Burns and Knieper have shown that a cylinder with no conjugate points, bounded cross-section, and a lower bound on curvature must be flat. This result extends to a surface which is almost a cylinder in the sense that its metric is almost-periodic. It is shown that if such a surface has no conjugate points, bounded cross-section, bounded asymptote, and a lower bound on sectional curvature, then it must be flat.
The surface is divided into strips perpendicular to the direction of almost-periodicity. Because the metric is almost-periodic, certain functions relating to the stable solution of the Jacobi equation are also almost-periodic. These functions are integrated on each strip to yield an almost-periodic sequence of integrals. Because this sequence is almost-periodic, it has a well-defined mean value.
Once this has been demonstrated, techniques of Burns and Knieper can be applied. From the bound on curvature, it follows that this mean value is finite. It can then be shown that this mean value can be made arbitrarily small. From almost-periodicity, it follows that the integrals are zero on each strip. This is enough to prove that the metric in question is in fact Euclidean.
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