The cogyrolines of Möbius gyrovector spaces are metric but not periodic

Demirel, Oğuzhan; Seyrantepe, Emine Soytürk

doi:10.1007/s00010-012-0134-1

Cited by 7 publications

(4 citation statements)

References 10 publications

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“…Möbius and Einstein gyrogroups themselves are of great importance in gyrogroup theory as they provide concrete models for an abstract gyrogroup. See for instance [3,6,8,14,[17][18][19][20].…”

Section: Examplesmentioning

confidence: 99%

Gyrogroup actions: A generalization of group actions

Suksumran

2016

Journal of Algebra

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This article explores the novel notion of gyrogroup actions, which is a natural generalization of the usual notion of group actions. As a first step toward the study of gyrogroup actions from the algebraic viewpoint, we prove three well-known theorems in group theory for gyrogroups: the orbit-stabilizer theorem, the orbit decomposition theorem, and the Burnside lemma (or the Cauchy-Frobenius lemma). We then prove that under a certain condition, a gyrogroup G acts transitively on the set G/H of left cosets of a subgyrogroup H in G in a natural way. From this we prove the structure theorem that every transitive action of a gyrogroup can be realized as a gyrogroup action by left gyroaddition. We also exhibit concrete examples of gyrogroup actions from the Möbius and Einstein gyrogroups.

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Section: Examplesmentioning

confidence: 99%

Gyrogroup actions: A generalization of group actions

Suksumran

2016

Journal of Algebra

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“…More precisely, the Möbius gyrovector space is associated with the Poincaré model of conformal geometry on the open unit ball in n-dimensional Euclidean space R n [11,38], and the Einstein gyrovector space is associated with the Beltrami-Klein model of hyperbolic geometry on the unit ball in R n [38,52,53,59,63].…”

Section: Introductionmentioning

confidence: 99%

The Algebra of Gyrogroups: Cayley’s Theorem, Lagrange’s Theorem, and Isomorphism Theorems

Suksumran

2016

Essays in Mathematics and Its Applications

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Using the Clifford algebra formalism, we show that the unit ball of a real inner product space equipped with Einstein addition forms a uniquely 2-divisible gyrocommutative gyrogroup or a B-loop in the loop literature. One notable result is a compact formula for Einstein addition in terms of Möbius addition. In the second part of this paper, we show that the symmetric group of a gyrogroup admits the gyrogroup structure, thus obtaining an analog of Cayley's theorem for gyrogroups. We examine subgyrogroups, gyrogroup homomorphisms, normal subgyrogroups, and quotient gyrogroups and prove the isomorphism theorems. We prove a version of Lagrange's theorem for gyrogroups and use this result to prove that gyrogroups of particular order have the Cauchy property.

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“…The resulting (i) Möbius gyrovector spaces form the algebraic setting for the Cartesian Poincaré ball model of analytic hyperbolic geometry, just as (ii) Einstein gyrovector spaces form the algebraic setting for the Cartesian Klein ball model of analytic hyperbolic geometry, just as (iii) vector spaces form the algebraic setting for the standard Cartesian model of analytic Euclidean geometry [1]. Interesting applications of gyrovector spaces are found, for instance, in [29][30][31][32][33][34][35][36][37][38][39][40].…”

Section: Möbius Addition and Scalar Multiplicationmentioning

confidence: 99%

The Hyperbolic Ptolemy’s Theorem in the Poincaré Ball Model of Analytic Hyperbolic Geometry

Ungar

2023

Symmetry

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Ptolemy’s theorem in Euclidean geometry, named after the Greek astronomer and mathematician Claudius Ptolemy, is well known. We translate Ptolemy’s theorem from analytic Euclidean geometry into the Poincaré ball model of analytic hyperbolic geometry, which is based on the Möbius addition and its associated symmetry gyrogroup. The translation of Ptolemy’s theorem from Euclidean geometry into hyperbolic geometry is achieved by means of hyperbolic trigonometry, called gyrotrigonometry, to which the Poincaré ball model gives rise, and by means of the duality of trigonometry and gyrotrigonometry.

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The cogyrolines of Möbius gyrovector spaces are metric but not periodic

Cited by 7 publications

References 10 publications

Gyrogroup actions: A generalization of group actions

Gyrogroup actions: A generalization of group actions

The Algebra of Gyrogroups: Cayley’s Theorem, Lagrange’s Theorem, and Isomorphism Theorems

The Hyperbolic Ptolemy’s Theorem in the Poincaré Ball Model of Analytic Hyperbolic Geometry

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