A Global Approach to Absolute Parallelism Geometry

In this paper we provide a global investigation of the geometry of parallelizable manifolds (or absolute parallelism geometry) frequently used in applications. We discuss 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.