An ambient approach to conformal geodesics
| Autor: | Joel Fine, Yannick Herfray |
|---|---|
| Jazyk: | francouzština |
| Rok vydání: | 2021 |
| Předmět: |
Mathematics - Differential Geometry
Geodesic General Mathematics FOS: Physical sciences Conformal map Riemannian geometry 01 natural sciences symbols.namesake FOS: Mathematics Computer Science::General Literature Géométries différentielle et infinitésimale 0101 mathematics Mathematical Physics Mathematical physics Mathematics Minimal surface Géométrie riemannienne intégrale symplectique et de poisson Computer Science::Information Retrieval Applied Mathematics 010102 general mathematics Astrophysics::Instrumentation and Methods for Astrophysics Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) Mathematical Physics (math-ph) Manifold 010101 applied mathematics Differential Geometry (math.DG) symbols Mathematics::Differential Geometry Conformal geometry Differential (mathematics) |
| Zdroj: | Communications in Contemporary Mathematics |
| Popis: | 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 $n$-manifold $X$ can be seen (formally at least) as the asymptotic boundary of a Poincar\'e--Einstein $(n+1)$-manifold $M$. We show that any curve $\gamma$ in $X$ has a uniquely determined extension to a surface $\Sigma_\gamma$ in $M$, which which we call the \emph{ambient surface of $\gamma$}. This surface meets the boundary $X$ in right angles along $\gamma$ and is singled out by the requirement that it it be a critical point of renormalised area. The conformal geometry of $\gamma$ is encoded in the Riemannian geometry of $\Sigma_\gamma$. In particular, $\gamma$ is a conformal geodesic precisely when $\Sigma_\gamma$ 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 $(n+2)$-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 characterisation of conformal geodesics. Comment: 23 pages, 3 figures. v2 acknowledgements added, v3 figures added, minor adjustments to the text to improve exposition, version accepted for publication in Communications in Contemporary Mathematics |
| Databáze: | OpenAIRE |
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