Foliated vector bundles and riemannian foliations

The purpose of this Note is to prove that each of the following conditions is equivalent to that of the foliation ${\cal F}$ is riemannian: 1) the lifted foliation ${\cal F}^{r}$ on the bundle of $r$-transverse jets is riemannian for an $r\geq 1$; 2) the foliation ${\cal F}_{0}^{r}$ on the slashed ${\cal J}_{0}^{r}$ is riemannian and vertically exact for an $r\geq 1$; 3) there is a positively admissible transverse lagrangian on ${\cal J}$, the $r$-transverse slashed jet bundle of a foliated bundle $$, for an $r\geq 1$.