Complete manifolds with nonnegative curvature operator

Abstract

In this short note, as a simple application of the strong result proved recently by Böhm and Wilking, we give a classification on closed manifolds with 2-nonnegative curvature operator. Moreover, by the new invariant cone constructions of Böhm and Wilking, we show that any complete Riemannian manifold (with dimension $\ge 3$) whose curvature operator is bounded and satisfies the pinching condition $R\ge δR_{I}>0$, for some $δ>0$, must be compact. This provides an intrinsic analogue of a result of Hamilton on convex hypersurfaces.