《微分流形》课程教学资源（英文讲义）PSet2-2 REGULAR VALUES

PROBLEMSET2,PART2:REGULARVALUESDUE: OCT.17(1) [Measure zero set in smooth manifolds](a) Prove: the phrase “measure zero" is well-defined on smooth manifolds.(b)DeduceSard'stheoremfromtheEuclideancase(c) Show that if f : M -→ N is a smooth map of constant rank r 0,there is a Morse function g E Co(U) so that |g - fl < e and all critical valuesof g are distinct.(d) (Not required) Extend the result in (c) to smooth functions defined on a compactmanifold.1

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

2PROBLEMSET2.PART2:REGULARVALUESDUE:OCT.17(4) [The Lagrange multiplier]Let M be a smooth manifold, and f e Coo(M) a smooth function. We would like tostudy the critical points of the function f := fls ECo(S) for a smooth submanifoldS C M. For simplicity, we suppose there is a smooth map g : M → N and a regularvalue p E N of g so that S = g-1(q). Prove: a point p e S is a critical point of fif there exists a linear function L:TqN-→ R, (called a Lagrange multiplier), so thatdfp=Lodgp.(5)[Propermaps]Recall that a map is called proper if the pre-image of any compact set is compact.Letf:M→Nbeasmoothandpropermap(a)Prove:If an injectiveimmersionf :M→N isproper,thenitisanembedding(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 (pi,..,pk), and there exist a neighborhood V of qin N and neighborhoods U; of p; in M such that.Ui,...,Uaredisjoint coordinatecharts in M. f-i(V)=UiU...UUk,. For each 1 ≤ i ≤ k, f is a diffeomorphism from U, onto V.(6) [The cotangent bundle]Let M be a smooth manifold of dimension n. Let T,M be the dual vector space ofT,M,with a dual basisdrl,-.,drn1 (whichisdefined locallyfora coordinate chartof M) which is defined to be the dual of [i,"-,On]. Let T*M =U, T*M be thedisjoint union of all T,M.Wewill call T*M the cotangent bundleof M.(a) Modify PSet2-1-3 to endow with T*M a topology so that it is a smooth manifoldof dimension 2n.(b) Prove: T*M is orientable.(c) (Not required) Prove: If f is a smooth function on M, then the map$f:M→T*M,p-(p,dfp)is an injective immersion and is proper. [In particular, its image is a smoothsubmanifold of T*M.](d)(Not required) For any (p,Sp) e T*M, the tangent space T(p.Ep)T*M ~ T,M T,'M

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