24 Aug 2000

Boris Kruglikov "Non-holonomic Riemannian geometry".

This geometry (also known as sub-Riemannian and Carnot-Caratheodory) arises from a distribution on manifold, which is equipped with an additional structure -- Riemannian metric. The asymptotic and dimensional study of such structures rely rather on the properties of distribution, so the Riemannian metric can be changed to any equivalent Riemannian (Finsler or even more general) metric. These asymptotics are connected to the dimensions of gradings coming from the derived flag of the distribution. More precise values (and shape) of  the metric are needed for Equivalence Problem and Control Theory.
We will show the connection of non-holonomic Riemannian geometry to such (seemingly) different fields as Dynamical Systems on metric spaces, Lagrange Mechanics and Partial Differential Equations. The basic and most important example of Heisenberg group H²ⁿ⁺¹ arises from study of the natural contact distribution in J¹(Rⁿ). We will consider also the higher analogs - Cartan distributions in J^k(Rⁿ).