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Math Help - Submanifold

  1. #1
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    Submanifold

    Hello,

    I have a exercise, but the exercise isn't written very well. Do you understand, what the statement should be?:

    "M is a smooth submanifold of N iff every point p \in M has an open nbh U_p \subset N , s.t. U_p \cap M is a smooth submanifold of U".

    It doesn't make sense to me, because p is a point in M and U_p a subset in N. And what about the set U?

    Our definition of a submanifold: M is a submanifold of N if there exist a smooth embedding and immersion f:M-> N

    thank you
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  2. #2
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    The last part of the statement should be "is a smooth submanifold of N", not "U".
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  3. #3
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    Ok. Thank you. But what about the set U_p? M isn't a subset of N.
    Does he identify the set M with f(M), i.e. U_p \cap M= U_p \cap f(M)?

    Regards
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  4. #4
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    It would seem to me that the author is making the identification you suggest (or the author simply made a typo and meant to write f(M) ).
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  5. #5
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    Hello,

    In my notes i have read, that:

    Let f:M->N is an injective immersion

    Then f is a topological embedding <=> f is a local topological embedding, i.e. for each p \in M there is a nbh. U_p s.t. U_p is hom÷omorph to f(U_p).


    Therefore i want to proof this statement. The direction => is easy. How can i show the other direction:

    "<="
    we know f:M->f(M) is bijective. Why it is continuous?

    Regards
    Last edited by Sogan; January 2nd 2011 at 09:29 AM.
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