$(\lambda,\lambda)$-eigenfunctions on compact manifolds

TL;DR

This paper investigates special eigenfunctions on compact manifolds, revealing their geometric structure as mapping tori and characterizing eigenfamilies as one-dimensional, with extensions to harmonic submersions to tori.

Contribution

It characterizes the structure of $(\lambda,\lambda)$-eigenfunctions on compact manifolds and relates generalized eigenfamilies to harmonic Riemannian submersions.

Findings

Manifolds with $(\lambda,\lambda)$-eigenfunctions are mapping tori.

$(\lambda,\lambda)$-eigenfamilies are one-dimensional.

Generalized eigenfamilies relate to harmonic Riemannian submersions to tori.

Abstract

In this note we study $(\lambda,\mu)$-eigenfamilies on compact Riemannian manifolds when $\lambda = \mu$. We show that any compact manifold admitting a $(\lambda,\lambda)$-eigenfunction is a mapping torus and that any $(\lambda,\lambda)$-eigenfamily is one dimensional. Additionally, we consider generalised eigenfamilies, which can have higher dimension, and relate these to harmonic Riemannian submersions to a torus.