Rigidity for odd-dimensional souls

Abstract

We prove a new rigidity result for an open manifold $M$ with nonnegative sectional curvature whose soul $\Sigma \subset M$ 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 $B_r\left(\Sigma \right)=\left\{p\in M\mid dist\left(p,\Sigma \right)=r\right\}$ contains an immersed flat cylinder, and thus could not have positive curvature.