Fat prolonged distributions of type (4,6)

Abstract

A distribution is a smooth sub-bundle of the tangent bundle of a given manifold. It can represent a physical system with restrictions on the degrees of freedom. Bracket generating distributions are a distinguished class of interest in control theory: when the restriction on the system is given by a bracket generating distribution, any configuration (e.g. position and orientation) can be obtained using the restricted directions only. This thesis focuses on fat distributions (also called strongly bracket generating distributions), which are, in a sense, the most extreme case of bracket generating distribution. A lot is known about co-rank 1 fat distributions (also called contact distributions), but much less is known for higher co-ranks. We focus on co-rank 2 distributions that are induced by the canonical distribution on the Grassmann bundle of 2-planes of a manifold. We define this class of distributions and refer to them as prolonged distributions. To be precise, we look at co-rank 2 sub-bundles of the Grassmann bundle 2-Gr(TX) of a 4-dimensional manifold X. We consider the canonical distribution on 2-Gr(TX) and restrict it to the given sub-bundle. The main question we investigate is under what conditions this restriction defines a fat distribution on the sub-bundle manifold. Our contributions go in two directions. First, we assume the 4-dimensional base manifold X to be endowed with an almost complex structure J. We consider the rank-2 sub-bundle of the Grassmann bundle consisting of the 2-planes invariant under the almost complex structure J. This sub-bundle forms a 6-dimensional manifold and the fibers are in fact complex Grassmannians. We show that the prolonged distribution of this sub-bundle is a fat distribution of co-rank 2. Furthermore, we consider a rank-2 fiber bundle M over a 4-dimensional base manifold X and a bundle map that maps M into the Grassmann bundle 2-Gr(TX); we identify necessary and sufficient local conditions for the bundle map to induce a fat prolonged distribution D of co-rank 2 on the fiber bundle M. More precisely, we show that requiring the prolonged distribution D on M to be fat is equivalent to requiring that the fibers of M –that map into the corresponding Grassmannian-fiber via the bundle map– are transverse to what we call the infinitesimal cone field on the Grassmannian. As a consequence, we show that, in this case, if M is closed, the fibers of M are either 2-spheres or projective planes, which is the main result of this thesis.