# Approximate expression for the metric in normal coordinates

In the Wikipedia article on Ricci curvature (here) it is mentioned that one can approximate the metric g in normal coordinates by $$g_{ij} = \delta_{ij} - \frac{1}{3} R_{ikjl} \,x^kx^l + \mathcal{O}(|x|^3)$$ 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 $$g^{ij} = \delta_{ij} - \frac{1}{3} R_{ikjl} \,x^kx^l + \mathcal{O}(|x|^3)$$

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 !

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The Ricci tensor only has two indices, so in your expression above $R_{ikjl}$ probably refers to the components of the curvature tensor. – treble Mar 12 '12 at 11:57
@treble oh, thanks for mentioning this, I have corrected it! – harlekin Mar 12 '12 at 12:00

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 $$(g^{-1})_{ij} = \delta_{ij} - h_{ijkl}x^kx^l + O(|x|^3)~.$$