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The Hyperbolic Derivative in the Poincaré Ball Model of Hyperbolic Geometry

✍ Scribed by Graciela S. Birman; Abraham A. Ungar

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
115 KB
Volume
254
Category
Article
ISSN
0022-247X

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✦ Synopsis

The generic Mobius transformation of the complex open unit disc induces a binary operation in the disc, called the Mobius addition. Following its introduction, ẗhe extension of the Mobius addition to the ball of any real inner product space änd the scalar multiplication that it admits are presented, as well as the resulting geodesics of the Poincare ball model of hyperbolic geometry. The Mobius gyrovec-´ẗor spaces that emerge provide the setting for the Poincare ball model of hyper-bolic geometry in the same way that vector spaces provide the setting for Euclidean geometry. Our summary of the presentation of the Mobius ball gyrovector spaces sets the stage for the goal of this article, which is the introduction of the hyperbolic derivative. Subsequently, the hyperbolic derivative and its application to geodesics uncover novel analogies that hyperbolic geometry shares with Euclidean geometry.

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## Abstract Let \documentclass{article}\usepackage{amssymb,amsmath,amsthm,amscd,amsxtra}\begin{document}\pagestyle{empty}$\mathbb {H}^n$\end{document} be the __n__‐dimensional hyperbolic space. It is well‐known that, if \documentclass{article}\usepackage{amssymb,amsmath,amsthm,amscd,amsxtra}\begin{