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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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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...

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## Extended Absolute Parallelism Geometry

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

36.61%

#Mathematics - Differential Geometry#General Relativity and Quantum Cosmology#Mathematical Physics#53B40, 53A40, 53B50

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...

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## On Finslerized Absolute Parallelism spaces

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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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

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