On the Poisson relation for flat and spherical space forms

TL;DR

This paper proves that for certain flat and spherical space forms, the wave trace singularities precisely match the lengths of closed geodesics, enabling the recovery of the length spectrum from the Laplace spectrum.

Contribution

It extends the Poisson relation equality to flat manifolds, homogeneous lens spaces, and specific three-dimensional lens spaces, broadening the class of spaces where this holds.

Findings

Poisson relation is an equality for all flat manifolds.

Poisson relation holds for homogeneous lens spaces.

Poisson relation is valid for three-dimensional lens spaces with prime order fundamental group.

Abstract

A general approach to proving that the length spectrum of a compact Riemannian manifold is an invariant of the Laplace spectrum comes from considering the wave trace, a spectrally determined tempered distribution. The Poisson relation states that the singularities of the wave trace can only occur at lengths of closed geodesics. Proving that the Poisson relation is an equality; i.e., every length of a closed geodesic is a singularity of the wave trace, yields an effective means of recovering the length spectrum from the Laplace spectrum. Regarding spaces of constant curvature this is known to be true for spaces of constant negative curvature. Continuing the study of space forms, we show that the Poisson relation is an equality for all flat manifolds, homogeneous lens spaces and three-dimensional lens spaces with fundamental group of prime order.