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.
Categories: math.DG