arXiv subjects
Publications and source records attributed to Nabil L. Youssef.
The $π$-exterior derivative $ød$, which is the Finslerian generalization of the (usual) exterior derivative $d$ of Riemannian geometry, is defined. The notion of a $ød$-closed vector field is introduced and investigated. Various characterizations of $ød$-closed vector fields are established. Some results concerning $ød$-closed vector fields in relation to certain special Finsler spaces are obtained.
In this paper we discuss curvature tensors in the context of Absolute Parallelism geometry. Different curvature tensors are expressed in a compact form in terms of the torsion tensor of the canonical connection. Using the Bianchi identities some other identities are derived from the expressions obtained. These identities, in turn, are used to reveal some of the properties satisfied by an intriguing fourth order tensor which we refer to as Wanas tensor. A further condition on the canonical connection is imposed, assuming it is semi-symmetric. The formulae thus obtained, together with other formulae (Ricci tensors and scalar curvatures of the different connections admitted by the space) are calculated under this additional assumption. Considering a specific form of the semi-symmetric connection causes all nonvanishing curvature tensors to coincide, up to a constant, with the Wanas tensor. Physical aspects of some of the geometric objects considered are mentioned.
The aim of the present paper is to establish a global theory of conformal changes in Finsler geometry. Under this change, we obtain the relationships between the most important geometric objects associated to $(M,L)$ and the corresponding objects associated to $(M,\tilde{L})$, $\widetilde{L}=e^{σ(x)} L$ being the Finsler conformal transformation. We have found explicit global expressions relating the two associated Cartan connections $\nabla$ and $\widetilde{\nabla}$, the two associated Berwald connections $D$ and $\widetilde{D}$ and the two associated Barthel connections $Γ$ and $\widetildeΓ$. The relationships between the corresponding curvature tensors have been also found. The relations thus obtained lead in turn to several interesting results. Among the results obtained, is a characterization of conformal changes, a characterization of homotheties, some conformal invariants and conformal $σ$-invariants. In addition, several useful identities have been found. Our global theory of conformal Finsler geometry is established within the Pull-back approach, making simultaneous use of the Klein-Grifone approach. This has been done via certain links we have found between both approaches (This shows that these two approaches to global Finsler geometry are not alternatives but rather complementary). Although our treatment is entirely global, the local expressions of the obtained results, when calculated, coincide with the existing classical local results.
In the present paper, we consider two different {\em Finsler} structures $L$ and $L^*$ on the same base manifold $M$, with no relation preassumed between them. \par Introducing the $π$-tensor field representing the difference between the Cartan connections associated with $L$ and $L^*$, we investigate the conditions, to be satisfied by this $π$-tensor field, for the geometric objects associated with $L$ and $L^*$ to have the same properties. Among the items investigated in the paper, we consider the properties of being a geodesic, a Jacobi field, a Berwald manifold, a locally Minkowskian manifold and a Landsberg manifold. \par It should be noticed that our approach is intrinsic, i.e., it does not make use of local coordinate techniques.
The aim of this paper is to generalize the theory of nonlinear connections of Grifone ([3] and [4]). We adopt the point of view of Anona [1] and continue developing the approach established by the first author in [10]. The first part of the work is devoted to the problem of associating to each $L$-regular linear connection on $M$ a nonlinear $L$-connection on $M$. The route we have followed is significantly different from that of Grifone. We introduce an almost-complex and an almost-product structures on $M$ by means of a given $L$-regular linear connection on $M$. The product of these two structures defines a nonlinear $L$-connection on $M$, which generalizes Grifone's nonlinear connection. The seconed part is devoted to the converse problem: associating to each nonlinear $L$-connection \G on $M$ an $L$-regular linear connection on $M$; called the $L$-lift of \G. The existence of this lift is established and the fundamental tensors associated with it are studied. In the third part, we investigate the $L$-lift of a homogeneous $L$-connection \G, called the Berwald $L$-lift of \G. Then we particularize our study to the $L$-lift of a conservative $L$-connection. This $L$-lift enjoys some interesting properties. We finally deduce various identities concerning the curvature tensors of such a lift. Grifone's theory can be retrieved by letting $M$ be the tangent bundle of a differentiable manifold and $L$ be the natural almost-tangent structure $J$ on $M$.
By a Randers' structure on a manifold $M$ we mean a Finsler structure $L^*=L+α$, where $L$ is a Riemannian structure and $α$ is a 1-form on $M$. This structure was first introduced by Randers ~\cite{[8]} from the standpoint of general relativity. In this paper, we replace $L$ by a Finsler structure, calling the resulting manifold a generalized Randers manifold. On one hand, we develop in some depth generalized Randers manifolds. On the other hand, we apply the results obtained in a foregoing paper ~\cite{[12]} to generalized Randers manifolds to obtain some new results in that domain. Among many results, we establish a necessary and sufficient condition for a generalized Randers manifold to be a general Landsberg manifold. It should be noticed that our approach is in general a global one.
The theory of connections in Finsler geometry is not satisfactorily established as in Riemannian geometry. Many trials have been carried out to build up an adequate theory. One of the most important in this direction is that of Grifone ([3] and [4]). His approach to the theory of nonlinear connections was accomplished in [3], in which his new definition of a nonlinear connection is easly handled from the algebraic point of view. Grifone's approach is based essentially on the natural almost-tangent structure $J$ on the tangent bundle $T(M)$ of a differentiable manifold $M$. This structure was introduced and investigated by Klein and Voutier [5]. Anona in [1] generalized the natural almost-tangent structure by considering a vector 1-form $L$ on a manifold $M$ (not on $T(M)$) satisfying certain conditions. He investigated the $d_L$-cohomology induced on $M$ by $L$ and generalized some of Grifone's results. \par In this paper, we adopt the point of view of Anona [1] to generalize Grifone's theory of nonlinear connections [3]: We consider a vector 1-form $L$ on $M$ of constant rank such that $[L,L]=0$ and that $Im(L_z)=Ker(L_z)$; $z\in M$. We found that $L$ has properties similar to those of $J$, which enables us to generalize systematically the most important results of Grifone's theory. \par The theory of Grifone is retrieved, as a special case of our work, by letting $M$ be the tangent bundle of a differentiable manifold and $L$ the natural almost-tangent structure $J$.
Adopting the global approach to tangent bundles of order two established in[1], we develop this approach to find new results. We also generalize various results of [3], [4] and [6] to the geometry of tangent bundles of order two.