Math @ Duke

Publications [#243401] of Robert Bryant
search www.ams.org.Papers Published
 with Bryant, RL; Hsu, L, Rigidity of integral curves of rank 2 distributions,
Inventiones Mathematicae, vol. 114 no. 1
(1993),
pp. 435461, ISSN 00209910 [MR94j:58003], [dvi], [doi]
(last updated on 2018/12/18)
Author's Comments: In this manuscript, Hsu and I show that, for the generic
2plane
field D on a manifold of dimension 4 or more, there
exist socalled 'rigid'
Dcurves, i.e., smooth curves tangent to the plane field
D with the property
that they admit no compactly supported smooth
variations through Dcurves
other than reparametrization. These curves will
therefore be abnormal extremals
for any variational problem for Dcurves.
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
subRiemannian 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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