Sprays and Connections on the Tangent Bundle of Order Two

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.Comment: 15 pages,laTeX fil

L-Connections and Associated Tensors

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 $J$.Comment: 11 pages, LaTeX fil

Characterization of Finsler Spaces of Scalar Curvature

The aim of the present paper is to provide an intrinsic investigation of two special Finsler spaces whose defining properties are related to Berwald connection, namely, Finsler space of scalar curvature and of constant curvature. Some characterizations of a Finsler space of scalar curvature are proved. Necessary and sufficient conditions under which a Finsler space of scalar curvature reduces to a Finsler space of constant curvature are investigated.Comment: LaTeX file, 10 page

Some Types of Recurrence in Finsler geometry

The pullback approach to global Finsler geometry is adopted. Three classes of recurrence in Finsler geometry are introduced and investigated: simple recurrence, Ricci recurrence and concircular recurrence. Each of these classes consists of four types of recurrence. The interrelationships between the different types of recurrence are studied. The generalized concircular recurrence, as a new concept, is singled out.Comment: LaTex file, 13 pages, Concluding remarks are changed, Last diagram is modifie

On Horizontal Recurrent Finsler Connections

In this paper we adopt the pullback approach to global Finsler geometry. We investigate horizontally recurrent Finsler connections. We prove that for each scalar ($\pi$)1-form $A$, there exists a unique horizontally recurrent Finsler connection whose $h$-recurrence form is $A$. This result generalizes the existence and uniqueness theorem of Cartan connection. We then study some properties of a special kind of horizontally recurrent Finsler connection, which we call special HRF-connection.Comment: 10 peges, LaTeX file, Few typos corrected, References adde

L-regular linear connections

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 $M$.Comment: 12 pages, LaTeX file, Minor change (concerning reference No. 10

Nullity distributions associated with Chern connection

The nullity distributions of the two curvature tensors \, \overast{R} and \overast{P} of the Chern connection of a Finsler manifold are investigated. The completeness of the nullity foliation associated with the nullity distribution $\N_{R^\ast}$ is proved. Two counterexamples are given: the first shows that $\N_{R^\ast}$ does not coincide with the kernel distribution of \, \overast{R}; the second illustrates that $\N_{P^\ast}$ is not completely integrable. We give a simple class of a non-Berwaldian Landsberg spaces with singularities.Comment: Major modifications, An Example added at the end of the paper, Some Maple calculations inserte

A note on "Sur le noyau de l'op\'{e}rateur de courbure d'une vari\'{e}t\'{e} finsl\'{e}rienne" [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.Comment: 4 pages, LaTeX file, small correction in the counterexample. Some modifications have been performed especially at the end of the pape

Computing nullity and kernel vectors using NF-package: Counterexamples

A computational technique for calculating nullity vectors and kernel vectors, using the new Finsler package, is introduced. As an application, three interesting counterexamples are given. The first counterexample shows that the two distributions $\mathrm{Ker}_R$ and $\N_R$ do not coincide. The second shows that the nullity distribution $\N_{P^\circ}$ is not completely integrable. The third shows that the nullity distribution $\N_\mathfrak{R}$ is not a sub-distribution of the nullity distribution $\N_{R^\circ}$.Comment: 10 pages, LaTeX file, More references added, some typos correcte

On Generalized Randers Manifolds

By a Randers' structure on a manifold $M$ we mean a Finsler structure $L^*=L+\alpha$, where $L$ is a Riemannian structure and $\alpha$ 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.Comment: 10 pages, LaTeX fil