Gradient flows within plane fields

Abstract

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 (i.e., a graph-manifold). Furthermore, there are strong restrictions on the types of gradient flows realized 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 taut foliations on surface bundles. The situation is completely different in the case of contact plane fields, however: it is easy to realize gradient fields within overtwisted contact structures (the nonintegrable analogue of a foliation with Reeb components).