# CR submanifold

Suppose that $M\subset {\u2102}^{N}$ is a real submanifold of real dimension $n.$ Take $p\in M,$ then let ${T}_{p}({\u2102}^{N})$ be the tangent vectors of ${\u2102}^{N}$ at the point $p.$ If we identify ${\u2102}^{N}$ with ${\mathbb{R}}^{2N}$ 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 $J:{T}_{p}({\u2102}^{N})\to {T}_{p}({\u2102}^{N})$ 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 ${T}_{p}({\u2102}^{N}).$ 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 ${T}_{p}({\u2102}^{N})$ 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 $\u2102{T}_{p}(M)$ and $\u2102{T}_{p}({\u2102}^{N})$ be the complexified vector spaces, by just allowing the coefficents of the vectors to
be complex numbers. That is for
${X=\sum {a}_{j}\frac{\partial}{\partial {x}_{1}}|}_{p}+{b}_{j}\frac{\partial}{\partial {x}_{1}}|{}_{p}$ we allow ${a}_{j}$ and ${b}_{j}$ to be complex numbers. Next we can extend the mapping $J$ to be $\u2102$-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 $\u2102{T}_{p}(M)$ corresponding to the eigenvalue $-i.$ That is

$${\mathcal{V}}_{p}:=\{X\in \u2102{T}_{p}(M)\mid J(X)=-iX\}.$$ |

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 $\mathrm{Im}{z}_{N}=0$ 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

$${{\frac{\partial}{\partial {\overline{z}}_{j}}|}_{p}:=\frac{1}{2}{\left(\frac{\partial}{\partial {x}_{j}}\right|}_{p}+i\frac{\partial}{\partial {y}_{j}}|}_{p}).$$ |

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

## 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 |