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 2 -nonnegative curvature operator. Moreover, by the new invariant cone constructions of Böhm and Wilking, we show that any complete Riemannian manifold (with dimension ≥ 3 \ge 3 ) whose curvature operator is bounded and satisfies the pinching condition R ≥ δ tr ⁡ ( R ) 2 n ( n − 1 ) I > 0 R\ge \delta \frac {\operatorname {tr}(R)}{2n(n-1)} \mathrm {I}>0 , for some δ > 0 \delta >0 , must be compact. This provides an intrinsic analogue of a result of Hamilton on convex hypersurfaces.