  In this paper we use symplectic reduction in an Uhlmann bundle to construct a
principal fiber bundle over a general space of unitarily equivalent mixed
quantum states. The bundle, which generalizes the Hopf bundle for pure states,
gives in a canonical way rise to a Riemannian metric and a symplectic structure
on the base space. With these we derive a geometric uncertainty relation for
observables acting on quantum systems in mixed states. We also give a geometric
proof of the classical Robertson-Schr\"{o}dinger uncertainty relation, and we
compare the two. They turn out not to be equivalent, because of the multiple
dimensions of the gauge group for general mixed states. We give examples of
observables for which the geometric relation provides a stronger estimate than
that of Robertson and Schr\"{o}dinger, and vice versa.
