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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 !

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    @treble oh, thanks for mentioning this, I have corrected it!2012-03-12

1 Answers 1

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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)~. $

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    ok that sounds good, I will have a look at these sources, many thanks !!2012-03-13