It is known that "The directions in the normal plane where the curvature takes its maximum and minimum values are always perpendicular, if k1 does not equal k2, a result of Euler (1760), and are called principal directions."
This is the graph of z = x^2 * y^2, it is continuous and differentiable. However it is not clear that the perpendicular planes shown here or any other perpendicular planes give a pair of maximum and minimum curvatures.
The minimum curvature here is in the x and y axis and the maximum curvature is along the lines x = y and x = -y. But, they are not perpendicular.
What is happening here?
Sorry for my bad english, Thanks.
Concretely, we have the global parametrization $X:\Bbb R^2 \to S$ given by $X(u,v) = (u,v,u^2v^2)$. It is easy to check that $X_u(0,0) = (1,0,0)$, $X_v(0,0) = (0,1,0)$, and so the normal vector at the origin is $N(X(0,0)) = (0,0,1)$. We have that the induced metric is Euclidean at the origin. And since all the second derivatives of $X$ vanish for $u=v=0$, we get that the origin is a flat point of the surface. Hence $$K(0,0, 0) = H(0,0,0) = k_1 (0,0,0) = k_2 (0,0,0)=0$$ and all directions are principal.