MTH3400 --- Geometry 1 --- Spring 1997

Homework #5

Prove that on any non-compact, connected differentiable manifold there exists an incomplete smooth vector field.
Give an example of a submersion , and a vector field which is not f-related to any .
Construct 3 linearly independent vector fields on , and calculate their Lie brackets.
Let H be the Heisenberg group
This group has natural coordinates , and it acts on itself by left translations. Let be the left-invariant vector-fields on H, with values at the identity , , and , respectively. Consider the 2-dimensional distributions E and F on H generated by and , respectively. Show that E is integrable and F is not.
Let . Consider the following 2-dimensional distributions on M:
1. , with ;
2. , with .
In each case, decide whether the distribution is integrable or not, and, if it is, describe the integral manifolds.