Simplifying terms of Ricci tensor

[Safinaz]

Homework Statement

Any help how to simplify these terms of Ricci tensor:

Relevant Equations

$R_{\alpha\mu} R_{\gamma \nu} g^{\alpha \gamma} + R_{\mu \beta} R_{\nu \delta} g^{\beta \delta} + g^{\alpha \gamma} g^{\beta \delta} \left( R_{\alpha\beta} \frac{ \nabla_\gamma \delta \Gamma^\rho_ {~ \delta \rho} - \nabla_\rho \delta \Gamma^\rho_ {~ \gamma \delta} }{ \delta g^{\mu\nu} } + R_{\gamma\delta} \frac{ \nabla_\alpha \delta \Gamma^\rho_ {~ \beta \rho} - \nabla_\rho \delta \Gamma^\rho_ {~ \alpha \beta} }{ \delta g^{\mu\nu} } \right)$

So that they become:

$g^{\sigma \rho} \nabla_\sigma \nabla_\rho R ~g_{\mu\nu} + R ~R_{\mu\nu} - \nabla_\mu \nabla_\nu R$

 

[sbrothy]

I'm not of much help but I'm sure that those who might help would like to see what work you've done so far...