Metric variation

For $h_{ab}=\delta g_{ab}$, variation of the inverse metric and metric volume form givesThe Levi-Civita connection varies by

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.

For $h_{ab}=\delta g_{ab}$, the Ricci scalar varies asThe last two terms form a covariant divergence and hence contribute only a boundary term to a compactly supported variation of the Einstein-Hilbert action.