Normal Coordinates on a Riemannian Manifold Consider Riemann Normal coordinates on a manifold. Consider a point other than the origin. Given that the metric has vanishing derivatives at this point, is it correct to deduce that the metric is Euclidean at this point? If the deduction is correct, how to prove/ argue this?

Clearly: $x_k=x^ig_{ik}$. Consider differentiating on both sides:  $\partial_j x_k=(\partial_j x^i)g_{ik}+x^i\partial_j g_{ik}$. Now if the metric derivatives vanish then the last term is zero and we have: $\partial_j x_k=(\partial_j x^i)g_{ik}\Rightarrow\delta_{jk}=\delta_j^ig_{ik}\Rightarrow\delta_{jk}=g_{jk}$. 
