Triple Products of Eigenfunctions and Spectral Geometry

TL;DR

This paper introduces a new global geometric invariant based on triple products of Laplace-Beltrami eigenfunctions, providing a precise characterization of isometric manifolds among isospectral Riemannian spaces.

Contribution

It presents a novel invariant derived from eigenfunction triple products that distinguishes isometric manifolds within isospectral classes.

Findings

The invariant effectively characterizes isometric manifolds.

Elementary techniques from geometric analysis and PDE are employed.

The approach links spectral data to geometric structure.

Abstract

Using elementary techniques from Geometric Analysis, Partial Differential Equations, and Abelian $C^*$ Algebras, we uncover a novel, yet familiar, global geometric invariant -- namely the indexed set of integrals of triple products of eigenfunctions of the Laplace-Beltrami operator, to precisely characterize which isospectral closed Riemannian manifolds are isometric.