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 < dimN, thenf(M) has measure zero.(2) [An counterexample to Sard's Theorem]HereisacounterexampletoSard'stheoremiff isnotsmoothenough(constructedby E. Grinberg). Let C C [0,1] be the Cantor set.(a) Construct a Cl function f : R -→ R such that the critical set of f contains C.(Hint: Denote [o,1] \C=U(ak,ba).Start with a *continuous bump function" fk on (ak,bk)withFfa(t)dt=bg-ak.)(b) Show that the function g : R2 → R defined by g(a,y) = f(r) + f(y) is Cl, andthe set of critical values contains an interval. (Hint: Show that C+C =[0,2].)(3)[Morse functionsLet U C Rn be an open set, and f e Co(U)..We say a critical point p e U of f is non-degenerate if the Hessian matrixa2fHessf(p)(p)Orioriis non-degenerate.·A function is called a Morse function if every critical point is non-degenerateProve:(a) Use inverse function theorem to prove that non-degenerate critical point mustbe isolated.(b) Given any f E C(U), for almost every a E Rn, the “linear perturbation"fa:U→R,aHf(r)+air'+...+ananof f is a Morse function on U.[Hint: Consider regular values of the map g= df = (,..,) :U→ R"](c) (Not required) Suppose U is bounded. Prove: for any f e C(U) and any e > 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 < dim N, then f(M) has measure zero. (2) [An counterexample to Sard’s Theorem] Here is a counterexample to Sard’s theorem if f is not smooth enough (constructed by E. Grinberg). Let C ⊂ [0, 1] be the Cantor set. (a) Construct a C 1 function f : R → R such that the critical set of f contains C. (Hint: Denote [0, 1] \ C = ∪(ak, bk). Start with a “continuous bump function” fk on (ak, bk) with R fk(t)dt = bk − ak.) (b) Show that the function g : R 2 → R defined by g(x, y) = f(x) + f(y) is C 1 , and the set of critical values contains an interval. (Hint: Show that C + C = [0, 2].) (3) [Morse functions] Let U ⊂ R n be an open set, and f ∈ C∞(U). • We say a critical point p ∈ U of f is non-degenerate if the Hessian matrix Hessf(p) = ∂ 2f ∂xi∂xj (p) is non-degenerate. • A function is called a Morse function if every critical point is non-degenerate. Prove: (a) Use inverse function theorem to prove that non-degenerate critical point must be isolated. (b) Given any f ∈ C∞(U), for almost every a ∈ R n , the “linear perturbation” fa : U → R, x 7→ f(x) + a1x 1 + · · · + anx n of f is a Morse function on U. [Hint: Consider regular values of the map g = df = ( ∂f ∂x1 , · · · , ∂f ∂xn ) : U → R n .] (c) (Not required) Suppose U is bounded. Prove: for any f ∈ C∞(U) and any ε > 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