Rank k vector distribution D on the manifold M is by definition a k-dimensional subbundle of the tangent bundle TM. In other words, for each point q in M a k-dimensional subspace D(q) of the tangent space T_qM is chosen and D(q) depends smoothly on q. Two vector distributions D_1 and D_2 are called locally equivalent at some point q_0 in M , if there exists a diffeomorphism F of some neighbourhood U of q_0, which transforms distribution D_1 to D_2, i.e., F_*D_1(q)=D_2 (F(q)) for all q in U. The question is when two distributions are locally equivalent? In the present talk we will restrict ourselves to the case k=2. If dim M=3 or 4, all generic germs of rank 2 distributions are equivalent (Darboux's and Engel's theorems). If dim M >= 5, the normal forms of these germs contain (dim M-4) functional parameters. Using the general theory of curves in the Lagrange Grassmannian developed in [1], we will describe the construction of basic invariant of the distribution, fundamental form, which is the obstacle to local equivalence. The case dim M=5 will be discussed in greater details. References [1] A. Agrachev, I. Zelenko, Geometry of Jacobi curves.I, J. Dynamical and Control Systems, 8(1):93-140,

## Fundamental form of rank 2 vector distribution.

Research Group:

Speaker:

Igor Zelenko

Institution:

Technion - Israel Institute of Technology - Haifa, Israel

Schedule:

Wednesday, March 7, 2001 - 06:30 to 07:30

Location:

Room L

Abstract: