Higher order transverse bundles and riemannian foliations

TL;DR

This paper establishes equivalent conditions for a foliation to be Riemannian, involving higher order transverse bundles, lifted foliations, and transverse Lagrangians, extending previous results on jet vector bundles.

Contribution

It introduces new equivalent criteria for Riemannian foliations using higher order transverse bundles and Lagrangian structures, generalizing earlier work on jet bundles.

Findings

Conditions for Riemannian foliations are equivalent to properties of lifted foliations.

Higher order transverse bundles can characterize Riemannian structures.

Extension of previous results on normal jet vector bundles.

Abstract

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