Distributivity on the Gyrovector Spaces

Abstract

Abstract. As a vector space provides a fundamental tool for the study of Euclideangeometry, a gyrovector space provides an algebraic tool for the study of hyperbolic ge-ometry. In general, the gyrovector spaces do not satisfy the distributivity with scalarmultiplication. In this article, we see under what condition the distributivity with scalarmultiplication is satis ed. 1. IntroductionIn order to provide an algebraic tool to study Einstein’s relativistic velocitysum, A. A. Ungar [2] has introduced a notion of gyrogroup and has developed to-gether the study of analytic hyperbolic geometry. The gyrogroup is a most naturalextension of a group into the nonassociative algebra. The associativity (and thecommutativity) of group operations is salvaged in a suitably modified form, calleda gyroassociativity (and a gyrocommutativity). In Section 2 we introduce a notionof (gyrocommutative) gyrogroup with its examples.Later on it is known that gyrocommutative gyrogroups are equivalent to Bruckloops (see [1]). To elaborate a precise language, we prefix a