Geometry & Topology 16 (2012) 957–962

DOI: 10.2140/gt.2012.16.957

Abstract

We prove a new rigidity result for an open manifold M with nonnegative sectional curvature whose soul Σ ⊂ M is odd-dimensional. Specifically, there exists a geodesic in Σ 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 B_r(Σ) = {p in M∣dist(p,Σ) = r} contains an immersed flat cylinder, and thus could not have positive curvature.

Keywords

Soul Theorem, nonnegative curvature, flat

Mathematical Subject Classification

Primary: 53C20

References

Publication

Received: 25 October 2011
Accepted: 10 March 2012
Published: 22 May 2012
Proposed: Dmitri Burago
Seconded: Yasha Eliashberg, Tobias H Colding

Authors