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The complex structure on a complex manifold pulls back to what's called a CR structure on any real codimension 1 submanifold. The structure induced on a submanifold of higher codimension is a CR structure if a non-degeneracy condition holds. It's possible to describe these structures intrinsically, without reference to an embedding. I don't know anything else.

I'd be happy with whatever kind of answer to the title question, but here are some more specific ones:

  1. Does CR stand for Cauchy-Riemann, or what?

  2. What kind of local invariants do CR manifolds have? Are there coordinates around every point that look like a real hyperplane in C^n? Or can there be some curvature or something.

  3. Can there be continuous families of CR structures on a given manifold? If the manifold is compact can these families (mod diffeomorphism) be infinite-dimensional?

  4. I have the impression, just from arxiv postings and seminar titles, of CR geometry being studied more in analysis than in softer geometric fields. Is that accurate, and if so what accounts for it?

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up vote 8 down vote accepted

CR does stand for Cauchy-Riemann.

CR structures on 3 dimensional manifolds arise as the boundaries of complex (or almost-complex) 4 manifolds; if these boundaries are strictly pseudo-convex (i.e. convex in "holomorphic directions") the CR structure on the 3-manifold is a contact structure (if the boundary is only (pseudo-)convex or (Levi) flat, the CR structure integrates to a confoliation or a foliation respectively). There can be infinite dimensional families of foliations on a 3-manifold; more generally, whenever the CR structure is "non-generic" or integrable, one has continuous moduli, otherwise (eg in the contact structure case) one has discrete moduli (to be explicit: what has discrete moduli is the contact structure, not the "CR+contact structure".)

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Is it special to three dimensions that the four-manifold it bounds only needs to by almost complex? It makes it sound like a CR structure is just an almost complex structure on M x R. – David Treumann Dec 2 '09 at 16:42
It is special of the case where HM (see my answer below) has real dimension 2. In this case indeed the integrability condition is automatically true. – user175348 Dec 4 '09 at 9:56

CR submanifolds of a complex manifold are defined as submanifolds M⊂X such that TM∩iTM⊂TX has constant rank (i is the imaginary unit). Note that the condition is automatically verified if M has codimension one; for higher codimension this is not true.

An abstract CR manifold is a real manifold M, with a distinguished subbundle HM⊂TM, corresponding to TM∩iTM, endowed with a linear endomorphism J with J2=-Id. The structure is furthermore required to satisfy a so called integrability condition: For all sections X,Y of HM:

  • [X,JY]+[JX,Y] is a section of HM

  • ([X,Y]-[JX,JY]) + J([X,JY]+[JX,Y]) = 0

Not every abstract CR manifold can be realized as a CR submanifold.

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Are there topological obstructions to realizing a CR manifold as a CR submanifold? (Or what kind of obstructions.) – David Treumann Dec 2 '09 at 16:45
There are local obstructions to the existence of CR functions (and hence to embeddability). For 3-dimensional CR manifolds "generically" (in the Baire sense) the only CR functions are the constants. – user175348 Dec 4 '09 at 9:58

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