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Abnormal Sub-Riemannian Geodesics: Morse Index and Rigidity

Summary: Abnormal Sub-Riemannian Geodesics:
Morse Index and Rigidity
A.A. Agrachev 1 A.V. Sarychev 2
Abstract. 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 2 M the problem of nding a minimal path out of those tangent to the distribution
(admissible) 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 Legendre-Jacobi-Morse-type theory of
2nd variation for abnormal geodesics we investigate some of their speci c properties such as
rigidity - isolatedness in the space of admissible paths connecting the two given points.
1 Introduction
The paper deals with abnormal sub-Riemannian geodesics. Let us remind that a sub-Riemannian
structure on a Riemannian manifold M is given by a completely non-integrable (or completely
non-holonomic, or possessing full Lie rank) distribution D on M: A locally Lipschitzian path
q() 2 W 1
1 [0; T ]) (W 1
1 [0; T ] denotes the space of Lipschitzian paths  ! q() on M) is called


Source: Agrachev, Andrei - Functional Analysis Sector, Scuola Internazionale Superiore di Studi Avanzati (SISSA)


Collections: Engineering; Mathematics