Local reduction theorems and invariants for singular contact structures

Abstract

A differential 1-form on a (2k+1)-dimensional manifolds M defines a singular contact structure if the set S of points where the contact condition is not satisfied, S={p∈M:(ω∧(dω) k (p)=0}, is nowhere dense in M. Then S is a hypersurface with singularities and the restriction of ω to S can be defined. Our first theorem states that in the holomorphic, real-analytic, and smooth categories the germ of Pfaffian equation (ω) generated by ω is determined, up to a diffeomorphism, by its restriction to S, if we eliminate certain degenerated singularities of ω (in the holomorphic case they form a set of infinite codimension). We also define other invariants of local singular contact structures: orientations, a line bundle, and a partial connection. We study the problem when these invariants, together with the hypersurface S and the restriction of the Pfaffian equation (ω) to S, form a complete set of local invariants. Our results include complete solutions to this problem in dimension 3 and in the case where S has no singularities.