  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
$\mathcal{M}$ is formulated as the Hamilton-Jacobi problem associated with a
canonically defined Lagrangian on $T\mathcal{M}$. 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 is outlined.
