arXiv subjects

Nabil L. Youssef

Publications and source records attributed to Nabil L. Youssef.

At least 37 records · Page 2Linked to original sources

arXiv2013

New Finsler Package

The book "Handbook of Finsler geometry" has been included with a CD containing an elegant Maple package, FINSLER, for calculations in Finsler geometry. Using this package, an example concerning a Finsler generalization of Einstein's vacuum field equations was treated. In this example, the calculation of the components of the hv-curvature of Cartan connection leads to wrong expressions. On the other hand, the FINSLER package works only in dimension four. We introduce a new Finsler package in which we fix the two problems and solve them. Moreover, we extend this package to compute not only the geometric objects associated with Cartan connection but also those associated with Berwald, Chern and Hashiguchi connections in any dimension. These improvements have been illustrated by a concrete example. Furthermore, the problem of simplifying tensor expressions is treated. This paper is intended to make calculations in Finsler geometry more easier and simpler.

math.DG↗

arXiv2013

A note on "Sur le noyau de l'opérateur de courbure d'une variété finslérienne" [C. R. Acad. Sci. Paris, t. 272 (1971), 807-810]

In this note, adopting the pullback formalism of global Finsler geometry, we show by a counterexample that the kernel $\text{Ker}_R$ of the h-curvature $R$ of Cartan connection and the associated nullity distribution $\N_R$ do not coincide, contrary to Akbar-Zadeh's result \cite{akbar.nul3.}. We give sufficient conditions for $\text{Ker}_R$ and $\N_R$ to coincide.

math.DG↗

arXiv2013

A Global Approach to Absolute Parallelism Geometry

In this paper we provide a \emph{global} investigation of the geometry of parallelizable manifolds (or absolute parallelism geometry) frequently used for application. We discuss the different linear connections and curvature tensors from a global point of view. We give an existence and uniqueness theorem for a remarkable linear connection, called the canonical connection. Different curvature tensors are expressed in a compact form in terms of the torsion tensor of the canonical connection only. Using the Bianchi identities, some interesting identities are derived. An important special fourth order tensor, which we refer to as Wanas tensor, is globally defined and investigated. Finally a double-view for the fundamental geometric objects of an absolute parallelism space is established: The expressions of these geometric objects are computed in the parallelization basis and are compared with the corresponding local expressions in the natural basis. Physical aspects of some geometric objects considered are pointed out.

gr-qc↗

arXiv2013

Concircular $π$-Vector Fields and Special Finsler Spaces

The aim of the present paper is to investigate intrinsically the notion of a concircular $π$-vector field in Finsler geometry. This generalizes the concept of a concircular vector field in Riemannian geometry and the concept of a concurrent vector field in Finsler geometry. Some properties of concircular $π$-vector fields are obtained. Different types of recurrence are discussed. The effect of the existence of a concircular $π$-vector field on some important special Finsler spaces is investigated. The whole work is formulated in a coordinate-free form.

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arXiv2012

Nullity distributions associated to Cartan connection

The Klein-Grifone approach to global Finsler geometry is adopted. The nullity distributions of the three curvature tensors of Cartan connection are investigated. Nullity distributions concerning certain relevant special Finsler spaces are considered. Concrete examples are given whenever the situation needs.

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arXiv2012

On Finslerized Absolute Parallelism spaces

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.

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arXiv2012

On Concircularly Recurrent Finsler Manifolds

Two special Finsler spaces have been introduced and investigated, namely $R^h$-recurrent Finsler space and consircularly recurrent Finsler space. The defining properties of these spaces are formulated in terms of the first curvature tensor of Cartan connection. The following three results constitute the main object of the present paper: 1. A concircularly flat Finsler manifold is necessarily of constant curvature (Theorem A); 2. Every $R^h$-recurrent Finsler manifold is concirculaly recurrent with the same recurrence form (Theorem B); 3. Every horizontally integrable concircularly recurrent Finsler manifold is $R^h$-recurrent with the same recurrence form (Theorem C). The whole work is formulated in a coordinate-free form.

