On the flatness of Riemannian cylinders without conjugate points

Details On the flatness of Riemannian cylinders without conjugate points On the flatness of Riemannian cylinders without conjugate points

2010-12-03

arxiv.org/abs/1012.0659v2

Mathematics

Differential Geometry

27 pages; v2: improved presentation in part (d) of the proof of Proposition 3.4, some stylistic changes; to appear in Communic

Scientific paper

What are appropriate geometric conditions ensuring that a complete Riemannian 2-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 measured by the length of horocycles. This is used to extend results by K. Burns and G. Knieper [9], and by H. Koehler [18], where the opening of the ends is measured in terms of shortest noncontractible loops.

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