Existence of Codimension-One Foliations

A codimension-k foliation of a manifold Mn is a geometric structure which is formally defined by an atlas {qf: U. - Mn}, with U c Rn-k x R , such that the transition functions have the form 9pj(x, y) = (f(x, y), g(y)), [x e Rnk, y e Rk]. Intuitively, a foliation is a pattern of (n - k)-dimensional stripes-i.e., submanifolds-on M, called the leaves of the foliation, which are locally well-behaved. See the survey article of Lawson [11], for basic examples and better explanations of the definitions. The tangent space to the leaves of a foliation If forms a vector bundle over Mz, denoted TR. The complementary bundle vf = TMn/TJY is the normal bundle of WF. We define a codimension-k Haefliger structure, SC, to be a k-dimensional Rn-bundle v(XJ) over Mn, together with a foliation Y(ThC) transverse to the fibers of >(UC). A foliation If has a Haefliger structure SCT naturally associated to it, with normal bundle >(ACE) = Iff). The foliation IF(UXW) is constructed via the exponential map, exp: vfy) - Mn, which is transverse to If in a neighborhood of the zero section so that it induces a foliation 2(XYQ) in some neighborhood isomorphic to the entire bundle. 7Cy has the special property