8 WENSHENG LIU AND HECTOR J. SUSSMANN
passes a locally rigid curve. Our results are related to those of [7]
as follows. The class of points p where our results apply is slightly
larger than that of [7], because in [7] it is assumed that the distribu-
tions E and [£?, E] are nowhere integrable, for which it is necessary
that dim[E,E}(p) = 3 and dim[£, [E, E]](p) dim[E,E}(p) (cf. [7],
Theorem 3.1), whereas our results apply at every point p such that
dim[£, [E,E\](p) dim[E,E](p), even if dim[E,E}(p) = 2. (This in-
cludes, in particular, the generic singularities of rank 2 distributions
in dimension 3.) On the other hand, for the points considered in
[7], our regular abnormal extremals are exactly the "projections of \&-
characteristics in Q*" of [7]. The main result of [7] is that these curves
are locally rigid. Our results —specialized to sub-Riemannian mani-
folds (M, £', G) for which E satisfies the nonintegrabihty hypotheses of
[7]— imply in particular that these curves are also local minimizers. In
Appendix D we show that by using essentially the same method as in
the proof of our local minimization theorem one can also establish the
local rigidity of regular abnormal extremals for rank 2 distributions,
thereby providing an alternative proof of the Bryant-Hsu result under
our slightly more general conditions. Since the curves studied here and
in [7] happen to be both locally rigid and locally minimizing, one might
suspect that there is a deeper relationship between rigidity and local
optimality, e.g. that perhaps the two properties are equivalent, or that
one of them might imply the other. The purpose of Appendix D is to
argue that this is not so, by showing that both implications are false in
general.* |
2 Three examples
Before discussing the general theory, we illustrate the main issues by
means of three examples. The first one is the classical theory of Rie-
*Our example of a nonrigid abnormal minimizer is for a rank 3 distribution.
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