Compact gradient Einstein-type manifolds with boundary satisfying the weakly Einstein condition

In this article, we study a family of compact gradient Einstein-type manifolds with boundary, introduced by Freitas and Gomes in [7] , satisfying the weakly Einstein condition. We prove the rigidity results for this case when the scalar curvature is constant. As a special case of gradient Einstein-type manifold, we also give the rigidity result of n ≥ 4 dimensional compact weakly Einstein ( m , ρ ) -quasi-Einstein manifold with boundary.