Gradient flows within plane fields
Joint work with Robert Ghrist.
Comm. Math. Helvetici 74 (1999), 507-529.

We consider the dynamics of vector fields on three-manifolds which are constrained to lie within a plane field, such as occurs in nonholonomic dynamics. On compact manifolds, such vector fields force dynamics beyond that of a gradient flow, except in cases where the underlying manifold is topologically simple (eg, a graph-manifold). Furthermore, there are strong restrictions on the types of gradient flows realised within plane fields: such flows lie on the boundary of the space of nonsingular Morse-Smale flows. This relationship translates to knot-theoretic obstructions for the link of singularities in the flow. In the case of an integrable plane field, the restrictions are even finer, forcing among other things the nonexistence of Reeb components in the foliation. The situation is completely different in the case of contact plane fields, however: it is easy to realise gradient fields within overtwisted contact structures (the analogue of a foliation with Reeb components).

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