A note on the uniqueness of quasi-Einstein metrics on $\Bbb{H}^{n}\times \Bbb{R}$

TL;DR

This paper explicitly describes and proves the uniqueness of certain quasi-Einstein metrics on hyperbolic space cross real line, and explores their connection to static metrics, providing new insights into their structure.

Contribution

It provides explicit examples and a uniqueness proof for quasi-Einstein metrics on b{H}^n imes \u00bb{R}, and clarifies their relation to static metrics.

Findings

Constructed two explicit quasi-Einstein metrics.

Proved the uniqueness of these metrics.

Described the relation to static metrics.

Abstract

The aim of this note is to give an explicit description of quasi-Einstein metrics on $\Bbb{H}^{n}\times \Bbb{R}.$ We shall construct two examples of quasi-Einstein metrics on this manifold and then we shall prove the uniqueness of these examples. Finally, we shall describe the closed relation between quasi-Einstein metrics and static metrics in the quoted space.