| Version 3 |
Version 2 |
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\theoremstyle{definition} |
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\newtheorem*{defn}{Definition} |
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| 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 |
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 $z_j = x_j + i y_j$, we can take the following vectors as our basis |
${\mathbb{R}}^{2N}$ by $z_j = x_j + i y_j$, we can take the following vectors as our basis |
| \begin{equation*} |
\begin{equation*} |
| \frac{\partial}{\partial x_1} \Bigg\rvert_p, |
\frac{\partial}{\partial x_1} \Bigg\rvert_p, |
| \frac{\partial}{\partial y_1} \Bigg\rvert_p, |
\frac{\partial}{\partial y_1} \Bigg\rvert_p, |
| \ldots, |
\ldots, |
| \frac{\partial}{\partial x_N} \Bigg\rvert_p, |
\frac{\partial}{\partial x_N} \Bigg\rvert_p, |
| \frac{\partial}{\partial y_N} \Bigg\rvert_p . |
\frac{\partial}{\partial y_N} \Bigg\rvert_p . |
| \end{equation*} |
\end{equation*} |
|
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| 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 |
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 |
| \begin{equation*} |
\begin{equation*} |
| J \left( |
J \left( |
| \frac{\partial}{\partial x_1} \Bigg\rvert_p |
\frac{\partial}{\partial x_1} \Bigg\rvert_p |
| \right) |
\right) |
| = \frac{\partial}{\partial y_1} \Bigg\rvert_p |
= \frac{\partial}{\partial y_1} \Bigg\rvert_p |
| \qquad \text{ and } |
\qquad \text{ and } |
| J \left( |
J \left( |
| \frac{\partial}{\partial y_1} \Bigg\rvert_p |
\frac{\partial}{\partial y_1} \Bigg\rvert_p |
| \right) |
\right) |
| = - \frac{\partial}{\partial x_1} \Bigg\rvert_p . |
= - \frac{\partial}{\partial x_1} \Bigg\rvert_p . |
| \end{equation*} |
\end{equation*} |
| 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. |
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 $M$). |
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 $M$). |
|
|
| \begin{defn} |
\begin{defn} |
| The subspace $T_p^c(M)$ defined as |
The subspace $T_p^c(M)$ defined as |
| \begin{equation*} |
\begin{equation*} |
| T_p^c(M) := \{ X \in T_p(M) \mid J(X) \in T_p(M) \} |
T_p^c(M) := \{ X \in T_p(M) \mid J(X) \in T_p(M) \} |
| \end{equation*} |
\end{equation*} |
| is called the {\em complex tangent space} of $M$ at the point $p$, and if the dimension of $T_p(M)$ is constant for all |
is called the {\em 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 |
$p \in M$ then the |
| corresponding vector bundle $T^c(M) := \bigcup_{p\in M} T_p^c(M)$ is called the {\em complex bundle} of $M$. |
corresponding vector bundle $T^c(M) := \bigcup_{p\in M} T_p^c(M)$ is called the {\em complex bundle} of $M$. |
| \end{defn} |
\end{defn} |
|
|
| Do note that the complex tangent space is a real (not complex) vector space, despite its rather unfortunate name. |
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 |
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 |
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 |
$X = \sum a_j \frac{\partial}{\partial x_1} \Big\rvert_p + b_j \frac{\partial}{\partial x_1} \Big\rvert_p$ 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 $-i$. |
that the operator $J$ has two eigenvalues, $i$ and $-i$. |
|
|
| \begin{defn} |
\begin{defn} |
| Let ${\mathcal{V}}_p$ be the eigenspace of ${\mathbb{C}} T_p(M)$ corresponding to the eigenvalue $-i$. That is |
Let ${\mathcal{V}}_p$ be the eigenspace of ${\mathbb{C}} T_p(M)$ corresponding to the eigenvalue $-i$. That is |
| \begin{equation*} |
\begin{equation*} |
| {\mathcal{V}}_p := \{ X \in {\mathbb{C}} T_p(M) \mid J(X) = -iX \} . |
{\mathcal{V}}_p := \{ X \in {\mathbb{C}} T_p(M) \mid J(X) = -iX \} . |
| \end{equation*} |
\end{equation*} |
| If the dimension of ${\mathcal{V}}_p$ is constant for all $p \in M$, then |
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 |
we get a corresponding vector bundle ${\mathcal{V}}$ which we call the |
| {\em CR bundle} of $M$. A smooth section of the CR bundle is then called |
{\em CR bundle} of $M$. A smooth section of the CR bundle is then called |
| a {\em CR vector field}. |
a {\em CR vector field}. |
| \end{defn} |
\end{defn} |
|
|
| \begin{defn} |
\begin{defn} |
| The submanifold $M$ is called a {\em CR submanifold} (or just {\em CR manifold}) if the dimension of ${\mathcal{V}}_p$ is constant for all $p \in M$. |
The submanifold $M$ is called a {\em CR submanifold} (or just {\em 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 |
The complex dimension of ${\mathcal{V}}_p$ will then be called the |
| {\em CR dimension} of $M$. |
{\em CR dimension} of $M$. |
| \end{defn} |
\end{defn} |
|
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| An example of a CR submanifold is for example a hyperplane defined by |
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. |
$\operatorname{Im} z_N = 0$ where the CR dimension is $N-1$. Another less trivial example is the Lewy hypersurface. |
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| Note that sometimes ${\mathcal{V}}_p$ is written as $T_p^{0,1} (M)$ and referred to as the {\em space of antiholomorphic vectors}, where an {\em antiholomorphic vector} is a tangent vector which can be written in terms of the basis |
Note that sometimes ${\mathcal{V}}_p$ is written as $T_p^{0,1} (M)$ and referred to as the {\em space of antiholomorphic vectors}, where an {\em antiholomorphic vector} is a tangent vector which can be written in terms of the basis |
| \begin{equation*} |
\begin{equation*} |
| \frac{\partial}{\partial \bar{z}_j} \Bigg\rvert_p := |
\frac{\partial}{\partial \bar{z}_j} \Bigg\rvert_p := |
| \frac{1}{2} |
\frac{1}{2} |
| \left( |
\left( |
| \frac{\partial}{\partial x_j} \Bigg\rvert_p + i |
\frac{\partial}{\partial x_j} \Bigg\rvert_p + i |
| \frac{\partial}{\partial y_j} \Bigg\rvert_p |
\frac{\partial}{\partial y_j} \Bigg\rvert_p |
| \right) . |
\right) . |
| \end{equation*} |
\end{equation*} |
|
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| The CR in the name refers to Cauchy-Riemann and that is because the vector space |
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 $\bar{z}_j$. |
${\mathcal{V}}_p$ corresponds to differentiating with respect to $\bar{z}_j$. |
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|
| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{ber:submanifold} |
\bibitem{ber:submanifold} |
| M.\@ Salah Baouendi, |
M.\@ Salah Baouendi, |
| Peter Ebenfelt, |
Peter Ebenfelt, |
| Linda Preiss Rothschild. |
Linda Preiss Rothschild. |
| {\em \PMlinkescapetext{Real Submanifolds in Complex Space and Their Mappings}}, |
{\em \PMlinkescapetext{Real Submanifolds in Complex Space and Their Mappings}}, |
| Princeton University Press, |
Princeton University Press, |
| Princeton, New Jersey, 1999. |
Princeton, New Jersey, 1999. |
| \end{thebibliography} |
\end{thebibliography} |
