Local eigenvectors frame of a isometric immersion

Question

I have a little (possible dumb) technical question about local eigenvectors frame of a isometric immersion.

Let $N^n$ a smooth manifold and $(M^{n+1},g)$ a smooth Riemannian manifold. Consider $\phi: N\to M$ a isometric immersion and let $S$ be the shape operator of $N$.

Given $p\in N$, we can always assume that exists a local orthonormal frame $\{e_1,e_2,\cdots,e_n\}$ on a neighborhood of $p$, such that diagonalizes $S$? Or we need to put the condition that $p$ is not a umbilical point?

Thanks!

Answer 1

At each point $p$, of course, there's always an orthonormal basis for $T_pN$ diagonalizing $S_p$. You may likely have local smoothness issues whenever there are repeated eigenvalues. However, in dimension $n>2$, it's not good enough to say there are no umbilic points; you actually need to require distinct eigenvalues.