Geometry of Parallelizable Manifolds in the Context of Generalized Lagrange Spaces

Details Geometry of Parallelizable Manifolds in the Context of Generalized Lagrange Spaces Geometry of Parallelizable Manifolds in the Context of Generalized Lagrange Spaces

2007-04-16

arxiv.org/abs/0704.2001v2

Balkan J. Geom. Appl., 13,2 (2008), 120-139.

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

20 pages, LaTeX file, Presented in "The International Conference on Finsler Extensions of Relativity Theory" held at Cairo, Eg

Scientific paper

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 $\pi^{-1}(TM)$ (the pullback of the tangent bundle $TM$ by $ \pi: 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 $\pi^{-1}(TM)$ 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.

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