Wendel, Eric Duong Ba (2015-08). On the Maximality of Class of Rank-2 Distributions with 5-Dimensional Cube. Master's Thesis.
This thesis is devoted to aspects of the local differential geometry of regular, bracket-generating distributions. The classical discrete in variants of such distributions at a point are its degree of nonholonomy and small growth vector, both of which encode how quickly the iterated Lie brackets of the distribution saturate the tangent space of the manifold at a point. Recently, Boris Doubrov and Igor Zelenko introduced a new discrete in variant of these distributions at a point called the Jacobi symbol, and constructed canonical frames for all distributions with given Jacobisymbol. Their constructions, however, require an additional generic assumption called the maximal class condition. This condition can be formulated in purely control-theoretic language as a property of the end point map along special curves called abnormal trajectories. There is a strong belief that the condition of maximal class is essentially redundant, which is to say that all bracket-generating distributions are of maximal class or become maximal class after a natural reduction process. The aim of this thesis is to develop general tools for proving this conjecture and to apply these tools for the verification of the conjecture for a number of cases. We begin by proving that the maximal class condition is essentially determined by its Tanaka symbol, which is the graded space associated with the filtration of the distribution. We are most interested in proving that rank-2 distributions of dimension n >= 6 and small growth vector (2, 3, 5, ...) are of maximal class, and make our first steps towards this goal by proving that all (2, n)-distributions with Tanaka symbol isomorphic to the free truncated graded Lie algebra with 2 generators and degree of nilpotency 4 or 5 are of maximal class. We then provide calculations proving that all (2, n)-distributions associated with Monge ODEs areof maximal class.