# A.A. Agrachev, J.-P. Gauthier, V. Zakalyukin, On sub-Riemannian caustics and wave fronts for contact distributions in the three-space

Title: On sub-Riemannian caustics and wave fronts for contact distributions in the three-space
Authors: A.A. Agrachev, J.-P. Gauthier, V. Zakalyukin
Journal title: Dynamical and Control Systems
Year: 2000
Volume: 6
Pages: 365-395
Abstract:

In a number of previous papers of the first and third authors, caustics, cut-loci, spheres, and wave fronts of a system of sub-Riemannian geodesics emanating from a point $q_0$ were studied. It turns out that only certain special arrangements of classical Lagrangian and Legendrian singularities occur outside $q_0$. As a consequence of this, for instance, the generic caustic is a globally stable object outside the origin $q_0$. Here we solve two remaining stability problems. The first part of the paper shows that in fact generic caustics have
moduli at the origin, and the first module that occurs has a simple geometric interpretation. On the contrary, the second part of the paper shows a stability result at $q_0$. We define the “big wave front”: it is the grap h of the multivalued function arclength → wave-front reparametrized in a certain way. This object is a three-dimensional surface that also has a natural structure of the wave front. The projection of the singular set of this “big wave front” on the 3-dimensional space is nothing else but the caustic. We show that in fact this big wave front is Legendre-stable at the origin.