Covariant derivatives of eigenfunctions along parallel tensors over space forms and a conjecture motivated by the vertex algebraic structure

TL;DR

This paper investigates how covariant derivatives of Laplace-Beltrami eigenfunctions behave along parallel tensors on space forms, revealing scalar multiples and polynomial relations, and proposes a conjecture inspired by vertex algebra structures.

Contribution

It demonstrates that covariant derivatives are scalar multiples of eigenfunctions along parallel tensors and introduces a conjecture linking these polynomials to vertex algebraic structures.

Findings

Covariant derivatives are scalar multiples of eigenfunctions along parallel tensors.

The scalar multiples are polynomials in the eigenvalue.

A conjecture relates these polynomials to vertex algebraic structures.

Abstract

We study the covariant derivatives of an eigenfunction for the Laplace-Beltrami operator on a complete, connected Riemannian manifold with nonzero constant sectional curvature. We show that along every parallel tensor, the covariant derivative is a scalar multiple of the eigenfunction. We also show that the scalar is a polynomial depending on the eigenvalue and prove some properties. A conjecture motivated by the study of vertex algebraic structure on space forms is also announced, suggesting the existence of interesting structures in these polynomials that awaits further exploration.