Weyl metrisability of two-dimensional projective structures

1 Introduction

In [11], Eisenhart and Veblen solve the Riemannian metrisability problem for a manifold equipped with a real analytic affine torsion-free connection \(\nabla\); i.e. they determine the necessary and sufficient conditions for \(\nabla\) to locally be a Levi-Civita connection or equivalently, the holonomy of \(\nabla\) being a subgroup of the orthogonal group. One can also ask to determine the necessary and sufficient conditions for \(\nabla\) to be projectively equivalent to a Levi-Civita connection. Recall that two affine connections on a manifold are said to be projectively equivalent if they have the same unparametrised geodesics. A projective equivalence class of affine torsion-free connections is called a projective structure and will be denoted by \([\nabla]\). Although known since Roger Liouville’s initial paper [18] which dates back to 1889, the projective local Riemannian metrisability problem has been solved only recently for real analytic projective structures on surfaces by Bryant, Dunajski and Eastwood [4]. A global characterisation of compact Zoll projective surfaces admitting a compatible Levi-Civita connection was given in [16]. In [10], the general case is shown to give rise to a linear pde system of finite type. An algorithmic procedure for checking if a given projective structure on a manifold contains a Levi-Civita connection is given in [23] (see also [19]). In [9], it was shown that locally the Riemannian metrisability problem for projective surfaces is equivalent to finding a Kähler metric on an associated conformal \(4\)-manifold of neutral signature.

There are two problems related to the projective Riemannian metrisability problem that are motivated by two different viewpoints:

First, the projective Riemannian metrisability problem may be thought of as an inverse problem in the calculus of variations, where one looks for a length functional whose (unparametrised) geodesics are prescribed. However, the functional is constrained to be the length functional of a Riemannian metric. Naturally one might look for a general length functional, more precisely a Finsler metric, whose geodesics are prescribed. This problem is studied in [1] and it is shown that locally on a surface every projective structure (or more generally path geometry) is Finsler metrisable.

Second, the projective Riemann metrisability problem may be thought of as looking for a connection \(\nabla\) in a projective equivalence class \([\nabla]\), whose parallel transport maps are linear isometries for some Riemannian metric \(g\). From this viewpoint one might also ask for existence of a connection \(\nabla \in [\nabla]\), whose parallel transport maps are merely linear conformal maps for some conformal structure \([g]\). It is this latter problem we investigate in this article. More precisely, we study the (projective) Weyl metrisability problem, i.e. the problem of finding an affine torsion-free connection preserving a conformal structure, a so-called Weyl connection, whose unparametrised geodesics are prescribed by some projective structure \([\nabla]\).

Weyl connections were introduced by Weyl [27] as an attempt to unify gravity and electromagnetism and are nowadays mainly studied in the context of the Einstein-Weyl equations in dimensions \(d\geq 3\) (see [8, 25] and references therein). Also, in [28] Wojtkowski observed a relation between Weyl connections and isokinetic dynamics as introduced by Hoover [13] and discussed by Gallavotti and Ruelle in the context of non equilibrium statistical mechanics [12].

This article is organised as follows. In §2 we use Cartan’s projective connection [6] and the theory of exterior differential systems [2, 14] to show that locally every smooth projective structure \([\nabla]\) on a surface is Weyl metrisable. In §3 we characterise the complex structure on the total space of the ‘twistor bundle’ [7, 24] of conformal inner products \(\mathcal{C}(M)\to M\) over an oriented surface \(M\) in terms of Cartan’s projective connection. We use this characterisation to prove the main result: A conformal structure \([g]\) on \(M\) is preserved by a \([\nabla]\)-representative if and only if \([g] : M \to \mathcal{C}(M)\) has holomorphic image. As an application of the main result we show in §4 that the Weyl connections on the \(2\)-sphere whose geodesics are the great circles are in one-to-one correspondence with the smooth quadrics \(C \subset \mathbb{CP}^2\) without real points.

The reader should note that in a certain sense the main results of this article generalise to higher dimensions in the context of Segre structures, see [20] for further details.

Acknowledgments

This article is based on the authors doctoral thesis; in this connection the author would like to thank his adviser Norbert Hungerbühler. Also, the author is very grateful to Robert L. Bryant for several stimulating conversations on the topic of this paper. Furthermore, the author would like to thank the referee for his detailed report and valuable suggestions.

Funding

Research for this article was partially supported by the Swiss National Science Foundation via the grants 200020-121506, 200020-107652, 200021-116165, 200020-124668 and the Postdoctoral Fellowship PBFRP2-133545.