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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
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 Σ 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 Br(Σ) = {p Mdist(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/