On the flatness of Riemannian cylinders without conjugate points

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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Published 3 February 2012