  It is known that the properties of almost all points of R^n being not very
well (multiplicatively) approximable are inherited by nondegenerate in R^n
(read: not contained in a proper affine subspace) smooth submanifolds. In this
paper we consider submanifolds which are contained in proper affine subspaces,
and prove that the aforementioned diophantine properties pass from a subspace
to its nondegenerate submanifold. The proofs are based on a correspondence
between multidimensional diophantine approximation and dynamics of lattices in
Euclidean spaces.
