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A classical morse theory question

  1. Sep 27, 2010 #1


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    Let M be a manifold and let f: m -> R a Morse function.
    Let x be a critical point of f and assume all critical points are non-degenerate.
    Let W^u(x) be th unstable manifold of x when considering the negative gradient flow on M.

    Why does the tangent space at x to W^u(x) = Eig^- H^2f(x)?

    Denote the Hessian by H^2f(x).

    I know that since the critical points are non-degenerate the Morse lemma gives a sort
    of quadratic decomposition of f. I also know the M can be written as the union over
    all x of the W^u(x).

    One of the problems is that I don't really understand the object T_xW^u(x). The tangent
    space is a vector space. So T_xW^u(x) consist of the points in the vector space that
    "repel" from x?
  2. jcsd
  3. Sep 28, 2010 #2
    It would be better to think that we find the equation of the tangent plane rather than points which 'repel' from x. We'd follow exactly the same steps.
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