December 24, 2003

Igor Zelenko

Title: Vector Distributions

Abstract:

Rank $k$ vector distribution on the $n$-dimensional manifold $M$ or $(k,n)$-distribution (where $k<n$) is by definition a $k$-dimensional subbundle of the tangent bundle $TM$. Two distributions are called (locally) equivalent, if one can be transformed to another by (local) diffeomorphism. Our talk is devoted to the equivalence problem of bracket-generating $(2,n)$-distributions. The case $n=5$ (the first case containibg functional parameters) was treated by E. Cartan in 1910 by ingenious use of his "reduction-prolongation" procedure. In particular, he found the covariant fourth-order tensor invariant (Cartan tensor) for such distributions. After the work of E. Cartan the following questions remained open: first the geometric reason for existence of Cartan's tensor was not clear (the tensor was obtained by very sophisticated algebraic manipulations) and the true analogs of this tensor in Riemannian geometry were not found; secondly it was not clear how to generalize this tensor to other classes of distributions ; finally there were no explicit formulas for computation of Cartan's tensor (in order to compute this tensor for concrete distribution, one had to repeat Cartan's "reduction-prolongation" procedure for this distribution from the very beginning, which is rather difficult task).

In the talk I will present alternative approach to equivalence problem, which allows to give the answers to the questions mentioned above. This approach is based on the theory of curves in the Lagrange Grassmannian, developed in our previous works with A. Agrachev. In this way we construct the fundamental form and the projective Ricci curvature of $(2,n)$-distributions for arbitrary $n\geq 5$. It turns out that in the case $n=5$ our fundamental form coincides with Cartan's tensor. In this case I will present explicit formulas for computation of our invariants. Also I will show, how to apply this formulas for the following natural examples: distributions, generated by rolling of two spheres without slipping and twisting; distributions, generated by curves of constant torsion on three-dimensional Riemannian manifolds of constant curvature; left-invariants distribution on Lie groups.