  Let (M,g) be a compact, connected and oriented Riemannian manifold. We denote
D the space of smooth probability density functions on M.
  In this paper, we show that the Frechet manifold D is equipped with a
Riemannian metric g^{D} and an affine connection \nabla^{D} which are infinite
dimensional analogues of the Fisher metric and exponential connection in the
context of information geometry. More precisely, we use Dombrowski's
construction together with the couple (g^{D},\nabla^{D}) to get a
(non-integrable) almost Hermitian structure on D, and we show that the
corresponding fundamental 2-form is a symplectic form from which it is possible
to recover the usual Schrodinger equation for a quantum particle living in M.
  These results echo a recent paper of the author where it is stressed that the
Fisher metric and exponential connection are related (via Dombrowski's
construction) to Kahler geometry and quantum mechanics in finite dimension.
