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

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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.