Fefferman-type metrics and the projective geometry of sprays in two dimensions

Abstract

A spray is a second-order differential equation field on the slit tangent bundle of a differentiable manifold, which is homogeneous of degree 1 in the fibre coordinates in an appropriate sense; two sprays which are projectively equivalent have the same base-integral curves up to reparametrization. We show how, when the base manifold is two-dimensional, to construct from any projective equivalence class of sprays a conformal class of metrics on a four-dimensional manifold, of signature (2, 2); the Weyl conformal curvature of these metrics is simply related to the projective curvature of the sprays, and the geodesics of the sprays determine null geodesics of the metrics. The metrics in question have previously been obtained by Nurowski and Sparling (Classical and Quantum Gravity20 (2003) 4995–5016), by a different method involving the exploitation of a close analogy between the Cartan geometry of second-order ordinary differential equations and of three-dimensional Cauchy–Riemann structures. From this perspective the derived metrics are seen to be analoguous to those defined by Fefferman in the CR theory, and are therefore said to be of Fefferman type. Our version of the construction reveals that the Fefferman-type metrics are derivable from the scalar triple product, both directly in the flat case (which we discuss in some detail) and by a simple extension in general. There is accordingly in our formulation a very simple expression for a representative metric of the class in suitable coordinates.

A Projective-to-Conformal Fefferman-Type Construction

“…Analogously to the discussion before Proposition 3.10 we may conclude: Remark 3.12. Conformal structures induced from 2-dimensional projective structures are wellstudied, see, e.g., [14,15,25]. Notably, the intermediate 3-dimensional Lagrangean contact structure can be equivalently viewed as a path geometry (or the geometry associated to second order ODEs modulo point transformations).…”

“…14. A split-signature (n, n) conformal spin structure c on a manifold M is (locally) induced by an n-dimensional projective structure via the Fefferman-type construction if and only if the following properties are satisfied:(a) M , c admits a nowhere-vanishing light-like conformal Killing field k such that the corresponding tractor endomorphism K = L A M 0 (k) is an involution, i.e., K 2 = id T .…”

“…14. The expansion of the prolonged conformal Killing equation (2.13) for k givesD a k b = µ ab + g ab ,(6.5)D a µ br = −2 P a[b k r] − W bras k s ,(6.6)according to (2.10) and(2.11).…”