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Smooth Charts on immersion image.

  1. Aug 12, 2007 #1


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    Hi, everyone:

    I have been reading Boothby's intro to diff. manifolds, and in def. 4.3,
    talking about 1-1 immersions F:N->M , n and m-mfld. and M an m-mfld.

    "...F establishes a 1-1 correspondence between N and the image N'=F(N)
    of M. If we use this correspondence to give N' a topology and a C^oo
    structure, then N' will be called a submanifold, and F:N->N' a diffeomorphism".

    I am just wondering how this topology and C^oo structure are defined.
    I imagine we are using pullbacks here, tho, since F is not necessarily
    a diffeom. , we cannot pullback by F, except maybe locally, using
    the inverse fn. theorem.
    And, re the topology on F'(N) , we cannot use subspace, since
    F may not be an embedding. Maybe we are using quotient topology,
    but it does not seem clear.

    Anyone know how the topology and C^oo structure are defined
    in N'=F(N)?

    I would appreciate any help.
  2. jcsd
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