The Regularity Problem for Sub-riemannian Geodesics
نویسنده
چکیده
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.
منابع مشابه
The regularity problem for sub-Riemannian geodesics
One of the main open problems in sub-Riemannian geometry is the regularity of length minimizing curves, see [12, Problem 10.1]. All known examples of length minimizing curves are smooth. On the other hand, there is no regularity theory of a general character for sub-Riemannian geodesics. It was originally claimed by Strichartz in [15] that length minimizing curves are smooth, all of them being ...
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