On metric structures of normed gyrogroups

TL;DR

This paper introduces the concept of normed gyrogroups, exploring their topological and geometric properties, and establishing their structure as homogeneous, left-invariant metric spaces with conditions for being topological gyrogroups.

Contribution

It defines normed gyrogroups inspired by Euclidean and hyperbolic geometries and analyzes their structure, including invariance properties and conditions for topological gyrogroup status.

Findings

Normed gyrogroups are homogeneous spaces.

They form left invariant metric spaces.

Conditions for being topological gyrogroups are established.

Abstract

In this article, we indicate that the open unit ball in $n$-dimensional Euclidean space $\mathbb{R}^n$ admits norm-like functions compatible with the Poincar\'e and Beltrami$-$Klein metrics. This leads to the notion of a normed gyrogroup, similar to that of a normed group in the literature. We then examine topological and geometric structures of normed gyrogroups. In particular, we prove that the normed gyrogroups are homogeneous and form left invariant metric spaces and derive a version of the Mazur$-$Ulam theorem. We also give certain sufficient conditions, involving the right-gyrotranslation inequality and Klee's condition, for a normed gyrogroup to be a topological gyrogroup.