Differentiable distance spaces

TL;DR

This paper explores a class of differentiable distance spaces on bR^n that are more general than Finsler spaces, analyzing their properties and geodesic-like curves without using variational calculus.

Contribution

It introduces and investigates differentiable distance spaces that are broader than Finsler spaces, focusing on their properties and geodesic-like curves through geometric methods.

Findings

Identified properties of curves in differentiable distance spaces.

Established relations between these spaces and Finsler spaces.

Described geodesic-like curves without variational calculus.

Abstract

The distance function $\varrho(p,q)$ (or $d(p,q)$) of a distance space (general metric space) is not differentiable in general. We investigate such distance spaces over $\mathbb R^n$, whose distance functions are differentiable like in case of Finsler spaces. These spaces have several good properties, yet they are no Finsler spaces (which are special distance spaces). They are situated between general metric spaces (distance spaces) and Finsler spaces. We will investigate such curves of differentiable distance spaces, which possess the same properties as geodesics do in Finsler spaces. So these curves can be considered as forerunners of Finsler geodesics. They are in greater plenitude than Finsler geodesics, but they become geodesics in a Finsler space. We show some properties of these curves, as well as some relations between differentiable distance spaces and Finsler spaces. We arrive to these curves and to our results by using distance spheres, and using no variational calculus. We often apply direct geometric considerations.