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.