Mobility spaces and their geodesic paths

TL;DR

This paper introduces a new algebraic framework called mobility spaces, modeling spaces with geodesic paths using algebraic structures called mobility algebras, and explores their properties and connections to existing geometric concepts.

Contribution

It defines mobility spaces based on mobility algebras, establishing a novel algebraic model for spaces with geodesic paths and analyzing their fundamental properties.

Findings

Defined axioms for mobility spaces

Connected algebraic structures to geodesic path spaces

Provided examples illustrating the concepts

Abstract

We introduce an algebraic system which can be used as a model for spaces with geodesic paths between any two of their points. This new algebraic structure is based on the notion of mobility algebra which has recently been introduced as a model for the unit interval of real numbers. Mobility algebras consist on a set $A$ together with three constants and a ternary operation. In the case of the closed unit interval $A=[0,1]$, the three constants are 0, 1 and 1/2 while the ternary operation is $p(x,y,z)=x-yx+yz$. A mobility space is a set $X$ together with a map $q\colon{X\times A\times X\to X}$ with the meaning that $q(x,t,y)$ indicates the position of a particle moving from point $x$ to point $y$ at the instant $t\in A$, along a geodesic path within the space $X$. A mobility space is thus defined with respect to a mobility algebra, in the same way as a module is defined over a ring. We introduce the axioms for mobility spaces, investigate the main properties and give examples. We also establish the connection between the algebraic context and the one of spaces with geodesic paths. The connection with affine spaces is briefly mentioned.