FIBRE BUNDLE MAPS AND COMPLETE SPRAYS IN FINSLERIAN SETTING

Title & Authors
FIBRE BUNDLE MAPS AND COMPLETE SPRAYS IN FINSLERIAN SETTING
Crasmareanu, Mircea;

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 $\small{Poincar{\acute{e}}}$ 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;
Language
English
Cited by
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