Foliated vector bundles and riemannian foliations

Abstract

The purpose of this Note is to prove that each of the following conditions is equivalent to that of the foliation $\mathcal{F}$ is riemannian: 1) the lifted foliation ${\mathcal{F}}^r$ on the bundle of $r$-transverse jets is riemannian for an $r\ge 1$; 2) the foliation ${\mathcal{F}}_0^r$ on the slashed ${\mathcal{J}}_0^r$ is riemannian and vertically exact for an $r\ge 1$; 3) there is a positively admissible transverse lagrangian on $\mathcal{J}$, the $r$-transverse slashed jet bundle of a foliated bundle , for an $r\ge 1$.

6 pages