An ambient approach to conformal geodesics - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Communications in Contemporary Mathematics Année : 2022

An ambient approach to conformal geodesics

Résumé

Conformal geodesics are distinguished curves on a conformal manifold, loosely analogous to geodesics of Riemannian geometry. One definition of them is as solutions to a third-order differential equation determined by the conformal structure. There is an alternative description via the tractor calculus. In this article, we give a third description using ideas from holography. A conformal [Formula: see text]-manifold [Formula: see text] can be seen (formally at least) as the asymptotic boundary of a Poincaré–Einstein [Formula: see text]-manifold [Formula: see text]. We show that any curve [Formula: see text] in [Formula: see text] has a uniquely determined extension to a surface [Formula: see text] in [Formula: see text], which we call the ambient surface of[Formula: see text]. This surface meets the boundary [Formula: see text] in right angles along [Formula: see text] and is singled out by the requirement that it be a critical point of renormalized area. The conformal geometry of [Formula: see text] is encoded in the Riemannian geometry of [Formula: see text]. In particular, [Formula: see text] is a conformal geodesic precisely when [Formula: see text] is asymptotically totally geodesic, i.e. its second fundamental form vanishes to one order higher than expected. We also relate this construction to tractors and the ambient metric construction of Fefferman and Graham. In the [Formula: see text]-dimensional ambient manifold, the ambient surface is a graph over the bundle of scales. The tractor calculus then identifies with the usual tensor calculus along this surface. This gives an alternative compact proof of our holographic characterization of conformal geodesics.

Dates et versions

hal-03883369 , version 1 (03-12-2022)

Identifiants

Citer

Joel Fine, Yannick Herfray. An ambient approach to conformal geodesics. Communications in Contemporary Mathematics, 2022, 24 (03), ⟨10.1142/S0219199721500097⟩. ⟨hal-03883369⟩

Collections

TDS-MACS
15 Consultations
0 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More