In the book I'm reading ("Differential Geometry Curves-Surfaces-Manifolds by Wolfgang Kuhnel") two definitions of prinicpal curvatures directions are presented:
- The extramum values of $II(X,X)$ under the constraint $I(X,X)=1$
- The eigenvectors of the shape operator $L$
I'm having some difficulty proving this equivalence. It was also mentioned that the gradient of $II(X,X)$ is simply $LX$ ($L$ operating on $X$). So I thought if I prove that, it follows that (1) holds iff the gradient of $II(X,X)$ is proportional to the gradient of $I(X,X)$ (which is clearly $X$) iff $LX$ and $X$ are proportional iff $X$ is an eigenvector.
- How do I compute the gradient of $II(X,X)$ and show it is equal to $LX$. It seems like it is a simple task that I'm missing some idea or technique.
- I'll be glad to hear of any interesting insights regarding prinicpal curvatures in order to have a better intuition.