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