1 Introduction

The structure of sub-Riemannian minimizers and of the corresponding

"geodesies" has recently attracted a great deal of attention (cf. [1], [5],

[6], [7], [9], [10], [11], [12], [14], [17], [18], [19], [22], [23], [27], [28]; see

Remark 1 below for a discussion of the use of the word "geodesic"),

due to the delicate issues that arise because of the possibility of the

existence of "abnormal" length-minimizing arcs.

This phenomenon, well known in Optimal Control Theory, was not

immediately recognized as possible in the more special case of sub-

Riemannian geometry. For example, in 1986 it was stated, in [22], that

all length-minimizing arcs for a sub-Riemannian manifold are charac-

teristics of the associated Hamiltonian (i.e. "normal extremals," in

our terminology). A proof was suggested for this result, relying on an

application of the Pontryagin Maximum Principle from Optimal Con-

trol Theory. It turns out, however, that the Maximum Principle only

makes it possible to draw the weaker conclusion that every minimize!

is either a characteristic of the Hamiltonian (i.e. a normal extremal)

or a member of another class of arcs known as "abnormal extremals."

The possibility that a minimizer might be an abnormal extremal can

easily be ruled out in the Riemannian case and, more generally, for the

special class —introduced by R. Strichartz in [23]— of sub-Riemannian

metrics defined on "strongly bracket-generating" distributions, but for

general sub-Riemannian metrics there is no obvious way to go beyond

the necessary conditions of the Maximum Principle and exclude ab-

normal extremals. This left open the question whether there can exist

sub-Riemannian minimizers that are not normal extremals ("strictly

abnormal minimizers," in the terminology introduced below). The sug-

gestion that this could indeed happen had in fact been made much

Work supported in part by the National Science Foundation under NSF grant

DMS92-02554.

Received by the editor November 1993, and in revised form February 1994.

1