# Diffeomorphism problem

1. Sep 15, 2007

### kakarukeys

1. The problem statement, all variables and given/known data
$$M$$ is a smooth manifold, $$U \subset M$$ is a proper open set.
Show that there exists a smooth non-trivial diffeomorphism from $$M$$ onto itself which restriction on $$M - U$$ is identity ("identity outside $$U$$").

2. Relevant equations

3. The attempt at a solution
If there exists a non-zero vector field that is zero outside $$U$$, in principle, its flow may be the required diffeomorphism. How do I construct the vector field? It is difficult because I need to work in local charts or is it trivial to see such vector field exists? a local flow is a local one-parameter group of local diffeomorphisms. When can it be extended to the whole $$M$$? Should the vector field be complete?

There are too many questions, I don't know how to start, please point me a right direction.

Last edited: Sep 15, 2007