Strong deformation retraction of the space of Zoll Finsler projective planes

TL;DR

This paper proves that the space of Zoll Finsler metrics on the projective plane can be smoothly deformed to the round metric, revealing its connectedness and describing geodesic behavior during the deformation.

Contribution

It introduces a strong deformation retraction of Zoll Finsler metrics to the round metric via geodesic flow deformation, a novel geometric construction.

Findings

The space of Zoll Finsler metrics is connected.

A smooth deformation of geodesic flows is constructed.

The deformation is induced by curvature flow and circle actions.

Abstract

We show that the infinite-dimensional space of Zoll Finsler metrics on the projective plane strongly deformation retracts to the canonical round metric. In particular, this space of Zoll Finsler metrics is connected. Moreover, the strong deformation retraction arises from a deformation of the geodesic flow of every 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 Zoll Finsler metrics along the retraction.