A closed geodesic is a nonconstant periodic geodesic on a Riemannian manifold. It is prime when parametrized with its least positive period, rather than as an iterate of a shorter closed geodesic. Geometric distinctness can be taken to mean distinct images in the manifold. Changing the initial point, reversing orientation, or traversing the same curve repeatedly then gives no new image. Great circles on a round sphere are examples.
The Berger geodesic conjecture predicts a common least period for all unit-speed geodesics on a simply connected Riemannian manifold when all geodesics are closed. Dittmer (2026) proves this assertion in dimension three.
