Deformations of the Veronese Embedding and Finsler 2-Spheres of Constant Curvature

1 Introduction

1.1 Background

Riemannian metrics of constant curvature on closed surfaces are fully understood, a complete picture in the case of Finsler metrics is however 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 also [16]). In the case of constant positive curvature a Finsler metric must still be Riemannian, provided it is reversible [11], but the situation turns out to be much more flexible in the non-reversible case.

Katok [22] gave the first examples (later analysed by Ziller [41]) of non-reversible Finsler metrics of constant positive curvature, though it was only realized later that Katok’s examples actually have constant curvature. Meanwhile, Bryant [8] gave another construction of non-reversible Finsler metrics of constant positive curvature on the \(2\)-sphere \(S^2\) and in subsequent work [9] classified all Finsler metrics on \(S^2\) having constant positive curvature and that are projectively flat. Bryant also observed that every Zoll metric on \(S^2\) with positive Gauss curvature gives rise to a Finsler metric on \(S^2\) with constant positive curvature [10]. Hence, already by the work of Zoll [42] from the beginning of the 20th century, the moduli space of constant curvature Finsler metrics on \(S^2\) is known to be infinite-dimensional. Its global structure is however not well understood.

1.2 A duality result

Recently in [6], Bryant et. al. inter alia showed that a Finsler metric on \(S^2\) with constant curvature \(1\) either admits a Killing vector field, or has all of its geodesics closed. Moreover, in the first case all geodesics become closed, and even of the same length, after a suitable (invertible) Zermelo transformation. Hence, in this sense the assumption that all geodesics are closed is not a restriction. However, in the second case the geodesics can in general have different lengths, unlike the geodesics of the Finsler metrics that arise from Bryant’s construction using Zoll metrics.

In this paper we generalise Bryant’s observation about Zoll metrics to a one-to-one correspondence which covers all Finsler metrics on \(S^2\) with constant curvature \(1\) and all geodesics closed. The correspondence arises from the classical notion of duality for so-called path geometries.

An oriented path geometry on an oriented surface \(M\) prescribes an oriented path \(\gamma\subset M\) for every oriented direction in \(TM\). This notion can be made precise by considering the bundle \(\pi : \mathbb{S}(TM):=\left(TM\setminus\{0_M\}\right)/\mathbb{R}^+ \to M\) which comes equipped with a tautological co-orientable contact distribution \(C\). An oriented path geometry is a one-dimensional distribution \(P\to \mathbb{S}(TM)\) so that \(P\) together with the vertical distribution \(L=\ker \pi^{\prime}\) span \(C\).

The orientation of \(M\) equips \(P\) and \(L\) with an orientation as well and following [9], a \(3\)-manifold \(N\) equipped with a pair of oriented one-dimensional distributions \((P,L)\) spanning a contact distribution is called an oriented generalized path geometry. In this setup the surface \(M\) is replaced with the leaf space of the foliation \(\mathcal{L}\) defined by \(L\) and the leaf space of the foliation \(\mathcal{P}\) defined by \(P\) can be thought of as the space of oriented paths of the oriented generalized path geometry \((P,L)\). We may reverse the role of \(P\) and \(L\) and thus consider the dual \((-L,-P)\) of the oriented generalized path geometry \((P,L)\), where here the minus sign indicates reversing the orientation.

The unit circle bundle \(\Sigma \subset TM\) of a Finsler metric \(F\) on an oriented surface \(M\) naturally carries the structure of an oriented generalized path geometry \((P,L)\). In the case where all geodesics are closed, the dual of the path geometry arising from a Finsler metric on the \(2\)-sphere with constant positive curvature arises from a certain generalization of a Besse \(2\)-orbifold [24] with positive curvature. Here a \(2\)-orbifold is called Besse if all its geodesics are closed. Namely, using the recent result [6] by Bryant et al. about such Finsler metrics (see Theorem 3.1 below), we show that the space of oriented geodesics is a spindle-orbifold \(\mathcal{O}\) – or equivalently, a weighted projective line – which comes equipped with a positive Besse–Weyl structure. By this we mean an affine torsion-free connection \(\nabla\) on \(\mathcal{O}\) which preserves some conformal structure – a so-called Weyl connection – and which has the property that the image of every maximal geodesic of \(\nabla\) is an immersed circle. Moreover, the symmetric part of the Ricci curvature of \(\nabla\) is positive definite. Conversely, having such a positive Besse–Weyl structure on a spindle orbifold, we show that the dual path geometry yields a Finsler metric on \(S^2\) with constant positive curvature all of whose geodesics are closed. More precisely, we prove the following duality result which generalizes [10] and [11] by Bryant:

