Lie Foliations

Abstract

The proofs of Theorems 10.17 and 10.18, as well as many other results on Riemannian foliations, rely substantially on the structure theory for Riemannian foliations developed by Molino [M 8] in arbitrary codimension. It is based on several fundamental observations. The first is that the canonical lift F of a Riemannian foliation F to the bundle P of orthonormal frames of (J, is a transversally parallelizable Riemannian foliation. The canonical lift F on P is a foliation of the same dimension as F on M, and invariant under the action of the orthogonal structural group of P. Now let M be closed and oriented, and consider on P the closures of the leaves of F. The second fundamental observation is that these closures form the fibers of a fibration ${X_0} \to P\mathop{ \to }\limits^{\pi } W$, over the space V of orbit closures, with typical fiber X₀. The foliation F induces on X₀, and on each fiber of π, a foliation with dense leaves, that is transversally modeled on a Lie group G with translations as transition functions. These are the Lie foliations previously studied by Fedida [F] and Molino [M 5,6]. The Lie algebra of this group G is another structural invariant of the foliation. With the help of this structure theorem, many questions on Riemannian foliations can be reduced to questions on Lie foliations, by passing to the bundle of transversal orthonormal frames.