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

我们一方面在具有恒定曲率 $1$ 且所有测地线闭合的 $2$ 球体上的 Finsler 结构之间建立一对一的对应关系，另一方面，在 具有对称 Ricci 曲率的某些主轴轨道上的 Weyl 连接是正定的，其测地线都是封闭的。作为对偶性结果的应用，我们证明了 Veronese 嵌入 $\mathbb {CP}(a_1,a_2)\rightarrow \mathbb {CP}(a_1,(a_1+a_2)/2,a_2)$ 的适当全纯变形加权射影空间提供了 Finsler $2$ 等曲率球体的例子，其测地线都是闭合的。