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A Gyrovector Space Approach to Hyperbolic GeometryGyrogroups

A Gyrovector Space Approach to Hyperbolic Geometry: Gyrogroups [Gyrogroups are generalized groups, which are best motivated by the algebra of Möbius transformations of the complex open unit disc. Groups are classified into commutative and non-commutative groups and, in full analogy, gyrogroups are classified into gyrocommutative and non-gyrocommutative gyrogroups. Some commutative groups admit scalar multiplication, giving rise to vector spaces. In full analogy, some gyrocommutative gyrogroups admit scalar multiplication, giving rise to gyrovector spaces. Furthermore, vector spaces form the algebraic setting for the standard model of Euclidean geometry and, in full analogy, gyrovector spaces form the algebraic setting for various models ofthe hyperbolic geometry of Bolyai and Lobachevsky.] http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

A Gyrovector Space Approach to Hyperbolic GeometryGyrogroups

Part of the Synthesis Lectures on Mathematics & Statistics Book Series
Ungar, Abraham Albert
Springer Journals — Jan 1, 2009
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31 pages

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Publisher
Springer International Publishing
Copyright
© Springer Nature Switzerland AG 2009
ISBN
978-3-031-01268-6
Pages
1 –32
DOI
10.1007/978-3-031-02396-5_1
Publisher site
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Abstract

[Gyrogroups are generalized groups, which are best motivated by the algebra of Möbius transformations of the complex open unit disc. Groups are classified into commutative and non-commutative groups and, in full analogy, gyrogroups are classified into gyrocommutative and non-gyrocommutative gyrogroups. Some commutative groups admit scalar multiplication, giving rise to vector spaces. In full analogy, some gyrocommutative gyrogroups admit scalar multiplication, giving rise to gyrovector spaces. Furthermore, vector spaces form the algebraic setting for the standard model of Euclidean geometry and, in full analogy, gyrovector spaces form the algebraic setting for various models ofthe hyperbolic geometry of Bolyai and Lobachevsky.]

Published: Jan 1, 2009

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