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Homework 1∗ I 1a: Let f : R2 → R2 be defined by f (x, y) = (x3 + 3xy 2 − 3x, y). What are the critical points of f ? For which critical points is the rank of df(x,y) one, and which have rank zero? B b: Let Σ ⊆ R2 be the critical points of f . In part a, you showed that Σ is a manifold. Therefore, we can look at the smooth map f |Σ : Σ → R2 . For which points p ∈ Σ is dfp injective? For which points is dfp = 0? B c: Draw f (Σ) ⊆ R2 , the set of critical values. Let S 1 (R) = {x2 + y 2 = 1} ⊆ R2 . For a small ε > 0, draw f (S 1 (1 + ε)) and f (S 1 (1 − ε)). I 2: Let M be an n-manifold. A k-dimensional submanifold is a subset N ⊆ M so that every point x ∈ N has a parametrization ϕ : U → M (U ⊆ Rn open) so that ϕ(U ∩ Rk ) = ϕ(U ) ∩ N (Rk = {xk+1 = . . . = xn = 0} ⊆ Rn ). If N is a submanifold then clearly the inclusion map ι : N → M has the property that dιx is an injection for every point x ∈ N . B a: Let c be an irrational number, and define f : R → T 2 = S 1 × S 1 by f (t) = (eit , eait ). Then f is an injection, and dft is an injection for all t ∈ R. Is f (R) a submanifold? B b: Let f : N → M be a smooth map, so that f and df are both injective. Assume that f is a proper map (that is, the preimage of compact sets in M are compact). Show that f (N ) is a submanifold. B c: Let N, Q ⊆ M be two submanifolds, N and Q intersect transversely if, for every point x ∈ N ∩ Q, T Nx + T Qx = T Mx . Assuming that N and Q intersect transversely, show that N ∩Q is a manifold, of dimension dim N +dim Q−dim M . B d: More generally, let N ⊆ M be a submanifold and f : Q → M a smooth map. We say f is transverse to N if, at every point x ∈ f −1 (N ), dfx (T Qx ) + T Nf (x) = T Mf (x) . Assuming that f is transverse to N , show that f −1 (N ) is a submanifold of Q of dimension dim N + dim Q − dim M . (Hint: if N is a point this just says the preimage of a regular value is a submanifold.) I 3: Let M be a manifold with boundary. Show that there is an open neighborhood U ⊆ M containing ∂M so that U is diffeomorphic to ∂M × [0, 1). (Hint: if you like, assume that ∂M is compact. Then use induction.) ∗ Due September 23rd 1