On prolongations of contact manifolds

TL;DR

This paper uses spectral sequences to analyze the existence and classification of n-fold prolongations of contact manifolds, revealing the need for additional topological data for complete classification.

Contribution

It introduces a topological obstruction and classification method for prolongations, refining previous assumptions about Engel structures on product manifolds.

Findings

Obstruction to n-fold prolongations derived

Classification requires fixing a cohomology class

Results modify previous classification assumptions

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 $M\times\mathbb{S}^1$ 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 $M$.