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 !

  • $\begingroup$ The Ricci tensor only has two indices, so in your expression above $R_{ikjl}$ probably refers to the components of the curvature tensor. $\endgroup$ – treble Mar 12 '12 at 11:57
  • $\begingroup$ @treble oh, thanks for mentioning this, I have corrected it! $\endgroup$ – 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 $$ (g^{-1})_{ij} = \delta_{ij} - h_{ijkl}x^kx^l + O(|x|^3)~. $$

  • $\begingroup$ Do you know a source where I can read about the process you describe specifically in this context, i.e. in the context of the expansion of the metric ? I would love to work through it so that I understand everything you say in detail. Many thanks! $\endgroup$ – harlekin Mar 13 '12 at 9:58
  • $\begingroup$ The general linear algebra fact is called "Neumann series approximation" (or something like that). You can find it in most linear algebra textbooks; Wikipedia suggests that this book contains the result. For the evaluation of the metric in normal coordinates, the result was already known to Riemann in 1854, and I believe a write-up is available in Spivak's Differential Geometry volume 2. Or you can unpack propositions IV.3.1 and IV.3.2 in Kobayashi-Nomizu Foundations of Differential Geometry vol 1 to get the result. $\endgroup$ – Willie Wong Mar 13 '12 at 10:19
  • $\begingroup$ Some other resources, you may want to look at this webpage and, IIRC, Chapter 4 of Jost's Riemannian Geometry and Geometric Analysis. Most geometry textbooks I am familiar with don't bother with a detailed explanation of the case of the inverse metric, since it is usually taken to be a simple consequence of the Neumann series approximation. $\endgroup$ – Willie Wong Mar 13 '12 at 10:31
  • $\begingroup$ ok that sounds good, I will have a look at these sources, many thanks !! $\endgroup$ – harlekin Mar 13 '12 at 13:02

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