On an odule in Hilbert geometry
type: Item
instance of: scholarly article
title: On an odule in Hilbert geometry
zbMATH Open document ID: 0857.53051
published in: Russian Mathematics
publication date: 1996-05-12T00:00:00Z
review text: The notion of odular structure was introduced by \textit{L. V. Sabinin} as an algebraic abstraction derived from the existence of local normal coordinates in affinely connected manifolds [Sov. Math., Dokl. 18, 515-518 (1977); translation from Dokl. Akad. Nauk SSSR 233, 800-803 (1977; Zbl 0375.53021)]. The present article concerns a metric space \((X,\rho)\) in which two distinct points can be joined by a unique shortest line and in which each shortest line can be uniquely extended to a geodesic line. An odular structure \(W\) on \((X,\rho)\) is then defined by the author; it consists of mappings \(L^p : X \times X \to X\) and \(\omega^p : \mathbb{R} \times X \to X\) associated to the points \(p\) of \(X\); the structure \(W\) is called topological if \(L^p\) and \(\omega^p\) are continuous for every \(p \in X\). If \((X,\rho)\) is finitely compact then \(W\) is topological (Theorem 1). The examples developed by the author are: the Hilbert geometry of a ball in a strictly convex Banach space, and the Beltrami-Klein model of the Lobachevskij geometry in a ball belonging to a Hilbert space.
Mathematics Subject Classification ID: 53C70
zbMATH DE Number: 857517
zbMATH Keywords: odular structure
zbMATH Keywords: metric space
zbMATH Keywords: Hilbert geometry
author: Evgenii N. Sosov
reviewed by: Marius Stoka
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 857517