The Diophantine inheritance meta-conjecture for nondegenerate smooth submanifolds

Let a Diophantine property be a property of vectors in an ambient space such as $\mathbb{R}^{n}$, and let $\mathcal{M}$ be a nondegenerate smooth submanifold of that space. Diophantine inheritance meta-conjecture. Any Diophantine property of vectors in the ambient space which holds for almost all points in that space should hold for generic points on $\mathcal{M}$. This meta-conjecture formulates the general theme of metric Diophantine approximation on manifolds: properties generic in the ambient space should be inherited by sufficiently curved submanifolds. The source gives no resolution of the general statement.