Abstract
We study a natural intrinsic definition of geometric simplices in Riemannian manifolds of arbitrary finite dimension, 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 the standard simplex under 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 nondegenerate if it has a positive quality measure, as in the Euclidean case. However, when the dimension is greater than two, nondegeneracy 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 nondegenerate Riemannian simplices leads to conditions which guarantee that a simplicial complex is homeomorphic to the manifold.
BibTeX  Entry
@InProceedings{dyer_et_al:LIPIcs:2015:5136,
author = {Ramsay Dyer and Gert Vegter and Mathijs Wintraecken},
title = {{Riemannian Simplices and Triangulations}},
booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)},
pages = {255269},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783939897835},
ISSN = {18688969},
year = {2015},
volume = {34},
editor = {Lars Arge and J{\'a}nos Pach},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2015/5136},
URN = {urn:nbn:de:0030drops51361},
doi = {10.4230/LIPIcs.SOCG.2015.255},
annote = {Keywords: Karcher means, barycentric coordinates, triangulation, Riemannian manifold, Riemannian simplices}
}
Keywords: 

Karcher means, barycentric coordinates, triangulation, Riemannian manifold, Riemannian simplices 
Seminar: 

31st International Symposium on Computational Geometry (SoCG 2015) 
Issue Date: 

2015 
Date of publication: 

11.06.2015 