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Homework Help: Diffeomorphism problem

  1. Sep 15, 2007 #1
    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.
    Last edited: Sep 15, 2007
  2. jcsd
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