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