Communications in Analysis and Geometry

Volume 19 (2011)

Number 4

On the flatness of Riemannian cylinders without conjugate points

Pages: 773 – 805



Victor Bangert (Mathematisches Institut, Universität Freiburg, Germany)

Patrick Emmerich (Mathematisches Institut, Universität Freiburg, Germany)


What are appropriate geometric conditions ensuring that acomplete Riemannian two-cylinder without conjugate points isflat? Examples with nonpositive curvature show that onehas to assume that the ends of the cylinder opensublinearly. We show that sublinear growth of the ends isindeed sufficient if it is measured by the length ofhorocycles. This is used to extend results by Burns andKnieper, and by Koehler, wherethe opening of the ends is measured in terms of shortestnoncontractible loops.

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