Foliated vector bundles and Riemannian foliations

Abstract: In this Note we prove the equivalence between the Riemannian foliation and each of the following conditions: 1) the lifted foliation on the bundle of r-transverse jets is Riemannian for ; 2) the foliation on the slashed is Riemannian and vertically exact for ; 3) there exists a positively admissible transverse Lagrangian on , the r-transverse slashed jet bundle of a foliated bundle , for . [Copyright &y& Elsevier]