Higher Order Transverse Bundles and Riemannian Foliations

The purpose of this paper is to prove that each of the following conditions is equivalent to that the foliation is riemannian: (1) the lifted foliation on the r-transverse bundle is riemannian for an r a parts per thousand yen 1; (2) the foliation on a slashed is riemannian and vertically exact for an r a parts per thousand yen 1; (3) there is a positively admissible transverse lagrangian on a , for an r a parts per thousand yen 1. Analogous results have been proved previously for normal jet vector bundles.