  Consider the sum of the first $N$ eigenspaces for the Laplacian on a
Riemannian manifold. A basis for this space determines a map to Euclidean space
and for $N$ sufficiently large the map is an embedding. In analogy with a
fruitful idea of K\"ahler geometry, we define (Riemannian) Bergman metrics of
degree $N$ to be those metrics induced by such embeddings. Our main result is
to identify a natural sequence of Bergman metrics approximating any given
Riemannian metric. In particular we have constructed finite dimensional
symmetric space approximations to the space of all Riemannian metrics. Moreover
the construction induces a Riemannian metric on that infinite dimensional
manifold which we compute explicitly.
