Geometry of Parallelizable Manifolds in the Context of Generalized Lagrange Spaces
| dc.creator | Wanas, M. I. | |
| dc.creator | Youssef, N. L. | |
| dc.creator | Sid-Ahmed, A. M. | |
| dc.date | 2007-04-16 | |
| dc.date | 2007-11-30 | |
| dc.date.accessioned | 2026-07-07T09:36:17Z | |
| dc.date.available | 2026-07-07T09:36:17Z | |
| dc.description | In this paper, we deal with a generalization of the geometry of parallelizable manifolds, or the absolute parallelism (AP-) geometry, in the context of generalized Lagrange spaces. All geometric objects defined in this geometry are not only functions of the positional argument $x$, but also depend on the directional argument $y$. In other words, instead of dealing with geometric objects defined on the manifold $M$, as in the case of classical AP-geometry, we are dealing with geometric objects in the pullback bundle $π^{-1}(TM)$ (the pullback of the tangent bundle $TM$ by $ π: T M\longrightarrow M$). Many new geometric objects, which have no counterpart in the classical AP-geometry, emerge in this more general context. We refer to such a geometry as generalized AP-geometry (GAP-geometry). In analogy to AP-geometry, we define a $d$-connection in $π^{-1}(TM)$ having remarkable properties, which we call the canonical $d$-connection, in terms of the unique torsion-free Riemannian $d$-connection. In addition to these two $d$-connections, two more $d$-connections are defined, the dual and the symmetric $d$-connections. Our space, therefore, admits twelve curvature tensors (corresponding to the four defined $d$-connections), three of which vanish identically. Simple formulae for the nine non-vanishing curvatures tensors are obtained, in terms of the torsion tensors of the canonical $d$-connection. The different $W$-tensors admitted by the space are also calculated. All contractions of the $h$- and $v$-curvature tensors and the $W$-tensors are derived. Second rank symmetric and skew-symmetric tensors, which prove useful in physical applications, are singled out. | |
| dc.description | 20 pages, LaTeX file, Presented in "The International Conference on Finsler Extensions of Relativity Theory" held at Cairo, Egypt, November 4-10,2006. AMS Subject Classification: 53B40, 53A40, 53B50 (References have been modified) | |
| dc.identifier | https://arxiv.org/abs/0704.2001 | |
| dc.identifier | http://arxiv.org/abs/0704.2001 | |
| dc.identifier | Balkan J. Geom. Appl., 13,2 (2008), 120-139. | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/160068 | |
| dc.subject | General Relativity and Quantum Cosmology | |
| dc.subject | Differential Geometry | |
| dc.title | Geometry of Parallelizable Manifolds in the Context of Generalized Lagrange Spaces | |
| dc.type | text |