Complete manifolds with nonnegative curvature operator

Details Complete manifolds with nonnegative curvature operator Complete manifolds with nonnegative curvature operator

2006-07-14

arxiv.org/abs/math/0607356v1

Mathematics

Differential Geometry

Scientific paper

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

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