  In this article, we establish an analogue of the dimension growth conjecture,
which is regarding the density of rational points on projective varieties, for
compact submanifolds of $\mathbb{R}^n$ with non-vanishing curvature. We also
establish the convergence theory for the set of simultaneously
$\psi$-approximable points lying on a generic hypersurface, thereby settling
the generalized Baker-Schmidt problem in the simultaneous setting for generic
hypersurfaces. These results are obtained as consequences of an optimal upper
bound for the density of rational points near manifolds of the form $\{
(\mathbf{x}, f(\mathbf{x})) \in \mathbb{R}^{d+1}: \mathbf{x} \in
B_{\nu}(\mathbf{x}_0) \}$ with non-zero Hessian matrix of $f$ at $\mathbf{x}_0$
and $\nu > 0$ sufficiently small.
