On local geometry of vector distributions with given Jacobi symbols

Abstract

Based on the ideas of Optimal Control, we introduce the new basic characteristic of a bracket generating distribution, the Jacobi symbol. In contrast to the classical Tanaka symbol, the set of Jacobi symbols is discrete and classifiable. We give an explicit and unified algebraic procedure for the construction of the canonical frames (the structure of absolute parallelism) for all distribution with given Jacobi symbol. We describe all Jacobi symbols for which this procedure ends up in a finite number of steps (i.e. all Jacobi symbols of finite type) and distinguish the symplectically flat distributions with given Jacobi symbol, which in the case of finite type are the maximally symmetric distribution among all distribution with given Jacobi symbol. Also, for the most of Jacobi symbols we relate the prolongation procedure and the resulting prolongation algebra, which is the algebra of infinitesimal symmetries of the corresponding symplectically flat distribution, to the standard (in the sense of Kobayashi and Sternberg) prolongation of certain subspace in the algebra of infinitesimal symmetries of the flat (unparametrized) curve of flags associated with the Jacobi symbol. Finally, we give an upper bounds for the size of the resulting prolongation algebra in terms of certain spaces of polynomials vanishing on certain projective varieties. For some class of Jacobi symbols we are able to identify the resulting prolongation algebra with such spaces of polynomials. In this way we get the description of the prolongation algebra for all Jacobi symbols of finite type appearing in rank 2 and rank 3 distributions and also for most of such Jacobi symbols appearing in rank 4 distributions. In particular, in the case of rank 3 distributions this description is given in terms of the tangential developable and the secant varieties of a rational normal curve.