Riemannian Geometry

  1. Hello.
    Let M,N be a connected smooth riemannian manifolds.
    I define the metric as usuall, the infimum of lengths of curves between the two points.
    (the length is defined by the integral of the norm of the velocity vector of the curve).

    Suppose phi is a homeomorphism which is a metric isometry.
    I wish to prove phi is a diffeomorphism.

    Please, anyone who can help.
    Thanks in advance,

    Roey
     
  2. jcsd
  3. Hurkyl

    Hurkyl 16,089
    Staff Emeritus
    Science Advisor
    Gold Member

    My instinct is to be lowbrow and just compute the derivative. Limit of ratios of distances, and all that.
     
  4. mathwonk

    mathwonk 9,814
    Science Advisor
    Homework Helper

    if you can embed them so that the metric is induced from that of euclidean space, wouldnt an isomoetry just be a restricted linearmap?

    that makes it seem as if klocally it is alkways true, and derivatives are local properties.
     
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