Complete manifolds with nonnegative curvature operator

Ni, Lei; Wu, Baoqiang

doi:10.1090/S0002-9939-06-08872-1

Complete manifolds with nonnegative curvature operator
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by Lei Ni and Baoqiang Wu PDF

Proc. Amer. Math. Soc. 135 (2007), 3021-3028

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 $R\ge \delta \frac {\operatorname {tr}(R)}{2n(n-1)} \mathrm {I}>0$, for some $\delta >0$, must be compact. This provides an intrinsic analogue of a result of Hamilton on convex hypersurfaces.

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