Inverse metric coefficients in normal coordinates

Question

I know that in normal coordinates centered at $p\in M$ the first order partial derivatives vanish at $p$, is the same thing true for the inverse metric coefficients, or is there a counterexample?

Answer 1

As $g^{-1} g = I$, differentiating gives

$$\partial_i g^{-1} g = - g^{-1} \partial_i g = 0 \Rightarrow \partial_i g^{jk} = 0$$

at the center of the normal coordinate too.