On the flatness of Riemannian cylinders without conjugate points

TL;DR

This paper investigates conditions under which a complete Riemannian 2-cylinder without conjugate points must be flat, demonstrating that sublinear growth of the ends measured by horocycles suffices, extending previous results.

Contribution

It introduces a new geometric condition based on horocycle length to ensure flatness, extending prior work that used shortest noncontractible loops.

Findings

Sublinear growth of ends measured by horocycles guarantees flatness.

Extends previous results by Burns, Knieper, and Koehler.

Provides new geometric criteria for flatness of Riemannian cylinders.

Abstract

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.