Analytic Geometry of Homogeneous Spaces

TL;DR

This paper develops a unified linear algebra-based framework for analyzing various nonlinear homogeneous spaces, enabling simultaneous study and comparison of different geometries through parameterization by space signature.

Contribution

It introduces a novel space parameterization method using space signature, allowing a unified and flexible approach to analyze multiple homogeneous spaces within a single framework.

Findings

Unified description of elliptic, hyperbolic, De Sitter, Anti de Sitter spaces

Parameterization by space signature simplifies comparison of different spaces

Framework applies to definitions, axioms, and theorems across all spaces

Abstract

The theory uses methods and language of linear algebra to study nonlinear spaces. These techniques can be used particularly to describe analytic geometry of non-linear elliptic, hyperbolic, De Sitter and Anti de Sitter spaces. The main innovation of elaborated theory is space parameterization by introduction of space signature. This parameterization allows studying of different homogeneous spaces in one global framework. When the parameters are used as variables in definitions, axioms, equations, theorems, proofs, all these have exactly the same form that describes the reality of all homogeneous spaces simultaneously. When it is necessary to describe some space particularities or to see the difference between two concrete spaces, the concrete values can be put in parameters of each definition, axiom, equation, theorem and proof.