Let $M$ be an $n$-dimensional smooth hypersurface in $\mathbb R^{n+1}.$ Is it always possible to find a local principal frame near a point $p\in M?$ To be more precise, could we always find a neighborhood $U$ of $p,$ on which there exists orthonormal vector fields $v_1,\cdots,v_n$ such that $$A(v_i,v_j)=-k_i\delta_{ij}$$ where $A$ is the second fundamental form and $k_i$'s are the principal curvatures?
I know that this could be done at a point $p,$ but I am not sure if this could be done locally, or if there exists counterexample.
Any comments or ideas are appreciated!