Contact structures of arbitrary codimension and idempotents in the Heisenberg algebra

Details Contact structures of arbitrary codimension and idempotents in the Heisenberg algebra Contact structures of arbitrary codimension and idempotents in the Heisenberg algebra

2010-01-29

arxiv.org/abs/1001.5426v2

Mathematics

Differential Geometry

13 pages. Second version contains major revision

Scientific paper

A contact manifold is a manifold equipped with a distribution of codimension one that satisfies a `maximal non-integrability' condition. A standard example of a contact structure is a strictly pseudoconvex CR manifold, and operators of analytic interest are the tangential Cauchy-Riemann operator and the Szego projector onto its kernel. The Heisenberg calculus is the natural pseudodifferential calculus developed originally for the analysis of these operators. We introduce a `non-integrability' condition for a distribution of arbitrary codimension that directly generalizes the definition of a contact structure. We call such distributions polycontact structures. We prove that the polycontact condition is equivalent to the existence of generalized Szego projections in the Heisenberg calculus, and explore geometrically interesting examples of polycontact structures.

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