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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 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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