6 WENSHENG LIU AND HECTOR J. SUSSMANN
here is almost exactly the same as that of [7]. For a discussion of the
precise relationship between the two classes, cf. Remark 2 below.) We
show, however, that an abnormal extremal need not be a local mini-
mizer, even if it is both smooth and rigid. (This is done in Appendix
D, where we exhibit a C1-rigid real-analytic abnormal extremal which
is not a local minimizer.)
So, abnormal extremals can be local minimizers, as in the examples
of Montgomery and Kupka, but need not be. This naturally leads to
the main question studied here, namely, which of the two possibilities
occurs for "typical" abnormal extremals. Our answer —for distribu-
tions E of rank two— is that local minimization is the most common
situation. We make this precise by first defining the class of "regular
abnormal biextremals." Our main result is then Theorem 5, which says
that regular abnormal extremals (i.e. projections of regular abnormal
biextremals) locally minimize length. In addition, we show that, when
the distribution E satisfies a very mild extra condition, then these reg-
ular abnormal biextremals are "typical" in the following precise sense:
if we use AEC(E) (the "abnormal extremal carrier of E) to denote
the set of nonzero members of the annihilator of [E, E], then: (i) every
nonconstant abnormal biextremal is contained in AEC{E)) (ii) there is
a relatively open dense subset RA(E) of AEC(E) which is a submani-
fold and carries a one-dimensional foliation FE whose leaves, properly
parametrized, are the regular abnormal biextremals, (iii) every locally
simple abnormal biextremal contained in RA(E) and parametrized by
arc-length is regular.
Remark 1 Arcs that minimize length are often called "geodesies." In
the Riemannian case, the precise definition of a "geodesic" is that it is
a curve that satisfies the necessary condition for length-minimization
given by the geodesic equation or, equivalently, a curve that locally
minimizes length. In the sub-Riemannian setting there is no unanimous
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