Generalized curvature tensor and the hypersurfaces of the Hermitian manifold for the class of Kenmotsu type

TL;DR

This paper explores the curvature properties of Kenmotsu type manifolds, establishing conditions for {\eta}-Einstein and constant G{\Phi}SH-curvature, and relates these manifolds as hypersurfaces within Hermitian manifolds.

Contribution

It introduces new curvature tensors and conditions for Kenmotsu type manifolds, linking their geometric properties to hypersurfaces in Hermitian manifolds.

Findings

Kenmotsu type manifolds are {\eta}-Einstein when the generalized curvature tensor is flat.

Necessary and sufficient conditions for constant G{\Phi}SH-curvature in Kenmotsu type manifolds.

Relationship between curvature tensors of Kenmotsu type hypersurfaces and ambient Hermitian manifolds.

Abstract

This paper determined the components of the generalized curvature tensor for the class of Kenmotsu type and established the mentioned class is {\eta}-Einstein manifold when the generalized curvature tensor is flat; the converse holds true under suitable conditions. It also introduced the notion of generalized {\Phi}-holomorphic sectional (G{\Phi}SH-) curvature tensor and thus found the necessary and sufficient conditions for the class of Kenmotsu type to be of constant G{\Phi}SH-curvature. In addition, the notion of {\Phi}-generalized semi-symmetric was introduced and its relationship with the class of Kenmotsu type and {\eta}-Einstein manifold established. Furthermore, this paper generalized the notion of the manifold of constant curvature and deduced its relationship with the aforementioned ideas. It finally showed that the class of Kenmotsu type exists as a hypersurface of the Hermitian manifold and derived a relation between the components of the Riemannian curvature tensors of the almost Hermitian manifold and its hypersurfaces.