Foliated vector bundles and riemannian foliations

TL;DR

This paper establishes equivalent conditions for a foliation to be Riemannian, involving properties of lifted foliations on jet bundles and the existence of a transverse Lagrangian, thus linking geometric structures with foliation theory.

Contribution

It introduces new equivalent characterizations of Riemannian foliations using jet bundle structures and transverse Lagrangians, expanding the theoretical framework.

Findings

Equivalence of Riemannian foliation conditions via lifted foliations

Characterization of Riemannian foliations through transverse Lagrangians

Connection between jet bundle structures and Riemannian properties

Abstract

The purpose of this Note is to prove that each of the following conditions is equivalent to that of the foliation ${\cal F}$ is riemannian: 1) the lifted foliation ${\cal F}^{r}$ on the bundle of $r$-transverse jets is riemannian for an $r\geq 1$; 2) the foliation ${\cal F}_{0}^{r}$ on the slashed ${\cal J}_{0}^{r}$ is riemannian and vertically exact for an $r\geq 1 $; 3) there is a positively admissible transverse lagrangian on ${\cal J}%_{0}^{r}E$, the $r$-transverse slashed jet bundle of a foliated bundle $% E\rightarrow M$, for an $r\geq 1$.