  We determine lower and exact estimates of Kolmogorov, Gelfand and linear
$n$-widths of unit balls in Sobolev norms in $L_{p}$-spaces on compact
Riemannian manifolds. As it was shown by us previously these lower estimates
are exact asymptotically in the case of compact homogeneous manifolds. The
proofs rely on two-sides estimates for the near-diagonal localization of
kernels of functions of elliptic operators.
