PROBLEMSET2,PART1:THEDIFFERENTIALDUE:OCT.17(1)[Not required] [Tangent vectors defined by curves]There is a more geometric way to define tangent vectors: Take a chart (o,U,V)around p. Let Cp be the set of all smooth maps : (-e,e) → U such that (O) = pDefine an equivalence relation on Cp byd((0) = (0(0),~dtdtDefine a natural vector space structure on Cp/~ so that T,M is isomorphic to Cp/ ~(2)[Product of manifolds-continued]Let M,Nbesmoothmanifoldsand considertheproductM xN.Prove:(a) The nature projections 1 : M × N-→ M and 2 : M × N → N are smooth.(b) [Universality] If P is any smooth manifold, then a map f :P→ M ×N is smoothifand onlyifbothπ1of and2ofaresmooth.(c) T(p1,p2)(Mi × M2) ~ Tp/Mi TpzM2.(d) More generally, let fi : Mi → Ni and f2 : M2 → N2 be smooth maps, andp1 E Mi,p2 E M2. Find the relation between d(fi x f2)(p1,p2) and (dfi)pi, i = 1,2.(3)[The tangent bundle]Let M be a smooth manifold of dimension n. Let TM = U,TpM be the disjointunion of all tangent vectors.Wewill call TM the tangent bundle of M.Wewilldenote an element X, in TM by (p,Xp), to emphasis its"base point".There is anatural projection mapπ:TM-→M,(p,X,)-→pFor each chart (,U,V) of M, we define a bijective mapTβ= (p0,d) : -1(U)→V × R", (p,Xp)- (0(p),dpp(Xp))We endow a topology on TM so that each T is a homeomorphism. Prove:(a) (To,π-1(U), V × Rn) is a chart of TM.(b) Charts of this type are compatible, so TM is a 2n-dimensional smooth manifold(c) No mater M is orientable or not, the tangent bundle TM is orientable.(d) The differential di(p,X,) : T(p,X,)TM → TpM is always surjective.(e) TS' is diffeomorphic to S' × R.(4) [Not required] [Submersions are open maps](a) Let f : M-→ N be a submersion. Prove: f is an open map.1
PROBLEM SET 2, PART 1: THE DIFFERENTIAL DUE: OCT. 17 (1) [Not required] [Tangent vectors defined by curves] There is a more geometric way to define tangent vectors: Take a chart {ϕ, U, V } around p. Let Cp be the set of all smooth maps γ : (−ε, ε) → U such that γ(0) = p. Define an equivalence relation on Cp by γα ∼ γβ ⇐⇒ d(ϕ ◦ γα) dt (0) = d(ϕ ◦ γβ) dt (0). Define a natural vector space structure on Cp/ ∼ so that TpM is isomorphic to Cp/ ∼. (2) [Product of manifolds – continued] Let M, N be smooth manifolds and consider the product M × N. Prove: (a) The nature projections π1 : M × N → M and π2 : M × N → N are smooth. (b) [Universality] If P is any smooth manifold, then a map f : P → M ×N is smooth if and only if both π1 ◦ f and π2 ◦ f are smooth. (c) T(p1,p2) (M1 × M2) ' Tp1M1 ⊕ Tp2M2. (d) More generally, let f1 : M1 → N1 and f2 : M2 → N2 be smooth maps, and p1 ∈ M1, p2 ∈ M2. Find the relation between d(f1×f2)(p1,p2) and (dfi)pi , i = 1, 2. (3) [The tangent bundle] Let M be a smooth manifold of dimension n. Let TM = S p TpM be the disjoint union of all tangent vectors. We will call TM the tangent bundle of M. We will denote an element Xp in TM by (p, Xp), to emphasis its “base point”. There is a natural projection map π : TM → M, (p, Xp) 7→ p. For each chart (ϕ, U, V ) of M, we define a bijective map T ϕ = (ϕ ◦ π, dϕ) : π −1 (U) → V × R n , (p, Xp) 7→ (ϕ(p), dϕp(Xp)). We endow a topology on TM so that each T ϕ is a homeomorphism. Prove: (a) (T ϕ, π−1 (U), V × R n ) is a chart of TM. (b) Charts of this type are compatible, so TM is a 2n-dimensional smooth manifold. (c) No mater M is orientable or not, the tangent bundle TM is orientable. (d) The differential dπ(p,Xp) : T(p,Xp)TM → TpM is always surjective. (e) T S1 is diffeomorphic to S 1 × R. (4) [Not required] [Submersions are open maps] (a) Let f : M → N be a submersion. Prove: f is an open map. 1
