Revisiting the foundations of Barbilian's metrization procedure
type: Item
instance of: scholarly article
title: Revisiting the foundations of Barbilian's metrization procedure
zbMATH Open document ID: 1221.53040
author: Wladimir G. Boskoff
author: Marian G. Ciucă
author: Bogdan D. Suceava
published in: Differential Geometry and its Applications
publication date: 2011-07-19T00:00:00Z
review text: The authors prove a natural extension of one of Barbilian's theorems [\textit{D. Barbilian}, Acad. Repub. Popul. Romine, Studii Cerc. Mat. 11, 7--44 (1960; Zbl 0112.37201)]. The extension supports two ideas: that all distances with constant curvature can be described by Barbilian's metrization principle, and that all Riemannian metrics corresponding to these distances can be obtained with the same unique procedure derived from the main result of the paper. It is shown that the hyperbolic metric of the disk, the hyperbolic metric on the exterior of the disk and the hyperbolic metric on the half-plane can be obtained in a similar way. As well, the authors obtain metrics corresponding to quadratic forms with signature that includes minus. Two oscillant distances are obtained in some subsets of the Lorentz or the Minkowski plane. They provide metrics of generalized Lagrange type on the corresponding subsets. A result concerning the distance induced by a Riemannian metric as a local Barbilian distance, is also proved.
reviewed by: Vladimir Balan
Mathematics Subject Classification ID: 53B40
Mathematics Subject Classification ID: 53C60
Mathematics Subject Classification ID: 51K05
zbMATH DE Number: 5925823
zbMATH Keywords: Barbilian's metrization procedure
zbMATH Keywords: Beltrami-Klein model
zbMATH Keywords: Poincaré metric
zbMATH Keywords: Apollonian metric
zbMATH Keywords: Lorentz geometry
zbMATH Keywords: Riemannian metric, generalized Lagrange structure.
Wikidata QID: Q115356851
MaRDI profile type: Publication
full work available at URL: j.difgeo.2011.04.038
OpenAlex ID: W2095368344
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DOI: 10.1016/J.DIFGEO.2011.04.038
description: scientific article; zbMATH DE number 5925823