A geometry where everything is better than nice

TL;DR

This paper introduces a novel Riemannian structure on the disk with unique geometric and spectral properties, including hypocycloid geodesics and integer spectrum Laplacian eigenvalues, with applications in acoustic scattering analysis.

Contribution

It presents a new Riemannian geometry on the disk featuring distinctive geodesics and spectral characteristics, linking geometry with acoustic scattering.

Findings

Geodesics are hypocycloids.

Laplacian has integer spectrum with multiplicities.

Eigenfunctions are orthogonal polynomials suited for acoustic analysis.

Abstract

We present a riemannian structure on the disk that has a remarkably rich structure. Geodesics are hypocycloids and the (negative of the) laplacian has integer spectrum with multiplicity the Dirichlet divisor function. Eigenfunctions of the laplacian are orthogonal polynomials naturally suited to the analysis of acoustic scattering in layered media.