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Many natural geometric structures on manifolds are given as sections of certain
bundles satisfying open relations at every point, depending on the derivatives of
these sections. Such relations are called open differential relations. Contact,
even-contact, and (exact) symplectic structures on manifolds can be described in this
way. The natural question is: do structures satisfying given open relations (called
the genuine solutions of the differential relation) exist on a given manifold?
Replacing all derivatives appearing in a differential relation by the additional
independent variables one obtains an open subset of the corresponding jet bundle. A
formal solution of the differential relation is a section of the jet bundle lying in
this open set. The existence of a formal solution is obviously a necessary condition
for the existence of the genuine one. One says that a differential relation satisfies
a (nonparametric) h-principle if any formal solution is homotopic to the genuine
solution in the space of formal solutions.
Versions of the h-principle have been successfully established for corank 1 distributions satisfying natural open relations. Such results are among the most remarkable advances in differential topology in the last four decades. However, very little is known about analogous results for other classes of distributions, e.g. generic distributions of corank 2 or higher (except the so-called Engel distributions, the smallest dimensional case of maximally nonholonomic distributions of corank 2 distributions on 4-dimensional manifolds). In my talk, I will show how to use the method of convex integration in order to establish all versions of the h-principle for corank 2 distributions of arbitrary odd rank satisfying a natural generic assumption on the associated pencil of skew-symmetric forms. During the talk, I will try to give all the necessary background. This is the joint work with Milan Jovanovic, Javier Martinez-Aguinaga, and Alvaro del Pino. |