MTH3400 — GEOMETRY 1 — SPRING 1997

HOMEWORK #5

1. Prove that on any non-compact, connected differentiable manifold there exists an incomplete smooth vector field.

2. Give an example of a submersion f : M → N, and a vector field X ∈ 𝔛(M) which is not f-related to any Y ∈ 𝔛(N).

3. Construct 3 linearly independent vector fields on S^{3}, and calculate their Lie brackets.

4. Let H be the Heisenberg group

H = \{(\begin{matrix}1 & x_{12} & x_{13} \\ 0 & 1 & x_{23} \\ 0 & 0 & 1 \end{matrix}) | x_{12}, x_{13}, x_{23} ∈ ℝ\}

This group has natural coordinates (x_{12}, x_{13}, x_{23}), and it acts on itself by left translations. Let v_{12}, v_{13}, v,_{23} be the left-invariant vector-fields on H, with values at the identity (1, 0, 0), (0, 1, 0), and (0, 0, 1), respectively. Consider the 2-dimensional distributions E and F on H generated by v_{12}, v_{13} and v_{12}, v_{23}, respectively. Show that E is integrable and F is not.

5. Let M = ℝ^{3} \ {0}. Consider the following 2-dimensional distributions on M: (i) E = ∪_{v∈M}E_{v}, with E_{v} = {w ∈ T_{v}M | w ⊥ v}; (ii) F = ∪_{v∈M}F_{v}, with F_{v} = {(a, b, c) ∈ T_{v}M | (b, c, a) ⊥ v}. In each case, decide whether the distribution is integrable or not, and, if it is, describe the integral manifolds.