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http://dx.doi.org/10.4134/JKMS.2009.46.3.551

FIBRE BUNDLE MAPS AND COMPLETE SPRAYS IN FINSLERIAN SETTING  

Crasmareanu, Mircea (FACULTY OF MATHEMATICS UNIVERSITY)
Publication Information
Journal of the Korean Mathematical Society / v.46, no.3, 2009 , pp. 551-560 More about this Journal
Abstract
A theorem of Robert Blumenthal is used here in order to obtain a sufficient condition for a function between two Finsler manifolds to be a fibre bundle map. Our study is connected with two possible constructions: 1) a Finslerian generalization of usually Kaluza-Klein theories which use Riemannian metrics, the well-known particular case of Finsler metrics, 2) a Finslerian version of reduction process from geometric mechanics. Due to a condition in the Blumenthal's result the completeness of Euler-Lagrange vector fields of Finslerian type is discussed in detail and two situations yielding completeness are given: one concerning the energy and a second related to Finslerian fundamental function. The connection of our last framework, namely a regular Lagrangian having the energy as a proper (in topological sense) function, with the celebrated $Poincar{\acute{e}}$ Recurrence Theorem is pointed out.
Keywords
fibre bundle; fibration; regular Lagrangian; energy; Euler-Lagrange equations; semispray; spray; Finsler fundamental function; complete vector field; proper function;
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