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