth immersion.
c. Let N be a smooth manifold, and letM be a subset ofN equipped with a given topology. Prove that there exists at most one smooth structure onM, up to equivalence, which makes(M, ι)an immersed submanifold of N, where ι : M → N is the inclusion. (Hint: Use Proposition 1.4.9.)
d. LetN be a smooth manifold, and letM be a subset ofN. Prove that there exists at most one structure of smooth manifold on M, up to equivalence, which makes(M, ι)an initial submanifold ofN, where ι:M →N is the inclusion. (Hint: Use Proposition 1.4.9.)
20 LetN be a smooth manifold of dimensionn+k. For a pointq ∈N and a subsetA⊂N, denote byCq(A)the set of all points ofAthat can be joined toqby a smooth curve inN whose image lies inA.
1.8. PROBLEMS 39 a. Prove that if(P, g)is an initial submanifold of dimensionnofN then for everyp∈P there exists a local chart(V, ψ)ofN aroundg(p)such that
ψ(Cg(p)(V ∩g(P))) =ψ(V)∩(Rn× {0}).
(Hint: Use Proposition 1.4.3.)
b. Conversely, assumeP is a subset ofN with the property that around any pointp ∈ P there exists a local chart(V, ψ) ofN aroundpsuch that
ψ(Cp(V ∩P)) =ψ(V)∩(Rn× {0}).
Prove that there exists a topology on P that makes each connected component ofP into an initial submanifold of dimensionnofN with respect to the inclusion. (Hint: Apply Proposition 1.2.10 to the re-strictionsψ|Cp(V∩P). Proving second-countability requires the follow-ing facts: for locally Euclidean Hausdorff spaces, paracompactness is equivalent to the property that each connected component is second-countable; every metric space is paracompact; the topology onP is metrizable since it is compatible with the Riemannian distance for the Riemannian metric induced from a given Riemannian metric on N; Riemannian metrics can be constructed onN using partitions of unity.)
21 Show that the product of any number of spheres can be embedded in some Euclidean space with codimension one.
§ 1.5
22 LetM be a smooth manifold of dimensionn. Givenp ∈M, construct a local chart(U, ϕ)ofM aroundpsuch thatϕis the restriction of a smooth mapM →Rn.
23 Letf : M → N be a map. Prove that f ∈ C∞(M, N) if and only if g◦f ∈C∞(M)for allg∈C∞(N).
24 LetM be a properly embedded submanifold ofN. Prove that the re-striction mapC∞(N)→ C∞(M)is well defined and surjective. Show that the result ceases to be true if: (i)Mis not closed; or (ii)M ⊂N is closed but merely assumed to be an immersed submanifold.
25 Prove that on any smooth manifoldM there exists a proper smooth mapf :M →R. (Hint: Useσ-compactness of manifolds and partitions of unity.)
§ 1.6
26 Determine the vector field onR2with flowϕt(x, y) = (xe2t, ye−3t).
27 Determine the flow of the vector fieldXonR2when:
a. X=y∂x∂ −x∂y∂ . b. X=x∂x∂ +y∂y∂ .
28 Given the following vector fields inR3,
X =y∂x∂ −x∂y∂ , Y =z∂y∂ −y∂z∂, Z = ∂x∂ +∂y∂ +∂z∂, compute their Lie brackets.
29 Show that the restriction of the vector field defined onR2n X=−x2∂x∂
1 +x1∂x∂
2 +· · · −x2n∂x∂
2n−1 +x2n−1∂x∂
2n
to the unit sphereS2n−1 defines a nowhere vanishing smooth vector field.
30 LetXandY be smooth vector fields onM andN with flows{ϕt}and {ψt}, respectively, and letf : M → N be smooth. Show thatXandY are f-related if and only iff◦ϕt=ψt◦f for allt.
31 Let M be a properly embedded submanifold ofN. Prove that every smooth vector field onMcan be smoothly extended to a vector field onN.
32 Construct a natural diffeomorphismT S1 ≈S1×Rwhich restricts to a linear isomorphismTpS1 → {p} ×Rfor everyp ∈ S1(we say that such a diffeomorphism maps fibers to fibers and is linear on the fibers).
33 Construct a natural diffeomorphismT(M×N)≈T M×T Nthat maps fibers to fibers and is linear on the fibers.
34 Construct a natural diffeomorphismTRn ≈Rn×Rnthat maps fibers to fibers and is linear on the fibers.
35 Show that T Sn×Ris diffeomorphic to Sn×Rn+1. (Hint: There are natural isomorphismsTpSn⊕R∼=Rn+1.)
36 A smooth manifold M of dimensionnis calledparallelizableif T M ≈ M×Rnby a diffeomorphism that maps fibers to fibers and is linear on the fibers. Prove thatM is parallelizable if and only if there exists a globally definedn-frame{X1, . . . , Xn}onM.
§ 1.7
1.8. PROBLEMS 41 37 Is there a non-constant smooth functionfdefined on a connected open subset ofR3such that
∂f
∂x−y∂f
∂z = 0 and ∂f
∂y +x∂f
∂z = 0?
