Geometric phases in the motion of rigid bodies

In this paper we make some differential-geometric observations on the kinematics of convex surfaces rolling along a fixed plane in ℝ³, and on the relationship of the problem with parallel transport and the Gauss-Bonnet formula. These ideas are then applied to recover "Berry's phase" of a free rigid body which was found by Montgomery using Stokes' Theorem. We also point out a new "twist" on this problem. As a second application, we give a solution of a problem posed by R. Brockett. As a third application, we give a geometrical description of "Berry's phase" in SO(3); this can be applied to various rigid-elastic systems to compute their geometric phases.