Geometry of projective Hilbert space

Geodesic equations determined by the metric in a projective Hilbert space openP are found. It is shown that connections in the tangent bundle over openP consistent with these geodesics can be introduced. A connection with zero torsion is proposed. Quantum-system dynamics follows the geodesics of another connection which is found for the case of the constant Hamiltonian. A set of geometrical quantities is defined, conserved by a quantum evolution.