CR submanifold

Suppose that $M\subset{\mathbb{C}}^{N}$ is a real submanifold of real dimension $n .$ Take $p\in M,$ then let $T_{p}({\mathbb{C}}^{N})$ be the tangent vectors of ${\mathbb{C}}^{N}$ at the point $p .$ If we identify ${\mathbb{C}}^{N}$ with ${\mathbb{R}}^{2N}$ by $z_{j}=x_{j}+iy_{j},$ we can take the following vectors as our basis

\frac{\partial}{\partial x_{1}}\Bigg{\rvert}_{p},\frac{\partial}{\partial y_{1% }}\Bigg{\rvert}_{p},\ldots,\frac{\partial}{\partial x_{N}}\Bigg{\rvert}_{p},% \frac{\partial}{\partial y_{N}}\Bigg{\rvert}_{p}.

We define a real linear mapping $J\colon T_{p}({\mathbb{C}}^{N})\to T_{p}({\mathbb{C}}^{N})$ such that for any $1\leq j\leq N$ we have

J\left(\frac{\partial}{\partial x_{1}}\Bigg{\rvert}_{p}\right)=\frac{\partial}% {\partial y_{1}}\Bigg{\rvert}_{p}\qquad\text{ and }J\left(\frac{\partial}{% \partial y_{1}}\Bigg{\rvert}_{p}\right)=-\frac{\partial}{\partial x_{1}}\Bigg{% \rvert}_{p}.

Where $J$ is referred to as the complex structure on $T_{p}({\mathbb{C}}^{N}).$ Note that $J^{2}=-I,$ that is applying $J$ twice we just negate the vector.

Let $T_{p}(M)$ be the tangent space of $M$ at the point $p$ (that is, those vectors of $T_{p}({\mathbb{C}}^{N})$ which are tangent to $M$ ).

Definition.

The subspace $T_{p}^{c}(M)$ defined as

T_{p}^{c}(M):=\{X\in T_{p}(M)\mid J(X)\in T_{p}(M)\}

is called the complex tangent space of $M$ at the point $p,$ and if the dimension of $T_{p}(M)$ is constant for all $p\in M$ then the corresponding vector bundle $T^{c}(M):=\bigcup_{p\in M}T_{p}^{c}(M)$ is called the complex bundle of $M$ .

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

Let ${\mathbb{C}}T_{p}(M)$ and ${\mathbb{C}}T_{p}({\mathbb{C}}^{N})$ be the complexified vector spaces, by just allowing the coefficents of the vectors to be complex numbers. That is for $X=\sum a_{j}\frac{\partial}{\partial x_{1}}\Big{\rvert}_{p}+b_{j}\frac{% \partial}{\partial x_{1}}\Big{\rvert}_{p}$ we allow $a_{j}$ and $b_{j}$ to be complex numbers. Next we can extend the mapping $J$ to be ${\mathbb{C}}$ -linear on these new vector spaces and still get that $J^{2}=-I$ as before. We notice that the operator $J$ has two eigenvalues, $i$ and $-i$ .

Definition.

Let ${\mathcal{V}}_{p}$ be the eigenspace of ${\mathbb{C}}T_{p}(M)$ corresponding to the eigenvalue $-i.$ That is

{\mathcal{V}}_{p}:=\{X\in{\mathbb{C}}T_{p}(M)\mid J(X)=-iX\}.

If the dimension of ${\mathcal{V}}_{p}$ is constant for all $p\in M,$ then we get a corresponding vector bundle ${\mathcal{V}}$ which we call the CR bundle of $M .$ A smooth section of the CR bundle is then called a CR vector field.

Definition.

The submanifold $M$ is called a CR submanifold (or just CR manifold) if the dimension of ${\mathcal{V}}_{p}$ is constant for all $p\in M.$ The complex dimension of ${\mathcal{V}}_{p}$ will then be called the CR dimension of $M .$

An example of a CR submanifold is for example a hyperplane defined by $\operatorname{Im}z_{N}=0$ where the CR dimension is $N-1.$ Another less trivial example is the Lewy hypersurface.

Note that sometimes ${\mathcal{V}}_{p}$ is written as $T_{p}^{0,1}(M)$ 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

\frac{\partial}{\partial\bar{z}_{j}}\Bigg{\rvert}_{p}:=\frac{1}{2}\left(\frac{% \partial}{\partial x_{j}}\Bigg{\rvert}_{p}+i\frac{\partial}{\partial y_{j}}% \Bigg{\rvert}_{p}\right).

The CR in the name refers to Cauchy-Riemann and that is because the vector space ${\mathcal{V}}_{p}$ corresponds to differentiating with respect to $\bar{z}_{j}.$

References

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