Group-Theoretic Matching of the Length and Equality Principles in Geometry

TL;DR

This paper reconciles two group-theoretic methods for describing Riemannian spaces by extending the deformed diffeomorphism group to include gauge rotations, thus unifying length and equality principles in geometry.

Contribution

It introduces a univocal extension of the deformed diffeomorphism group to incorporate gauge rotations, unifying two foundational geometric principles.

Findings

Reconciliation of two methods of group-theoretical description of Riemannian spaces.

Extension of the group to include gauge rotations that preserve vector lengths.

Implementation of Klein's principle of equality alongside the length principle.

Abstract

Deformed generalized gauge groups, whch were created from physical considerations and made it possible to clarify some long-standing problems in physics, such as the problem of motion and the problem of the energy of the gravitational field, simultaneously carried out the implementation of the Klein's Erlangen program for spaces with variable curvature, and for Riemannian spaces even in two different ways. In this paper, this issue is considered from a geometric point of view and these two methods of group-theoretical description of Riemannian spaces are reconciled. The paper deals with the canonical deformed group of diffeomorphisms with a given length scale which describes the motion of unit scales in a Riemannian space. This allows one to measure the lengths of arbitrary curves implementing the length principle which was laid by B. Riemann at the foundation of geometry. We present a method of univocal extension of this group to a group which contains gauge rotations of vectors (the group of parallel transports) whose transformations leave unchanged the lengths of vectors and the corners between them, implementing for Riemannian spaces the Klein's principle of equality and matching both principles of the foundations of geometry, thus overcoming the Riemann-Klein antagonism.