Palatini identity ($\delta R_{ab}={\nabla }_c\delta {\Gamma }^c{}_{ab}-{\nabla }_b\delta {\Gamma }^c{}_{ac}$)

For the Levi-Civita connection, the difference of two connections is a tensor. Varying the coordinate expression for the Ricci tensor then gives the displayed covariant identity. It is the starting point for the metric variation of scalar curvature.