Hausdorff approximations on Hadamard manifolds and their ideal boundaries
type: Item
instance of: scholarly article
title: Hausdorff approximations on Hadamard manifolds and their ideal boundaries
zbMATH Open document ID: 0895.53043
author: Fumiko Ohtsuka
published in: Tsukuba Journal of Mathematics
publication date: 1998-09-24T00:00:00Z
review text: The author considers the ideal boundary of a Hadamard manifold (a complete, simply connected Riemannian manifold \(M\) with nonpositive sectional curvature). The ideal boundary \(M(\infty)\) consists of asymptote classes of unit speed geodesics of \(M\), and for each point \(p\) of \(M\) and each point \(x\) of \(M(\infty)\) there exists a unique geodesic \(\gamma_{px}\) that belongs to \(x\) and satisfies \(\gamma_{px}(0) = p.\) There exists a natural sphere topology on \(M(\infty)\) such that the map \(x\to \gamma_{px}'(0)\) is a homeomorphism of \(M(\infty)\) onto the unit sphere in \(T_pM\) for every point \(p\) in \(M\). Gromov introduced a pseudometric \(\text{Td}\) on \(M(\infty)\), usually called the Tits metric, which may be defined as follows. Fix a point \(p\) of \(M\). For points \(z,z^*\) in \(M(\infty)\) define \(\ell(z,z^*)= \lim_{t\to\infty}\ell_t(z,z^*)\), where \(\ell_t (z,z^*)= (1/t)d(\gamma_{pz}(t),\gamma_{pz^*}(t))\), and define \(\text{Td}\) to be the inner metric on \(M(\infty)\) determined by \(\ell\), which is independent of \(p\). Even if \(\text{Td}\) is finite on \(M(\infty)\), the metric space \((M(\infty), \text{Td})\) is rarely compact; if \(M\) is a symmetric space, then \((M(\infty), \text{Td})\) is compact if and only if \(M\) is isometric to flat Euclidean space. For two metric spaces \(X,Y\) a map \(\varphi:X\to Y\) is an \((\alpha,\Delta)\) rough isometry if every point of \(Y\) lies in the closed \(\Delta\)-neighborhood of \(\varphi(X)\) and if \((1/\alpha)d(x,x^*)-\Delta\leq d(\varphi(x),\varphi(x^*)\leq \alpha d(x,x^*)+\Delta\) for all points \(x,x^*\) in \(X\). For \(t > 0\) and \(z,z^*\) in \(M(\infty)\) define \(\alpha_t(z,z^*) = \ell_t(z,z^*)/\ell(z,z^*)\) and \(\alpha_t = \inf\{\alpha_t(z,z^*): z,z^*\) are arbitrary in \(M(\infty)\) with \(z\neq z^*\}\). The numbers \(\alpha_t\) are nondecreasing in \(t\) and bounded above by 1. If \(\alpha=\lim_{t\to\infty} \alpha_t\) is positive, then the \(\text{Td}\) and sphere topolo…
Mathematics Subject Classification ID: 53C23
zbMATH DE Number: 1102953
zbMATH Keywords: Hadamard manifold
zbMATH Keywords: ideal boundary
zbMATH Keywords: Tits metric
zbMATH Keywords: rough isometry
zbMATH Keywords: condition (E)
reviewed by: Patrick Eberlein
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496163092
OpenAlex ID: W645494107
DOI: 10.21099/TKBJM/1496163092
description: scientific article; zbMATH DE number 1102953