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.
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