99 research outputs found
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 on
the tangent bundle of a differentiable manifold . 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 on a
manifold (not on ) satisfying certain conditions. He investigated the
-cohomology induced on by 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 on of constant rank such that and that
; . We found that has properties similar to those
of , 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 be the tangent bundle of a
differentiable manifold and the natural almost-tangent structure .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 ()1-form , there exists a unique horizontally recurrent Finsler
connection whose -recurrence form is . 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
-regular linear connection on a nonlinear -connection on . The
route we have followed is significantly different from that of Grifone. We
introduce an almost-complex and an almost-product structures on by means of
a given -regular linear connection on . The product of these two
structures defines a nonlinear -connection on , which generalizes
Grifone's nonlinear connection.
The seconed part is devoted to the converse problem: associating to each
nonlinear -connection \G on an -regular linear connection on ;
called the -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 -lift of a homogeneous
-connection \G, called the Berwald -lift of \G. Then we particularize our
study to the -lift of a conservative -connection. This -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 be the tangent bundle of a
differentiable manifold and be the natural almost-tangent structure on
.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 is proved. Two counterexamples are given: the first
shows that does not coincide with the kernel distribution of \,
\overast{R}; the second illustrates that 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 of the h-curvature
of Cartan connection and the associated nullity distribution do not
coincide, contrary to Akbar-Zadeh's result \cite{akbar.nul3.}. We give
sufficient conditions for and 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 and do not coincide. The second shows
that the nullity distribution is not completely integrable. The
third shows that the nullity distribution is not a
sub-distribution of the nullity distribution .Comment: 10 pages, LaTeX file, More references added, some typos correcte
On Generalized Randers Manifolds
By a Randers' structure on a manifold we mean a Finsler structure
, where is a Riemannian structure and is a 1-form on
. This structure was first introduced by Randers ~\cite{[8]} from the
standpoint of general relativity. In this paper, we replace 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
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