# CR manifold

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In mathematics, a CR manifold is a differentiable manifold together with a geometric structure modeled on that of a real hypersurface in a complex vector space, or more generally modeled on an edge of a wedge.

Formally, a CR manifold is a differentiable manifold M together with a preferred complex distribution L, or in other words a subbundle of the complexified tangent bundle CTM = TMC such that

• $[L,L]\subseteq L$ (L is formally integrable)
• $L\cap\bar{L}=\{0\}$ (L is almost Lagrangian).

The bundle L is called a CR structure on the manifold M.

The abbreviation CR stands for Cauchy-Riemann or Complex-Real.

## Introduction and motivation

The notion of a CR structure attempts to describe intrinsically the property of being a hypersurface in complex space by studying the properties of holomorphic vector fields which are tangent to the hypersurface.

Suppose for instance that M is the hypersurface of C2 given by the equation

$F(z,w) := |z|^2+|w|^2=1,$

where z and w are the usual complex coordinates on C2. The holomorphic tangent bundle of C2 consists of all linear combinations of the vectors

$\frac{\partial}{\partial z},\quad \frac{\partial}{\partial w}.$

The distribution L on M consists of all combinations of these vectors which are tangent to M. In detail, the tangent vectors must annihilate the defining equation for M, so L consists of complex scalar multiples of

$\bar{w}\frac{\partial}{\partial z}-\bar{z}\frac{\partial}{\partial w}.$

Note that L gives a CR structure on M, for [L,L] = 0 (since L is one-dimensional) and $L\cap\bar{L}=\{0\}$ since ∂/∂z and ∂/∂w are linearly independent of their complex conjugates.

More generally, suppose that M is a real hypersurface in Cn, with defining equation F(z1, ..., zn) = 0. Then the CR structure L consists of those linear combinations of the basic holomorphic vectors on Cn:

$\frac{\partial}{\partial z_1}, ..., \frac{\partial}{\partial z_n}$

which annihilate the defining function. In this case, $L\cap\bar{L}=\{0\}$ for the same reason as before. Moreover, [L,L] ⊂ L since the commutator of vector fields annihilating F is again a vector field annihilating F.

### Embedded and abstract CR manifolds

There is a sharp contrast between the theories of embedded CR manifolds (hypersurface and edges of wedges in complex space) and abstract CR manifolds (those given by the Lagrangian distribution L). Many of the formal geometrical features are similar. These include:

Embedded CR manifolds possess some additional structure, though: a Neumann and Dirichlet problem for the Cauchy-Riemann equations.

This article first treats the geometry of embedded CR manifolds, shows how to defined these structures intrinsically, and then generalizes these to the abstract setting.

## Embedded CR manifolds

### Preliminaries

Embedded CR manifolds are, first and foremost, submanifolds of Cn. Define a pair of subbundles of the complexified tangent bundle C ⊗ TC'n by:

$T^{(1,0)}\mathbb{C}^n = \mathrm{span}\left(\frac{\partial}{\partial z_1},\dots,\frac{\partial}{\partial z_n}\right).$
• T(0,1)Cn consists of the complex vectors annihilating the holomorphic functions. In coordinates:
$T^{(0,1)}\mathbb{C}^n = \mathrm{span}\left(\frac{\partial}{\partial \bar{z}_1},\dots,\frac{\partial}{\partial \bar{z}_n}\right).$

Also relevant are the characteristic annihilators from the Dolbeault complex:

• Ω(1,0)Cn = (T(0,1)Cn). In coordinates,
$\Omega^{(1,0)}\mathbb{C}^n = \mathrm{span}(dz_1,\dots,dz_n).$
• Ω(0,1)Cn = (T(1,0)Cn). In coordinates,
$\Omega^{(0,1)}\mathbb{C}^n = \mathrm{span}(d\bar{z}_1,\dots,d\bar{z}_n).$

The exterior products of these are denoted by the self-evident notation Ω(p,q), and the Dolbeault operator and its complex conjugate map between these spaces via

$\partial : \Omega^{(p,q)} \rightarrow \Omega^{(p+1,q)}$
$\bar{\partial} : \Omega^{(p,q)} \rightarrow \Omega^{(p,q+1)}$

Furthermore, there is a decomposition of the usual exterior derivative via $d = \partial + \bar{\partial}$.

### Real submanifolds of complex space

Let M ⊂ Cn be a real submanifold, defined locally as the locus of a system of smooth real-valued functions

F1 = 0, F2 = 0, ..., Fk = 0.

