Sard property in Carnot groups

In this work we introduce the topic of sub-Riemannian geometry from an elementary viewpoint. Sub-Riemannian geometry is a quite modern field of differential geometry. The subject has been studied by that name from the 90s, however, several key ideas of Sub-Riemannian geometry are antecedent, e.g. the concept of sub-Riemannian distance, firstly denoted as Carnot-Carathéodory distance. The main objective of this thesis is to provide a first description of abnormal curves, which are particular curves on a sub-Riemannian manifold which exhibit an anomalous (and hopefully rare) behaviour. Abnormal curves are related to many open problems in sub-Riemannian geometry such as the regularity of the sub-Riemannian distance, the homotopy of small sub-Riemannian balls and the study of the sub-Laplacian which is related to the heat diffusion on sub-Riemannian manifolds. Our main concern will be the study of the Sard problem for the end-point map: we will see how abnormal curves are related to critical values of a specific map from a functional space to a sub-Riemannian manifold, then we will be interested to determine the structure of this set of critical values and, in particular, whether or not its measure is always zero.