A class of semibasic vector 1-forms on Finsler manifolds
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A class of semibasic vector 1-forms on Finsler manifolds Akbar Tayebi · Mansoor Barzegari
Published online: 9 October 2014 © Akadémiai Kiadó, Budapest, Hungary 2014
Abstract In this paper, we define conservative semibasic vector 1−forms on the tangent bundle of a Finsler manifold. Using these vector 1−forms, we characterize conservative L−Ehresmann connections with respect to the energy function. Then we find a correspondence between torsion-free semibasic vector 1−forms and the subset of vertical vector fields. Taking into account this correspondence, we construct a class of semisprays that generates the Ehresmann connections mentioned above. Keywords
Finsler manifold · Ehresmann connection · Semibasic vector 1-form
Mathematics Subject Classification
53C07 · 53C60
1 Introduction The difference of two Ehresmann connections over a manifold M is a semibasic vector 1−form on T M. Motivated by this fact, it is natural to define a conservative semibasic vector 1−form on a Finsler manifold with respect to an energy function induced by a Finsler metric: a semibasic vector 1−form is conservative on a Finsler manifold (M, E) if it can be expressed as a difference of two conservative Ehresmann connections on M. The conservativity property of a semibasic vector 1−form L on T M implies the conservativity of an L−Ehresmann connection, which is defined by h L := h 0 + L + [J, (d L E)# ], where h 0 is the Berwald connection. It is well known that this class of Ehresmann connections contains the Wagner connection of the Finsler manifold (M, E) and among them the conservative ones are conformally closed. For more details see [2] and [3]. A. Tayebi · M. Barzegari (B) Faculty of Science, Department of Mathematics, University of Qom, Qom, Iran e-mail: [email protected] A. Tayebi e-mail: [email protected]
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In Sect. 4, using conservative semibasic vector 1−forms, we characterize conservative L−Ehresmann connections with respect to the energy function E. The weak torsion of an L−Ehresmann connection is in the following form t L = [J, L]. Thus, it is natural to define torsion-freeness of a semibasic vector 1−form L on the tangent manifold T M by requiring the condition [J, L] = 0. In Sect. 5, we establish a correspondence between the subset of vertical vector fields Xv (T M) and the set of torsion–free semibasic vector 1−forms on T M. Taking into account this correspondence, in Subsect. 4.1, we introduce a set of semisprays as follows V
S = S0 + 2 V + 2 (d [J,V ] E)# ,
where V is a vertical vector field on T M. If V is a two-homogeneous vertical vector field, V
then the semispray S is a spray and we can find a projectively related relation between two sprays in this form. In [9], Vincze presents the theory of conservative semisparays on a Finsler manifold (M, E) with respect to the energy function E. Then he defines conservative vertical vector fields on a Finsler manifold. In this paper, using conservativity and torsion-freeness of L−Ehresmann connections, we find a class of conservative
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