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Conformal Hyperbolic Numbers and Two-dimensional Finsler Geometry

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Abstract

An abelian group of two-dimensional conformal hyperbolic numbers is investigated. A characteristic equation of a hyperbolic number is derived using the theory of permanents. A conformal multiplier, which depends on components of a hyperbolic number or its hyperbolic angle, is defined, and generalized hyperbolic functions are considered. A geometric representation of the group is introduced. It is shown that modulus of a hyperbolic number gives a metric for a two-dimensional Finsler geometry with a quadratic form and a conformal multiplier. A one-dimensional parameter is derived by requiring that the form of the metric be invariant. A group of parameters and functions of hyperbolic numbers are also investigated.

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  1. Institute of Mechanics and Engineering, Kazan Science Center, Russian Academy of Sciences, Lobachevsky str. 2/31, Kazan, 420111, Russia

    Rinat G. Zaripov

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  1. Rinat G. Zaripov
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Zaripov, R.G. Conformal Hyperbolic Numbers and Two-dimensional Finsler Geometry. Adv. Appl. Clifford Algebras 27, 1741–1760 (2017). https://doi.org/10.1007/s00006-016-0680-z

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  • DOI: https://doi.org/10.1007/s00006-016-0680-z

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