On models of non-Eucludian spaces generated by associative algebras

TL;DR

This paper demonstrates how to generate non-Euclidean geometries using associative unital algebras, presenting two models of degenerate non-Euclidean space and analyzing their properties.

Contribution

It introduces a novel approach to constructing non-Euclidean geometries from associative algebras and develops two explicit models with detailed geometric analysis.

Findings

Two models of degenerate non-Euclidean space are constructed.

The conformal model uses stereographic-like mapping.

The projective model employs Norden normalization to find geodesics.

Abstract

We present the non-trivial example how to generate non-Euclidean geometries from associative unital algebras. We consider bundles of the sphere of the degenerate non-Eucleadian space and its two models. The first (conformal) model is obtained by the mapping S onto a plane pass through the origin. It is analogous to the stereographic mapping. The second model (projective) is con- structed by the Norden normalization method, where we project the sphere onto a plane of normalization defining the metric and Christoffel symbols which allow us to find geodesic curves.