Lightlike hypersurfaces in indefinite $\mathcal{S}$-manifolds

TL;DR

This paper investigates the geometric properties of lightlike hypersurfaces within indefinite $ ext{S}$-manifolds, focusing on their structure, decompositions, and conditions for special types like totally umbilical hypersurfaces.

Contribution

It introduces new decompositions of tangent bundles and screen distributions in lightlike hypersurfaces, and explores the existence of indefinite $ ext{S}$-structures and totally umbilical hypersurfaces.

Findings

Decomposition of tangent bundle and screen distribution invariant under structure

Existence of indefinite $ ext{S}$-structure on hypersurface leaves

Characterization of totally umbilical lightlike hypersurfaces

Abstract

In a metric $g.f.f$-manifold we study lightlike hypersurfaces $M$ tangent to the characteristic vector fields, and owing to the presence of the $f$-structure, we determine some decompositions of $TM$ and of a chosen screen distribution obtaining two distributions invariant with respect to the structure. We discuss the existence of a $g.f.f$-structure on a lightlike hypersurface and, under suitable hypotheses, we obtain an indefinite $\mathcal{S}$-structure on the leaves of an integrable distribution. The existence of totally umbilical lightlike hypersurfaces of an indefinite $\mathcal{S}$-space form is also discussed. Finally, we explicitely describe a lightlike hypersurface of an indefinite $\mathcal{S}$-manifold.