  We develop a general method to calculate entropy numbers of standard
Sobolev's classes on an arbitrary compact homogeneous Riemannian manifold. Our
method is essentially based on a detailed study of geometric characteristics of
norms induced by subspaces of harmonics. The method's possibilities are not
confined to the statements proved but can be applied in studying more general
problems such as entropy of multiplier operators. As an application, we
establish sharp orders of entropy of Sobolev's classes and respective n-widths.
