Harmonic Integrals on Foliated Manifolds

In the paper [5], we considered harmonic integrals on local product manifolds, that is, manifolds having two families of submanifolds in complementary dimensions, such that locally they look like the product of two euclidean spaces. The metric was assumed to be such that this local product could be taken in the sense of Riemannian manifolds. The results obtained were such as to suggest that analogous theorems could be proved if we assunmed only one family of submanifolds, with a suitable choice of metric. This is indeed the case. We shall show in ? 4 that on compact manifolds, the cohomology of base-like differential forms (defined in ? 2) is isomorphic to the harmonic space of a certain semi-definite Laplacian (defined in ?3). The metric is assumed to be bundle-like in the sense of [6] ; that paper mav be referred to for examples of foliated manifolds possessing such a metric. In ? 5, we discuss the meaning of our harmonic integral theorem for these examples. 1. Definitions. By a manifold I1, we mean a C- differentiable manifold; topologically, it is a connected, orientable, separable, locally euclidean Hausdorff space. We shall assurme given on M (of dimension n) a C- completely integrable q form ?, that is, a locally decomposable, non-zero q form