  In this paper we propose a geometrization of the non-relativistic quantum
mechanics for mixed states. Our geometric approach makes use of the Uhlmann's
principal fibre bundle to describe the space of mixed states and as a novelty
tool, to define a dynamic-dependent metric tensor on the principal manifold,
such that the projection of the geodesic flow to the base manifold gives the
temporal evolution predicted by the von Neumann equation. Using that approach
we can describe every conserved quantum observable as a Killing vector field,
and provide a geometric proof for the Poincare quantum recurrence in a physical
system with finite energy levels.
