  For a closed Riemannian manifold $\mathcal{M}$ and a metric space $S$ with a
small Gromov$\unicode{x2013}$Hausdorff distance to it, Latschev's theorem
guarantees the existence of a sufficiently small scale $\beta>0$ at which the
Vietoris$\unicode{x2013}$Rips complex of $S$ is homotopy equivalent to
$\mathcal{M}$. Despite being regarded as a stepping stone to the topological
reconstruction of Riemannian manifolds from a noisy data, the result is only a
qualitative guarantee. Until now, it had been elusive how to quantitatively
choose such a proximity scale $\beta$ in order to provide sampling conditions
for $S$ to be homotopy equivalent to $\mathcal{M}$. In this paper, we prove a
stronger and pragmatic version of Latschev's theorem, facilitating a simple
description of $\beta$ using the sectional curvatures and convexity radius of
$\mathcal{M}$ as the sampling parameters. Our study also delves into the
topological recovery of a closed Euclidean submanifold from the
Vietoris$\unicode{x2013}$Rips complexes of a Hausdorff close Euclidean subset.
As already known for \v{C}ech complexes, we show that
Vietoris$\unicode{x2013}$Rips complexes also provide topologically faithful
reconstruction guarantees for submanifolds.
