Sphere Theorems for <i>σ_k</i>-Einstein Manifolds

Abstract

A problem that geometers have always been concerned with is when a closed manifold is isometric to a round sphere. A classical result shows that a closed locally conformally flat Einstein manifold is always isometric to a quotient of a round sphere. In this note, we provide the definitions of σk-curvatures and σk-Einstein manifolds, and we show that a closed σk-Einstein manifold under certain pinching conditions of a Weyl curvature and Einstein curvature is isometric to a quotient of a round sphere.

Keywords

• scalar curvature
• Einstein manifold
• sphere theorem
• <i>σ_k</i>-Einstein manifolds
• <i>σ_k</i>-curvatures