Primitive closed geodesic ($\gamma \ne {\eta }^m(m\ge 2)$)

A primitive closed geodesic is a closed geodesic with its smallest positive period, rather than a repeated traversal of a shorter periodic geodesic. A degree-$m$ lift of a primitive base geodesic is still primitive upstairs when its monodromy permutation has a cycle of exact length $m$: no smaller number of traversals returns to the starting sheet.