TITLE: orthonormal vector fields
QUESTION [9 upvotes]: In a Riemannian manifold $(M^n,g)$, is it true that given a point $p$ we can find $n$ vector fields $X_i$ on a neighborhood $U$ of $p$ such that $g(X_i,X_j)(x)=\delta_{ij}$ with $x \in U$?
It seems that if I do Gram-Schmidt to the coordinate basis would work, but then the sectional cuvature $K$ will be $0$ and as it is intrinsic, it will imply that the surface (in the case $n=2$) is locally isometric to a plane which obviously is not true in general

REPLY [4 votes]: Another way to think about this: 
1.) if the manifold is flat then you can find coordinate vector fields which are locally orthogonal. 
2.) if the manifold is not flat then you can find non-coordinate fields which are locally orthogonal. In physics, in general relativity, these non-coordinate vector fields are sometimes taken to be locally orthogonal to the minkowski metric. The curvature of spacetime is then encoded in the details of how this non-coordinate frame varies from point to point.
Some of the formulas you have at your disposal are likely tacitly assuming a coordinate basis. If you forget that and assume the non-coordinate basis is appropriate for such formulas then you get contradictions like you mention.
A vaguely similar problem: we cannot find a set of orthonormal eigenvectors for a non-symmetric real matrix. Even if the matrix is diagonalizable the eigenbasis will not remain an eigenbasis after it is put through Gram-Schmidt. 
To summarize: one way to understand the other answers thus far given is that they serve to show you why the Gram-Schmidt process need not produce coordinate vector fields.