Conjugate and conformally conjugate parallelisms on Finsler manifolds
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Conjugate and conformally conjugate parallelisms on Finsler manifolds Bernadett Aradi1 · Mansoor Barzegari2 · Akbar Tayebi2
© Akadémiai Kiadó, Budapest, Hungary 2016
Abstract In this paper we study conjugate parallelisms and their conformal changes on Finsler manifolds. We provide sufficient conditions for a Finsler manifold endowed with two conjugate (resp. conformally conjugate) covering parallelisms to become a Berwald (resp. Wagner) manifold. As an application for Lie groups, we give a new proof for a theorem of Latifi and Razavi about bi-invariant Finsler functions being Berwald. By introducing the concept of a conformal change of a parallelism, we also obtain a conceptual proof of a theorem of Hashiguchi and Ichijy¯o: the class of generalized Berwald manifolds is closed under conformal change. Keywords Conjugate parallelisms · Conformal change · Generalized Berwald manifold · Wagner manifold · Berwald manifold · Lie group Mathematics Subject Classification 53B40 · 53C60
1 Introduction Generalized Berwald manifolds have been studied from many aspects, see, for example, [2,7, 10,11]. In reference [2], the authors, using parallelisms, proved the following characterization of generalized Berwald manifolds: a Finsler manifold is a generalized Berwald manifold if,
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Bernadett Aradi [email protected] Mansoor Barzegari [email protected] Akbar Tayebi [email protected]
1
MTA-DE Research Group “Equations, Functions and Curves”, Hungarian Academy of Sciences and Institute of Mathematics, University of Debrecen, Debrecen P.O. Box 400, 4002, Hungary
2
Faculty of Science, Department of Mathematics, University of Qom, Qom, Iran
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B. Aradi et al.
and only if, the Finsler function is compatible with a covering parallelism. Our aim in this paper is to find a counterpart of this theorem for the more special cases of Berwald and Wagner manifolds. We managed to find appropriate sufficient conditions, and it turned out that in these cases we need two covering parallelisms on the base manifold. For stating our results, we generalized the notion of conjugate parallelisms [4], obtaining conjugate covering parallelisms, and defined the conformal change of parallelisms, as well as conformally conjugate parallelisms. Since on a Lie group we canonically have two conjugate parallelisms, the left and right parallelisms, it is natural to test the obtained results in this setting. Our theorem provides a new proof for the fact that a Lie group endowed with a bi-invariant Finsler function is a Berwald manifold [8], and it also turns out that if the Lie group is Abelian, then the Finsler manifold is locally Minkowskian. The paper is organized as follows. In Sect. 2 we summarize the tools used in the paper and recall the concepts which play key roles in our arguments. Then, in Sect. 3, we turn to conjugate parallelisms; after their introduction and an auxiliary lemma we are ready to state one of our main results (Theorem 3.3): a condition on a Finsler manifold (in terms of covering parallelisms) which makes
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