Conformal Hyperbolic Numbers and Two-dimensional Finsler Geometry

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.