Abnormal sub-Riemannian geodesics: Morse index and rigidity

Considering a smooth manifold M provided with a sub-Riemannian structure, i.e. with Riemannian metric and nonintegrable vector distribution, we set a problem of finding for two given points q(0), q(1) is an element of M a length minimizer among Lipschitzian paths tangent to the vector distribution (admissible) and connecting these points. Extremals of this variational problem are called sub-Riemannian geodesics and we single out the abnormal sub-Riemannian geodesics, 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 vector distribution and, in fact, are singular points in the set of admissible paths connecting q(0) and q(1). Developing the Legendre-Jacobi-Morse-type theory of 2nd variation for abnormal geodesics we investigate some of their specific properties such as weak minimality and rigidity-isolatedness in the space of admissible paths connecting the two given points.