The Alpha Group Tensorial Metric

TL;DR

This paper introduces the Alpha Group tensorial metric, a novel geometric framework for hypercomplex space in four dimensions, unifying Riemannian and Euclidean metrics through a new tensorial approach.

Contribution

It presents a new tensorial metric formula based on the Alpha Group, offering a fresh interpretation of hypercomplex space and its geometric and topological properties.

Findings

Riemannian and Euclidean distances are special cases of the Alpha Group metric.

Provides a new geometric interpretation of $ ext{R}^4$ space at infinity.

Introduces a tensorial framework unifying different distance metrics.

Abstract

The Alpha Group is an abstract geometry group in $\mathbb{R}^4$. The way it was conceived allows a new interpretation of the structure of hypercomplex space, with a new geometry and spatial topology, and a meaning for the geometric representation of $\mathbb{R}^4$ space to infinity. Therefore, it has been described as the tensorial metric formula in the Alpha Group. It will be shown that the Riemannian and Euclidean distance metrics between infinitesimal surfaces are represented as special cases of the metric of the Alpha Group.