Rigidity for odd-dimensional souls

We prove a new rigidity result for an open manifold M with nonnegative sectional curvature whose soul Sigma subset of M is odd-dimensional. Specifically, there exists a geodesic in Sigma 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(Sigma) = {p is an element of M vertical bar dist(p, Sigma) = r g contains an immersed flat cylinder, and thus could not have positive curvature.