dg.differential geometry - What are CR manifolds like?

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.