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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

Complete manifolds with nonnegative curvature operator


Authors: Lei Ni and Baoqiang Wu
Journal: Proc. Amer. Math. Soc. 135 (2007), 3021-3028
MSC (2000): Primary 58J35
Published electronically: November 29, 2006
MathSciNet review: 2511306
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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)}\operatorname{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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Additional Information

Lei Ni
Affiliation: Department of Mathematics, University of California at San Diego, La Jolla, California 92093
Email: lni@math.ucsd.edu

Baoqiang Wu
Affiliation: Department of Mathematics, Xuzhou Normal University, Xuzhou, Jiangsu, People’s Republic of China
Email: wubaoqiang@xznu.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08872-1
PII: S 0002-9939(06)08872-1
Received by editor(s): June 22, 2006
Received by editor(s) in revised form: August 16, 2006
Published electronically: November 29, 2006
Additional Notes: The first author was supported in part by NSF Grants and an Alfred P. Sloan Fellowship
Communicated by: Jon G. Wolfson
Article copyright: © Copyright 2006 by the authors