Put $h_{ab}=\delta g_{ab}$. Varying $g^{ac}g_{cb}=\delta^a_b$ gives
$$
\boxed{\delta g^{ab}=-g^{ac}g^{bd}h_{cd}=-h^{ab}}.
$$
The determinant identity $\delta\log|\det g|=g^{ab}h_{ab}$ gives
$$
\boxed{\delta(d\operatorname{vol}_g)
=\frac12g^{ab}h_{ab}\,d\operatorname{vol}_g}.
$$
Finally, varying the formula for the <Christoffel symbols> and rewriting partial derivatives covariantly gives the tensor
$$
\boxed{\delta\Gamma^a{}_{bc}
=\frac12g^{ad}(\nabla_bh_{cd}+\nabla_ch_{bd}-\nabla_dh_{bc})}.
$$
These are the basic <metric variation> identities.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 309 / 3 / a / i / Solution

For $h_{ab}=\delta g_{ab}$, variation of the inverse metric and metric volume form gives
$$
\delta g^{ab}=-h^{ab},
\qquad
\delta(d\operatorname{vol}_g)=\frac12g^{ab}h_{ab}\,d\operatorname{vol}_g.
$$
The <Levi-Civita connection> varies by
$$
\delta\Gamma^a{}_{bc}=\frac12g^{ad}
(\nabla_bh_{cd}+\nabla_ch_{bd}-\nabla_dh_{bc}).
$$

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