is there a reference on the structure of the space of metrics on a compact manifold that induce a given measure $\mu $?

i have a given manifold $M$, a given measure $\mu$ with an everywhere positive $C^{\infty}$ density with respect to Lebesgue. i want to construct a metric in this class satisfying some additional properties.

is there any way to express, say, $$ \frac{d}{dt} \nabla _{g(t)} f $$ where $g(t)$ is a family of metrics inducing the same measure, $\nabla _{g(t)} $ is the gradient w.r.t. the metric, and $f$ is an eigenvector of the Laplacian-Beltrami operator associated to the metric $g = g(0)$? I am looking for coordinate-free (if possible) calculations including this kind of objects.

thanks, nikos