with Lucas Hsu, Rigidity of integral curves of rank 2 distributions,
Invent. Math., vol. 114 no. 2
(1993),
pp. 435–461 [MR94j:58003], [dvi]
(last updated on 2004/04/06)
Author's Comments:
In this manuscript, Hsu and I show that, for the generic
2-plane
field D on a manifold of dimension 4 or more, there
exist so-called 'rigid'
D-curves, i.e., smooth curves tangent to the plane field
D with the property
that they admit no compactly supported smooth
variations through D-curves
other than reparametrization. These curves will
therefore be abnormal extremals
for any variational problem for D-curves.
We investigate related phenomena, such as locally
rigid curves that
are not globally rigid, and compute several examples
drawn from geometry
and mechanics. For example, we analyze the
mechanical system of one surface
rolling over another without twisting or slipping (the
case where the surfaces
are a plane and a sphere had already been treated by
Brockett and Dai)
as well as the geometry of space curves of constant
curvature (but variable
torsion).
Since our paper, quite a lot of work has appeared
about rigid curves
and abnormal extremals in the context of
sub-Riemannian geometry, particularly,
see the recent works of H. Sussman and W. Liu,
Agrachev and Sarychev, Milyutin,
and Dmitruk.
This paper is also available as a .pdf
file
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