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

The aim of this note is to give an explicit description of quasi-Einstein metrics on ${\mathbb{H}}^n\times \mathbb{R}\mathrm{.}$ 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.