Strong deformation retraction of the space of Zoll Finsler projective planes (Q2318838)

A Zoll metric on a closed manifold is a Riemannian or Finsler metric all of whose geodesics are simple closed curves of the same length. Zoll manifolds have finite fundamental groups, thus, in the two-dimensional case they are diffeomorphic to either the sphere or the projective plane. The canonical round metric on the two-sphere or the projective plane is a Zoll Riemannian metric. However, there exist Zoll Riemannian two-spheres which are not round. In fact, Zoll Riemannian metrics on the two-sphere (modulo isometries and rescaling) form an infinite-dimensional space. In contrary, a Riemannian metric on the projective plane is a Zoll metric if and only if it has constant curvature. However, this does not hold in the Finsler case, indeed, Zoll Finsler metrics on the projective plane (modulo isometries and rescaling) form an infinite-dimensional space. The goal of the present paper is to study the space of Zoll Finsler metrics on the projective plane and the dynamics of their geodesic flow. The main result is the following theorem. Theorem 1.1. The space of Zoll Finsler metrics on the projective plane whose geodesic length is equal to \(\pi\) strongly deformation retracts to the canonical round metric on the projective plane. In particular, this result provides the first example of a closed manifold whose space of all Zoll Finsler metrics is non-empty and connected. The proof is constructive. The main part of the argument deals with closed geodesics of Zoll Finsler two-spheres and the curvature flow. Here one needs to assume that the geodesics of the Zoll Finsler sphere divide the sphere into two domains of the same area, which is the case on the orientable double cover of a Zoll Finsler projective plane. The final step of the proof is a construction of a natural deformation of Zoll Finsler metrics on \(S^2\) to the canonical round metric by applying the Crofton formula to the orbits of the converging family of the circle actions given by the curvature flow on the canonical round two-sphere.