Sufficient Conditions for a Bundle-Like Foliation to Admit a Riemannian Submersion onto its Leaf Space

Abstract

This note furnishes a necessary and sufficient condition for a bundlelike foliation to be defined globally by a Riemannian submersion.Introduction.The purpose of this note is to furnish sufficient conditions for a foliated manifold with a bundle-like metric to admit a Riemannian submersion onto its leaf space in a natural way.The main result, Theorem 2.2, says that this will occur whenever all the leaves are closed and the holonomy of each leaf, with respect to the foliation, is trivial.Since a foliation with a bundle-like metric can be thought of as one being defined locally by Riemannian submersions, this note provides, on the basis of a result of Hermann, a necessary and sufficient condition for the foliation to be defined globally by a Riemannian submersion.Several applications of the result to recover known theorems conclude the paper.This work was done while the author was on sabbatical from Canisius College.He is grateful to the college for that opportunity.