 
Summary: Strong Minimality of Abnormal Geodesics for 2Distributions
Andrei A. Agrachev Andrei V. Sarychev y
Abstract
We investigate the local length minimality (by the W 1;1 or H 1 topology) of abnor
mal subRiemannian geodesics for rank 2 distributions. In particular, we demonstrate
that this kind of local minimality is equivalent to the rigidity for generic abnormal
geodesics, and introduce an appropriate Jacobi equation in order to compute conjugate
points. As a corollary, we obtain a recent result of Sussmann and Liu about the global
length minimality of short pieces of the abnormal geodesics.
1 Introduction
In this paper we study abnormal subRiemannian geodesics. Let us recall that a sub
Riemannian structure on a Riemannian manifold M is dened by a bracket generating (or
a possessing full Lie rank) distribution D on M: A locally Lipschitzian path q() ( 2 [0; T ])
is admissible if its tangents lie in D for almost all 2 [0; T ]: Given two points q 0 and q 1
one can set out the problem of nding minimal (i.e. lengthminimizing) admissible path
connecting q 0 with q 1 :
An essential distinction of this setting from the classical Riemannian case is that the
space of all locally Lipschitzian paths connecting q 0 with q 1 has a structure of Banach
manifold with minimal paths being critical points of the length functional, or Riemannian
geodesics on the manifold M , whereas the space of admissible paths is not, in general,
