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
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
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: