Foliated vector bundles and Riemannian foliations

In this Note we prove the equivalence between the Riemannian foliation and each of the following conditions: 1) the lifted foliation F-r on the bundle of r-transverse jets is Riemannian for r >= 1; 2) the foliation F-0(r) on the slashed I-0(r) is Riemannian and vertically exact for r >= 1; 3) there exists a positively admissible transverse Lagrangian on (I0E)-E-r, the r-transverse slashed jet bundle of a foliated bundle E -> M, for r >= 1. (C) 2011 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.