PROBLEM SET 4, PART 1: VECTOR FIELDS DUE: NOV. 14 (1) [Coefficients of smooth vector fields] Prove Lemma 3.2.6 and Proposition 3.1.7. (2) [The commutator of two vector fields] Prove Proposition 3.1.13 in three different ways: (a) Method 1: show that in a local chart (ϕ, U, V ), if X = PXi∂i and Y = PY j∂j , then we have [X, Y ] = P i,j (Xi∂iY j − Y i∂iXj )∂j , which is a vector field. (b) Method 2: Check that Xp ◦ Y − Yp ◦ X is a tangent vector at p which depends smoothly on p. (c) Method 3: Show that [X, Y ] is a derivation on C∞(M). (3) [Γ∞(TM) as a Lie algebra] Prove Proposition 3.1.17. (4) [Examples of smooth vector fields] Construct smooth vector fields satisfying the given constraints: (a) A smooth vector field X on S 2n+1 such that Xp 6= 0 for all p. (b) A smooth vector field X on T n such that Xp 6= 0 for all p. (c) A smooth vector field X on S 2 such that Xp = 0 for exactly two points. (d) [Not required] A smooth vector field X on S 2 such that Xp = 0 for exactly one point. (e) Three smooth vector fields on S 3 that are everywhere linearly independent. (5) [The sum of two complete vector fields may fail to be complete] Let X1 = y 2 ∂ ∂x and X2 = x 2 ∂ ∂y be two vector fields on R 2 . (a) Prove: X1 and X2 are complete. (b) Prove: X1 + X2 is not complete. [Hint: Consider the integral curve starting at some point (c, c) with c 6= 0.] (6) [Any derivation on continuous functions] (Not required) Can one find a nontrivial map D : C 0 (M) → C 0 (M) such that D(fg) = (Df)g + f(Dg)? Justify your conclusion. (7) [The Hessian] (Not required) (a) Given any Xp ∈ TpM. Prove: there exists X ∈ Γ∞(TM) so that X(p) = Xp. 1
2 PROBLEM SET 4, PART 1: VECTOR FIELDS DUE: NOV. 14 (b) Let f ∈ C∞(M), and suppose p is a critical point of f. Define Hessf : TpM × TpM → R, (Xp, Yp) → Yp(Xf), where X ∈ Γ∞(TM) is any vector field so that X(p) = Xp. Prove: Hessf is well-defined, bilinear and symmetric. (c) For M = R n , what is Hessf ? (d) A critical point p is called non-degenerate if Hessf is non-degenerate at p. f is called a Morse function if every critical point is non-degenerate. Prove: On any smooth manifold, one can find Morse functions. (8) [First integral] (Not required) Let X be a smooth vector field on M and let γp : Jp → M be the maximal integral curve of X passing p. (a) Suppose for any (a, b) ⊂ Jp, γp((a, b)) lies in a compact subset of M. Prove: Jp = R. (b) Suppose f ∈ C∞(M) is a first integral of X, i.e. Xf ≡ 0. Prove: γp ⊂ f −1 (f(p)). (c) Suppose X admits a first integral whose level sets are all compact. Prove: X is complete