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
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