Local criteria for triangulation of manifolds

TL;DR

This paper introduces local criteria to verify when a simplicial complex provides a valid triangulation of a manifold, applicable without differentiability or explicit metrics, aiding algorithms in manifold triangulation.

Contribution

It offers new local criteria for manifold triangulation that do not depend on differentiability or Delaunay properties, facilitating practical algorithmic verification.

Findings

Criteria are presented in local coordinate charts.

Criteria do not require differentiability or explicit metrics.

Guarantee triangulation in algorithms working with local patches.

Abstract

We present criteria for establishing a triangulation of a manifold. Given a manifold M, a simplicial complex A, and a map H from the underlying space of A to M, our criteria are presented in local coordinate charts for M, and ensure that H is a homeomorphism. These criteria do not require a differentiable structure, or even an explicit metric on M. No Delaunay property of A is assumed. The result provides a triangulation guarantee for algorithms that construct a simplicial complex by working in local coordinate patches. Because the criteria are easily verified in such a setting, they are expected to be of general use.