Math 6510 Take home final, due Dec 23, 2012.
This is a realistic half-prelim. Try to do it in 90 minutes without looking at
any books or notes. After that look at the books as necessary to complete it,
but record how far you got in 90 minutes.
(1) Let S 1 denote the unit circle in C and define f : S 1 × S 1 → R by f (z, w) =
Re(z) + Re(w). Show that f is a Morse function and compute the indices
of the critical points of f .
(2) Let X be any subset of Rn and f : X → R a function with the following
property. For every x ∈ X there is a neighborhood Ux of x in Rn and a
smooth function fx : Ux → R such that f = fx on X ∩ Ux . Show that there
is an open set U in Rn containing X and a smooth function g : U → R such
that g|X = f .
(3) Let X, Y ⊂ R3 be two 1-dimensional submanifolds. Show that there is
v ∈ R3 such that X is disjoint from Y + v := {y + v|y ∈ Y }.
(4) Give an explicit example of a closed 1-form on R2 − {0} which is not exact,
and prove both properties.
(5) (a) Give a careful definition of the tangent bundle of a manifold X. You
may assume that X is a submanifold of Rn .
(b) Show that the total space of the tangent bundle of S 2 is not diffeomorphic to S 2 × R2 . (Be careful: this is more than showing the bundle is
nontrivial.)
(6) (a) State the tubular neighborhood theorem for a compact submanifold
X ⊂ Rn .
(b) Give an outline of the proof.
∂
∂
+ y ∂y
and ω = x dy be a vector field and a 1-form
(7) (a) Let X = x ∂x
2
respectively in R . Compute the Lie derivative LX ω.
(b) Prove Weinstein’s lemma: If X, Y are vector fields tangent to a plane
field ∆ and Yp = 0 then [X, Y ]p ∈ ∆. Hint: Compute LX Y at p.
(8) (a) Define the index of an isolated zero of a smooth vector field v. You may
assume that dv is invertible.
(b) State the Poincaré-Hopf theorem.
(c) Let X and Y be two closed oriented manifolds. Prove that Euler characteristic is multiplicative, i.e. that
χ(X × Y ) = χ(X)χ(Y )
You may not appeal to the Künneth formula, but rather use PoincaréHopf.
1