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

Rigidity for Odd-Dimensional Souls