Parallelism Structure on a Smooth Manifold

TL;DR

This paper develops a theory of parallelism structures on smooth manifolds using extensor theory, introducing Cartan connections and exploring deformed and relative parallelism relevant to gravitational geometry.

Contribution

It presents a novel intrinsic framework for parallelism on manifolds, including new Cartan connection operators and their applications to gravitational theories.

Findings

Introduction of two types of Cartan connection operators

Intrinsic versions of Cartan structure equations

Analysis of deformed and relative parallelism structures

Abstract

Using the theory of extensors developed in a previous paper we present a theory of the parallelism structure on arbitrary smooth manifold. Two kinds of Cartan connection operators are introduced and both appear in intrinsic versions (i.e., frame independent) of the first and second Cartan structure equations. Also, the concept of deformed parallelism structures and relative parallelism structures which play important role in the understanding of geometrical theories of the gravitational field are investigated.