Higher Order Transverse Bundles and Riemannian Foliations

Abstract

The purpose of this paper is to prove that each of the following conditions is equivalent to that the foliation ${{\mathcal{F}}}$ is riemannian: (1) the lifted foliation ${{\mathcal{F}}^{r}}$ on the r-transverse bundle ${\nu^{r}{\mathcal{F}}}$ is riemannian for an r ≥  1; (2) the foliation ${{\mathcal{F}}_{0}^{r}}$ on a slashed ${\nu _{\ast }^{r}{\mathcal{F}}}$ is riemannian and vertically exact for an r ≥  1; (3) there is a positively admissible transverse lagrangian on a ${\nu _{\ast }^{r}{\mathcal{F}}}$, for an r ≥  1. Analogous results have been proved previously for normal jet vector bundles.