Random \v{C}ech Complexes on Riemannian Manifolds

TL;DR

This paper investigates the homology of random Cech complexes generated by Poisson processes on compact Riemannian manifolds, extending known results from Euclidean spaces to curved geometries.

Contribution

It generalizes homological connectivity results from flat spaces to Riemannian manifolds and develops a framework for translating Euclidean random geometric results to curved spaces.

Findings

Identifies phase transition for homological connectivity on Riemannian manifolds

Extends previous Euclidean results to general compact Riemannian manifolds

Provides a framework for applying Euclidean geometric graph results to Riemannian settings

Abstract

In this paper we study the homology of a random Cech complex generated by a homogeneous Poisson process in a compact Riemannian manifold M. In particular, we focus on the phase transition for "homological connectivity" where the homology of the complex becomes isomorphic to that of M. The results presented in this paper are an important generalization of [7], from the flat torus to general compact Riemannian manifolds. In addition to proving the statements related to homological connectivity, the methods we develop in this paper can be used as a framework for translating results for random geometric graphs and complexes from the Euclidean setting into the more general Riemannian one.