2PROBLEMSET2,PART1:THEDIFFERENTIALDUE:OCT.17(b) Prove: If M is compact and N is connected, then any submersion f : M -→ N issurjective. Conclude that there exists no submersion from any compact smoothmanifold M to any connected noncompact smooth manifold (e.g. Rn).(5)[Hadamard's global inverse function theorem]Let M, N be connected smooth manifolds, and f e Co(M, N).(a) Prove:If f is proper (i.e. the pre-images of compact subsets are compact), then it isclosed (i.e: the image of closed subsets are closed).(b)[Not required] Prove:If f is proper and local diffeomorphism everywhere, thenitisacoveringmap(c) Finally conclude that if f is proper and local diffeomorphism everywhere, andN is simply connected, then f is a diffeomorphism.[Compositions of submersion/immersions/constantrankmaps](6）Consider the compositions of submersions/immersions/constant rank maps. [We as-sume all maps are smooth.](a) Is it true that the composition of two submersions is still submersion? Whatabout the composition of immersions? What about constant rank maps?(b) What if we composite a constant rank map with a submersion? With an immer-sion?A submersion with an immersion?(c) If the composition g o f is an immersion, can we conclude f is an immersion?(d) If the composition g o f is a submersion, can we conclude g is a submersion? Ifnot, what extra condition do we need?(7)[Matrix Lie groups]Recall that GL(n, R) is an n? dimensional smooth manifold.(a) Consider the map det : GL(n,R) → R.(i)Prove:detisasmoothfunction.(ii) For any X E GL(n, R), what is Tx(GL(n, R)?(ii) Show that (ddet)x(A) = (detX)tr(X-i A)(iv) Show that det is a submersion.(b)[Not required] Consider themapf: GL(n,R) →GL(n,R), X -f(X)=XTx.(i) Prove: f is smooth, and dfx(A) = XTA+ATX.(ii) Prove: f has constant rank n(n +1)/2.(8) [Not required] [Germ and the category of pointed manifolds]Suppose you are a teacher. Write down lecture notes (no more than 1 page), startingwith the definition of germ below (and explore some properties of germ that you mayneed), then explain what is "the category of pointed smooth manifolds" and why"taking differential at a point" is a functor.Definition of germ: Let M,N be smooth manifolds and p E M. Letfi:U,→N be smooth maps, where U; are neighborhoods p.For such
2 PROBLEM SET 2, PART 1: THE DIFFERENTIAL DUE: OCT. 17 (b) Prove: If M is compact and N is connected, then any submersion f : M → N is surjective. Conclude that there exists no submersion from any compact smooth manifold M to any connected noncompact smooth manifold (e.g. R n ). (5) [Hadamard’s global inverse function theorem] Let M, N be connected smooth manifolds, and f ∈ C∞(M, N). (a) Prove: If f is proper (i.e. the pre-images of compact subsets are compact), then it is closed (i.e. the image of closed subsets are closed). (b) [Not required] Prove: If f is proper and local diffeomorphism everywhere, then it is a covering map. (c) Finally conclude that if f is proper and local diffeomorphism everywhere, and N is simply connected, then f is a diffeomorphism. (6) [Compositions of submersion/immersions/constant rank maps] Consider the compositions of submersions/immersions/constant rank maps. [We assume all maps are smooth.] (a) Is it true that the composition of two submersions is still submersion? What about the composition of immersions? What about constant rank maps? (b) What if we composite a constant rank map with a submersion? With an immersion? A submersion with an immersion? (c) If the composition g ◦ f is an immersion, can we conclude f is an immersion? (d) If the composition g ◦ f is a submersion, can we conclude g is a submersion? If not, what extra condition do we need? (7) [Matrix Lie groups] Recall that GL(n, R) is an n 2 dimensional smooth manifold. (a) Consider the map det : GL(n, R) → R. (i) Prove: det is a smooth function. (ii) For any X ∈ GL(n, R), what is TX(GL(n, R))? (iii) Show that (d det)X(A) = (detX)tr(X−1A). (iv) Show that det is a submersion. (b) [Not required] Consider the map f : GL(n, R) → GL(n, R), X 7→ f(X) = XT X. (i) Prove: f is smooth, and dfX(A) = XT A + AT X. (ii) Prove: f has constant rank n(n + 1)/2. (8) [Not required] [Germ and the category of pointed manifolds] Suppose you are a teacher. Write down lecture notes (no more than 1 page), starting with the definition of germ below (and explore some properties of germ that you may need), then explain what is “the category of pointed smooth manifolds” and why “taking differential at a point” is a functor. Definition of germ: Let M, N be smooth manifolds and p ∈ M. Let fi : Ui → N be smooth maps, where Ui are neighborhoods p. For such
3PROBLEMSET2.PART1:THEDIFFERENTIAIDUE: OCT. 17functions, define an equivalence relation byfi~ f2 3U C UinU2s.t.filu = f2lu.The equivalence class containing f is called an germ of smooth maps at pand is denoted by [f]p.Here are some necessary definitions of category and functor you may assume withoutrepeating:定义1.148.（范畴）一个范畴C包含1.一个类Ob(C)，其中的元素称为对象，2.一个类Mor(C)，其中的元素称为对象间的态射，满足。每个态射f都有一个始对象 X E Ob(C)和一个终对象 YEOb(C).我们记f：X-→Y并称“f是从X到Y的态射”，我们将从X到Y的态射全体记为Mor(X,Y）a。态射f:X→Y和g:Y→Z的复合是态射gofX→Z，且满足(a).结合性）设f:X→Yg:Y-→Z和h:Z-W则ho(gof)=(hog)of,(b).（单位元）对XEOb(C)，存在单位态射Idx：X→X，使得对于任意态射f:Z→X和g:X→Y，都有Idxof=f且goIdx=g“对于一些对象X和Y，可能并不存在从X到Y的态射，此时Mor（X,Y）=0.定义3.2.20.（函子）设C,D是范畴.从C到D的（协变）函子"F是一个对应，使得(I)将Ob(C)中的每个对象X都对应到Ob(D)中的某个对象F(X)(2)将范畴C中的每个态射fEMor(X,Y）都对应到范畴D中的某个态射F(f)EMor(F(X),F(Y)),且该对应满足：。对范畴C中的任意对象XEOb(C)，有F(Idx) = Id(x);。对范畴C中的任意态射fEMor(XY）和gEMor(Y,Z)都有F(gof)=F(g)oF(f).“类似地我们可以定义反变函子的概念：反变函子也是一个对应，它将每个态射fEMor(X，Y）对应到态射F(f)EMor(F(Y),F(X)并且满足F(Idx)=Idp(x)和F(gf)=F(f)F(g)
PROBLEM SET 2, PART 1: THE DIFFERENTIAL DUE: OCT. 17 3 functions, define an equivalence relation by f1 ∼ f2 ⇐⇒ ∃U ⊂ U1 ∩ U2s.t.f1|U = f2|U . The equivalence class containing f is called an germ of smooth maps at p and is denoted by [f]p. Here are some necessary definitions of category and functor you may assume without repeating: