Hamilton-Jacobi approach to potential functions in information geometry

The search for a potential function S allowing us to reconstruct a given metric tensor g and a given symmetric covariant tensor T on a manifold M is formulated as the Hamilton-Jacobi problem associated with a canonically defined Lagrangian on T M . The connection between this problem, the geometric structure of the space of pure states of quantum mechanics, and the theory of contrast functions of classical information geometry are outlined.