An Osserman-type condition on $g.f.f$-manifolds with Lorentz metric

TL;DR

This paper introduces and studies a new Osserman-type condition called $$-null Osserman in Lorentz $f$-manifolds, extending known results and providing characterizations with explicit examples.

Contribution

It defines the $$-null Osserman condition in Lorentz $f$-manifolds, proves its relation to constant $$-sectional curvature, and offers characterizations and examples in this geometric setting.

Findings

Lorentz $$-manifolds with constant $$-sectional curvature are $$-null Osserman.

The $$-null Osserman condition is natural in Lorentz $$-manifolds, supported by explicit examples.

Characterizations of $$-null Osserman $$-manifolds are provided.

Abstract

A condition of Osserman type, called $\phi$-null Osserman condition, is introduced and studied in the context of Lorentz globally framed $f$-manifolds. An explicit example shows the naturalness of this condition in the setting of Lorentz $\mathcal{S}$-manifolds. We prove that a Lorentz $\mathcal{S}$-manifold with constant $\phi$-sectional curvature is $\phi$-null Osserman, extending a result stated for Lorentz Sasaki space forms. Then we state some characterizations for a particular class of $\phi$-null Osserman $\cal{S}$-manifolds. Finally, some examples are examined.