Geometric analysis on quaternion \(\mathbb H\)-type groups
type: Item
instance of: scholarly article
title: Geometric analysis on quaternion \(\mathbb H\)-type groups
zbMATH Open document ID: 1093.53042
DOI: 10.1007/BF02922116
author: Irina Markina
author: Der-Chen Chang
published in: The Journal of Geometric Analysis
publication date: 2006-03-28T00:00:00Z
review text: The authors consider seven examples of \(\mathbb{H}\)-type groups, which are related to the quaternion numbers. Three of them have a one-dimensional center, and actually, are isomorphic to the Heisenberg group \(\mathbb{H}^2\). Three others have a two-dimensional center and the last one has a three dimensional center which is isomorphic to imaginary quaternions. In each case the authors construct the Hamiltonian function associated with the sub-Laplacian. They solve the Hamiltonian system of differential equations and give exact solutions describing the geodesics on the groups. Considering different positions of points, the number of geodesics connecting these points is studied. Kepler's law is proved, that gives a nice geometric interpretation of the length of a part of a horizontal curve belonging to a center of the group and the area of a certain surface. The authors also consider complex geodesics and find a relation between the complex action function and the Carnot-Carathéodory length of geodesics.
reviewed by: Marian Ioan Munteanu
Mathematics Subject Classification ID: 53C22
Mathematics Subject Classification ID: 53C17
Mathematics Subject Classification ID: 35H20
zbMATH DE Number: 5015571
zbMATH Keywords: quaternions
zbMATH Keywords: Carnot-Carathéodory metrics
zbMATH Keywords: nilpotent Lie groups
zbMATH Keywords: Hamiltonian formalism
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02922116
OpenAlex ID: W2116314144
description: scientific article; zbMATH DE number 5015571