PROBLEM SET 2, PART 2: REGULAR VALUES DUE: OCT. 17 (1) [Measure zero set in smooth manifolds] (a) Prove: the phrase “measure zero” is well-defined on smooth manifolds. (b) Deduce Sard’s theorem from the Euclidean case. (c) Show that if f : M → N is a smooth map of constant rank r 0, there is a Morse function g ∈ C∞(U) so that |g − f| < ε and all critical values of g are distinct. (d) (Not required) Extend the result in (c) to smooth functions defined on a compact manifold. 1
2 PROBLEM SET 2, PART 2: REGULAR VALUES DUE: OCT. 17 (4) [The Lagrange multiplier] Let M be a smooth manifold, and f ∈ C∞(M) a smooth function. We would like to study the critical points of the function ˜f := f|S ∈ C∞(S) for a smooth submanifold S ⊂ M. For simplicity, we suppose there is a smooth map g : M → N and a regular value p ∈ N of g so that S = g −1 (q). Prove: a point p ∈ S is a critical point of ˜f if there exists a linear function L : TqN → R, (called a Lagrange multiplier ), so that dfp = L ◦ dgp. (5) [Proper maps] Recall that a map is called proper if the pre-image of any compact set is compact. Let f : M → N be a smooth and proper map. (a) Prove: If an injective immersion f : M → N is proper, then it is an embedding. (b) Now suppose dim M = dim N, and suppose q ∈ f(M) be a regular value of f. Prove: f −1 (q) is a finite set {p1, · · · , pk}, and there exist a neighborhood V of q in N and neighborhoods Ui of pi in M such that • U1, · · · , Uk are disjoint coordinate charts in M, • f −1 (V ) = U1 ∪ · · · ∪ Uk, • For each 1 ≤ i ≤ k, f is a diffeomorphism from Ui onto V . (6) [The cotangent bundle] Let M be a smooth manifold of dimension n. Let T ∗ p M be the dual vector space of TpM, with a dual basis {dx1 , · · · , dxn} (which is defined locally for a coordinate chart of M) which is defined to be the dual of {∂1, · · · , ∂n}. Let T ∗M = S p T ∗ p M be the disjoint union of all T ∗ p M. We will call T ∗M the cotangent bundle of M. (a) Modify PSet2-1-3 to endow with T ∗M a topology so that it is a smooth manifold of dimension 2n. (b) Prove: T ∗M is orientable. (c) (Not required) Prove: If f is a smooth function on M, then the map sf : M → T ∗M, p 7→ (p, dfp) is an injective immersion and is proper. [In particular, its image is a smooth submanifold of T ∗M.] (d) (Not required) For any (p, ξp) ∈ T ∗M, the tangent space T(p,ξp)T ∗M ' TpM ⊕ T ∗ p M