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C R Submanifold

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

$\displaystyle \frac{\partial}{\partial x_1} \Bigg\rvert_p, \frac{\partial}{\par... ...ial}{\partial x_N} \Bigg\rvert_p, \frac{\partial}{\partial y_N} \Bigg\rvert_p .$

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

$\displaystyle J \left( \frac{\partial}{\partial x_1} \Bigg\rvert_p \right) = \frac{\partial}{\partial y_1} \Bigg\rvert_p$ and $\displaystyle J \left( \frac{\partial}{\partial y_1} \Bigg\rvert_p \right) = - \frac{\partial}{\partial x_1} \Bigg\rvert_p .$

Where

is referred to as the complex structure on $T_p({\mathbb{C}}^N).$ Note that

that is applying

twice we just negate the vector.

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

Definition 1 The subspace

defined as

$\displaystyle T_p^c(M) := \{ X \in T_p(M) \mid J(X) \in T_p(M) \}$

is called the complex tangent space of

at the point

and if the dimension of

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

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 and to be complex numbers. Next we can extend the mapping to be ${\mathbb{C}}$ -linear on these new vector spaces and still get that as before. We notice that the operator has two eigenvalues, and .

Definition 2 Let ${\mathcal{V}}_p$ be the eigenspace of ${\mathbb{C}} T_p(M)$ corresponding to the eigenvalue

That is

$\displaystyle {\mathcal{V}}_p := \{ X \in {\mathbb{C}} T_p(M) \mid J(X) = -iX \} .$

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

A smooth section of the CR bundle is then called a CR vector field.

Definition 3 The submanifold

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

An example of a CR submanifold is for example a hyperplane defined by $\operatorname{Im} z_N = 0$ where the CR dimension is 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

$\displaystyle \frac{\partial}{\partial \bar{z}_j} \Bigg\rvert_p := \frac{1}{2} ... ...al x_j} \Bigg\rvert_p + i \frac{\partial}{\partial y_j} \Bigg\rvert_p \right) .$

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 $\bar{z}_j.$

Bibliography

1: M. Salah Baouendi, Peter Ebenfelt, Linda Preiss Rothschild. Real Submanifolds in Complex Space and Their Mappings, Princeton University Press, Princeton, New Jersey, 1999.
2: Albert Boggess. CR Manifolds and the Tangential Cauchy Riemann Complex, CRC, 1991.

1	Point
1	Linear Transformation
1	Vector Space
1	Vector
1	Subspace Topology
1	Complex Number
1	Complex
1	Basic Polynomial
1	Real Number
1	Mapping
1	Operator
2	Tangent Line
2	Characteristic Values And Vectors Of A Matrix
2	Basis
3	Tangent Space
3	Dimension Of A Poset
4	Eigenvalue
4	Linear Manifold
4	Submanifold
5	Vector Bundle
6	Section Of A Fiber Bundle
7	Reduction Of Structure Group
9	Eigenspace
21	Lewy Hypersurface