CR submanifolds of a complex manifold are defined as submanifolds M⊂X$M \subseteq X$ such that TM∩iTM⊂TX$TM \cap iTM \subseteq TX$ has constant rank (i$i$ is the imaginary unit). Note that the condition is automatically verifiedsatisfied if MM$$ has codimension one; for higher codimension this is not true.
An abstract CR manifold is a real manifold M$M$, with a distinguished subbundle HM⊂TM$HM \subseteq TM$, corresponding to TM∩iTM$TM \cap iTM$, endowed with a linear endomorphism J$J$ with J2=-Id$J^2=-Id$. The structure is furthermore required to satisfy a so called integrability condition: For all sections X,Y$X,Y$ of HM$HM$:
[X,JY]+[JX,Y]$[X,JY]+[JX,Y]$ is a section of HM$HM$
([X,Y]-[JX,JY]) + J([X,JY]+[JX,Y]) = 0$([X,Y]-[JX,JY]) + J([X,JY]+[JX,Y]) = 0$
Not every abstract CR manifold can be realized as a CR submanifold.
