On a compact Riemannian manifold
of dimension
,
we present a rigorous construction of the renormalized partition function
of a massive Gaussian free field where we explicitly determine the
local counterterms using microlocal methods. Then we show that
determines the
Laplace spectrum of
and hence imposes some strong geometric constraints on the Riemannian structure of
. From
this observation, using classical results in Riemannian geometry, we illustrate how the
partition function allows us to probe the Riemannian structure of the underlying manifold
.