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A contact manifold is a manifold equipped with a distribution of codimensionone that satisfies a `maximal non-integrability' condition. A standard exampleof a contact structure is a strictly pseudoconvex CR manifold, and operators ofanalytic interest are the tangential Cauchy-Riemann operator and the Szegoprojector onto its kernel. The Heisenberg calculus is the naturalpseudodifferential calculus developed originally for the analysis of theseoperators.  We introduce a `non-integrability' condition for a distribution of arbitrarycodimension that directly generalizes the definition of a contact structure. Wecall such distributions polycontact structures. We prove that the polycontactcondition is equivalent to the existence of generalized Szego projections inthe Heisenberg calculus, and explore geometrically interesting examples ofpolycontact structures.

Contact structures of arbitrary codimension and idempotents in the Heisenberg algebra