Closed generalized Einstein manifolds with radially flat Ricci curvature

TL;DR

This paper classifies closed generalized Einstein manifolds with specific curvature conditions, showing they are isometric to spheres or certain product manifolds, especially under positive isotropic curvature assumptions.

Contribution

It provides a classification of closed generalized Einstein manifolds with radially flat Ricci curvature, identifying them as spheres or products with specific geometric properties.

Findings

Manifolds are isometric to spheres or products with positive Ricci curvature.

Under positive isotropic curvature, manifolds are spheres or circle-sphere products.

Results hold up to finite cover and rescaling.

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.