Abstract
We say that a nonnegatively curved manifold (M, g) has quarter-pinched flag curvature if for any two planes which intersect in a line the ratio of their sectional curvature is bounded above by 4. We show that these manifolds have nonnegative complex sectional curvature. By combining with a theorem of Brendle and Schoen it follows that any positively curved manifold with strictly quarter-pinched flag curvature must be a space form. This in turn generalizes a result of Andrews and Nguyen in dimension 4. For odd-dimensional manifolds we obtain results for the case that the flag curvature is pinched with some constant below one quarter, one of which generalizes a recent work of Petersen and Tao.
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Acknowledgments
The first author thanks Nolan Wallach for some helpful discussions. His research is partially supported by a NSF grant DMS-0805834. The paper was written when the second author visited UCSD during February–March of 2009. The second author thanks the mathematical department of UCSD for its hospitality.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Ni, L., Wilking, B. Manifolds with 1/4-Pinched Flag Curvature. Geom. Funct. Anal. 20, 571–591 (2010). https://doi.org/10.1007/s00039-010-0068-5
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DOI: https://doi.org/10.1007/s00039-010-0068-5