A.A. Agrachev, A.V. Sarychev, Abnormal sub-Riemannian geodesics: Morse index and rigidity

Title: Abnormal sub-Riemannian geodesics: Morse index and rigidity
Authors: A.A. Agrachev, A.V. Sarychev
Journal title: Annales de l'Institut Henri Poincare - Analyse non lineaire
Year: 1996
Volume: 13
Pages: 635-690

Considering a smooth manifold M provided with a sub_riemannian structure, i.e. with Riemannian metric and completely nonintegrable distribution, we set for two given points q^0, q^1 \in M the problem of finding a minimal path out of those tangent to the distribution and connecting these points. Extremals of this variational problem are called sub-Riemannian geodesics and we single out the abnormal ones which correspond to the vanishing Lagrange multiplier for the length functional. These abnormal geodesics are not related to the Riemannian structure but only to the distribution and, in fact, are singular points in the set of admissible paths connecting q^0 and q^1. Developing the Lagrange-Jacobi-Morse-type theory of 2nd variation for abnormal geodesics we investigate some of their specific properties such as rigidity - isolatedness in the space of admissible paths connecting the two given points.