Structure of Riemannian Foliations

Abstract

For Riemannian foliations on closed manifolds, Molino has found a remarkable structure theorem [Mo 8,10]. This theorem is based on several fundamental observations. The first is that the canonical lift \(\hat{\mathcal{F}}\) of a Riemannian foliation F to the bundle \(\hat{M}\) of orthonormal frames of Q is a transversally parallelizable Riemannian foliation. The canonical lift \(\hat{\mathcal{F}}\) on \(\hat{M}\) is a foliation of the same dimension as F on M, and invariant under the action of the orthogonal structural group of \(\hat{M}\) Now let M be closed and oriented, and consider on \(\hat{M}\) the closures of the leaves of \(\hat{\mathcal{F}}\) The second fundamental observation is that these closures form the fibers of a fibration \({{X}_{0}} \to \hat{M}\xrightarrow{{\hat{\pi }}}\hat{W}\) over the space \(\hat{W}\) of orbit closures, with typical fiber Xo. The foliation \(\hat{\mathcal{F}}\) induces on X0, and on each fiber of \(\hat{\pi }\), a Lie foliation with dense leaves. The Lie algebra of the model group G of this Lie foliation 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. We state all this in more detail, but without proof, which is the main topic of Molino’s book [Mo 10].