On the Finsler group and an almost symplectic structure on a tangent bundle
type: Item
instance of: scholarly article
title: On the Finsler group and an almost symplectic structure on a tangent bundle
zbMATH Open document ID: 0654.53037
author: Yoshihiro Ichijyō
published in: Journal of Mathematics of Kyoto University
publication date: 1988-01-01T00:00:00Z
review text: In a previous paper [Finsler geometry, Proc. Rom.-Jap. Colloq., Braşov, Iaşi, Bucureşti/Rom. 1984, 91-106 (1985; Zbl 0602.53043)] the author has found a Lie group F(n) which is called the Finsler group and has investigated tangent bundles T(M) admitting an F(n)-structure in the sense of the theory of G-structures. An F(n)- structure which is defined on T(M) as a reduction of the standard tangent structure has been called an almost Finsler structure. In the present paper the author first studies almost Finsler structures without the assignment of a nonlinear connection, and finds a necessary and sufficient condition for T(M) to admit an almost Finsler structure, which is expressed in terms of some quantities in the base manifold M. Based on the fact that the Finsler group F(n) is a subgroup of the symplectic group, and consequently T(M) admits an almost symplectic structure if it admits an almost Finsler structure, the author introduces a special 2-form on T(M), which plays an important role in the development of the theory of almost Finsler structures. In {\S} 2 the author is concerned with this 2-form and deals with the case where the 2- form is closed or has an integrating factor. Finally, {\S} 3 is devoted to the consideration on almost Hamiltonian vectors with respect to the almost symplectic structure derived from the almost Finsler structure. In the case of Finsler manifolds, Hamiltonian vectors and Hamiltonian systems are treated and Hamiltonian functions are shown concretely.
reviewed by: F. C. Klepp
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53B40
zbMATH DE Number: 4067788
zbMATH Keywords: almost Finsler structures
zbMATH Keywords: Finsler group
zbMATH Keywords: almost symplectic structure
zbMATH Keywords: almost Hamiltonian vectors
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250520565
OpenAlex ID: W2170550964
DOI: 10.1215/KJM/1250520565
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description: scientific article; zbMATH DE number 4067788