On the geometry at infinity of the universal covering of \(\text{Sl}(2,\mathbb{R})\)
type: Item
instance of: scholarly article
title: On the geometry at infinity of the universal covering of \(\text{Sl}(2,\mathbb{R})\)
zbMATH Open document ID: 1025.53020
author: Marcos Salvai
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 2003-04-22T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108541
review text: Given a Riemannian manifold \((M,g)\), two geodesics \(\gamma\) and \(\gamma'\) are said to be asymptotic, \(\gamma \sim \gamma'\), if there exists a constant \(C\) such that \(d(\gamma (t),\gamma' (t))\leq C\), for any \(t \geq 0\). Furthermore, \(A_{\infty}(M)\) denotes the set of equivalence classes determined by the asymptoticity relation between geodesics. Clearly, when \(M\) is a Hadamard manifold, for each \(p \in M\), \(A_{\infty}(M)\) is bijective to \(T^1_p (M)\), the set of unit tangent vectors at \(p\). The paper under review focuses on the universal covering of \(\text{Sl}(2,\mathbb{R})\) with its canonical metric. The author considers the group \(G= \text{PSl}(2,\mathbb{R})= \text{Sl}(2, \mathbb{R})/\{\pm I\}\) endowed with the metric \(\langle\;,\;\rangle\) such that \(\langle X,Y\rangle=2 \operatorname {tr}(XY^t)\), \(X, Y\) in the Lie algebra \(\mathfrak g\) of \(G\), and denotes by \(\widetilde{\pi} : \widetilde{G} \rightarrow G\) the Riemannian universal covering of \(G\). Moreover, setting \(K= \text{PSO}(2)\) and \(H= G/K\) with the Riemannian metric making the canonical projection \(\pi:G \rightarrow H\) a Riemannian submersion, \(H\) is the hyperbolic plane of constant curvature -1. In this context, the author examines the asymptoticity of geodesics in \(\widetilde G\) with respect to the projection \(\pi \circ \widetilde \pi\) and with respect to a fixed point in \(\widetilde G\). He proves that a geodesic in \(\widetilde G\) projects on \(H\) to a point or to a curve of constant geodesic curvature. Recalling that a non constant complete curve \(c\) in \(H\) having constant geodesic curvature \(k\) is: a circle if \(|k |> 1\), a horocycle if \(|k |=1\), and there exists \(\lim_{t\to \infty}c(t) =c(\infty)\) if \(|k |< 1\). The author states the following: Theorem. Let \(\gamma\) and \(\gamma'\) be geodesics in \(\widetilde G\), which project to curves \(c\) and \(c'\) in \(H\) with constant geodesic curvatures \(k\) and \(k'\) respectively, i…
reviewed by: Anna Maria Pastore
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 53C70
Mathematics Subject Classification ID: 57M50
zbMATH DE Number: 1898321
zbMATH Keywords: asymptotic geodesics
zbMATH Keywords: hyperbolic plane
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1898321