Hamilton-Jacobi approach to Potential Functions in Information Geometry

The search for a potential function S allowing 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 TM. 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 is outlined.