Manifolds with 1/4-Pinched Flag Curvature

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...

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Bibliographic Details
Published inGeometric and functional analysis Vol. 20; no. 2; pp. 571 - 591
Main Authors Ni, Lei, Wilking, Burkhard
Format Journal Article
LanguageEnglish
Published Basel SP Birkhäuser Verlag Basel 01.08.2010
Subjects
Analysis
Mathematics
Mathematics and Statistics
Ricci flow
Primary 53C20
Secondary 53C44
curvature pinching
complex sectional curvature
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Summary: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.
ISSN:1016-443X
1420-8970
DOI:10.1007/s00039-010-0068-5