Thread: Principle curvature as a continuous function

1. Principle curvature as a continuous function

Dear all, as we know, principle curvature $\displaystyle k_1, k_2$ (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?

2. 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.

3. 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.