Relevant Article: https://projecteuclid.org/journals/michigan-mathematical-journal/volume-20/issue-4/The-volume-of-a-small-geodesic-ball-of-a-Riemannian/10.1307/mmj/1029001150.full
From this paper on the finding of a series expansion formula for the volume of Geodesic balls on a Riemannian Manifold, the following formula is used:
$$S (r) =\int_{S^{n-1}(1)} r^{n-1} \omega_{1…n} (\text{exp}(ru)) \, du $$
$S(r)$ denotes the desired volume of the sphere at some point in an analytic Riemannian manifold, $S^{n-1}(1)$ is the volume of an $r=1$, n-1 sphere, $\omega_{1…n}$ is the volume element (alternatively $\omega = \sqrt{\text{det}(g)}$), and $\text{exp}(ru)$ is a factor introduced by a change of variables that utilizes the exponential mapping ($j:S^{n-1}(1) \rightarrow \text{exp}(S^{n-1}(r))$ with $j(u) = \text{exp}(ru)$). I suggest reading the paper to see where this formula comes from more precisely(section 2 and 3 especially).
From these sections, my question is how to find the variational/functional derivative of the volume function in terms of the metric tensor.
$$\frac{\delta S(r)}{\delta g^{ab}} = \text{??}$$.
Using the series expansion one can find the variational derivative of each term fine as the Riemann Tensor's derivatives and scalars can be written out explicitly, but having a general formula similar to equation 9 or 10 from section 3 is what I'm looking for.
Fairly lost besides the brute force term by term approach.
Best Answer
Here's a sketch of a possible approach to deriving an ODE that $S(r)$ satisfies in terms of the variation of the curvature tensor. The idea is that if $\partial_1, \dots, \partial_n$ are the standard coordinate vector fields on $\mathbb{R}^n$, then $$ J = (\exp_p)_*\partial_i $$ are Jacobi fields. They satisfy $$ g_{ij} = J_i\cdot J_j, $$ where $g_{ij}\,dx^i\,dx^j = (\exp_p)^*g$, and $$ J_i(0) = 0,\ \nabla_rJ_i(0) = \partial_i $$ and $$ \nabla_r\nabla_rJ_i = K(\partial_r, J)\partial_r. $$ Now the idea is to write the volume of the ball in terms of the Jacobi fields and differentiating it (possibly twice) with respect to the variational parameter. For a fixed value of the parameter, the Christoffel symbols will satisfy equation coming from the fact that radial lines are geodesics. This will in turn give formulas for the variation of the Christoffel symbols, which will be needed.
You might need to calculate $S'(r)$ instead.