On prolongations of contact manifolds

Abstract

Abstract. We apply spectral sequences to derive both an obstruction to the existence of n-fold prolongations and a topological classification. Prolongations have been used in the literature in an attempt to prove that every Engel structure on M × S 1 with characteristic line field tangent to the fibers is determined by the contact structure induced on a cross section and the twisting of the Engel structure along the fibers. Our results show that this statement needs some modification: to classify the diffeomorphism type of the Engel structure we additionally have to fix a class in the first cohomology of M .

Isotopy Classification of Engel Structures on Circle Bundles

“…However, the definition of the class provided there was on a very abstract level using spectral sequences and Čech cohomology. The following result shows that our geometrically defined class d(φ 1 , φ 2 ) reduced modulo n equals the class defined in [6] (see [6,Theorem 3.3] and cf. [6, Corollary 4.1]).…”

“…In [6] we already identified a cohomology class which provides a classification of fiberwise covering maps up to isomorphism of coverings. However, the definition of the class provided there was on a very abstract level using spectral sequences and Čech cohomology.…”

“…For the class of Engel structures we are considering, Engel structures and their development maps can be considered as equivalent objects. Development maps are fiberwise covering maps as observed in [6] and, therefore, to understand Engel structures on circle bundles we can equivalently try to understand fiberwise covering maps. To this end, in §3.1 we define a homotopy invariant of fiberwise covering maps we call the horizontal distance (see Definition 3.1).…”

“…The approach to the homotopical classification of fiberwise coverings chosen in this paper is very intuitive (see §3.1 and §3.2). As a small detour, we included §3.3 in which we discuss the relation of the present results with the results from [6]. In [6] we provided a classification of fiberwise coverings up to isomorphism of coverings in a very abstract way using spectral sequences and Čech cohomology.…”

“…As a small detour, we included §3.3 in which we discuss the relation of the present results with the results from [6]. In [6] we provided a classification of fiberwise coverings up to isomorphism of coverings in a very abstract way using spectral sequences and Čech cohomology. With our geometrically intuitive construction in the present article we are able to recover these abstractly derived results (see §3.3) using a more hands-on approach.…”