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

A condition of Osserman type, called $ϕ$-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 $ϕ$-sectional curvature is $ϕ$-null Osserman, extending a result stated for Lorentz Sasaki space forms. Then we state some characterizations for a particular class of $ϕ$-null Osserman $\cal{S}$-manifolds. Finally, some examples are examined.

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