  In the present paper, we show that on a compact Riemannian manifold $(M,g)$
of dimension $d\leqslant 4$ whose metric has negative curvature, the
renormalized partition function $Z_g(\lambda)$ of a massive Gaussian Free Field
determines the length spectrum of $(M,g)$ and imposes some strong geometric
constraints on the Riemannian structure of $(M,g)$. In any finite dimensional
family of Riemannian metrics of negative sectional curvature bounded from below
and above and whose isometry group is trivial, there is only a \textbf{finite
number of isometry classes} of metrics with given partition function
$Z_g(\lambda)$. When $d<4$, the same result holds true if the random variable
$\int_M:\phi^2:dv$ has given probability distribution and without the lower
bound on the sectional curvatures.
