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Generalized gyrovector spaces revisited

Toshikazu Abe, Osamu Hatori

Abstract

We revise a proof of a Mazur-Ulam theorem for generalized gyrovector spaces.

Generalized gyrovector spaces revisited

Abstract

We revise a proof of a Mazur-Ulam theorem for generalized gyrovector spaces.
Paper Structure (7 sections, 10 theorems, 81 equations)

This paper contains 7 sections, 10 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Definition of a generalized gyrovector space revisited
  4. A Mazur-Ulam theorem for GGVs
  5. Preparations of the proof
  6. Proof of Theorem \ref{['GMU']}
  7. Proof of Corollary \ref{['gl']}

Key Result

Proposition 2.1

Let $(G,\oplus)$ be a gyrocommutative gyrogroup with the map $\otimes:\mathbb{R}\times G\to G$. We denote $\hbox{\boldmath $e$}$ the unit of $(G,\oplus)$. Let $\phi$ be an injection from $G$ into a real normed space $(\mathbb V,\|\cdot\|)$. We assume Then we have the following.

Theorems & Definitions (24)

  • Proposition 2.1
  • proof
  • Definition 3.1: A generalized gyrovector space
  • Proposition 3.2
  • proof
  • Example 3.3
  • Definition 3.4: gyrometric
  • Proposition 3.5
  • proof
  • Proposition 3.6
  • ...and 14 more