(Hint: Consider a regular level set off.)
38 Consider the first order system of partial differential equations
∂z
∂x =α(x, y, z), ∂z
∂y =β(x, y, z)
whereα,βare smooth functions defined on an open subset ofR3.
a. Show that iff is a solution, then the smooth vector fieldsX = ∂x∂ + α∂z∂ eY = ∂y∂ +β∂z∂ span the tangent space to the graph off at all points.
b. Prove that the system admits local solutions if and only if
∂β
∂x +α∂β
∂z = ∂α
∂y +β∂α
∂z.
39 Prove that there exists a smooth functionf defined on a neighborhood of(0,0)inR2such thatf(0,0) = 0and∂f∂x =ye−(x+y)−f,∂f∂y =xe−(x+y)−f.
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) over V. Elements ofT(V)are called tensors onV. Later we will apply these constructions to the tangent space TpMof 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 ofV and W isthan what itdoes. Namely, a tensor product ofV andW is a vector spaceV ⊗W together with a bilinear maph: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˜◦h=B.
V ⊗W
V ×W h∧
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, h1),(V⊗2W, h2) are two tensor products, then there exists an isomorphismℓ : V ⊗1W → V ⊗2W such thatℓ◦h1=h2.
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
43
the linear map on V∗ given byf 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 maph:V ×W →Bil(V, W)∗given byh(v, w)(b) = b(v, w)for b∈Bil(V, W). We claim that(Bil(V, W)∗, h)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˜◦h=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(ek, fℓ)is1if(k, ℓ) = (i, j) and0 otherwise. Prove that{bij} is a basis ofBil(V, W). Prove also that {h(ei, fj)}is the dual basis ofBil(V, W)∗. Deduce that the image ofhspans 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=h(v, w)forv∈V andw∈W. As a consequence of Exercise 2.1.3, note thatdimV ⊗W = (dimV)(dimW).
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).
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{ei⊗fj : i= 1, . . . , mandj = 1, . . . , n}is a basis forV ⊗W.
2.1.8 Exercise LetA= (aij)be a realm×nmatrix, viewed as an element of Hom(Rn,Rm). Use the canonical inner product inRnto identify(Rn)∗ ∼= Rn. What is the element ofRn⊗Rmthat corresponds toA?
2.1. MULTILINEAR ALGEBRA 45 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 the convention that⊗0V =R. Thetensor algebraT(V)overV is the direct sum
T(V) = M
r,s≥0
Vr,s where
Vr,s= (⊗rV)⊗(⊗sV∗)
is called thetensor space of type(r, s). The elements ofT(V)are calledtensors, and those ofVr,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⊗ · · · ⊗vs∗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:
Vr1,s1 ⊗Vr2,s2 ⊂Vr1+r2,s1+s2.
Note thatV0,0 = R, V1,0 = V, V0,1 = 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≥0Vk,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∧· · ·∧
vk. This is also a graded algebra, where the space of elements of degreekis Λk(V) =Vk,0/I∩Vk,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∈Λk(V),β ∈Λℓ(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∧ · · · ∧uk, β = v1 ∧ · · · ∧vℓ. Now
α∧β = u1∧ · · · ∧uk∧v1∧ · · · ∧vℓ
= −u1∧ · · · ∧uk−1∧v1∧uk∧v2· · · ∧vℓ
= u1∧ · · · ∧uk−1∧v1∧v2∧uk∧v3· · · ∧vℓ
= · · ·
= (−1)ℓu1∧ · · · ∧uk−1∧v1∧ · · · ∧vℓ∧uk
= (−1)2ℓu1∧ · · · ∧uk−2∧v1∧ · · · ∧vℓ∧uk−1∧uk
= · · ·
= (−1)kℓβ∧α,
as we wished.
2.1.12 Lemma IfdimV =n, thendim Λn(V) = 1andΛk(V) = 0fork > n.
Proof. Let{e1, . . . , en}be a basis ofV. Since
(2.1.13) {ei1⊗ · · · ⊗eik : i1, . . . , ik∈ {1, . . . , n}}
is a basis ofVk,0 (see Exercise 2.1.7), the image of this set under the pro-jectionVk,0 → Λk(V)is a set of generators of Λk(V). Taking into account Proposition 2.1.11 yieldsΛk(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 ∈ Vk,0, β ∈ Vℓ,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.en−ℓ+1⊗ · · · ⊗en). It follows that we can write (2.1.14) e1⊗ · · · ⊗en=
n−2X
k=0
cke1⊗ · · · ⊗ek⊗vk⊗vk⊗ek+3⊗ · · · ⊗en whereck∈Randvk∈V for allk. Finally, writevk =Pn
i=1aikeiforak∈R. Form= 0, . . . , n−2, the coefficient of
e1⊗ · · · ⊗em⊗em+2⊗em+1⊗em+3⊗ · · · ⊗en on the right hand side of (2.1.14) is
cmam+2,mam+1,m,
2.2. TENSOR BUNDLES 47