On para-Kenmotsu manifolds

TL;DR

This paper investigates the geometric properties of para-Kenmotsu manifolds, characterizing their curvature and tensor equations, and specifically studies 3-dimensional cases with special curvature conditions.

Contribution

It provides new characterizations of para-Kenmotsu manifolds through tensor equations and explores their curvature properties, especially in three dimensions.

Findings

Conformally flat para-Kenmotsu manifolds have constant negative curvature -1.

Space with constant $$-para-holomorphic sectional curvature H is of constant curvature with H=-1.

3-dimensional para-Kenmotsu manifolds with $$-parallel Ricci tensor have constant scalar curvature.

Abstract

In this paper we study para-Kenmotsu manifolds. We characterize this manifolds by tensor equations and study their properties. We are devoted to a study of $\eta-$Einstein manifolds. We show that a conformally flat para-Kenmotsu manifold is a space of constant negative curvature $-1$ and we prove that if a para-Kenmotsu manifold is a space of constant $\varphi-$para-holomorphic sectional curvature $H$, then it is a space of constant curvature and $H=-1$. Finally the object of the present paper is to study a 3-dimensional para-Kenmotsu manifold, satisfying certain curvature conditions. Among other, it is proved that any 3-dimensional para-Kenmotsu manifold with $\eta-$parallel Ricci tensor is of constant scalar curvature and any 3-dimensional para-Kenmotsu manifold satisfying cyclic Ricci tensor is a manifold of constant negative curvature $-1$.