THE PROBLEM OF GEODESICS IN SINGULAR SUB-RIEMANNIAN GEOMETRY

This Note announces and summarizes the results to appear in [1]. The aim was to gather geometric results ([2] to [5]) of the regular case, and probability results ([6], [7]), and to create a unified framework which gives account of both regular and singular cases. In the regular case any possible metric on the plane distribution provides a distance ([4]. [8]). In the singular case, we exhibit a family of horizontal metrics such that, between two points, the distance exists and is achieved, but whatever this metric there does not exist any Riemannian extending it. Contrary to the Riemannian case, where the energy minimizing curves are characterized as solution of a differential system (G), here both notions can be generalized, but they are no longer equivalent (regular case [4]). Finally, it is the application of the Maximum Principle of the Control Theory, which will give account of the whole of the length minimizing curves.