#### Volume 16, issue 2 (2012)

 Recent Issues
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Rigidity for odd-dimensional souls

### Kristopher Tapp

Geometry & Topology 16 (2012) 957–962
##### 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.

##### Keywords
Soul Theorem, nonnegative curvature, flat
Primary: 53C20
##### Publication
Accepted: 10 March 2012
Published: 22 May 2012
Proposed: Dmitri Burago
Seconded: Yasha Eliashberg, Tobias H Colding
##### Authors
 Kristopher Tapp Department of Mathematics Saint Joseph’s University 5600 City Avenue Philadelphia PA 19131 USA http://www.sju.edu/~ktapp/