Surfaces in which every point sounds the same

TL;DR

This paper investigates the spectral properties of surfaces where all points sound identical, revealing that such surfaces have highly symmetric isometry groups and demonstrating this phenomenon on Klein bottles and hyperbolic quotients.

Contribution

It characterizes surfaces with uniform point soundness, showing they possess transitive isometry groups, and extends the analysis to Klein bottles and hyperbolic quotients.

Findings

Surfaces with identical point sound have transitive isometry groups.

You can determine your location on Klein bottles from spectral data.

Lengths and multiplicities of geodesics are audible on hyperbolic quotients.

Abstract

We address a maximally structured case of the question, "Can you hear your location on a manifold," posed in arXiv:2304.04659 for dimension $2$. In short, we show that if a compact surface without boundary sounds the same at every point, then the surface has a transitive action by the isometry group. In the process, we show that you can hear your location on Klein bottles and that you can hear the lengths and multiplicities of looping geodesics on compact hyperbolic quotients.