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Math Help - Principle curvature as a continuous function

  1. #1
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    Principle curvature as a continuous function

    Dear all, as we know, principle curvature k_1, k_2 (the eigenvalue of the Weingarten transform) is a continous function near an umbilical point. However, I could not give an example showing that this could not be improved (to be a smooth function!)

    Thank you very much, and would you show me an example --- that means, a surface with the principle curvature being exactly a continous function, but not a differential one?
    Last edited by Ackbeet; November 8th 2011 at 06:22 AM.
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  2. #2
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    Re: Principle curvature as a continuous function

    For example, take any function z=f(x) that has only 2 order derivatives, that is f is C^2 but not C^3.
    Try to compute the principle curvature of a surface parametrized by (x,y) -> ( x, y, f(x) )
    k1 is a function involving f'' and f', k2 is 0.
    Since f'' is not differentiable, k1 is only continuous.
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  3. #3
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    Re: Principle curvature as a continuous function

    3x. However, the principle curvature then is not smooth near a point which is not umbilical...

    What I want in a problem is that find an example, where the principle curvature is smooth near points which are not umbilical, and is only continuous near umbilical points.
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