Closed generalized Einstein manifolds with radially flat Ricci curvature

Abstract

In this paper, we show that a closed $n$-dimensional generalized $(\hat{I}»,n+m)$-Einstein manifold of constant scalar curvature with weakly radially zero Ricci curvature is isometric to either a sphere ${\mathbb{S}}^n$, or a product ${\mathbb{S}}^1\times \hat{I}{\pounds }^{n-1}$ 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 ${\mathbb{S}}^n$, or a product ${\mathbb{S}}^1\times {\mathbb{S}}^{n-1}$ of a circle with an $(n-1)$-sphere.

21 pages