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 $r\ge 1$; 2) the foliation ${\mathcal{F}}_0^r$ on a slashed ${\nu }_{\ast }^r\mathcal{F}$ is riemannian and vertically exact for an $r\ge 1$; 3) there is a positively admissible transverse lagrangian on a ${\nu }_{\ast }^r\mathcal{F}$, for an $r\ge 1$. Analogous results have been proved previously for normal jet vector bundles.

13 pages

References in corpus (3)

• Affine Hamiltonians in higher order geometry
• Hamiltonian structures on foliations
• Foliated vector bundles and riemannian foliations