# Deformations of the Veronese embedding and Finsler 2-spheres of constant curvature

## TL;DR

This paper establishes a duality between certain Finsler structures on the 2-sphere with all geodesics closed and Weyl connections on spindle orbifolds, using deformations of the Veronese embedding to construct examples.

## Contribution

It introduces a duality between Finsler 2-spheres with constant curvature and Weyl connections on orbifolds, and constructs explicit examples via holomorphic deformations of the Veronese embedding.

## Key findings

- Established a one-to-one correspondence between Finsler structures and Weyl connections.
- Constructed examples of Finsler 2-spheres with all geodesics closed.
- Demonstrated the use of holomorphic deformations of the Veronese embedding for constructing such structures.

## Abstract

We establish a one-to-one correspondence between Finsler structures on the $2$-sphere with constant curvature $1$ and all geodesics closed on the one hand, and Weyl connections on certain spindle orbifolds whose symmetric Ricci curvature is positive definite and all of whose geodesics are closed on the other hand. As an application of our duality result, we show that suitable holomorphic deformations of the Veronese embedding $\mathbb{CP}(a_1,a_2)\to \mathbb{CP}(a_1,(a_1+a_2)/2,a_2)$ of weighted projective spaces provide examples of Finsler $2$-spheres of constant curvature whose geodesics are all closed.