Geodesics on the moduli space of oriented circles in \(\mathbb{S}^{3}\)
type: Item
instance of: scholarly article
title: Geodesics on the moduli space of oriented circles in \(\mathbb{S}^{3}\)
zbMATH Open document ID: 1232.53038
author: Linyuan Fan
author: Jingyang Zhong
author: Ying Lu
author: Changping Wang
published in: Results in Mathematics
publication date: 2011-06-22T00:00:00Z
review text: The Grassmannian of oriented circles in the conformal \(3\)-sphere carries a natural complex structure. The authors describe this space of circles in terms of a quadric in projective \(4\)-space and equip it with a hermitian metric \(h\) of signature \((2,1)\) that is invariant under the action of the conformal group of \(S^3\). Hence they study the geodesics with respect to the real part \(g=\text{Re}\,h\) of this metric. The two main theorems of the text are Theorem 1.1, which provides a geometric interpretation of the geodesics as, up to trivial cases, families of circles on helicoids in space form subgeometries of conformal geometry; Theorem 1.2, which states that there exist countably many geodesics in the space of circles connecting any two given circles.
reviewed by: Udo Hertrich-Jeromin
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53A30
Mathematics Subject Classification ID: 53C35
zbMATH DE Number: 5911458
zbMATH Keywords: geodesic
zbMATH Keywords: helicoid
zbMATH Keywords: circle foliation
zbMATH Keywords: Grassmannian
MaRDI profile type: Publication
full work available at URL: s00025-011-0113-1
OpenAlex ID: W1992503387
cites work: Q4431277
cites work: Complete minimal surfaces in \(S^ 3\)
cites work: Rotation Hypersurfaces in Spaces of Constant Curvature
cites work: Q3333688
DOI: 10.1007/S00025-011-0113-1
description: scientific article; zbMATH DE number 5911458