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arXiv2011

Telleparallel Lagrange Geometry and a Unified Field Theory: Linearization of the Field Equations

The present paper is a natural continuation of our previous paper: "Teleparallel Lagrange geometry and a unified field theory, Class. Quantum Grav., 27 (2010), 045005 (29pp)" \cite{WNA}. In this paper, we apply a linearization scheme on the field equations obtained in \cite{WNA}. Three important results under the linearization assumption are accomplished. First, the vertical fundamental geometric objects of the EAP-space loose their dependence on the positional argument $x$. Secondly, our linearized theory in the Cartan-type case coincides with the GFT in the first order of approximation. Finally, an approximate solution of the vertical field equations is obtained.

gr-qc↗

arXiv2010

Generalized $β$-conformal change and special Finsler spaces

In this paper, we investigate the change of Finslr metrics $$L(x,y) \to\bar{L}(x,y) = f(e^{σ(x)}L(x,y),β(x,y)),$$ which we refer to as a generalized $β$-conformal change. Under this change, we study some special Finsler spaces, namely, quasi C-reducible, semi C-reducible, C-reducible, $C_2$-like, $S_3$-like and $S_4$-like Finsler spaces. We also obtain the transformation of the T-tensor under this change and study some interesting special cases. We then impose a certain condition on the generalized $β$-conformal change, which we call the b-condition, and investigate the geometric consequences of such condition. Finally, we give the conditions under which a generalized $β$-conformal change is projective and generalize some known results in the literature.

math.DG↗

arXiv2010

Teleparallel Lagrange Geometry and a Unified Field Theory

In this paper, we construct a field theory unifying gravity and electromagnetism in the context of Extended Absolute Parallelism (EAP-) geometry. This geometry combines, within its structure, the geometric richness of the tangent bundle and the mathematical simplicity of Absolute Parallelism (AP-) geometry. The constructed field theory is a generalization of the Generalized Field Theory (GFT) formulated by Mikhail and Wanas. The theory obtained is purely geometric. The horizontal (resp. vertical) field equations are derived by applying the Euler-Lagrange equations to an appropriate horizontal (resp. vertical) scalar Lagrangian. The symmetric part of the resulting horizontal (resp. vertical) field equations gives rise to a generalized form of Einstein's field equations in which the horizontal (resp. vertical) energy-momentum tensor is purely geometric. The skew-symmetric part of the resulting horizontal (resp. vertical) field equations gives rise to a generalized form of Maxwell equations in which the electromagnetic field is purely geometric. Some interesting special cases, which reveal the role of the nonlinear connection in the obtained field equations, are examined. Finally, the condition under which our constructed field equations reduce to the GFT is explicitly established.

gr-qc↗

arXiv2009

Concurrent $π$-vector fields and energy beta-change

The present paper deals with an \emph{intrinsic} investigation of the notion of a concurrent $π$-vector field on the pullback bundle of a Finsler manifold $(M,L)$. The effect of the existence of a concurrent $π$-vector field on some important special Finsler spaces is studied. An intrinsic investigation of a particular $β$-change, namely the energy $β$-change ($\widetilde{L}^{2}(x,y)=L^{2}(x,y)+ B^{2}(x,y) with \ B:=g(\barζ,\barη)$; $\barζ $ being a concurrent $π$-vector field), is established. The relation between the two Barthel connections $Γ$ and $\widetildeΓ$, corresponding to this change, is found. This relation, together with the fact that the Cartan and the Barthel connections have the same horizontal and vertical projectors, enable us to study the energy $β$-change of the fundamental linear connection in Finsler geometry: the Cartan connection, the Berwald connection, the Chern connection and the Hashiguchi connection. Moreover, the change of their curvature tensors is concluded. It should be pointed out that the present work is formulated in a prospective modern coordinate-free form.

math.DG↗

arXiv2009

Generalized $β$-conformal change of Finsler metrics

In this paper, we introduce and investigate a general transformation or change of Finsler metrics, which is referred to as a generalized $β$-conformal change: $$L(x,y) \longrightarrow\overline{L}(x,y) = f(e^{σ(x)}L(x,y),β(x,y)).$$ This transformation combines both $β$-change and conformal change in a general setting. The change, under this transformation, of the fundamental Finsler connections, together with their associated geometric objects, are obtained. Some invariants and various special Finsler spaces are investigated under this change. The most important changes of Finsler metrics existing in the literature are deduced from the generalized $β$-conformal change as special cases.

math.DG↗

arXiv2009

Conformal change of special Finsler spaces

The present paper is a continuation of a foregoing paper [Tensor, N. S., 69 (2008), 155-178]. The main aim is to establish \emph{an intrinsic investigation} of the conformal change of the most important special Finsler spaces, namely, $C^{h}$-recurrent, $C^{v}$-recurrent, $C^{0}$-recurrent, $C_{2}$-like, quasi-$C$-reducible, $C$-reducible, Berwald space, $S^{v}$-recurrent, $P^*$-Finsler manifold, $R_{3}$-like, $P$-symmetric, Finsler manifold of $p$-scalar curvature and Finsler manifold of $s$-$ps$-curvature. Necessary and sufficient conditions for such special Finsler manifolds to be invariant under a conformal change are obtained. Moreover, the conformal change of Chern and Hashiguchi connections, as well as their curvature tensors, are given.

