Overview

Some gyrocommutative gyrogroups admit scalar multiplication, turning them into gyrovector spaces. The latter, in turn, are analogous to vector spaces just as gyrogroups are analogous to groups. Indeed, gyrovector spaces provide the setting for hyperbolic geometry just as vector spaces provide the setting for Euclidean geometry.

The elements of a gyrovector space are called points. Any two points of a gyrovector space give rise to a gyrovector. Points give rise to geodesies and cogeodesics that share analogies with Euclidean geodesies, the straight lines.

6.1 Definition and First Gyrovector Space Theorems

Definition 6.1: (Real Inner Product Vector Spaces).

A real inner product vector space ( , +, ) (vector space, in short) is a real vector space together with a map

(6.1)

called a real inner product, satisfying the following properties for all u, v, and r ? ?:

1. v v ? 0, with equality if, and only if, v = 0.

2. u v = v u

3. ( u + v) w = u w + v w

4. ( r u) v = r( u v)

The norm ? v ? of is given by the equation ? v ? ² = v v.

Note that the properties of vector spaces imply (i) the Cauchy-Schwarz inequality u v ? ? u ?? v ? for all u, ; and (ii) the