Kolmogorov and Linear Widths of Balls in Sobolev Spaces on Compact Manifolds

We determine upper asymptotic estimates of Kolmogorov and linear $n$-widths of unit balls in Sobolev 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, cubature formulas with positive coefficients and uniform estimates on Clebsch-Gordon coefficients on general compact homogeneous manifolds.