MTH3400 — GEOMETRY 1 — SPRING 1997

HOMEWORK #4

1. (i) Show that T^{n} is parallelizable, for all n ≥ 1.

(ii) Show that S^{n} × S^{k} is parallelizable, for all n odd, and k ≥ 1. (iii) Show that S^{4n−1} has 3 independent vector fields, for all n ≥ 1.

2. Let X ∈ 𝔛(ℝ) be the vector field

X = e^{t} \frac{d}{dt} .

For a ∈ ℝ, compute the integral curve s_{a} : (−ϵ_{a}, δ_{a}) → ℝ to X through a. (Be sure to specify its domain.) Find the flow determined by X. Is X a complete vector field?

3. Let X, Y ∈ 𝔛(ℝ) be smooth vector fields given by

X = x^{2} cos^{2}(ϕ(x))\frac{d}{dx}
Y = x^{2} sin^{2}(ϕ(x))\frac{d}{dx}

where ϕ : ℝ → [0, \frac{π}{2}] is a smooth map such that ϕ ≡ 0 on [0, \frac{1}{5}] ∪ [\frac{4}{5}, 1], ϕ ≡ \frac{π}{2} on [\frac{2}{5}, \frac{3}{5}], and ϕ(x + 1) = ϕ(x). (Such a map exists by the smooth Urysohn lemma.) Prove that:

(i) X and Y are complete vector fields. (ii) X + Y is not complete.

4. Let Φ : ℝ × M → M be a (global) flow. A subset C ⊂ M is said to be Φ-invariant if Φ_{t}(x) ∈ C, for all x ∈ C, t ∈ ℝ. Let R_{x} = {Φ_{t}(x)}_{t∈ℝ} be the flow line through x. Show that the closure \overline{R}_{x} is a Φ-invariant set.

5. Given A ∈ Mat(n), view the right translation

R_{A} : GL(n) → Mat(n)

as a vector field R_{A} ∈ 𝔛(GL(n)). (The value of this vector field at B ∈ GL(n) is the matrix BA ∈ Mat(n) = T_{B}(GL(n)).) For

A = (\begin{matrix}0 & 1 \\ 0 & 0 \end{matrix})

find the (global) flow Φ : ℝ × GL(2) → GL(2) generated by R_{A}.

6. Problem 13 from Chapter 5 in Spivak’s book.