  In the present paper we have discussed the mechanics of incompressible test
bodies moving in Riemannian spaces with non-trivial curvature tensors. For
Hamilton's equations of motion the solutions have been obtained in the
parametrical form and the special case of the purely gyroscopic motion on the
sphere has been discussed. For the geodetic case when the potential is equal to
zero the comparison between the geodetic and geodesic solutions have been done
and illustrated in the case of a particular choice of the constants of motion
of the problem.
  The obtained results could be applied, among others, in geophysical problems,
e.g., for description of the motion of a drop of fat or a spot of oil on the
surface of the ocean (e.g., produced during some "ecological disaster") or the
motion of continental plates, or generally in biomechanical problems, e.g., for
description of the motion of objects with internal structure on different
curved two-dimensional surfaces (e.g., transport of proteins along the curved
biological membranes).
