Density of wave fronts

TL;DR

This paper proves that wave fronts on a flat torus become dense, implying similar density results for related geometries like square billiards and geodesic flows on certain flat surfaces.

Contribution

It establishes the density of wave fronts on flat tori and related surfaces, extending understanding of wave behavior in flat geometries.

Findings

Wave fronts on a flat torus are dense.

Density results extend to square billiards and geodesic flows on flat Klein bottles.

Provides a unified approach to wave front density in flat geometries.

Abstract

We prove that wave fronts on a flat torus become dense. As a corollary, wave fronts become dense for a square billiard or for the geodesic flow on the flat Klein bottle or the cube surface.