Rigidity of (m,ρ)-quasi Einstein manifolds

Abstract

This paper deals with the study on (m,ρ)‐quasi Einstein manifolds. First, we give some characterizations of an (m,ρ)‐quasi Einstein manifold admitting closed conformal or parallel vector field. Then, we obtain some rigidity conditions for this class of manifolds. We prove that an (m,ρ)‐quasi Einstein manifold with a closed conformal vector field has a warped product structure of the form I×eq/2M∗, where I is a real interval, (M∗,g∗) is an (n−1)‐dimensional Riemannian manifold and q is a smooth function on I. Finally, a non‐trivial example of an (m,ρ)‐quasi Einstein manifold verifying our results in terms of the potential function is presented.

The Existence of Gradient Yamabe Solitons on Spacetimes

“…After the Ricci flow theory had been introduced and some substantial amount of progress had been made to classify the Riemannian manifolds and generalize the Ricci solitons, almost gradient Ricci solitons, quasi-Einstein manifolds and generalized quasi-Einstein manifolds were introduced and studied extensively. For further details, we refer to [9][10][11][12][13] and many others. Analogously, some generalizations of the self-similar solutions of Yamabe flow are also defined in the related literature.…”

Ricci almost solitons on semi‐Riemannian warped products

“…Motivated by these relations with geometric flows, Ricci solitons and Einstein manifolds we are interested in investigating the geometry of such manifolds and we consider problems such as when a Ricci almost soliton becomes a Ricci soliton or even an Einstein manifold. In [1], [3], [4], [6], [21], [24], [37] and [40] the authors proved that under certain geometric constraints a Ricci almost soliton becomes a Ricci soliton or an Einstein manifold carrying a conformal field.…”

“…More examples can be found in [16], [18], [38] and in references therein. It is important to note that warped products naturally arose in the studies of related structures on semi-Riemannian manifolds, as one can see for instance in [1], [24] and [40].…”

η-Ricci Solitons on Kenmotsu 3-Manifolds

“…In [15], De and De investigated (m, ρ)-quasi-Einstein solitons in the frame-work of paracontact manifolds. Demirbeg and Güler [20] studied rigidity of (m, ρ)-quasi-Einstein manifolds. As the study of almost co-Kähler manifolds with (m, ρ)-quasi-Einstein solitons is still pending, we want to fill this gap in this paper.…”