PlanetMath: CR submanifold

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CR submanifold

(Definition)

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.

"CR submanifold" is owned by jirka.

(view preamble)

Other names:

CR manifold, Cauchy-Riemann submanifold

Also defines:

CR bundle, CR vector field, complex tangent space, complex bundle, space of antiholomorphic vectors, antiholomorphic vector, CR dimension

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Cross-references: terms, Lewy hypersurface, hyperplane, submanifold, smooth section, eigenvalue, eigenspace, eigenvalues, operator, mapping, complex numbers, vector space, complex, vector bundle, subspace, tangent, tangent space, complex structure, linear mapping, basis, vectors, point, tangent vectors, dimension, real, real submanifold
There are 9 references to this entry.

This is version 6 of CR submanifold, born on 2004-11-16, modified 2007-12-04.
Object id is 6479, canonical name is CRSubmanifold.
Accessed 7923 times total.

Classification:

AMS MSC:

32V05 (Several complex variables and analytic spaces :: CR manifolds :: CR structures, CR operators, and generalizations)

Pending Errata and Addenda

None.

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