On quasi-Einstein manifolds with constant scalar curvature
Johnatan Costa, Ernani Ribeiro, Márcio Santos
TL;DR
This work investigates $m$-quasi-Einstein manifolds with constant scalar curvature, introducing and exploiting the $T$-tensor to relate Cotton and Weyl curvature. It proves that $T$-flat noncompact examples with constant $R$ must fall into rigid warped-product models or Einstein-type geometries, and it provides a new explicit noncompact example with $ ext{\lambda}<0$ and $\mu<0$. In dimension three, the authors achieve a complete classification of simply connected (possibly with boundary) $m$-quasi-Einstein manifolds with constant $R$, showing they are isometric to standard models such as hemispheres, hyperbolic spaces, cylinders, or their warped-product variants. The results extend the understanding of rigidity phenomena and the geometry of quasi-Einstein spaces, highlighting the pivotal role of the tensorial framework involving $T$, Cotton, Weyl, and $P$-structures. The findings have implications for warped-product Einstein bases, smooth metric measure spaces, and related geometric analysis problems.
Abstract
In this article, we study quasi-Einstein manifolds with constant scalar curvature. We provide a classification of compact and noncompact (possibly with boundary) $T$-flat quasi-Einstein manifolds with constant scalar curvature, where the $T$-tensor is directly related to the Cotton and Weyl tensors. Moreover, we construct new explicit examples of noncompact quasi-Einstein manifolds. In addition, we prove a complete classification of compact and noncompact (possibly with boundary) $3$-dimensional $m$-quasi-Einstein manifolds with constant scalar curvature.