Rigidity for odd-dimensional souls

摘要

We prove a new rigidity result for an open manifold M with nonnegative sectional curvature whose soul Σ is contained in 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 ∈ M | dist(p, Σ) = r} contains an immersed flat cylinder, and thus could not have positive curvature.