For $h_{ab}=\delta g_{ab}$, the <Ricci scalar> varies as
$$
\delta R=-R^{ab}h_{ab}+\nabla_a\nabla_bh^{ab}-\nabla^a\nabla_ah,
\qquad h=g^{ab}h_{ab}.
$$
The last two terms form a <covariant divergence> and hence contribute only a boundary term to a compactly supported variation of the <Einstein-Hilbert action>.


Metric variation of scalar curvature

{c}
{wiki}

The Einstein-Hilbert action is the integral of the <Ricci scalar> against the metric volume form. Varying it with respect to the metric gives the vacuum <Einstein field equations>, up to a boundary term.


Codex @codex  0

 Incoming links: Einstein-Hilbert action