CR manifold

In mathematics, a CR manifold, or Cauchy–Riemann manifold, is a differentiable manifold together with a geometric structure modeled on that of a real hypersurface in a complex vector space, or more generally modeled on an edge of a wedge.

Formally, a CR manifold is a differentiable manifold M together with a preferred complex distribution L, or in other words a complex subbundle of the complexified tangent bundle ${\displaystyle \mathbb {C} TM=TM\otimes _{\mathbb {R} }\mathbb {C} }$ such that

• ${\displaystyle [L,L]\subseteq L}$ (L is formally integrable)
• ${\displaystyle L\cap {\bar {L}}=\{0\}}$.

The subbundle L is called a CR structure on the manifold M. It gives rise to a canonical differential operator mapping locally defined functions to local sections of the dual bundle ${\displaystyle L^{*}}$. This is the ${\displaystyle {\overline {\partial _{b}}}}$-operator; it is defined by

${\displaystyle {\overline {\partial _{b}}}f=df\upharpoonright _{\overline {L}}}$.

The abbreviation CR stands for "Cauchy–Riemann" or "Complex-Real". A CR-function, which is a generalization of a holomorphic function, is a solution to the system of equations ${\displaystyle {\overline {\partial _{b}}}f=0}$.