Chow Theorem and structure of Carnot-Caratheodory balls

This thesis is meant to be a study on the role of the C-C metrics in the Chow theorem and on the structure of C-C balls. It opens with a section in which we give the definitions of vector field and bracket and we state the properties of the latter. Then we give the definition of the most important object of this work: the Carnot-Caratheodory metrics and we state some basic propositions on its behavior, hinting also to some metrics which are equivalent to it. We now proceed giving preliminary notions about exponential maps on a compact set of R^n, we state the Campbell-Hausdorff formula for two smooth vector fields and we introduce an object defined as a composition of exponentials E(J,t). We finally state the Chow theorem for m smooth vector fields in R^n satisfying the Chow-Hoermander condition. We prove it using the Campbell-Hausdorff formula and defining through E(J,t) n approximated exponential maps whose composition provides us with a local diffeomorphism. This theorem is particularly important because it tells us that under Chow-Hoermander condition we can regain directions that would be otherwise prohibited. We introduce the doubling metric spaces and in particular we state a theorem on the structure of C-C balls: the Nagel-Stein-Wainger theorem, which provides us with the size of the C-C balls and tells us they are represented as the image of rectangles under the exponential map. At last, we close this work with a variant of the structure theorem.