Résume | A tangent distribution is a subbundle of the tangent bundle. There is a control theoretical motivation for considering such objects: The manifold represents the possible states of a particle and the distribution represents the admissible directions of motion.
Much like metrics, distributions have plenty of local invariants that measure how far the distribution is from being isomorphic to the "flat model". In this case, the flat model is a foliation and the invariants are thus a measure of non-involutivity.
In this talk, we will discuss bracket-generating distributions. This means that any vector in the manifold is a linear combination of iterated lie brackets applied to vector fields tangent to the distribution. This condition implies non-involutivity and it says that, infinitesimally, we can use curves tangent to the distribution ("horizontal curves") to move in any direction.
Imagine we take a horizontal curve and we vary its endpoint. We may ask whether we can follow this variation by a variation of the curve itself, keeping its initial point fixed. Gromov conjectured that this can always be achieved, but Bryant and Hsu showed that this need not be the case in general. The study of such curves has been a driving problem in SubRiemannian Geometry for the last 30 years. |