Journal of the
Korean Mathematical Society JKMS

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\'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

MSC numbers: 55R10, 53C60, 83E15