  We determine upper asymptotic estimates of Kolmogorov and linear $n$-widths
of unit balls in Sobolev and Besov norms in $L_{p}$-spaces on smooth compact
Riemannian manifolds. For compact homogeneous manifolds, we establish estimates
which are asymptotically exact, for the natural ranges of indices. The proofs
heavily rely on our previous results such as: estimates for the near-diagonal
localization of the kernels of elliptic operators, Plancherel-Polya
inequalities on manifolds of bounded geometry, cubature formulas with positive
coefficients and uniform estimates on Clebsch-Gordon coefficients on general
compact homogeneous manifolds.
