# structure of metrics on a compact manifold

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

• variation of the laplacian? do you have a reference for this? Mar 20 '15 at 13:54
• Two follow up questions: (1) is it true that, as your notations indicated, that you want those $n$ eigenvalues to be the same? (2) Can I assume that $f'_i$ should be obtained by $A_{ij} f_j$ where $A_{ij}$ is some (given, fixed) invertible matrix? Mar 20 '15 at 16:01
• 1. corrected the misprint, 2. the matrix A is indeed invertible and constant. Mar 20 '15 at 17:15