Geodesic space, orthogonality between geodesics and nonexistence of local points in Hadamard spaces
type: Item
instance of: scholarly article
title: Geodesic space, orthogonality between geodesics and nonexistence of local points in Hadamard spaces
zbMATH Open document ID: 1035.53095
author: Taoufik Bouziane
published in: Boletín de la Sociedad Matemática Mexicana. Third Series
publication date: 2002-01-01T00:00:00Z
review text: Let \((X, d)\) be a metric space. Following mainly Alexandrov and Busemann, the author recalls first the notion of geodesics or triangles on \(X\), and injective radius and then defines the notion of the orthogonality between geodesics on \(X\). Next, the author defines for a triangle on \(X\), the notion of \(\text{CAT}_k\) triangle for \(k\in \mathbb{R}\) and the notion of curvature bounded by \(k\), denoted by \(K_X\leq k\). The space \(X\) is called a geodesic space if for any two points \(p\) and \(q\in X\) there exists a minimal geodesic connecting \(p\) and \(q\). The author proves that if \(X\) is a simply connected locally compact complete geodesic space with injectivety radius \(\delta\) then for any \(\delta\)-minimal complete geodesic \(\sigma: \mathbb{R}\to X\) and any point \(p\not\in\sigma(\mathbb{R})\) with \(d(p,\sigma(\mathbb{R}))< \delta/2\), there exists a geodesic \(\sigma_1\) through \(p\), which is orthogonal to \(\sigma\). Finally, the author defines the notion of a focal point, for a geodesic on \(X\) and proves that if \(X\) is a complete geodesic space with \(K_X\leq 0\) then the universal covering of \(X\) has no focal point.
Mathematics Subject Classification ID: 53C45
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 1933100
zbMATH Keywords: geodesic space
zbMATH Keywords: focal points
reviewed by: Akihiko Morimoto
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1933100