This paper is a sequel to [4]. Its purpose is to show that the concept of isometric foldings of Riemannian manifolds can be extended to a much wider class of manifolds without losing the main structure theorem. We present here what we believe to be a definitive form of the folding concept for smooth manifolds.

The theory discussed here is based on the idea of a 1-spread [2], where the role of geodesies on a Riemannian manifold is assumed by smooth, unoriented and unparametrised curves on a smooth manifold. The absence of metrical structure forces a fresh approach to the basic definitions. A crucial feature of the Riemannian theory does survive, however, in this general setting: a 1-spread on a sufficiently smooth manifold M induces a 1-spread on sufficiently small spheres surrounding any point of M. With the help of this fact, we are able to construct an inductive definition of “star folding” f:M → N between smooth manifolds M and N, and to retain the theorem that the manifold M is stratified by the “folds”, each of which has the character of a “totally geodesic” submanifold with respect to the above-mentioned curves.