Sphere Theorems for σk-Einstein Manifolds - Jingyang Zhong; Xinran Mu (Axioms)
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 <semantics>σk</semantics>-curvatures and <semantics>σk</semantics>-Einstein manifolds, and we show that a closed <semantics>σk</semantics>-Einstein manifold under certain pinching conditions of a Weyl curvature and Einstein curvature is isometric to a quotient of a round sphere.