Smooth local frame extends to a local chart

Question

This is kind of a dumb question, but I was wondering if given a local orthonormal frame,$s_i$, in a Riemannian manifold, does it extend to a local chart $\varphi(p)=(x^1(p),...,x^n(p))$ such that $\frac{\partial}{\partial x^i}=s^i$. I know that every local frame induces a local triviliazation, but I am not sure if it induces a chart. Would someone tell me if this is true, or if there is a counterexample. Thanks.

Answer 1

No, due to the fact that the lie bracket $[s_i, s_j]$ may not be zero, whereas $$\left[\frac{\partial}{\partial x_i}, \frac{\partial}{\partial x_j}\right] = 0.$$