Diophantinc approximation by linear forms on manifolds

The following Khintchine-type theorem is proved for manifolds M embedded in R ^k which satisfy some mild curvature conditions. The inequality |q·x| <?(|q|) where ?(r) ? 0 as r ? 8 has finitely or infinitely many solutions qeZ ^k for almost all (in induced measure) points x on M according as the sum S _r ^{= 1/8} ?(r)r ^k-2 converges or diverges (the divergent case requires a slightly stronger curvature condition than the convergent case). Also, the Hausdorff dimension is obtained for the set (of induced measure 0) of point in M satisfying the inequality infinitely often when ?(r) =r ^-t . t >k - 1. © 1990 Indian Academy of Sciences.