Hyperbolic geometry for non-differential topologists

TL;DR

This paper provides a non-differential approach to hyperbolic spaces, demonstrating their unique properties and challenging classical Euclidean postulates within a purely metric framework.

Contribution

It introduces a differential-free presentation of hyperbolic spaces and proves their uniqueness among certain geodesic metric spaces.

Findings

Fifth Euclid's postulate is invalid in hyperbolic spaces

Spheres with great-circle distances are characterized within hyperbolic geometry

Hyperbolic and Euclidean spaces are the only locally compact three-point homogeneous geodesic spaces

Abstract

A soft presentation of hyperbolic spaces, free of differential apparatus, is offered. Fifth Euclid's postulate in such spaces is overthrown and, among other things, it is proved that spheres (equipped with great-circle distances) and hyperbolic and Euclidean spaces are the only locally compact geodesic (i.e., convex) metric spaces that are three-point homogeneous.