Curvature properties of $\phi$-null Osserman Lorentzian $\mathcal{S}$-manifolds

TL;DR

This paper investigates the curvature and eigenvalue properties of Jacobi operators on Lorentzian $ ext{S}$-manifolds, generalizing previous results to cases with multiple characteristic vector fields.

Contribution

It extends curvature characterizations of $ ext{S}$-manifolds to include arbitrary numbers of characteristic vector fields, broadening the understanding of their geometric structure.

Findings

Eigenvalues of Jacobi operators are characterized under specific conditions.

Generalization of curvature results to manifolds with multiple characteristic vector fields.

Relationships between null and spacelike Jacobi operators are established.

Abstract

We expound some results about the relationships between the Jacobi operators with respect to null vectors on a Lorentzian $\mathcal{S}$-manifold $M$ and the Jacobi operators with respect to particular spacelike unit vectors on $M$. We study the number of the eigenvalues of such operators in a $\phi$-null Osserman Lorentzian $\mathcal{S}$-manifold, under suitable assumptions on the dimension of the manifold. Then, we generalize a curvature characterization, previously obtained by the first author for Lorentzian $\phi$-null Osserman $\mathcal{S}$-manifolds with exactly two characteristic vector fields, to the case of those with an arbitrary number of characteristic vector fields.