Almost Kenmotsu Manifolds Admitting Certain Critical Metric

TL;DR

This paper introduces a new critical metric equation on almost contact metric manifolds, specifically studying its implications on almost Kenmotsu manifolds, leading to classification results and an example.

Contribution

It defines the $ ext{ extsterling}$-Miao-Tam critical equation on almost contact metric manifolds and characterizes solutions on almost Kenmotsu manifolds, showing they are $ ext{ extsterling}$-Ricci flat and locally product spaces.

Findings

Manifolds satisfying the $ ext{ extsterling}$-Miao-Tam critical equation are $ ext{ extsterling}$-Ricci flat.

Such manifolds are locally isometric to a product of a constant curvature space and a flat space.

An explicit example supports the theoretical results.

Abstract

In the present paper, we introduce the notion of $\ast$-Miao-Tam critical equation on almost contact metric manifolds and studied on a class of almost Kenmotsu manifold. It is shown that if the metric of a $(2n + 1)$-dimensional $(k,\mu)'$-almost Kenmotsu manifold $(M,g)$ satisfies the $\ast$-Miao-Tam critical equation, then the manifold $(M,g)$ is $\ast$-Ricci flat and locally isometric to the Riemannian product of a $(n + 1)$-dimensional manifold of constant sectional curvature $-4$ and a flat $n$-dimensional manifold. Finally, an illustrative example is presented to support the main theorem.