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We determine upper asymptotic estimates of Kolmogorov and linear $n$-widthsof unit balls in Sobolev and Besov norms in $L_{p}$-spaces on smooth compactRiemannian manifolds. For compact homogeneous manifolds, we establish estimateswhich are asymptotically exact, for the natural ranges of indices. The proofsheavily rely on our previous results such as: estimates for the near-diagonallocalization of the kernels of elliptic operators, Plancherel-Polyainequalities on manifolds of bounded geometry, cubature formulas with positivecoefficients and uniform estimates on Clebsch-Gordon coefficients on generalcompact homogeneous manifolds.

Kolmogorov and Linear Widths of Balls in Sobolev and Besov Norms on Compact Manifolds