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

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.