| dc.description.abstract | The thesis is devoted to the study of the fundamental system of invariants of bracket-generating rank 2 distributions, i.e. fields of planes, on n-dimensional manifolds (shortly, (2, n)-distributions). Distributions are considered modulo the natural action of the group of diffeomorphisms. In the case of n = 5, the minimal n for which there are locally nonequivalent maximally nonholonomic (2, n)-distributions, E. Cartan constructed the fundamental invariant called the Cartan tensor and showed that vanishing of this invariant implies that the distribution is locally equivalent to the maximally symmetric one. In a series of works in the 2000s A . Agrachev, B. Doubrov, and I. Zelenko discovered the relation between local geometry of (2, n)-distributions and classical geometry of self-dual curves in (2n−7)-dimensional projective space. In particular, from their theory, it follows that a collection of n − 4 fundamental invariants for such self-dual curves, constructed by E. Wilczynski in 1905 gives rise to the new invariants of (2, n)-distributions, called the generalized Wilczynski invariants. I. Zelenko proved that in the case of n = 5, wherein there is only one generalized Wilczynski invariant, it coincides with the Cartan tensor. Further, B. Doubrov and I. Zelenko showed that there exists exactly one, up to local equivalence, maximally symmetric bracket-generating (2, n)-distribution with the 5-dimensional cube, called the symplectically flat distribution, and for such distribution, all generalized Wilczynski invariants vanish identically. The natural question is whether or not the symplectically flat distribution is the only, up to local equivalence, distribution with this property. For n = 5 the answer to this question is positive due to E. Cartan. The main result of the thesis is that the answer is also positive in the case of left-invariant maximally nonholonomic (2, 6)-distributions on Lie groups. We also give an example of a left-invariant distribution with both Wilczynski invariants equal to zero on a 6-dimensional Lie group which is not isomorphic to the nilpotent Lie group on which the symplectically flat distribution naturally lives. Among other new results, we clarify the algebraic structure of the generalized Wilczynski invariants and their dependence on the Tanaka symbol of a distribution. | |
