2023/05/26 by Majhi, Sushovan
#Algebraic Topology (math.AT) #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2305.17288
For a closed Riemannian manifold M and a metric space S with a small Gromov\unicodex2013Hausdorff distance to it, Latschev's theorem guarantees the existence of a sufficiently small scale β>0 at which the Vietoris\unicodex2013Rips complex of S is homotopy equivalent to 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 β in order to provide sampling conditions for S to be homotopy equivalent to M. In this paper, we prove a stronger and pragmatic version of Latschev's theorem, facilitating a simple description of β using the sectional curvatures and convexity radius of M as the sampling parameters. Our study also delves into the topological recovery of a closed Euclidean submanifold from the Vietoris\unicodex2013Rips complexes of a Hausdorff close Euclidean subset. As already known for Čech complexes, we show that Vietoris\unicodex2013Rips complexes also provide topologically faithful reconstruction guarantees for submanifolds.