Ricci tensor

$R_{\alpha\beta} = {R^\lambda}_{\alpha \lambda\beta}.$

Variation

Starting with the variation of the Riemann tensor,

$\delta R^\rho{}_{\sigma\mu\nu} = \nabla_\mu (\delta \Gamma^\rho_{\nu\sigma}) - \nabla_\nu (\delta \Gamma^\rho_{\mu\sigma})\,$ ,

the variation of the Ricci tensor is found by contracting indices:

$\delta R_{\mu\nu} = \delta R^\rho{}_{\mu\rho\nu} = \nabla_\rho (\delta \Gamma^\rho_{\nu\mu}) - \nabla_\nu (\delta \Gamma^\rho_{\rho\mu})\,$ .

$R_{\mu\nu} = \frac{R}{2} g_{\mu\nu}\,$ (see Riemann tensor).