Global behaviours of circles in a complex hyperbolic space
type: Item
instance of: scholarly article
title: Global behaviours of circles in a complex hyperbolic space
zbMATH Open document ID: 0891.53036
author: Toshiaki Adachi
author: Sadahiro Maeda
published in: Tsukuba Journal of Mathematics
publication date: 1998-07-19T00:00:00Z
review text: A unit speed curve \(\gamma: \mathbb{R}\to M\) in a Riemannian manifold \(M\) is called a circle iff there exists \(k\in \mathbb{R}_+\) and a unit vector field \(Y\) along the curve such that \(\nabla_\partial \dot\gamma =kY\) and \(\nabla_\partial Y= -k\dot \gamma\), where \(\partial\) denotes the canonical unit vector field of \(\mathbb{R}\). The behaviour of circles in real space forms is well-known. After \textit{T. Adachi}, \textit{S. Maeda} and \textit{S. Udagawa} have investigated circles in the complex projective space [Osaka J. Math. 32, 709-719 (1995; Zbl 0857.53034)], the present authors treat now the problem in the \(n\)-dimensional complex hyperbolic space \(\mathbb{C} H^n\). Here the circles are characterized (up to holomorphic isometry) by their curvature \(k \) and their ``complex torsion'' \(\tau: =\langle \dot\gamma, JY\rangle \in[-1,1]\). By means of these entities the authors can characterize bounded and unbounded circles, in the case of bounded circles the periodic ones and in the case of unbounded ones their behaviour at \(\pm\infty\).
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C55
zbMATH DE Number: 1102908
zbMATH Keywords: complex torsion
zbMATH Keywords: circles
zbMATH Keywords: complex hyperbolic space
reviewed by: Helmut Reckziegel
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496163159
OpenAlex ID: W2249495090
DOI: 10.21099/TKBJM/1496163159
description: scientific article; zbMATH DE number 1102908