Rigidity for Odd-Dimensional Souls

TL;DR

This paper establishes a new rigidity property for open manifolds with nonnegative curvature and odd-dimensional souls, showing the existence of flat cylinders in small distance spheres under certain symmetry conditions.

Contribution

It introduces a novel rigidity theorem for manifolds with odd-dimensional souls and demonstrates the existence of flat cylinders in small spheres under rotational symmetry assumptions.

Findings

Existence of a geodesic and parallel vertical plane field with constant vertical curvature.

Presence of an immersed flat cylinder in small distance spheres.

Implication that such spheres cannot have positive curvature.

Abstract

We prove a new rigidity result for an open manifold M with nonnegative sectional curvature whose soul S is odd-dimensional. Specifically, there exists a geodesic in S 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 of radius r about S contains an immersed flat cylinder, and thus could not have positive curvature.