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Inclusions of submflds. and diffeomorphisms.

  1. Mar 17, 2009 #1

    WWGD

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    Hi, everyone:

    I am a little confused about the issue of the inclusion map on submanifolds.

    AFAIK, if S is a submanifold of M , then if we give S the weak (or initial) topology

    of the inclusion, then the inclusion map is a homeomorphism.(his is the way I understand,

    of making the inclusion map of a subspace into a space, into a homeomorphism. If

    S is open in M, with this initial topology, then I think inclusion is also a diffeomorphism.)


    Question: under what conditions on S is the inclusion map a diffeomorphism?.

    I think that if S is closed in the topology of M, or at least not open in M , and

    given the weak topology, then the inclusion may not be a diffeomorphism.

    Is this correct?.

    And when is a (topological) subspace S of M a submanifold, other than when S

    is open in M, i.e., S is open as a subset of M?



    Thanks.
     
  2. jcsd
  3. Mar 17, 2009 #2
    You have to distinguish between immersed and embedded submanifolds (see also http://en.wikipedia.org/wiki/Submanifold" [Broken]). For immersed submanifolds the inclusion is an injective immersion, but not necessarily a homeomorphism onto its image (i.e. an embedding). If it is an embedding then the submanifold is an emdedded submanifold and the inclusion is also a diffeomorphism onto its image, this is not completely trivial and requires the implicit function theorem.

    If one can find charts of M such that S looks locally like R^k in R^n.
     
    Last edited by a moderator: May 4, 2017
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