On a Riemannian manifold $(M,g)$, there always exists local orthonormal frame $\{E_i\}$ with respect to the metric $g$. But there does not necessarily exist orthonormal coordinate frame $\{ \frac{\partial}{\partial x^i} \}$ in a small neighborhood around a point $p$, where $\{x^i:U \rightarrow \mathbb{R}$ } are coordinate maps.
My question is the follow: instead of local orthonormal coordinate frame, is it possible to find local orthogonal coordinate frame? i.e. with respect to some coordinate system, the metric g can be written locally as $$g= g_{ii} {dx^{i}}dx^i$$
For surface, I know this is true because of something stronger, the existence of isothermal coordinate. I wonder whether this is possible for higher dimension.
