MTH3400 --- Geometry 1 --- Spring 1997

Homework #4

- Show that Tⁿ is parallelizable, for all .
- Show that is parallelizable, for all n odd, and .
- Show that has 3 independent vector fields, for all .
Let be the vector field
For , compute the integral curve to X through a. (Be sure to specify its domain.) Find the flow determined by X. Is X a complete vector field?
Let X and Y be smooth real vector fields, given by

where is a smooth map such that on , on , and . (Such a map exists by the smooth Urysohn lemma.) Prove that:
- X and Y are complete vector fields.
- X+Y is not complete.
Let be a (global) flow. A subset is said to be -invariant if , for all , . Let be the flow line through x. Show that the closure is a -invariant set.
Given , view the right translation
as a vector field . (The value of this vector field at is the matrix .) For
find the (global) flow generated by .
Problem 13 from Chapter 5 in Spivak's book.