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I was wondering if there is a straightforward way to compute the Ricci curvature of a metric that has the form (à la Kaluza-Klein):

$g_{MM}\equiv\begin{pmatrix}g_{\mu\nu}&g_{\mu i}\\g_{i\nu}&g_{ij}\end{pmatrix}$

Ie, finding an expression of the form:

$\mathcal{R}(G_{MN})=R(g_{\mu\nu})+R(g_{ij})+(\text{scalar depending on both metrics})$

I tried using the usual formula

$R^M_{NPQ}=2\partial_{[Q}\Gamma^M_{P]N}+2\Gamma^S_{M[P}\Gamma^M_{Q]S}$

where $A_{[MN]}=\frac{1}{2}(A_{MN}-A_{NM})$. But this becomes really quickly a mess due to the different indices.

Does anyone has an idea how to find the result quickly? For a kaluza klein metric where $g_{ij}$ is a scalar and $g_{\mu j}$ is a vector this is straightforward, but I can't find a nice solution.

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  • $\begingroup$ There is no nice way to do this. You'll have to go through the mess :) $\endgroup$ – Prahar Jan 14 '14 at 18:20
  • $\begingroup$ Well, let me rephrase. There is a somewhat nice way to do it. It is known that $R[g_{\mu\nu}]$ is a scalar. It is therefor possible to write down the most general form of $R[g_{\mu\nu}]$ in terms of all the quantities in the problem. Use gauge symmetries of $g_{\mu j}$ to simplify some expressions. The coefficients of various terms can be fixed by looking at special forms of the metric. $\endgroup$ – Prahar Jan 14 '14 at 18:22

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