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