TL;DR
This paper proves that two-dimensional extra-large metrics, characterized by cone angles greater than 2pi and long geodesics, can be triangulated with vertices at cone points, with implications for higher dimensions.
Contribution
It establishes a triangulation result for extra-large cone metrics in 2D, Euclidean, hyperbolic, and higher dimensions, highlighting the necessity of the extra-large condition.
Findings
Triangulation with cone points only for 2D extra-large metrics
Extension of results to Euclidean and hyperbolic cone metrics
Potential generalization to higher-dimensional extra-large metrics
Abstract
An extra large metric is a spherical cone metric with all cone angles greater than 2 pi and every closed geodesic longer than 2pi. We show that every two-dimensional extra large metric can be triangulated with vertices at cone points only. The argument implies the same result for Euclidean and hyperbolic cone metrics, and can be modified to show a similar result for higher-dimensional extra-large metrics. The extra-large hypothesis is necessary.
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Taxonomy
TopicsGeometric and Algebraic Topology · Geometric Analysis and Curvature Flows · Soft tissue tumor case studies