The Berger geodesic conjecture asserts that on a simply connected Riemannian manifold all of whose geodesics are closed, every unit-speed geodesic has the same least period (Besse 1978, Radeschi and Wilking 2017). Equivalently, the effective circle action induced by the geodesic flow using its least common period on the unit tangent bundle has no nontrivial isotropy group.
Dittmer (2026) proves the conjecture for three-dimensional manifolds. His argument combines the topology of the geodesic
flow with the symplectic geometry of its quotient.
The quotient is a smooth symplectic
manifold symplectomorphic to 
with a split symplectic form. This establishes
the three-dimensional case, rather than the conjecture in every dimension.
