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Is there a complete Riemannian metric on the cylinder such that the metric is not flat but the cylinder is foliated by closed geodesics with the same length?

A possibility non complete metric with the above properties is introduced in the "Remark" of the following answer:

A curvature description for center condition for quadratic vector field

Ali Taghavi
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The standard non-flat metric on a torus $T^2$ is foliated by closed geodesics. It can be unwrapped to give an example of what you want.

Specifically, let $\gamma_1$ and $\gamma_2$ be the standard generators for $\pi_1(T^2)$, where there is a foliation of $T^2$ by closed geodesics freely homotopic to $\gamma_1$ (but there isn't one for $\gamma_2$). The covering space $\mathbb{R}^2 / <\!\!\gamma_1\!\!> \to T^2$ is a complete cylinder foliated by closed geodesics that isn't flat.

James Dibble
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  • +1 and thank you for your very interesting answer. I am thinking to the "complete part of the metric. as you indicated in your answer. BTW you are very well come to MO! – Ali Taghavi Jun 29 '18 at 22:55
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    Implicitly, I mean that the cylinder should be endowed with the covering metric, which makes it complete. Of course, every compact Riemannian manifold is complete, and every cover of a complete manifold is complete. – James Dibble Jul 03 '18 at 02:00