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Given a metric tensor $\gamma_{ij}$ (where $i, j = 1, 2, 3$; the metric tensor of 3- dimensional space is denoted by $\gamma_{ij}$ to distinguish it from the metric tensor $g_{\mu\nu}$ of 4-dimensional space-time), show that the Riemann tensor is equal to

\begin{equation} ^{(3)}R_{ijkl}=\frac{\xi}{R^2}\left(\gamma_{ik}\gamma_{jl}-\gamma_{il}\gamma_{jk}\right),\end{equation}

and

\begin{equation}^{(3)}R_{ij}=2\frac{\xi}{R^2}\gamma_{ij}\end{equation} where $\xi=0,\pm 1$, distinguishes 3-plane, 3-sphere and 3-hyperboloid.

The first equation above looks quite simple, but I have no idea how to derive it. Can you please suggest me how to start deriving the above expressions?

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    $\begingroup$ This should be in any textbook on general relativity or differential geometry, such as the freely available lecture notes by S. Caroll. $\endgroup$
    – Neuneck
    Commented Jun 6, 2014 at 6:47
  • $\begingroup$ Hint: use the Cartan formalism on the general metric, using the structure equations. It's probably the fastest method. $\endgroup$
    – JamalS
    Commented Jun 6, 2014 at 7:44

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