On Einstein-type manifold with cyclic parallel Ricci tensor

TL;DR

This paper derives an integral formula for compact Einstein-type manifolds with constant scalar curvature and classifies three-dimensional cases with cyclic parallel Ricci tensor, extending previous rigidity results.

Contribution

It introduces a new integral formula and provides a classification of three-dimensional Einstein-type manifolds with cyclic parallel Ricci tensor, unifying prior results.

Findings

Derived an integral formula involving tensor D_{ijk}

Classified 3D Einstein-type manifolds with cyclic parallel Ricci tensor

Extended and unified previous rigidity results

Abstract

In this article, we derive an integral formula involving the tensor $D_{ijk}$ for compact Einstein-type manifolds with constant scalar curvature. As an application, we classify three-dimensional compact Einstein-type manifolds satisfying the cyclic parallel Ricci tensor condition, obtaining rigidity results that extend and unify previous work in the literature.