Riemannian simplices and triangulations

TL;DR

This paper introduces a new intrinsic way to define and analyze simplices in Riemannian manifolds using Karcher means, providing criteria for triangulating such manifolds and ensuring non-degeneracy of simplices.

Contribution

It develops a novel intrinsic definition of Riemannian simplices via Karcher means and establishes conditions for their non-degeneracy and for triangulating manifolds.

Findings

Criteria for smooth embedding of barycentric coordinate maps

Conditions for non-degeneracy based on curvature and size

Guarantees for triangulating manifolds with simplicial complexes

Abstract

We study a natural intrinsic definition of geometric simplices in Riemannian manifolds of arbitrary dimension $n$, and exploit these simplices to obtain criteria for triangulating compact Riemannian manifolds. These geometric simplices are defined using Karcher means. Given a finite set of vertices in a convex set on the manifold, the point that minimises the weighted sum of squared distances to the vertices is the Karcher mean relative to the weights. Using barycentric coordinates as the weights, we obtain a smooth map from the standard Euclidean simplex to the manifold. A Riemannian simplex is defined as the image of this barycentric coordinate map. In this work we articulate criteria that guarantee that the barycentric coordinate map is a smooth embedding. If it is not, we say the Riemannian simplex is degenerate. Quality measures for the "thickness" or "fatness" of Euclidean simplices can be adapted to apply to these Riemannian simplices. For manifolds of dimension 2, the simplex is non-degenerate if it has a positive quality measure, as in the Euclidean case. However, when the dimension is greater than two, non-degeneracy can be guaranteed only when the quality exceeds a positive bound that depends on the size of the simplex and local bounds on the absolute values of the sectional curvatures of the manifold. An analysis of the geometry of non-degenerate Riemannian simplices leads to conditions which guarantee that a simplicial complex is homeomorphic to the manifold.