A Hamiltonian version of a result of Gromoll and Grove

Abstract

The theorem that if all geodesics of a Riemannian two-sphere are closed they are also simple closed is generalized to real Hamiltonian structures on $\mathbb{R}{P}^3$. For reversible Finsler $2$-spheres all of whose geodesics are closed this implies that the lengths of all geodesics coincide.

Comments are welcome. v2: Improvement of the result by removal of an unnecessary assumption