Is  there  a  **complete** Riemannian metric  on the cylinder such  that 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:

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