Motion of test rigid bodies in Riemannian spaces

We derive and discuss equations of motion of rigid bodies of infinitesimal size in a Riemann space. The treatment is nonrelativistic. The rigorous meaning of “infinitisemal size” consists in replacing an extended body by the structured material point with internal degrees of freedom (co-moving orthonormal frame). The special case of constant curvature two-dimensional spaces is discussed and that of three-dimensional ones mentioned. The problem reduces then to Hamiltonian systems with the higher-dimensional rotation and Lorentz groups as configuration spaces.