Quasi-Einstein Metrics and a curvature identity assoicated to the Ricci flow
Atreyee Bhattacharya, Sayoojya Prakash
TL;DR
This work studies rigidity of closed quasi-Einstein manifolds under a curvature-identity framework derived from the Ricci flow. By analyzing the evolution equation $\frac{\partial \mathcal{R}}{\partial t} = \Delta \mathcal{R} + Q(\mathcal{R})$ and the fixed-point relations $\Delta \mathcal{R} + Q(\mathcal{R}) = 2\lambda \mathcal{R}$ or $= \mu \mathcal{R}$, it derives conditions under which a quasi-Einstein metric must be Einstein (rigid). The main results show rigidity under Curvature Identities I and II, including cases with harmonic Weyl tensor or Bach-flatness, and establish an integral criterion $\int ((h-2\lambda)s) \, dV_g = 0$ that characterizes rigidity when a curvature identity with a function $h$ holds; the paper also identifies broad classes of admissible $h$ (polynomials, exponentials, logs, trigonometric forms) that guarantee rigidity and relates quasi-Einstein metrics to warped-product Einstein metrics via a warped-product correspondence. These findings provide a curvature-flow-based route to rigidity beyond Einstein metrics and connect to warped-product constructions and curvature operator identities.
Abstract
Quasi-Einstein manifolds are well-studied generalizations of Einstein manifolds. This includes gradient Ricci solitons and has a natural correspondence with the warped product Einstein manifolds. A quasi-Einstein metric is said to be rigid when it reduces to an Einstein metric. On a different note, Einstein metrics can be viewed as fixed points of the Ricci flow up to homothety. While gradient Ricci solitons are generalized fixed points of the Ricci flow, not much is known, in general, about the evolution of quasi-Einstein metrics under the Ricci flow. In this paper, we employ an identity associated to the evolution of curvature along the Ricci flow, to conclude the rigidity of certain closed quasi-Einstein manifolds.