In the Wikipedia article on Ricci curvature (here) it is mentioned that one can approximate the metric g in normal coordinates by
\begin{equation}
g_{ij} = \delta_{ij} – \frac{1}{3} R_{ikjl} \,x^kx^l + \mathcal{O}(|x|^3)
\end{equation}
where $\delta_{ij}$ is the Kroenecker delta and $R_{ijkl}$ denotes the components of the curvature tensor in local coordinates.
Now, I have an article that states the same holds true for $g^{ij}$, the inverse of the metric. That is, I have the approximation
\begin{equation}
g^{ij} = \delta_{ij} – \frac{1}{3} R_{ikjl} \,x^kx^l + \mathcal{O}(|x|^3)
\end{equation}
That confuses me because I thought as the inverse it cannot look the same. If anyone could point to an explanation of this that would be great, many thanks !
Best Answer
There is either a sign problem, or (more likely) Wikipedia is using a different convention of the Riemann curvature then your article is (some people write $$ R_{ijkl}X^iY^jz^kW^l = \langle [\nabla_X,\nabla_Y]Z - \nabla_{[X,Y]}Z,W\rangle $$ and some people write it as the negative of that expression [or, with the spots of $Z$ and $W$ swapped on the right hand side]).
Ignoring the sign issue, what you have is the classic asymptotic expansion that for a matrix $A$ and $\epsilon$ sufficiently small, $$ (I + \epsilon A)^{-1} = I - \epsilon A + O(\epsilon^2) $$ (this is just the Taylor expansion of $B\mapsto B^{-1}$ near the point $B = I$). So if $$ g_{ij} = \delta_{ij} + h_{ijkl}x^kx^l + O(|x|^3) $$ you must have, for $|x|$ sufficiently small $$ (g^{-1})_{ij} = \delta_{ij} - h_{ijkl}x^kx^l + O(|x|^3)~. $$