Local geometry of random geodesics on negatively curved surfaces

TL;DR

This paper demonstrates that long geodesics on negatively curved surfaces induce a tessellation resembling a Poisson line process locally, with global statistics converging to those of the Poisson process, revealing universal geometric properties.

Contribution

It establishes a connection between geodesic tessellations on negatively curved surfaces and Poisson line processes, providing new insights into their local and global geometric behavior.

Findings

Local tessellation resembles Poisson line process

Global tessellation statistics converge to Poisson process

Fraction of triangles approaches Poisson process predictions

Abstract

It is shown that the tessellation of a compact, negatively curved surface induced by a typical long geodesic segment, when properly scaled, looks locally like a Poisson line process. This implies that the global statistics of the tessellation -- for instance, the fraction of triangles -- approach those of the limiting Poisson line process.