Homework 3∗
I 1: In R3 with the standard metric, let M be the manifold parametrized
by (x, y, z) = (r cos(θ), r sin(θ), θ), and let N be the manifold parametrized by
(x, y, z) = (cosh(u) cos(v), cosh(u) sin(v), u).
B a: Sketch M and N .
B b: Calculate the restriction of h·, ·i and II for M and N , with respect to their
coordinates (r, θ) and (u, v).
B c: Show that the map M → N given by θ = v and r = sinh(u) is an isometry.
B b: Show that there is no isometry of R3 taking M to N , even locally.
I 2a: Give RN the standard metric. Let M ⊆ RN be a submanifold, and
assume that II is identically zero. Show that M is an affine linear subspace.
B b: Suppose that q ∈ M is a point where II is not identically zero. Show that
there is a point p ∈ RN \ M so that q is a degenerate critical point of d2p .
B c: Let M ⊆ R3 be the helix curve, given by (x, y, z) = (cos(t), sin(t), t). Find
all points p ∈ R3 so that d2p is not Morse on M .
P
I 3a: Let f : M → R be a Morse function. Show that χ(M ) = p∈crit(f ) (−1)ind(p) .
B b: Let Σ be a closed surface with χ(Σ) ≤ 1. Show that any Morse function
on Σ must have at least |χ(Σ)| + 2 critical points.
B c: Let M be a closed orientable 3-manifold, and let f : M → R be a Morse
function with four critical points, which has distinct critical values. Show that
there is an embedded circle S 1 ⊆ M , so that M \ S 1 ∼
= S 1 × R2 . You may assume two facts without proof: the only boundaryless surface which is homotopy
equivalent to a point is R2 , and every orientation preserving diffeomorphism of
R2 is isotopic to the identity. (Hint: If c ∈ R has two critical values below it
and two above it, what is the topology of f −1 ((−∞, c])?. Use cases.)
B d: Give an example of a closed 3-manifold M , so that any Morse function on
M has at least 6 critical points.
∗ Due
November 13th
1