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

2020/06/30 by Fei Qi, Qi, Fei
Mathematics · Medicine · #Advanced Neuroimaging Techniques and Applications #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Representation Theory (math.RT) #Tensor decomposition and applications

paper · pdf · doi:10.48550/arxiv.2006.16704

openalex publication_date 2020/06/30 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

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.

Related

• Some Properties of the Eigenfunctions of The Laplace-Operator on Riemannian Manifolds
• Estimates for eigensections of Riemannian vector bundles
• Laplace eigenfunctions on Riemannian symmetric spaces and Borel-Weil Theorem
• Linear Independence of Covariant Derivatives and Space-Curvatures
• Gradient estimate of an eigenfunction on a compact Riemannian manifold without boundary