On the geometry at infinity of the universal covering of \(\text{Sl}(2,\mathbb{R})\)
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, if they are not constant curves. Then one has: a) If \(c\) is constant, then \(\gamma \sim \gamma'\) if \(c'\) is constant. b) If \(|k |>1\), then \(\gamma \sim \gamma'\) if and only if \(k=k'\). c) If \(|k |=1\), then \(\gamma \sim \gamma'\) if and only if there exists \(t_0 \in \mathbb{R} \) such that \(c'(t)=c(t+t_0)\). d) If \(|k |<1\), then \(\gamma \sim \gamma'\) if and only if \(k=k'\) and \(c'(\infty)=c(\infty)\). Moreover, \(\gamma\) is a line, i.e. \(\gamma\) minimizes the distance between any two of its points, if and only if \(|k |\leq 1\). After that, considering the Cartan decomposition \({\mathfrak g}=\mathbb{R} Z + {\mathfrak p}\) associated to \(K\), where \[ Z=\frac 12 \begin{pmatrix} 0& 1 \\ -1 & 0 \end{pmatrix} \] and looking at \(Z\) as North pole, the author proves that, given distinct unit tangent vectors \(X\), \(Y\) at \(\widetilde e \in \widetilde G\), the geodesics \(\gamma_X\), and \(\gamma_Y\) are asymptotic if and only if \(X\) and \(Y\) lie in the same parallel at a distance less than \(\pi/4\) from the poles. Furthermore, there exist geodesics in \(\widetilde G\) which are not asymptotic to any geodesic through \(\widetilde e\) and they correspond, in a canonical way, to the horocycles of \(H\) not containing \(eK\). Then he considers the Riemannian invariant defined by \textit{J. E. D'Atri} [Proc. Am. Math. Soc. 116, 813-817 (1992; Zbl 0792.53045)], which reflects the long-time behavior of geodesics, and proves that the spread of \(\widetilde G\) vanishes. Finally, concerning the topology on \(A_{\infty}(\widetilde G)\) coming from the identification with \(T^1 (\widetilde G) /\sim\), where \(X \sim Y\) if and only if \(\gamma_X\) and \(\gamma_Y\) are asymptotic, the author proves that it is neither Hausdorff nor compact.
scientific article; zbMATH DE number 1898321
scientific article