Abstract
AbstractWe 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.
Highlights
Using the recent result [11] by Bryant et al about such Finsler metrics, we show that the space of oriented geodesics is a spindle-orbifold O – or equivalently, a weighted projective line – which comes equipped with a positive Besse– Weyl structure
The symmetric part of the Ricci curvature of ∇ is positive definite. Having such a positive Besse–Weyl structure on a spindle orbifold, we show that the dual path geometry yields a Finsler metric on S2 with constant positive curvature whose geodesics are all closed
In [31], it is shown that Weyl connections with prescribed geodesics on an oriented surface M are in one-to-one correspondence with certain holomorphic curves into the ‘twistor space’ over M
Summary
Background Riemannian metrics of constant curvature on closed surfaces are fully understood, but a complete picture in the case of Finsler metrics is still lacking. Akbar-Zadeh [2] proved a first key result by showing that on a closed surface, a Finsler metric of constant negative curvature must be Riemannian, and locally Minkowskian in the case where the curvature vanishes identically (see [16]). In the case of constant positive curvature, a Finsler metric must still be Riemannian, provided it is reversible [10], but the situation turns out to be much more flexible in the nonreversible case. Bryant [7] gave another construction of Downloaded from https://www.cambridge.org/core.
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