Heisenberg uncertainty relations and average space curvature in geometric quantum mechanics

A relationship between the position root-mean-square deviation of a quantum particle and average space curvature is derived in the framework of non-relativistic geometric quantum mechanics. It is shown that particle localization forces the space to be curved. Heisenberg's uncertainty relations are proved to be a consequence of the more fundamental position-curvature relation, provided the operational prescriptions of standard quantum mechanics are used.