Transversally affine foliations

Abstract

Let ℱ be a smooth foliation of codimension p on a smooth manifold Mm. We can define ℱ by an atlas of coordinate charts (U, (x, y)), called leaf charts, where (x, y): U → Rm−p × Rp are coordinate functions for which the leaves of ℱ are given by y1 constant,…,yp constant, in U. Clearly, on the overlap of two such leaf charts (U, (x, y)) and (U′, (x′, y′)) we have a coordinate transformation of the formIf y′ is always affine in y, i.e.where and Bi are constants, we shall say that ℱ is a transversally affine foliation. This notion is, in a sense, dual to that of affine foliation, see [2], in which x′ is affine in x and each leaf has an induced flat affine structure.