Geodesics and curvature of Möbius invariant metrics
type: Item
instance of: scholarly article
title: Geodesics and curvature of Möbius invariant metrics
zbMATH Open document ID: 1171.30015
DOI: 10.1216/RMJ-2008-38-3-891
author: David A. Herron
author: Zair Ibragimov
author: D. Minda
published in: Rocky Mountain Journal of Mathematics
publication date: 2009-04-07T00:00:00Z
review text: A region \(\Omega\) on the Riemann sphere \(\hat{\mathbb{C}}\) is called a quasi-hyperbolic region if \(\hat{\mathbb{C}}\setminus \Omega\) contains at least two points. In this paper the authors examine the Ferrand metric and the Kulkarni-Pinkall metric on \(\Omega\). The Ferrand metric \(\varphi_{\Omega}(z)|dz|\) is defined by \[ \varphi_{\Omega}(z)=\sup_{a,b\in \Omega^{c}}\tau_{ab}(z) \;\text{for} \;z\in \Omega \cap \mathbb{C}, \] where \(\tau_{ab}(z)=|a-b|/(|z-a||z-b|)\) is the density of the quasi-hyperbolic metric on the region \(\hat{\mathbb{C}}\setminus \{a,b\}\). The Kulkarni-Pinkall metric \(\mu_{\Omega}(z)|dz|\) is defined by \[ \mu_{\Omega}(z)=\inf\{\lambda_{D}(z); z\in D\subset\Omega, \;D \;\text{is a disk on} \;\hat{\mathbb{C}} \} \;\text{for} \;z\in \Omega \cap \mathbb{C}, \] where \(\lambda_{D}(z)\) is the density of the hyperbolic metric on \(D\). It is shown that these metrics are calculated in terms of Euclidean diameters and circumdiameters. One of the main results in this paper is that circular geodesics are also geodesics for these two metrics. Another result is that every Kulkarni-Pinkall isometry between two regions with a certain property, especially nonsimply and nondoubly connected regions, is a Möbius transformation. The other results concern the generalized Gaussian curvatures of these two metrics. The authors study the possible values of the curvatures of these metrics, and determine the quasi-hyperbolic regions on which the curvatures are constant. For many fundamental regions such as strips, infinite sectors, the punctured disk and so on, these metrics and the generalized Gaussian curvatures are presented exactly.
reviewed by: Gou Nakamura
Mathematics Subject Classification ID: 30F45
Mathematics Subject Classification ID: 30C55
Mathematics Subject Classification ID: 30F30
zbMATH DE Number: 5541249
zbMATH Keywords: Poincaré hyperbolic metric
zbMATH Keywords: Kulkarni-Pinkall metric
zbMATH Keywords: Ferrand metric
zbMATH Keywords: quasi-hyperbolic metric
MaRDI profile type: Publication
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full work available at URL: rmj-2008-38-3-891
OpenAlex ID: W1976240867
description: scientific article; zbMATH DE number 5541249