A Characterization of Quasi-Einstein Metrics
Antonio Airton Freitas Filho
TL;DR
The article introduces modified Ricci solitons and almost solitons as Einstein-type metrics tied to $n$-quasi-Einstein structures, and connects them to warped-product Ricci solitons and the Ricci-harmonic flow. It establishes a Lichnerowicz–Obata–type rigidity result for compact manifolds with constant scalar curvature, via an eigenvalue bound and a gradient-soliton reduction. It develops a DeTurck-based short-time existence theory for the modified Ricci-harmonic flow and defines modified Ricci-harmonic solitons as special flow solutions, with equivalence between the soliton notion and a special flow. Finally, it provides explicit constructions of a modified Ricci-harmonic soliton and its associated evolving metrics, including a nongradient example on a warped product.
Abstract
We study the modified Ricci solitons as a new class of Einstein type metrics that contains both Ricci solitons and $n$-quasi-Einstein metrics. This class is closely related to the construction of the Ricci solitons that are realised as warped products. A modified Ricci soliton appears as part of a special solution of the modified Ricci-harmonic flow, which result a new characterization of $n$-quasi-Einstein metrics. We also study the modified Ricci almost solitons. In the spirit of the Lichnerowicz and Obata first eigenvalue theorems, we prove that in the class of compact Riemannian manifolds with constant scalar curvature the standard sphere with a structure of gradient modified Ricci almost soliton is rigid under some specific geometric conditions. Moreover, we display an example of modified Ricci-harmonic soliton.