Foliated Bundles, Invariant Measures and Flat Manifolds

Abstract

Let e = (p, E, M) be a smooth (= C1) bundle. A foliation of e is a foliation i of E where leaves are transverse to the fibres and of complementary dimension. The leaves are required to be smooth, and their tangent planes must vary continuously on M but the foliation may be merely C0. Such foliated bundles occur in several geometrical situations. Suppose a manifold M has an affine connection of zero curvature, i.e., M is a flat manifold; then its tangent sphere bundle has a foliation. Or if a compact Riemannian manifold has negative sectional curvature, then its tangent sphere bundle is foliated. The normal sphere bundle of a leaf of a foliation is foliated. The purpose of this paper is to show that under certain assumptions about the holonomy homomorphism of a foliated bundle, there are strong restrictions on the homomorphisms induced by the bundle projection in real homology and cohomology. In some cases it follows that the bundle has a section. A typical application of our results is: