# How to prove the derive the expression for space part of Riemann tensor for homogeneous and isotropic space-time?

It's not a homework!!

For spheric, hyperbolic and flat case $$dl^{2} = R^{2}\left(d \psi^{2} + sin^{2}(\psi )(d \theta^{2} + sin^{2}(\theta )d \varphi^{2})\right),$$

$$dl^{2} = R^{2}\left(d \psi^{2} + sh^{2}(\psi )(d \theta^{2} + sin^{2}(\theta )d \varphi^{2})\right),$$

$$dl^{2} = dx^2 + dy^{2} + dz^{2},$$ or, in generalized form, $$dl^{2} = \frac{dr^{2}}{1 - \kappa \frac{r^{2}}{R^{2}}} + r^{2}(d \theta^{2} + sin^{2}(\theta )d \varphi^{2}), \quad \kappa = 1, -1, 0 ,$$ the space part of Riemann tensor is given by $$R_{ijkl} = \frac{\kappa}{R^{2}}\left( g_{ik}g_{jl} - g_{il}g_{jk}\right).$$ How to prove it by using an expression for $R_{ijkl}$ via $$R_{iklm} = \frac{1}{2}\left( \partial_{k}\partial_{l}g_{im} + \partial_{i}\partial_{m}g_{kl} - \partial_{k}\partial_{m}g_{il} - \partial_{i}\partial_{l}g_{km}\right) + g_{np}(\Gamma^{n}_{kl}\Gamma^{p}_{im} - \Gamma^{n}_{km}\Gamma^{p}_{il})?$$ I tried to do it with hyperbolic metric, but I failed.

• Hi PhysiXxx. If you haven't already done so, please take a minute to read the definition of when to use the homework tag, and the Phys.SE policy for homework-like problems. Whether it is actual homework is irrelevant. – Qmechanic Apr 29 '13 at 23:00
• Also please do not post the same question in a new entry. Re-edit the old entry instead. – Qmechanic Apr 30 '13 at 19:41