2012/09/30 by Nabil L. Youssef, Waleed A. Elsayed
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Black Holes and Theoretical Physics #Tensor decomposition and applications #gr-qc #math.DG #msc:51P05 #msc:53A40 #msc:53C05
paper · pdf · doi:10.1016/s0034-4877(13)00016-5
published as Rep. Math. Phys., 72, 1 (2013), 1-23 · LateX-file, 23 pages, small modifications have been made, some typos corrected
arxiv created 2013/07/15 · openalex publication_date 2013/08/01 · arxiv updated 2015/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we provide a 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.