Journal of Symplectic Geometry

Volume 17 (2019)

Number 2

Strong deformation retraction of the space of Zoll Finsler projective planes

Pages: 443 – 476



Stéphane Sabourau (Laboratoire d’Analyse et Mathématiques Appliquées, Université Paris-Est, Créteil, France)


We show that the infinite-dimensional space of reversible Zoll Finsler metrics on the projective plane strongly deformation retracts to the canonical round metric. In particular, this space of reversible Zoll Finsler metrics is connected. Moreover, the strong deformation retraction arises from a deformation of the geodesic flow of every reversible Zoll Finsler projective plane to the geodesic flow of the round metric through a family of smooth free circle actions induced by the curvature flow of the canonical round projective plane. This construction provides a description of the geodesics of the reversible Zoll Finsler metrics along the retraction.

Partially supported by the ANR grants Finsler and Min-Max.

Received 28 March 2016

Accepted 11 July 2018

Published 26 July 2019