In mathematics, a CR manifold, or Cauchy–Riemann 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 complex subbundle of the complexified tangent bundle such that
(L is formally integrable) .
The subbundle L is called a CR structure on the manifold M. It gives rise to a canonical differential operator mapping locally defined functions to local sections of the dual bundle . This is the -operator; it is defined by
.
The abbreviation CR stands for "Cauchy–Riemann" or "Complex-Real". A CR-function, which is a generalization of a holomorphic function, is a solution to the system of equations .
Introduction and motivation The notion of a CR structure attempts to describe intrinsically the property of being a hypersurface (or certain real submanifolds of higher codimension) 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 given by the equation
where z and w are the usual complex coordinates on . The holomorphic tangent bundle of consists of all linear combinations of the vectors
The distribution L on M consists of all combinations of these vectors which are tangent to M. The tangent vectors must annihilate the defining equation for M, so L consists of complex scalar multiples of
In particular, L consists of the holomorphic vector fields which annihilate F. Note that L gives a CR structure on M, for [L,L] = 0 (since L is one-dimensional) and since ∂/∂z and ∂/∂w are linearly independent of their complex conjugates.
More generally, suppose that M is a real hypersurface in with defining equation F(z1, ..., zn) = 0. Then the CR structure L consists of those linear combinations of the basic holomorphic vectors on :
which annihilate the defining function. In this case, for the same reason as before. Moreover, [L,L] ⊂ L since the commutator of holomorphic vector fields annihilating F is again a holomorphic 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 complex distribution L). Many of the formal geometrical features are similar. These include: A notion of convexity (supplied by the Levi form) A canonical differential operator, analogous to the Dolbeault operator, and an associated cohomology (the tangential Cauchy–Riemann or -complex).
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 define these structures intrinsically, and then generalizes these to the abstract setting.
Embedded CR manifolds
Preliminaries Embedded CR manifolds are, first and foremost, submanifolds of Define a pair of subbundles of the complexified tangent bundle by:
consists of the complex vectors annihilating the antiholomorphic functions. In the holomorphic coordinates:
consists of the complex vectors annihilating the holomorphic functions. In coordinates:
Also relevant are the characteristic annihilators from the Dolbeault complex:
In coordinates,
In coordinates,
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:
Furthermore, there is a decomposition of the usual exterior derivative via .
Real submanifolds of complex space Let be a real submanifold, defined locally as the locus of a system of smooth real-valued functions
Suppose that the complex-linear part of the differential of this system has maximal rank, in the sense that the differentials satisfy the following independence condition:
Note that this condition is strictly stronger than needed to apply the implicit function theorem: in particular, M is a manifold of real dimension We say that M is a generic embedded CR submanifold of CR codimension k. The adjective generic indicates that the tangent space spans the tangent space of over complex numbers. In most applications, k = 1, in which case the manifold is said to be of hypersurface type.
Let be the subbundle of vectors annihilating all of the defining functions 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,See Levi 909 page 207: the Levi form is the differential form associated to the differential operator C, according to Levi's notation. is the Hermitian 2-form
This determines a metric on L. M is said to be strictly pseudoconvex (from the side F<0) 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 pseudoconvexity.
This nomenclature comes from the study of pseudoconvex domains: M is the boundary of a (strictly) pseudoconvex domain in if and only if it is (strictly) pseudoconvex as a CR manifold from the side of the domain. (See plurisubharmonic functions and Stein manifold.)
Abstract CR structures
An abstract CR structure on a real manifold M of real dimension n consists of a complex subbundle L of the complexified tangent bundle which is formally integrable, in the sense that [L,L] ⊂ L, which has zero intersection with its complex conjugate. The CR codimension of the CR structure is where dim L is the complex dimension. In case k = 1, the CR structure is said to be of hypersurface type. Most examples of abstract CR structures are of hypersurface type.
The Levi form and pseudoconvexity Suppose that M is a CR manifold of hypersurface type. The Levi form is the vector valued 2-form, defined on L, with values in the line bundle
given by
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 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
For each local section α ∈ Γ(H0M), let
The form hα is a complex-valued hermitian form associated to α. The 1-form is real; the complex hyperplane bundle is the complexification of the real hyperplane bundle, . In the strictly pseudoconvex case this bundle defines a contact structure on , with a contact form. The conformal class of the contact form is well defined.
A choice of contact form defines a vector field, transverse to the contact structure, . This vector field, called the Reeb vector field, is defined by
.
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.
On abstract CR manifolds, of strongly pseudo-convex type, the Levi form gives rise to a pseudo-Hermitian metric on . If , then we define the hermitian inner product by setting
.
This only defines a hermitian metric for the holomorphic tangent vectors and so is degenerate. We can extend it to define a Riemannian metric on the real tangent space, , by observing that any vector field can be expressed as
; if .
This defines a Riemannian metric on . The Levi form gives the length of the projection of in the -direction.
One can then define a connection and torsion and related curvature tensors for example the Ricci curvature and scalar curvature using this metric. This gives rise to an analogous CR Yamabe problem first studied by David Jerison and John Lee. The connection associated to CR manifolds was first defined and studied by Sidney M. Webster in his thesis on the study of the equivalence problem and independently also defined and studied by Tanaka. Accounts of these notions may be found in the articles. A strictly pseudoconvex CR-manifold also has a canonical Cartan connection, which is defined independently of the choice of contact form. This connection was introduced in 3-dimensions by Cartan, and in higher dimensions by S.S. Chern and J. Moser. The Cartan-Chern-Moser connection and its curvature are invariant under CR-diffeomorphisms.
One of the basic questions of CR Geometry is to ask when a smooth manifold endowed with an abstract CR structure can be realized as an embedded manifold in some . Thus not only are we embedding the manifold, but we also demand for global embedding that the map embedding the abstract manifold in must pull back the induced CR structure of the embedded manifold( coming from the fact that it sits in ) so that the pull back CR structure agrees with the abstract CR structure. The coordinate functions of such a map are CR-functions, that is in the null-space of the -operator. It is easy to show that this null--space is an algebra with point-wise multiplication. One can consider both the questions of local embeddability and of global embeddability.
Louis Boutet de Monvel proved that an abstractly defined, compact strongly pseudoconvex CR-manifold, of real dimension 5 or higher is globally embeddable. His work shows that if the -operator, (see below) acting on functions, has a closed range in , then the CR-manifold is globally embeddable.
In dimension 3, there are obstructions to global embeddability. The standard CR-structure on the three sphere is spanned by the complex vector field, . For , with , the complex vector fields define abstract CR structures on the 3-sphere, which are not globally embeddable. These are often called the Rossi example, though they go back to earlier work of Hans Grauert and appear in a paper by Aldo Andreotti and Yum-Tong Siu. These examples are locally embeddable in a neighborhood of every point, but not globally embeddable: the algebra of CR-functions does not separate points. As shown by Burns all solutions to are even functions.
Joseph J. Kohn showed that global embeddability implies that the Kohn Laplacian, , (see below) acting on functions has a closed range in . Combining this with the result of Boutet de Monvel, it follows that global embeddability of a compact strictly pseudoconvex CR-manifold is equivalent to the closedness of the range of , acting on functions. Thus the closed range property for is CR-invariant.