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.


more precisely, for a given metric $g$ inducing $\mu $, i want to keep a finite number $n$ of eigenfunctions $\{ f\_i \}$ of the laplacian,
$$
\Delta \_{g} f\_i = \lambda \_i f\_i
$$
Then, i generate another family of $n$ functions $\{ f'\_i \}$ as linear combinations of the $\{ f\_i \}$. What i want to do is to construct a metric $g'$ inducing the same measure such that
$$
\Delta \_{g'} f'\_i = \lambda ' \_i f'\_i
$$


I am looking for coordinate-free (if possible) calculations including this kind of objects.


thanks,
nikos



structure of metrics on a compact manifold
Have a look at 


* Martin Bauer, Philipp Harms, Peter W. Michor: Sobolev metrics on the manifold of all Riemannian metrics. Journal of Differential Geometry 94, 2 (2013), 187-208. [(pdf)](http://www.mat.univie.ac.at/~michor/rie-met2.pdf).


