Página 1 dos resultados de 3 itens digitais encontrados em 0.003 segundos

Geometry of Parallelizable Manifolds in the Context of Generalized Lagrange Spaces

Wanas, M. I.; Youssef, N. L.; Sid-Ahmed, A. M.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.61%
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...

Extended Absolute Parallelism Geometry

Youssef, Nabil. L.; Sid-Ahmed, A. M.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
36.61%
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...

On Finslerized Absolute Parallelism spaces

Youssef, Nabil L.; Sid-Ahmed, Amr M.; Taha, Ebtsam H.
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.61%
The aim of the present paper is to construct and investigate a Finsler structure within the framework of a Generalized Absolute Parallelism space (GAP-space). The Finsler structure is obtained from the vector fields forming the parallelization of the GAP-space. The resulting space, which we refer to as a Finslerized Parallelizable space, combines within its geometric structure the simplicity of GAP-geometry and the richness of Finsler geometry, hence is potentially more suitable for applications and especially for describing physical phenomena. A study of the geometry of the two structures and their interrelation is carried out. Five connections are introduced and their torsion and curvature tensors derived. Some special Finslerized Parallelizable spaces are singled out. One of the main reasons to introduce this new space is that both Absolute Parallelism and Finsler geometries have proved effective in the formulation of physical theories, so it is worthy to try to build a more general geometric structure that would share the benefits of both geometries.; Comment: Some references added and others removed, PACS2010, Typos corrected, Amendemrnts and revisions performed