Suppose that this system has maximal rank, in the sense that the differentials satisfy the following independence condition:

$\partial F_1\wedge\bar{\partial} F_1\wedge\dots \wedge \partial F_k\wedge \bar{\partial} F_k \not= 0.$

Note that this condition is strictly stronger than needed to apply the implicit function theorem: in particular, M is a manifold of real dimension 2n − k. We say that M is an embedded CR manifold of CR codimension k. In most applications, k = 1, in which case the manifold is said to be of hypersurface type.

Let L ⊂ T(1,0)Cn|M be the subbundle of vectors annihilating all of the defining functions F1, ..., Fk. Note that, by the usual considerations for integrable distributions on hypersurfaces, L is involutive. Moreover, the independence condition implies that L is a bundle of constant rank n − k.

Henceforth, suppose that k = 1 (so that the CR manifold is of hypersurface type), unless otherwise noted.

### The Levi form

Let M be a CR manifold of hypersurface type with single defining function F = 0. The Levi form of M, named after Eugenio Elia Levi,[1] is the Hermitian 2-form

$h=i\partial\bar{\partial}F|_L.$

This determines a metric on L. M is said to be strictly pseudoconvex if h is positive definite (or pseudoconvex in case h is positive semidefinite). Many of the analytic existence and uniqueness results in the theory of CR manifolds depend on the strict pseudoconvexity of the Levi form.

This nomenclature comes from the study of pseudoconvex domains: M is the boundary of a (strictly) pseudoconvex domain in Cn if and only if it is (strictly) pseudoconvex as a CR manifold. (See plurisubharmonic functions and Stein manifold.)

## Abstract CR structures

An abstract CR structure on a manifold M of dimension n consists of a subbundle L of the complexified tangent bundle which is formally integrable, in the sense that [L,L] ⊂ L, which is linearly independent of its complex conjugate. The CR codimension of the CR structure is k = n - 2 dim L. In case k = 1, the CR structure is said to be of hypersurface type. Most examples of abstract CR structures are of hypersurface type, unless otherwise made explicit.

### The Levi form and pseudoconvexity

Suppose that M is a CR manifold of hypersurface type. The Levi form is the vector valued form, defined on L, with values in the line bundle

$V = \frac{TM\otimes{\mathbb C}}{L\oplus\bar{L}}$

given by

$h(v,w) = \frac{1}{2i}[v,\bar{w}] \mod L\oplus\bar{L},\quad v,w\in L.$

h defines a sesquilinear form on L since it does not depend on how v and w are extended to sections of L, by the integrability condition. This form extends to a hermitian form on the bundle $L\oplus\bar{L}$ by the same expression. The extended form is also sometimes referred to as the Levi form.

The Levi form can alternatively be characterized in terms of duality. Consider the line subbundle of the complex cotangent bundle annihilating V

$H_0M = V^* = (L\oplus\bar{L})^\perp\sub T^*M\otimes{\mathbb C}.$

For each local section α∈Γ(H0M), let

$h_\alpha(v,w) = d\alpha(v,\bar{w}) = -\alpha([v,\bar{w}]),\quad v,w\in L\oplus\bar{L}.$

The form hα is a complex-valued hermitian form associated to α.

Generalizations of the Levi form exist when the manifold is not of hypersurface type, in which case the form no longer assumes values in a line bundle, but rather in a vector bundle. One may then speak, not of a Levi form, but of a collection of Levi forms for the structure.

## Examples

The canonical example of a CR manifold is the real $2n+1$ sphere as a submanifold of $\mathbb{C}^{n+1}$. The bundle $L$ described above is given by

$L = \mathbb{C}TS^{2n+1} \cap T^{1,0}\mathbb{C}^{n+1}$

where $T^{1,0}\mathbb{C}^{n+1}$ is the bundle of holomorphic vectors. The real form of this is given by $P=\Re (L\oplus \bar{L})$, the bundle given at a point $p\in S^{2n+1}$ concretely in terms of the complex structure, $I$, on $\mathbb{C}^{n+1}$ by

$P_p = \{ X\in T_pS^{2n+1} : IX \in T_pS^{2n+1}\subset T_p\mathbb{C}^{n+1}\},$

and the almost complex structure on $P$ is just the restriction of $I$.

The Heisenberg group also is an example of a CR manifold.[2]

## Notes

1. ^ See (Levi 909, p. 207): the Levi form is the differential form associated to the differential operator C, according to Levi's notation.
2. ^ Capogna, Luca; Danielli, Donatella; Pauls, Scott; Tyson, Jeremy (2007). Applications of Heisenberg Geometry. "An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem". Progress in Mathematics 259 (Berlin: Birkhauser). pp. 45–48.