CR structure

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 subbundle of the complexified tangent bundle CTM = TM ⊗ C such that

The bundle 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 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 C² given by the equation

where z and w are the usual complex coordinates on C². The holomorphic tangent bundle of C² 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 [L,L] = 0 (since L is one-dimensional) and ${\displaystyle L\cap {\bar {L}}=\{0\}}$ since ∂/∂z and ∂/∂w are linearly independent of their complex conjugates.

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