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 F is riemannian: 1) the lifted foliation F r on the r-transverse bundle ν r F is riemannian for an r ≥ 1; 2) the foliation F r 0 on a slashed ν r * F is riemannian and vertically exact for an r ≥ 1; 3) there is a positively admissible transverse lagrangian on a ν r * F, for an r ≥ 1. Analogous results have been proved previously for normal jet vector bundles.