1. The problem statement, all variables and given/known data [tex]M[/tex] is a smooth manifold, [tex]U \subset M[/tex] is a proper open set. Show that there exists a smooth non-trivial diffeomorphism from [tex]M[/tex] onto itself which restriction on [tex]M - U[/tex] is identity ("identity outside [tex]U[/tex]"). 2. Relevant equations 3. The attempt at a solution If there exists a non-zero vector field that is zero outside [tex]U[/tex], 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 [tex]M[/tex]? Should the vector field be complete? There are too many questions, I don't know how to start, please point me a right direction.