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Immersion and Manifold Question

  1. Oct 20, 2010 #1
    1. The problem statement, all variables and given/known data
    Let's assume that M is a compact n-dimensional manifold,
    then from Whitney's Immersion Theorem,
    we know that there's an immersion, f: M -> R_2n, and
    let's define f*: TM --> R_2n such that
    f* sends (p, v) to df_x (v).
    Since f is an immersion, it's clear that f* must be one-to-one by definition of immersion.
    let "x" be a regular value of f*, then how would you show that
    the inverse image of x (with respect to f, not f*) consists of finitely many points?


    2. Relevant equations



    3. The attempt at a solution

    I reduced the problem to this. Once I show that there are only finitely many preimages (with respect to f, not f*) of
    x in the compact set C = {(p, v) in TM : |v| <= 1}, I'm done.
    I tried to prove it using proof by contradiction, so
    I assumed that there are infinitely many points in that set, then we obviously,
    there's a subsequence such that
    pi --> p vi --> w for some (p, w) in C
    by properties of a compact set.
    Then once I show that df_p(w) = 0, I'm done, but
    I'm struggling to show how that would work.
     
    Last edited: Oct 20, 2010
  2. jcsd
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