Theorem A

There is a one-to-one correspondence between Finsler structures on \(S^2\) with constant Finsler–Gauss curvature \(1\) and all geodesics closed on the one hand, and positive Besse–Weyl structures on spindle orbifolds \(S^2(a_1,a_2)\) with \(c:=\gcd(a_1,a_2)\in \{1,2\}\), \(a_1\geqslant a_2\), \(2|(a_1+a_2)\) and \(c^3|a_1a_2\) on the other hand. More precisely,

  1. such a Finsler metric with shortest closed geodesic of length \(2\pi\ell \in (\pi,2\pi]\), \(\ell=p/q\in (\frac{1}{2},1]\), \(\gcd(p,q)=1\), gives rise to a positive Besse–Weyl structure on \(S^2(a_1,a_2)\) with \(a_1=q\) and \(a_2=2p-q\), and

  2. a positive Besse–Weyl structure on such a \(S^2(a_1,a_2)\) gives rise to such a Finsler metric on \(S^2\) with shortest closed geodesic of length \(2\pi\left(\frac{a_1+a_2}{2a_1}\right) \in (\pi,2\pi]\),

and these assignments are inverse to each other. Moreover, two such Finsler metrics are isometric if and only if the corresponding Besse–Weyl structures coincide up to a diffeomorphism.

1.3 Construction of examples

In [31], it is shown that Weyl connections with prescribed (unparametrised) geodesics on an oriented surface \(M\) are in one-to-one correspondence with certain holomorphic curves into the “twistor space” over \(M\). In 4 we make use of this observation to construct deformations of positive Besse–Weyl structures on the weighted projective line \(\mathbb{CP}(a_1,a_2)\) in a fixed projective class, by deforming the Veronese embedding of \(\mathbb{CP}(a_1,a_2)\) into the weighted projective plane with weights \((a_1,(a_1+a_2)/2,a_2)\). Applying our duality result, we obtain a corresponding real two-dimensional family of non-isometric, rotationally symmetric Finsler structures on the \(2\)-sphere with constant positive curvature and all geodesics closed, but not of the same length. The length of the shortest closed geodesic of the resulting Finsler metric is unchanged for our family of deformations and so it is of different nature than the Zermelo deformation used by Katok in the construction of his examples [22]. Moreover, we expect that not all of these examples are of Riemannian origin in the following sense (cf. Remark 4.13).

The construction of rotationally symmetric Zoll metrics on \(S^2\) can be generalized to give an infinite-dimensional family of rotationally symmetric Riemannian metrics on spindle orbifolds all of whose geodesics are closed [4, 24]. Since every Levi-Civita connection is a Weyl connection, we obtain an infinite-dimensional family of rotationally symmetric positive Riemannian Besse–Weyl structures.

Furthermore, in [27, 28] LeBrun–Mason construct a Weyl connection \(\nabla\) on the \(2\)-sphere \(S^2\) for every totally real embedding of \(\mathbb{RP}^2\) into \(\mathbb{CP}^2\) which is sufficiently close to the standard real linear embedding. The Weyl connection has the property that all of its maximal geodesics are embedded circles and hence defines a Besse–Weyl structure. In addition, they show that every such Weyl connection on \(S^2\) is part of a complex \(5\)-dimensional family of Weyl connections having the same unparametrised geodesics (see also [32]). In particular, the Weyl connections of LeBrun–Mason that arise from an embedding of \(\mathbb{RP}^2\) that is sufficiently close to the standard embedding provide examples of positive Besse–Weyl structures. The corresponding dual Finsler metrics on \(S^2\) will have geodesics that are all closed and of the same length.

A complete local picture of the space of Finsler \(2\)-spheres of constant positive curvature and with all geodesics closed likely requires extending the work of LeBrun–Mason to the orbifold setting. Our results in 4 lay the foundation for such an extension. We hope to be able to built upon it in future work.

Acknowledgements

The authors would like to thank Vladimir Matveev for drawing their attention to the work of Bryant and its possible extension as well as for helpful correspondence. CL was partially supported by the DFG funded project SFB/TRR 191. A part of the research for this article was carried out while TM was visiting FIM at ETH Zürich and Mathematisches Institut der Universität zu Köln. TM thanks FIM for its hospitality and the DFG for travel support via the project SFB/TRR 191. TM was partially funded by the priority programme SPP 2026 “Geometry at Infinity” of DFG.