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.
Cite this article
Fei Qi, Covariant derivatives of eigenfunctions along parallel tensors over space forms and a conjecture motivated by the vertex algebraic structure. J. Noncommut. Geom. 16 (2022), no. 2, pp. 717–759DOI 10.4171/JNCG/472