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](https://mathoverflow.net/questions/277481/a-curvature-description-for-center-condition-for-quadratic-vector-field/278766#278766)



A foliation of the cylinder by closed geodesics of the same length when the metric is complete but non flat
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.


