2008/05/31 by Nabil. L. Youssef, NABIL. L. YOUSSEF, A. M. Sid-Ahmed +1 · 15 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Connection (principal bundle) #Curvature #Differential geometry #Geometric Analysis and Curvature Flows #Holonomy #Manifold (fluid mechanics) #Noncommutative and Quantum Gravity Theories #Parallel transport #Parallelism (grammar) #Tangent #Tangent bundle #Torsion (gastropod) #gr-qc #math-ph #math.DG #math.MP #msc:53A40 #msc:53B40 #msc:53B50
paper · pdf · doi:10.1142/s0219887808003235
published in International Journal of Geometric Methods in Modern Physics 05(07), 1109-1135 (World Scientific) · 27 pages, LaTeX-file, The last version of this paper was replaced by mistake (by arXiv: 0905.0209[gr-qc])
openalex publication_date 2008/11/01 · arxiv created 2009/08/05 · arxiv updated 2015/05/12 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
In this paper, we study Absolute Parallelism (AP-) geometry on the tangent bundle TM of a manifold M. Accordingly, all geometric objects defined in this geometry are not only functions of the positional argument x, but also depend on the directional argument y. Moreover, many new geometric objects, which have no counterpart in the classical AP-geometry, emerge in this different framework. We refer to such a geometry as an Extended Absolute Parallelism (EAP-) geometry. The building blocks of the EAP-geometry are a nonlinear connection (assumed given a priori) and 2n linearly independent vector fields (of special form) defined globally on TM defining the parallelization. Four different d-connections are used to explore the properties of this geometry. Simple and compact formulae for the curvature tensors and the W-tensors of the four defined d-connections are obtained, expressed in terms of the torsion and the contortion tensors of the EAP-space. Further conditions are imposed on the canonical d-connection assuming that it is of Cartan type (resp. Berwald type). Important consequences of these assumptions are investigated. Finally, a special form of the canonical d-connection is studied under which the classical AP-geometry is recovered naturally from the EAP-geometry. Physical aspects of some of the geometric objects investigated are pointed out and possible physical implications of the EAP-space are discussed, including an outline of a generalized field theory on the tangent bundle TM of M.