Differential Topology — Spring 2014

Gerald Hoehn

Problem sheet 9 April 24, 2014

Problem 1:

Let M ⊂ R^{n−1} be an (n − 1)-dimensional submanifold of R^{n}. Assume that there exists a differentiable map ν : M ⟶ R^{n} such that ν(p) ∉ T_{p}M for all p ∈ M. Show that M is orientable.

Problem 2: Consider

ω = 3z dy ∧ dz + (x^{2} + y^{2}) dz ∧ dx + xz dx ∧ dy

on R^{3} and the 2-dimensional submanifold

M = {(x, y, z) ∈ R^{3} | z = xy}.

Determine ∫_{A} ω, where A = {(x, y, z) ∈ M | |x| ≤ 1, |y| ≤ 1}.

(Note: Choose an orientation on M; it is uncritical that the boundary of A is not smooth at the four corners.)

Problem 3:

(a) Let f_{1}, …, f_{n} be homogenous polynomials of degree m in the coordinates x_{1}, …, x_{n} of R^{n} and

ω := ∑_{i=1}^{n}(−1)^{i−1}f_{i} dx_{1} ∧ … ∧ d̂x_{i} ∧ … ∧ dx_{n}.

Let S^{n−1} ⊂ R^{n} be the unit sphere. Show that if m is even, then ∫_{S^{n−1}} ω = 0. (Hint: Consider the antipodal map x ↦ −x of R^{n}.)

(b) Prove that the (n − 1)-dimensional real projective space RP^{n−1} is orientable if and only if n is even. (Hint: Oberserve that the antipodal map on the (n − 1)-sphere S^{n−1} is orientation preserving if and only if n is even.)

Problem 3: (Spherical coordinate system)

Let Φ : R^{3} ⟶ R^{3} be defined by

(r, ϑ, φ) ↦ (x, y, z) = (r sin ϑ cos φ, r sin ϑ sin φ, r cos ϑ).

It defines a diffeomorphism

Φ : T := R_{+}^{∗} × (0, π) × (−π, π) ⟶ Ω := R^{3} \ {(x, y, z) ∈ R^{3} | x < 0, y = 0}.

We also denote by r, ϑ, φ → R the components of the inverse of Φ.

a) Show that the function ϑ and the differential form dϕ can be extended from Ω to R^{3} \ (0 × 0 × R) and the differential form r^{3} sin ϑ dϑ ∧ dφ can be extended to the whole R^{3}.

b) Show that for a compact regular domain K ⊂ R^{3} one has

Vol_{3}(K) = \frac{1}{3} ∫_{∂K} r^{3} sin ϑ dϑ ∧ dφ.

(Here, the volume Vol_{3}(K) is defined as ∫_{K} dx ∧ dy ∧ dz.)

c) Let A ⊂ S^{2} be a compact regular domain inside the 2-sphere which does neither contain the north pole P_{N} = (0, 0, 1) not the south pole P_{S} = (0, 0, −1). Show that

Vol_{2}(A) = 2kπ − ∫_{∂A} cos ϑ dφ

where k = 0, 1, 2, depending on A containing none, one or two of the points {P_{N}, P_{S}}, respectivly. (Here, the volume Vol_{2}(A) is defined as

c · ∫_{A} xdy ∧ dz − ydx ∧ dz + z dx ∧ dy

with the constant c chosen such that Vol_{2}(S^{2}) = 4π.)