ETD

• Carnot groups
• Chow-Rashevsky theorem
• Goh conditions
• Hakavuori-Le Donne theorem
• Liu-Sussmann minimality theorem
• normal and abnormal extremals
• Sub-Riemannian manifolds

After a brief introduction to sub-Riemannian manifolds, we give a first order classification of geodesics, which can be normal or abnormal, as well as the second order Goh conditions. Explicit examples of strictly abnormal minimizers are provided, using a result by Liu-Sussmann.
We examine closely the model case of Carnot groups, reviewing here a new result by Hakavuori-Le Donne which excludes corner-like singularities. Finally, we obtain an interesting quantitative refinement, which excludes a wider class of singularities in Carnot groups of rank 2.