Rigidity for odd-dimensional souls

Tapp, Kristopher

doi:10.2140/gt.2012.16.957

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Rigidity for odd-dimensional souls

Kristopher Tapp

Geometry & Topology 16 (2012) 957–962

DOI: 10.2140/gt.2012.16.957

Abstract

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

Keywords

Soul Theorem, nonnegative curvature, flat

Mathematical Subject Classification 2010

Primary: 53C20

References

Publication

Received: 25 October 2011

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/

Volume 16, issue 2 (2012)