CR submanifold

Suppose that MN is a real submanifold of real dimension n. Take pM, then let Tp(N) be the tangent vectors of N at the point p. If we identify N with 2N by zj=xj+iyj, we can take the following vectors as our basis


We define a real linear mapping J:Tp(N)Tp(N) such that for any 1jN we have

J(x1|p)=y1|p   and J(y1|p)=-x1|p.

Where J is referred to as the complex structureMathworldPlanetmath on Tp(N). Note that J2=-I, that is applying J twice we just negate the vector.

Let Tp(M) be the tangent space of M at the point p (that is, those vectors of Tp(N) which are tangent to M).


The subspacePlanetmathPlanetmathPlanetmath Tpc(M) defined as


is called the complex tangent space of M at the point p, and if the dimensionPlanetmathPlanetmathPlanetmath of Tp(M) is constant for all pM then the corresponding vector bundleMathworldPlanetmath Tc(M):=pMTpc(M) is called the complex bundle of M.

Do note that the complex tangent space is a real (not complex) vector spaceMathworldPlanetmath, despite its rather unfortunate name.

Let Tp(M) and Tp(N) be the complexified vector spaces, by just allowing the coefficents of the vectors to be complex numbers. That is for X=ajx1|p+bjx1|p we allow aj and bj to be complex numbers. Next we can extend the mapping J to be -linear on these new vector spaces and still get that J2=-I as before. We notice that the operator J has two eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath, i and -i.


Let 𝒱p be the eigenspaceMathworldPlanetmath of Tp(M) corresponding to the eigenvalue -i. That is


If the dimension of 𝒱p is constant for all pM, then we get a corresponding vector bundle 𝒱 which we call the CR bundle of M. A smooth section of the CR bundle is then called a CR vector field.


The submanifold M is called a CR submanifold (or just CR manifold) if the dimension of 𝒱p is constant for all pM. The complex dimension of 𝒱p will then be called the CR dimension of M.

An example of a CR submanifold is for example a hyperplane defined by ImzN=0 where the CR dimension is N-1. Another less trivial example is the Lewy hypersurface.

Note that sometimes 𝒱p is written as Tp0,1(M) and referred to as the space of antiholomorphic vectors, where an antiholomorphic vector is a tangent vector which can be written in terms of the basis


The CR in the name refers to Cauchy-Riemann and that is because the vector space 𝒱p corresponds to differentiating with respect to z¯j.


  • 1 M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. , Princeton University Press, Princeton, New Jersey, 1999.
  • 2 Albert Boggess. , CRC, 1991.
Title CR submanifold
Canonical name CRSubmanifold
Date of creation 2013-03-22 14:49:04
Last modified on 2013-03-22 14:49:04
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 9
Author jirka (4157)
Entry type Definition
Classification msc 32V05
Synonym CR manifold
Synonym Cauchy-Riemann submanifold
Related topic GenericManifold
Related topic TotallyRealSubmanifold
Related topic TangentialCauchyRiemannComplexOfCinftySmoothForms
Related topic ACRcomplex
Defines CR bundle
Defines CR vector field
Defines complex tangent space
Defines complex bundle
Defines space of antiholomorphic vectors
Defines antiholomorphic vector
Defines CR dimension