CR submanifold

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

$${{{\frac{\partial }{\partial x_1}|}_p,\frac{\partial }{\partial y_1}|}_p,\mathrm{\dots },\frac{\partial }{\partial x_N}|}_p,\frac{\partial }{\partial y_N}|{}_p.$$

We define a real linear mapping $J:T_p({ℂ}^N)\to T_p({ℂ}^N)$ such that for any $1\le j\le N$ we have

$${J{\left(\frac{\partial }{\partial x_1}\right|}_p)=\frac{\partial }{\partial y_1}|}_p\mathit{\hspace{1em}\hspace{1em}}\text{ and }J{\left(\frac{\partial }{\partial y_1}\right|}_p)=-\frac{\partial }{\partial x_1}|{}_p.$$

Where $J$ is referred to as the complex structure on $T_p({ℂ}^N).$ 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 $T_p({ℂ}^N)$ 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 thecorresponding 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 $ℂT_p(M)$ and $ℂT_p({ℂ}^N)$ be the complexified vector spaces, by just allowing the coefficents of the vectors tobe complex numbers. That is for${X=\sum a_j\frac{\partial }{\partial x_1}|}_p+b_j\frac{\partial }{\partial x_1}|{}_p$ we allow $a_j$ and $b_j$ to be complex numbers. Next we can extend the mapping $J$ to be $ℂ$-linear on these new vector spaces and still get that $J^2=-I$ as before. We noticethat the operator $J$ has two eigenvalues, $i$ and $-i$.

Definition.

Let ${𝒱}_p$ be the eigenspace of $ℂT_p(M)$ corresponding to the eigenvalue $-i.$ That is

$${𝒱}_p:=\{X\in ℂT_p(M)\mid J(X)=-iX\}.$$

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

Definition.

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

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

Note that sometimes ${𝒱}_p$ 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

$${{\frac{\partial }{\partial {\overline{z}}_j}|}_p:=\frac{1}{2}{\left(\frac{\partial }{\partial x_j}\right|}_p+i\frac{\partial }{\partial y_j}|}_p).$$

The CR in the name refers to Cauchy-Riemann and that is because the vector space${𝒱}_p$ corresponds to differentiating with respect to ${\overline{z}}_j.$