What are CR manifolds like?

Question

The complex structure on a complex manifold pulls back to what's called a CR structure on any real codimension 1 submanifold. The structure induced on a submanifold of higher codimension is a CR structure if a non-degeneracy condition holds. It's possible to describe these structures intrinsically, without reference to an embedding. I don't know anything else.

I'd be happy with whatever kind of answer to the title question, but here are some more specific ones:

1. Does CR stand for Cauchy-Riemann, or what?

2. What kind of local invariants do CR manifolds have? Are there coordinates around every point that look like a real hyperplane in C^n? Or can there be some curvature or something.

3. Can there be continuous families of CR structures on a given manifold? If the manifold is compact can these families (mod diffeomorphism) be infinite-dimensional?

4. I have the impression, just from arxiv postings and seminar titles, of CR geometry being studied more in analysis than in softer geometric fields. Is that accurate, and if so what accounts for it?

Answer 1

CR does stand for Cauchy-Riemann.

CR structures on 3 dimensional manifolds arise as the boundaries of complex (or almost-complex) 4 manifolds; if these boundaries are strictly pseudo-convex (i.e. convex in "holomorphic directions") the CR structure on the 3-manifold is a contact structure (if the boundary is only (pseudo-)convex or (Levi) flat, the CR structure integrates to a confoliation or a foliation respectively). There can be infinite dimensional families of foliations on a 3-manifold; more generally, whenever the CR structure is "non-generic" or integrable, one has continuous moduli, otherwise (eg in the contact structure case) one has discrete moduli (to be explicit: what has discrete moduli is the contact structure, not the "CR+contact structure".)

Answer 2

The complex structure on a complex manifold pulls back to what's called a CR structure on any real codimension 1 submanifold. The structure induced on a submanifold of higher codimension is a CR structure if a non-degeneracy condition holds.

The CR structure is induced on arbitrary real CR submanifolds $M$ in a complex manifold $X$, that is submanifolds for which the intersection $H^{01}$ of the bundle $T^{01}X$ of all $(0,1)$ vectors with the complexified tangent bundle $\mathbb C\otimes TM$ of $M$ is of constant dimension along $M$. In particular, a real codimension $1$ submanifold, i.e. a real hypersurface, is always a CR submanifold.

The induced CR structure of a CR submanifold is then defined by that intersection subbundle $H^{01}\subset \mathbb C\otimes TM$. (Equivalently, $(1,0)$ vectors can be used instead of $(0,1)$, but the latter is more convenient e.g. in the context of the $\bar\partial$ problem.)

I have never seen the condition being a CR submanifold called a "non-degeneracy condition", and wouldn't do it, because that condition is generally not stable under small perturbation. For instance, a complex line inside $\mathbb C^2$ is a CR submanifold that can be perturbed into a non CR submanifold.

The special property of a codimension $1$ submanifold is that it is always a CR submanifold. The same is not true in general for submanifolds of higher codimension.

It's possible to describe these structures intrinsically, without reference to an embedding.

Yes, a CR structure on a real manifold $M$ is defined by any complex subbundle $V=H^{01}$ of the complexification $\mathbb C\otimes TM$ satisfying $V\cap \bar V=\{0\}$ and the integrability $[V, V]\subset V$. If only the first condition is assumed, $V$ defines an almost CR structure. There is also the intermediate partial integrability condition $[V, V]\subset V\oplus \bar V$.

Equivalently, an almost CR structure can be defined without complexification, by a pair $(H,J)$ of a real subbundle $H\subset TM$ and a complex structure $J\colon H\to H$, see e.g. this answer for more details. However, the integrability condition $[V, V]\subset V$ becomes more verbose, when written in terms of $H$ and $J$.

The CR codimension of an almost CR structure is defined intrinsically as the complex codimension of $H^{10}\oplus H^{01}$ in $\mathbb C\otimes TM$, or equivalently, the real codimension of $H$ in $TM$, where $H = (H^{10}\oplus H^{01}) \cap TM$ and $H^{10}=\overline{H^{01}}$.

1. Does CR stand for Cauchy-Riemann, or what?

It stands for both Cauchy-Riemann and Complex-Real.

2. What kind of local invariants do CR manifolds have? Are there coordinates around every point that look like a real hyperplane in C^n? Or can there be some curvature or something.

The lowest order invariant, the Levi form, is of the 2nd order. It is possible to choose local coordinates, the only 2nd order terms are the ones from the Levi form.

In contrast to complex structures (corresponding to CR structures of CR codimension $0$), for a general CR structure of positive CR codimension, there are infinitely many higher order local invariants.

See e.g. my paper, Normal forms for almost non-integrable CR structures, Amer. J. of Math., 134 (2012), no. 4, 915-947, also available on the arxiv.org, for a complete intrinsic normal form, including the non-integrable case.

3. Can there be continuous families of CR structures on a given manifold? If the manifold is compact can these families (mod diffeomorphism) be infinite-dimensional?

Yes, see above. Also, infinite-dimensional families of non-CR-equivalent CR structures can be obtained even locally using the Chern-Moser normal form or Cartan connection, see e.g.

Chern, S. S.; Moser, J. K. Real hypersurfaces in complex manifolds. Acta Math. 133 (1974), 219–271

4. I have the impression, just from arxiv postings and seminar titles, of CR geometry being studied more in analysis than in softer geometric fields. Is that accurate, and if so what accounts for it?

You can find CR structures in both analysis and "softer" geometry such as compatible CR structures with contact structures, fillable CR structures etc, for instance in this book:

MR3012475 Cieliebak, Kai; Eliashberg, Yakov. From Stein to Weinstein and back. Symplectic geometry of affine complex manifolds. American Mathematical Society Colloquium Publications, 59. American Mathematical Society, Providence, RI, 2012.

Answer 3

CR submanifolds of a complex manifold are defined as submanifolds $M \subseteq X$ such that $TM \cap iTM \subseteq TX$ has constant rank ($i$ is the imaginary unit). Note that the condition is automatically satisfied if M$$ has codimension one; for higher codimension this is not true.

An abstract CR manifold is a real manifold $M$, with a distinguished subbundle $HM \subseteq TM$, corresponding to $TM \cap iTM$, endowed with a linear endomorphism $J$ with $J^2=-Id$. The structure is furthermore required to satisfy a so called integrability condition: For all sections $X,Y$ of $HM$:

• $[X,JY]+[JX,Y]$ is a section of $HM$

• $([X,Y]-[JX,JY]) + J([X,JY]+[JX,Y]) = 0$

Not every abstract CR manifold can be realized as a CR submanifold.

Answer 4

As matter of fact, the abbreviation CR stands for Complex-Real or Cauchy–Riemann.

In mathematics, the CR manifold is a differentiable manifold together with a geometric structure modeled on that of a "real" hypersurface in a "complex" vector space.

Poincare raised the concept of C^2 but CR firstly appeared in Cartan's work. Chern and Morse generalized the C^2 to C^∞.

In 1907, H. Poincaré wrote a seminal paper, [a6], in which he showed that two real hypersurfaces in C^2 are, in general, biholomorphically inequivalent

Cartan called his geometry "conformal geometry" in order to emphasize that what he was studying was a generalization of the theory of one complex variable (conformal geometry). The term "CR manifold" was first used in [Gr].

A second solution was given by Chern-Moser in 1973. In joint work with Chern [CM] it was generalized, along with Cartan's original solution, to dimensions greater than two.