Transversally affine foliations

Let ℱ be a smooth foliation of codimension p on a smooth manifold M m . We can define ℱ by an atlas of coordinate charts ( U , ( x , y )), called leaf charts , where ( x , y ): U → R m−p × R p are coordinate functions for which the leaves of ℱ are given by y 1 constant,…, y p 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 form If y ′ is always affine in y , i.e. where and B i 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.