Khintchine-type theorems on manifolds: the convergence case for standard   and multiplicative versions
  An analogue of the convergence part of the Khintchine-Groshev theorem, as well as its multiplicative version, is proved for nondegenerate smooth submanifolds in $\mathbb{R}^n$. The proof combines methods from metric number theory with a new approach involving the geometry of lattices in Euclidean spaces. 
V. Bernik, D. Kleinbock, and G. A. Margulis