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REMARKS ON SQUARE-INTEGRABLE BASIC COHOMOLOGY SPACES ON A FOLIATED RIEMANNIAN MANIFOLD

Let M be an n dimensional C°° manifold which, topologically, is connected, orientable, paracompact Hausdorff space. We shall assume given on M a foliation E of codimension q, and we may find about each point a coordinate neighbourhood with coordinates (x, •••, x, y, •••, y) (n^p+q) such that (i) \x*\^l, \y*\^l, (ii) The integral manifolds of E are given locally by y=c, •••, y=c for constants c satisfying | c α | ^ l . (Here and hereafter, Latin indices run from 1 to p, and Greek indices from 1 to q.) Such a coordinate neighbourhood will be called flat, while each of slices given by a set of equations y=c will be called a plaque. If U is a flat neighbourhood, the quotient space of U by its plaques will be called the local base and be denoted by Uy. We may assume that there exist in U differential forms w and vectors va such that (i) {d/dx} forms the base for the space of cross-sections of E in U at each point,