Tits geometry associated with 4-dimensional closed real-analytic manifolds of nonpositive curvature
type: Item
instance of: scholarly article
title: Tits geometry associated with 4-dimensional closed real-analytic manifolds of nonpositive curvature
zbMATH Open document ID: 0920.53021
author: Christoph Hummel
author: Viktor Schroeder
published in: Journal of Differential Geometry
publication date: 1999-09-19T00:00:00Z
review text: Let \(X\) denote a complete, simply connected Riemannian manifold of nonpositive sectional curvature, and let \(X(\infty)\) denote the boundary sphere consisting of equivalence classes of asymptotic geodesics of \(X\). The Tits pseudometric Td is defined on \(X\) as follows. Fix a basepoint \(p\) in \(X\), and let points \(x\) and \(y\) in \(X(\infty)\) be given. If \(\gamma_{px}\) and \(\gamma_{py}\) denote the unit speed geodesics starting at \(p\) that belong to \(x\) and \(y\), then \(\text{Td}(x,y) =\lim_{t\to\infty} d_S(\gamma_{px}(t), \gamma_{py} (t))/t\), where \(d_S(\gamma_{px} (t),\gamma_{py}(t))\) denotes the distance between \(\gamma_{px}(t)\) and \(\gamma_{py}(t)\) in the sphere \(S\) of radius \(t\) centered at \(p\). The value of \(\text{Td}(x,y)\) does not depend on the basepoint \(p\), and in general it may equal \(+\infty\). For example, the Tits distance between any two distinct points of \(X(\infty)\) is \(+\infty\) if the sectional curvature is bounded above by a negative constant. The pseudometric Td is always a finite metric if \(X\) is a symmetric space of rank at least two. Define an equivalence relation on \(X( \infty)\) in which two points \(x\) and \(y\) are equivalent if \(\text{Td}(x,y)\) is finite. An equivalence class of points in \(X (\infty)\) will be called a component of \((X(\infty),\text{Td})\). Any two points in a component of \((X(\infty),\text{Td})\) can be joined by a Tits geodesic, which is unique if \(\text{Td}(x,y)<\pi \). The collection of components of \((X (\infty), \text{Td})\) that contain more than one point is denoted \(\partial_TX\). In this article, the authors consider the case that \(X\) is 4-dimensional and real analytic, and they investigate the components of \((X(\infty), \text{Td})\), which they classify into standard and nonstandard types. \(Y\) is a called a higher rank submanifold of \(X\) if \(Y\) is a totally geodesic submanifold that is isometric to a 2- or 3-dimensional Euclidean space (a 2- or 3-fla…
reviewed by: Patrick Eberlein
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 57M05
zbMATH DE Number: 1268528
zbMATH Keywords: nonpositive curvature
zbMATH Keywords: Tits boundary
zbMATH Keywords: nonstandard components
zbMATH Keywords: quasiflats
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214460862
OpenAlex ID: W1576586517
Wikidata QID: Q115174686
DOI: 10.4310/JDG/1214460862
description: scientific article; zbMATH DE number 1268528