Rigidity of quasi-Einstein metrics

Abstract

We call a metric quasi-Einstein if the m-Bakry-Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of Einstein metrics, which contains gradient Ricci solitons and is also closely related to the construction of the warped product Einstein metrics. We study properties of quasi-Einstein metrics and prove several rigidity results. We also give a splitting theorem for some Kähler quasi-Einstein metrics.

Rigid properties of quasi-Einstein metrics

“…This fact is very useful in studying the geometry of complete gradient shrinking Ricci solitons [10,11,12]. The authors of [5] obtained some estimates of the scalar curvature for an m−dimensional quasi-Einstein metric on closed manifolds. In [10,11], the authors considered the estimate of scalar curvature for a gradient Ricci soliton on noncompact manifolds.…”

“…Similar to the Einstein metric, we call a metric g m−dimensional quasi-Einstein with potential function f if for some constant λ, This definition can be found in [5,6]. A quasi-Einstein metric becomes Einstein when the potential function is constant.…”

On gradient solitons of the Ricci–Harmonic flow

“…The following definition is taken from [11]. Recently, there have been some papers generalizing gradient Ricci solitons, for instance [2,12] on almost Ricci solitons, [3] on quasi-Einstein manifolds and [7] on generalized quasi-Einstein manifolds. The gradient RH soliton is a generalization of gradient Ricci solitons in a different way.…”

ON THE SOLUTIONS OF THE (λ, n + m)-EINSTEIN EQUATION

“…Recently, there has been an increasing interest in the study of the extension of the Ricci tensor called the m-BakryEmery Ricci tensor (c.f. [3], [5]). It is given by It is well known that when m is a positive integer, the m-quasi Einstein metrics correspond to certain warped product Einstein metrics (c.f.…”

“…It is given by It is well known that when m is a positive integer, the m-quasi Einstein metrics correspond to certain warped product Einstein metrics (c.f. [3], [5]). It is clear that if we take m to infinity, we obtain the gradient Ricci soliton equation…”