Title: How to Describe Infinitesimally the Symmetric Product? (½ hour)
Abstract: The notion of Lie bracket of vector fields is well-understood in differential geometry. Infinitesimally speaking, this bracket measures how far the flows of these two vector fields are from commuting. However, there is no such infinitesimal description for the symmetric product of vector fields. So far this product has been used to describe the geodesic invariance of a distribution infinitesimally [1].
In this talk we describe the symmetric product infinitesimally by means of the parallel transport. These results are useful to determine the accessibility distribution not only restricted to the zero section of the tangent bundle, as described in [2].
References:
[1] A. D. Lewis, Affine connections and distributions with applications to
nonholonomic mechanics, Reports on Mathematical Physics
42(1/2), pp. 135-164, 1998 (Proceedings for the workshop on
Non-Holonomic Constraints in Dynamics held in Calgary in August 1997).
[2] A. D. Lewis, R. M. Murray, Configuration controllability of simple
mechanical control systems, SIAM Journal on Control and Optimization
35(3), pp. 766-790, May 1997.