Inserted: 5 may 2014
Last Updated: 25 jul 2017
Journal: Geometric measure theory and real analysis, CRM Series, Ed. Norm., Pisa.
Lecture notes based on a course given by the author on the occasion of the ERC School "Geometric Measure Theory and Real Analysis" held at the Centro De Giorgi, Pisa, in October 2013.
We study the regularity problem for sub-Riemannian geodesics, i.e., for those curves that minimize length among all curves joining two fixed endpoints and whose derivatives are tangent to a given, smooth distribution of planes with constant rank. We review necessary conditions for optimality and we introduce extremals and the Goh condition. The regularity problem is nontrivial due to the presence of the so-called abnormal extremals, i.e., of certain curves that satisfy the necessary conditions and that may develop singularities. We focus, in particular, on the case of Carnot groups and we present a characterization of abnormal extremals, that was recently obtained in collaboration with E. Le Donne, G. P. Leonardi and R. Monti, in terms of horizontal curves contained in certain algebraic varieties. Applications to the problem of geodesics' regularity are provided.