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Math Help - Closed Submanifold, in Spivak's Vol 1, chap 2, exercise 24(b)

  1. #1
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    Closed Submanifold, in Spivak's Vol 1, chap 2, exercise 24(b)

    The problem is indeed an exercise in spivak's 'a comprehensive introduction to differential geometry' vol 1, chap 2, exercise 24.
    I can do 24(a) and a half part of (b), but got stuck on the other half....
    I first gave the result of 24(a), you can just use it:
    If M is a C^\infty manifold, a set M_1 in M can be made into a k-dimensional submanifold of M if and only if around each point in M_1 there is a coordinate system (x,U) on M such that M_1 \cap U=\{p:\quad x^{k+1}(p)=\cdots=x^n(p)=0\}.

    Now here is the second part of the question which I got stuck:

    If the subset M_1 can be made into a closed submanifold, then the coordinate systems mentioned above exist around every point of M (not just M_1).

    My idea is that suppose M is connected, then let W= \bigcup U, where U is the coordinates mentioned above, then W is an also a submanifold in M containing M_1, W is open obviously, I want to show W is also closed so that W=M, then we are done. But I got stuck here, maybe my idea was wrong.....

    Please note that in Spivak's book, a closed submanifold M_1 in M is defined as:
    (a) an immersed submanifold whose inclusion map is also an embedding, and:
    (b) the subset M_1 is a closed subset of M.

    Namely, a submanifold is just an embedding submanifold, and a closed submanifold is an embedding submanifold with itself is a closed set in the whole space.

    Thank you in advance! Hope some guys who are familar with Spivak's book can help me~
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  2. #2
    Super Member Rebesques's Avatar
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    You can always assume your manifolds are connected, as you can always work on a connected component.


    About the answer. Collect all neighbourhoods of points in M with that property. Their union is exactly M_1 but also an open set in  M. So M_1 is open and closed in M, giving M=M_1.
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