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

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