math.DG↗

arXiv2009

Intrinsic Theory of Projective Changes in Finsler Geometry

The aim of the present paper is to provide an intrinsic investigation of projective changes in Finlser geometry, following the pullback formalism. Various known local results are generalized and other new intrinsic results are obtained. Nontrivial characterizations of projective changes are given. The fundamental projectively invariant tensors, namely, the projective deviation tensor, the Weyl torsion tensor, the Weyl curvature tensor and the Douglas tensor are investigated. The properties of these tensors and their interrelationships are obtained. Projective connections and projectively flat manifolds are characterized. The present work is entirely intrinsic (free from local coordinates).

math.DG↗

arXiv2008

Geometric objects associated with the fundamental connections in Finsler geometry

The aim of the present paper is to provide an \emph{intrinsic} investigation of the properties of the most important geometric objects associated with the fundamental linear connections in Finsler geometry. We investigate intrinsically the most general relations concerning the torsion tensor fields and the curvature tensor fields associated with a given regular connection on the pullback bundle of a Finsler manifold. These relations, in turn, play a key role in obtaining other interesting results concerning the properties of the most important geometric objects associated with the fundamental canonical linear connections on the pullback bundle of a Finsler manifold, namely, the Cartan connection, the Berwald connection, the Chern (Rund) connection and the Hashiguchi connection. For the sake of completeness and for comparison reasons, we provide an appendix presenting a global survey of canonical linear connections in Finsler geometry and the fundamental geometric objects associated with them.

math.DG↗

arXiv2008

A Global Approach to the Theory of Connections in Finsler Geometry

Adopting the pullback approach to global Finsler geometry, the aim of the present paper is to provide intrinsic (coordinate-free) proofs of the existence and uniqueness theorems for the Chern (Rund) and Hashiguchi connections on a Finsler manifold. To accomplish this, we introduce and investigate the notions of semispray and nonlinear connection associated with a given regular connection, in the pullback bundle. Moreover, it is shown that for the the Chern (Rund) and Hashiguchi connections, the associated semispray coincides with the canonical spray and the associated nonlinear connection coincides with the Barthel connection. Explicit intrinsic expressions relating these connections and the Cartan connection are deduced. Although our investigation is entirely global, the local expressions of the obtained results, when calculated, coincide with the existing classical local results. We provide, for the sake of completeness and for comparison reasons, two appendices, one of them presenting a global survey of canonical linear connections in Finsler geometry and the other presenting a local survey of our global approach.

math.DG↗

arXiv2007

Cartan and Berwald connections in the pullback formalism

Adopting the pullback approach to global Finsler geometry, the aim of the present paper is to provide new intrinsic (coordinate-free) proofs of intrinsic versions of the existence and uniqueness theorems for the Cartan and Berwald connections on a Finsler manifold. To accomplish this, the notions of semispray and nonlinear connection associated with a given regular connection, in the pullback bundle, is introduced and investigated. Moreover, it is shown that for the Cartan and Berwald connections, the associated semispray coincides with the canonical spray and the associated nonlinear connection coincides with the Barthel connection. An explicit intrinsic expression relating both connections is deduced.

math.DG↗

arXiv2007

A Global Approach to the Theory of Special Finsler Manifolds

The aim of the present paper is to provide a global presentation of the theory of special Finsler manifolds. We introduce and investigate globally (or intrinsically, free from local coordinates) many of the most important and most commonly used special Finsler manifolds: locally Minkowskian, Berwald, Landesberg, general Landesberg, $P$-reducible, $C$-reducible, semi-$C$-reducible, quasi-$C$-reducible, $P^{*}$-Finsler, $C^{h}$-recurrent, $C^{v}$-recurrent, $C^{0}$-recurrent, $S^{v}$-recurrent, $S^{v}$-recurrent of the second order, $C_{2}$-like, $S_{3}$-like, $S_{4}$-like, $P_{2}$-like, $R_{3}$-like, $P$-symmetric, $h$-isotropic, of scalar curvature, of constant curvature, of $p$-scalar curvature, of $s$-$ps$-curvature. The global definitions of these special Finsler manifolds are introduced. Various relationships between the different types of the considered special Finsler manifolds are found. Many local results, known in the literature, are proved globally and several new results are obtained. As a by-product, interesting identities and properties concerning the torsion tensor fields and the curvature tensor fields are deduced. Although our investigation is entirely global, we provide; for comparison reasons, an appendix presenting a local counterpart of our global approach and the local definitions of the special Finsler spaces considered.

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