PROBLEM SET 4, PART 2: FLOWS, DISTRIBUTIONS DUE: NOV. 14 (1) [Plenty of diffeomorphisms] Let M be a connected smooth manifold of dimension m ≥ 2, and let {p1, · · · , pn} and {q1, · · · , qn} be two sets of points in M. (a) Prove: there exists a diffeomorphism Φ such that Φ(pi) = qi holds for all i. (b) Let Xi ∈ TpiM and Yi ∈ TqiN (1 ≤ i ≤ n) be nonzero vectors. Prove: the diffeomorphism Φ above can be chosen so that dΦpi (Xi) = Yi holds for all i. (2) [ϕ-related vector fields] (a) Let ϕ : M → N be a smooth map. We say two vector fields X ∈ Γ∞(TM) and Y ∈ Γ∞(T N) are ϕ-related, if for any p ∈ M, dϕp(Xp) = Yϕ(p) . Prove: (i) If X and Y are ϕ-related, then X(ϕ ∗ g) = ϕ ∗ (Y g) holds for any g ∈ C∞(N). (ii) If Xi are ϕ-related to Yi for i = 1, 2, then [X1, X2] is ϕ-related to [Y1, Y2]. (iii) If X and Y are ϕ-related, γ : I → M is an integral curve of X, then ϕ ◦ γ : I → N is an integral curve of Y . (b) Now let ϕ : M → N be a diffeomorphism, one can define a “push-forward” operator ϕ∗ : Γ∞(TM) → Γ∞(T N) by (ϕ∗X)ϕ(p) = dϕp(Xp). Prove: (iv) (ϕ∗X)g = (ϕ −1 ) ∗Xϕ∗ g for any g ∈ C∞(N). (v) If X, Y ∈ Γ∞(TM), then ϕ∗([X, Y ]) = [ϕ∗X, ϕ∗Y ]. (3) [Time dependent vector fields]] (NOT required) Read Remark 3.3.5. Rewrite it to be a subsection entitled “Time dependent vector fields”, including necessary definition(s), theorem(s) and proof(s). (4) [The Frobenius condition] (a) Let f, g ∈ C∞(M), X, Y ∈ Γ∞(TM). Show [fX, gY ] = fg[X, Y ] + f(Xg)Y − g(Y f)X. (b) Prove Lemma 3.4.11. (5) [Commuting vector fields] (a) Let X1, · · · , Xk be smooth vector fields on M that are pointwise linearly independent on a region W of M, so that [Xi , Xj ] = 0 for all 1 ≤ i, j ≤ k. Prove: 1
2 PROBLEM SET 4, PART 2: FLOWS, DISTRIBUTIONS DUE: NOV. 14 For any point p ∈ W, one can find a local chart (ϕ, U, V ) near p so that Xi = ∂i on U for all 1 ≤ i ≤ k. (b) Find a coordinate chart on R 3 \ {(0, 0, z) | z ∈ R} so that ∂1 = x ∂ ∂x + y ∂ ∂y , ∂2 = x ∂ ∂y − y ∂ ∂x, ∂3 = ∂ ∂z . (c) [Not required] Let {φ X t } and {φ Y t } be the one-parameter group of diffeomorphisms generated by the vector field X and Y respectively. Prove: [X, Y ] = 0 if and only if φ X t ◦ φ Y s = φ Y s ◦ φ X t holds for all t, s ∈ R. (6) [A distribution and its integral manifolds] Consider the distribution V on R 3 spanned by V = x ∂ ∂x + ∂ ∂y + x(y + 1) ∂ ∂z and W = ∂ ∂x + y ∂ ∂z . (a) Show that V is involutive. (b) Consider the projection map π : R 3 → R 2 ,(x, y, z) 7→ (x, y). Show that X = ∂ ∂x + y ∂ ∂z and Y = ∂ ∂y + x ∂ ∂z are the vector fields spanning V that are π-related to ∂ ∂x and ∂ ∂y . (c) Find the integral curves of X and Y respectively. (d) What are the integral manifolds of V? (7) [Morse theory] (NOT required) (a) [Morse Lemma] Suppose p is a non-degenerate critical point of a Morse function f (c.f. PSet4-1-7(d)). Prove: There exists a coordinate chart on which f can be written as f(x) = f(p) − x 2 1 − · · · − x 2 r + x 2 r+1 + · · · + x 2 n . (b) [Reeb’s theorem] Suppose f is a Morse function on a closed manifold M of dimension n. If f admits exactly two critical points, then M is homeomorphic to S n . (8) [Foliation] (NOT required) Let F be a collection of disjoint connected immersed k-dimensional submanifolds of M whose union is M. If for any p ∈ M, there is a local chart (ϕ, U, V ) near p so that V is a cube, and each submanifold in F intersects U in either the empty set, or a countable union of k-dimensional slices of the form x k+1 = c k+1 , · · · , xm = c m, then we say F is a foliation on M, and call the submanifolds in F the leaves of the foliation. Prove: (a) If F is a foliation, then the tangent spaces to the leaves of F form an involutive distribution V on M. (b) Conversely, if V is an integrable distribution, then the maximal connected integral manifolds of V form a foliation on M