On the flatness of Riemannian cylinders withoutconjugate points

摘要

What are appropriate geometric conditions ensuring that a com-plete Riemannian two-cylinder without conjugate points is flat? Examples with nonpositive curvature show that one has to assume that the ends of the cylinder open sublinearly. We show that sublinear growth of the ends is indeed sufficient if it is mea-sured by the length of horocycles. This is used to extend results by Burns and Knieper [9], and by Koehler [18], where the opening of the ends is measured in terms of shortest noncontractible loops.