# Local eigenvectors frame of a isometric immersion

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!

• You mean $\{e_1,e_2,\dots,e_n\}$, of course. Commented Oct 24, 2018 at 16:50
• @TedShifrin, yes, of course. Commented Oct 24, 2018 at 20:55

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.
• Is it really necessary to require all eigenvalues to be distinct? I think repeated eigenvalues don't automatically give issues. For instance: the $n$-sphere $S^n$ in $\mathbb{R}^{n+1}$ is totally umbilical and a frame in a point can be extended smoothly to a neighbourhood of that point. Or do I miss something? Commented Oct 24, 2018 at 20:42