Rigidity for Odd-Dimensional Souls

Details Rigidity for Odd-Dimensional Souls Rigidity for Odd-Dimensional Souls

2011-09-23

arxiv.org/abs/1109.5150v1

Mathematics

Differential Geometry

Scientific paper

We prove a new rigidity result for an open manifold M with nonnegative sectional curvature whose soul S is odd-dimensional. Specifically, there exists a geodesic in S and a parallel vertical plane field along it with constant vertical curvature and vanishing normal curvature. Under the added assumption that the Sharafutdinov fibers are rotationally symmetric, this implies that for small r, the distance sphere of radius r about S contains an immersed flat cylinder, and thus could not have positive curvature.

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