The Critical Point Equation on Kenmotsu and almost Kenmotsu manifolds

TL;DR

This paper investigates the critical point equation on Kenmotsu and almost Kenmotsu manifolds, proving Einstein and hyperbolic space properties under certain conditions and providing explicit potential functions and examples.

Contribution

It establishes that complete Kenmotsu manifolds satisfying the CPE are Einstein and hyperbolic, and explicitly determines potential functions for Kenmotsu manifolds.

Findings

Complete Kenmotsu manifolds satisfying CPE are Einstein.

Such manifolds are locally isometric to hyperbolic space H2n+1.

Explicit potential functions are determined for Kenmotsu manifolds.

Abstract

In this paper, we have studied the critical point equation (shortly, CPE) within the frame-work of Kenmotsu and almost Kenmotsu manifold satisfying certain nullity conditions. First, we prove that a complete Kenmotsu metric satisfies the CPE is Einstein and locally isometric to the hyperbolic space H2n+1. In case of Kenmotsu manifolds, it is possible to determine the potential function explicitly (locally). We also provide some examples of Kenmotsu and almost Kenmotsu manifolds that satisfies the CPE.