A Global Approach to Absolute Parallelism Geometry

TL;DR

This paper offers a comprehensive global analysis of absolute parallelism geometry, introducing a unique canonical connection, exploring curvature tensors, and examining the Wanas tensor, with implications for physical theories.

Contribution

It provides a global existence and uniqueness theorem for the canonical connection and expresses curvature tensors solely in terms of torsion, advancing the understanding of absolute parallelism geometry.

Findings

Derived identities from Bianchi identities.

Defined and analyzed the Wanas tensor.

Compared global and local geometric object expressions.

Abstract

In this paper we provide a \emph{global} investigation of the geometry of parallelizable manifolds (or absolute parallelism geometry) frequently used for application. We discuss the different linear connections and curvature tensors from a global point of view. We give an existence and uniqueness theorem for a remarkable linear connection, called the canonical connection. Different curvature tensors are expressed in a compact form in terms of the torsion tensor of the canonical connection only. Using the Bianchi identities, some interesting identities are derived. An important special fourth order tensor, which we refer to as Wanas tensor, is globally defined and investigated. Finally a double-view for the fundamental geometric objects of an absolute parallelism space is established: The expressions of these geometric objects are computed in the parallelization basis and are compared with the corresponding local expressions in the natural basis. Physical aspects of some geometric objects considered are pointed out.