Sub-Riemannian structures are not Euclidean, or even Riemannian, at any scale. These structures are highly non isotropic in the sense that at every point only some direction of the motion are allowed. These directions are described by vector fields, which play the same role as the partial derivatives in the Euclidean setting. Hence all the geometric and analytic concepts have to be rephrased in terms of vector fields. We will need to define from a purely mathematical point of view, the main properties of the objects of the space, with instruments of Sub-Riemannian differential geometry, and introduce PDEs (partial differential equations) in Lie groups, which can describe their motion.

We will see that these instruments can be used to formalize models of anisotropy systems. In particular we will be interested in the modellization of the strong anysotropy structure of the visual cortex. These studies can justify pehnomena of subjective completion or visual allucination.