Entropy of Sobolev's classes on Compact Homogeneous Riemannian Manifolds

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.