Closed generalized Einstein manifolds with radially flat Ricci curvature

Abstract: In this paper, we show that a closed $n$-dimensional generalized $(\lambda, n+m)$-Einstein manifold of constant scalar curvature with weakly radially zero Ricci curvature is isometric to either a sphere ${\Bbb S}^n$, or a product ${\Bbb S}^{1} \times \Sigma^{n-1}$ of a circle with an $(n-1)$-dimensional Einstein manifold of positive Ricci curvature, up to finite cover and rescaling. Furthermore, if we assume $(M, g)$ has positive isotropic curvature, $M$ must be isometric to either a sphere ${\Bbb S}^n$, or a product ${\Bbb S}^{1} \times {\Bbb S}^{n-1}$ of a circle with an $(n-1)$-sphere.