CR manifold

In mathematics, a CR 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. In mathematics, a CR 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 C T M = T M ⊗ R C {displaystyle mathbb {C} TM=TMotimes _{mathbb {R} }mathbb {C} } such that The subbundle L is called a CR structure on the manifold M. The abbreviation CR stands for Cauchy–Riemann or Complex-Real. The notion of a CR structure attempts to describe intrinsically the property of being a hypersurface (or certain real submanifolds of higher codimension) in complex space by studying the properties of holomorphic vector fields which are tangent to the hypersurface. Suppose for instance that M is the hypersurface of C 2 {displaystyle mathbb {C} ^{2}} given by the equation where z and w are the usual complex coordinates on C 2 {displaystyle mathbb {C} ^{2}} . The holomorphic tangent bundle of C 2 {displaystyle mathbb {C} ^{2}} consists of all linear combinations of the vectors The distribution L on M consists of all combinations of these vectors which are tangent to M. The tangent vectors must annihilate the defining equation for M, so L consists of complex scalar multiples of In particular, L consists of the holomorphic vector fields which annihilate F. Note that L gives a CR structure on M, for = 0 (since L is one-dimensional) and L ∩ L ¯ = { 0 } {displaystyle Lcap {ar {L}}={0}} since ∂/∂z and ∂/∂w are linearly independent of their complex conjugates.