First, we show that a warped product of line with Riemannian manifold (fiber) is weakly conformally flat and quasi Einstein if only the fiber in which case Bach flat. Finally, characterize classify contact manifolds satisfying (and doubly weakly) quasi-Einstein ( $$\eta $$ -Einstein) conditions.

We numerically calculate Perelman’s entropy for a variety of canonical metrics on CP-bundles over products of Fano Kähler-Einstein manifolds. The metrics investigated are Einstein metrics, Kähler-Ricci solitons and quasi-Einstein metrics. The calculation of the entropy allows a rough picture of how the Ricci flow behaves on each of the manifolds in question.

N(k)-quasi Einstein manifolds are introduced and studied.

In this note we adapt the work of Hall to find quasi-Einstein metrics on sphere bundles over products Fano Kaehler-Einstein manifolds, as well where only one end is blown down.

The goal of this article is to study compact quasi-Einstein manifolds with boundary. We provide boundary estimates for similar previous results obtained static and V-static spaces. In addition, we show that connected satisfying a suitable pinching condition must be isometric, up scaling, the standard hemisphere S + n $\mathbb {S}_{+}^{n}$ .