On a generalization of the Ricci solitons or m-quasi-Einstein manifolds.

What is it about?

We study generalized quasi-Einstein manifolds, or briefly, GQE manifolds. Here, we present relations between the Bach, Cotton and D tensors on GQE manifolds. Next, a 3-tensor E which measures the deviation of m-quasi-Einstein manifolds from GQE manifolds is introduced. Among others in dimension 3, it is shown that Bach-flatness implies locally conformally flatness. Furthermore, it is proved that, around a regular point of the fourth-order divergence free Weyl tensor, a GQE manifold is a locally warped product manifold with (n − 1)-dimensional Einstein fibers in suitable cases.

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Why is it important?

Ricci solitons are self-similar solutions of the Ricci flow equation.