Calculate the Riemann tensor and Ricci tensor [closed]

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

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

and

$$^{(3)}R_{ij}=2\frac{\xi}{R^2}\gamma_{ij}$$ 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?

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