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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.

For a compact hyperbolic surface, the Selberg trace formula equates a sum over Laplace-Beltrami operator eigenvalues to an area term and a sum over primitive closed geodesics and their iterates. It proves that the Laplacian spectrum and the unmarked length spectrum, with multiplicities, determine each other.