A Global Approach to Absolute Parallelism Geometry

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.

LateX-file, 23 pages, small modifications have been made, some typos corrected

References in corpus (5)

• Vacuum Non Singular Black Hole Solutions in Tetrad Theory of Gravitation
• Charged Axially Symmetric Solution, Energy and Angular Momentum in Tetrad Theory of Gravitation
• Linear Connections and Curvature Tensors in the Geometry of Parallelizable Manifolds
• The Accelerating Expansion of the Universe and Torsion Energy
• Absolute Parallelism Geometry: Developments, Applications and Problems

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