SUB-RIEMANNIAN METRICS
7
agreement in the literature on the proper use of the word "geodesic."
There is an obvious sub-Riemannian analogue of the geodesic equation,
namely, the equation of non-null bicharacteristics of the Hamiltonian
of the sub-Riemannian metric. This suggests the possibility of calling
these bicharacteristics —or their projections on the base manifold—
"geodesies." This would specialize in the Riemannian case to the usual
definition of geodesies, but in the more general sub-Riemannian setting
it would no longer be true that every minimizer is a geodesic. Al-
ternatively, we could choose to define a "geodesic" to be an arc that
locally minimizes length. (Some authors —e.g. R. Montgomery in [17],
whose first version was pointedly entitled "Geodesies that do not sat-
isfy the geodesic equations"— appear to be leaning in this direction,
and Hamenstadt in [12] explicitly states that "geodesies" are "locally
minimizing curves.") This would also reduce in the Riemannian case
to the usual definition of geodesies, and would in addition have the
desired property that all minimizers are geodesies. On the other hand,
the new definition would have the drawback of making geodesies hard
to characterize by means of differential equations. (Such a character-
ization has not yet been found, and may very well not exist.) In this
work we choose to avoid the word "geodesic" altogether. Arcs that
minimize length are called "minimizers." Arcs that satisfy the neces-
sary conditions for minimization given by the Maximum Principle of
Optimal Control Theory are "extremals." Extremality is a necessary
condition for an arc to be a minimizer, and has an explicit character-
ization in terms of constrained ordinary differential equations, but in
the sub-Riemannian case an extremal need not be a local minimizer,
as explained above. |
Remark 2 In [7], Bryant and Hsu studied rigid curves for smooth dis-
tributions, and proved that, if a rank 2 distribution E on M satisfies
a nonintegrability condition, then through every point p of M there
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