CR-manifold

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

In 1907, H. Poincaré wrote a seminal paper, [a6], in which he showed that two real hypersurfaces in $\mathbf C ^ {2}$ are, in general, biholomorphically inequivalent (cf. Biholomorphic mapping; Hypersurface). Later, E. Cartan [a10], [a11] found all the invariants that distinguish one real hypersurface from another. The general solution for complex dimensions greater than two was given by S.S. Chern and J. Moser [a3] and N. Tanaka [a8], [a7].

The concept of a CR-manifold (CR-structure) has been defined having in mind the geometric structure induced on a real hypersurface of $\mathbf C ^ {n}$, $n \geq 2$.

Let $M$ be a real differentiable manifold and $TM$ the tangent bundle of $M$. One says that $M$ is a CR-manifold if there exists a complex subbundle $H$ of the complexified tangent bundle $\mathbf C \otimes TM$ satisfying the conditions:

$H \cap {\overline{H}\; } = \{ 0 \}$;

$H$ is involutive, i.e., for any complex vector fields $U$ and $V$ in $H$ the Lie bracket $[ U,V ]$ is also in $H$.

Alternatively, by using real vector bundles it can be proved (cf. [a1]) that $M$ is a CR-manifold if and only if there exists an almost-complex distribution $( D,J )$ on $M$( i.e., $D$ is a vector subbundle of $TM$ and $J$ is an almost-complex structure on $D$) such that

$[ JX,JY ] - [ X,Y ]$ lies in $D$;

$[ JX,JY ] - [ X,Y ] - J ( [ JX,Y ] + [ X,JY ] ) = 0$ for any real vector fields $X$, $Y$ in $D$.

Thus the CR-structure on $M$ is determined either by the complex vector bundle $H$ or by the almost-complex distribution $( D,J )$. The abbreviation CR refers to A.L. Cauchy and B. Riemann, because, for $M$ in $\mathbf C ^ {n}$, $H$ consists of the induced Cauchy–Riemann operators (cf. Cauchy-Riemann equations).

A $C ^ {1}$- function $f : {( M,H ) } \rightarrow \mathbf C$ is called a CR-function if $Lf = 0$ for all complex vector fields $L$ in $H$. A $C ^ {1}$- mapping $F : {( M,H ) } \rightarrow {( {\widetilde{M} } , {\widetilde{H} } ) }$ is said to be a CR-mapping if $F _ {*} H \subset {\widetilde{H} }$, where $F _ {*}$ is the tangent mapping of $F$. In particular, if $F$ is a diffeomorphism, one says that $F$ is a pseudo-conformal mapping and that $M$ and ${\widetilde{M} }$ are CR-diffeomorphic or, briefly, that they are equivalent. A CR-structure on $M$ is said to be realizable if $M$ is equivalent to some real hypersurface of a complex Euclidean space.

Let $\pi : {\mathbf C \otimes TM } \rightarrow {( \mathbf C \otimes TM ) / ( H \oplus {\overline{H}\; } ) }$ be the natural projection mapping. Then the Levi form for $M$ is the mapping

$$h : H \rightarrow {( \mathbf C \otimes TM ) / ( H \oplus {\overline{H}\; } ) } ,$$

$$h ( U ) = { \frac{1}{2i } } \pi ( [ U, {\overline{U}\; } ] ) ,$$

for any complex vector field $U$ in $H$. If $M$ is the real hypersurface in $\mathbf C ^ {n}$ given by the equation $g ( z ) = 0$, where $g : {\mathbf C ^ {n} } \rightarrow \mathbf R$ is smooth, then the Levi form for $M$ is identified with the restriction of the complex Hessian of $g$ to $H$( cf. also Hessian matrix). When $h$ is positive- or negative-definite on $M$, one says that $M$ is strictly pseudo-convex.

The differential geometry of CR-manifolds (cf. [a4]) has potential applications to both partial differential equations (cf. [a2]) and mathematical physics (cf. [a5] and [a9]).

References

 [a1] A. Bejancu, "Geometry of CR submanifolds" , Reidel (1986) [a2] A. Boggess, "CR manifolds and tangential Cauchy–Riemann complex" , CRC (1991) [a3] S.S. Chern, J. Moser, "Real hypersurfaces in complex manifolds" Acta Math. , 133 (1974) pp. 219–271 [a4] H. Jacobowitz, "An introduction to CR structures" , Math. Surveys and Monographs , 32 , Amer. Math. Soc. (1990) [a5] R. Penrose, "Physical space-time and non-realizable CR structures" , Proc. Symp. Pure Math. , 39 , Amer. Math. Soc. (1983) pp. 401–422 [a6] H. Poincaré, "Les functions analytiques de deux variables et la représentation conforme" Rend. Circ. Mat. Palermo , 23 (1907) pp. 185–220 [a7] N. Tanaka, "On the pseudo-conformal geometry of hypersurfaces of the space of complex variables" J. Math. Soc. Japan , 14 (1962) pp. 397–429 [a8] N. Tanaka, "On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections" Japan J. Math. (N.S.) , 2 (1976) pp. 131–190 [a9] J.R. Wells, Jr., "Complex manifolds and mathematical physics" Bull. Amer. Math. Soc. (N.S.) , 1 (1979) pp. 296–336 [a10] É. Cartan, "Sur l'équivalence pseudo-conforme des hypersurfaces de l'espace de deux variables complexes I." Ann. Mathém. , 11 (1932) pp. 17–90 [a11] É. Cartan, "Sur l'équivalence pseudo-conforme des hypersurfaces de l'espace de deux variable complexes II." Ann. Scuola Norm. Sup. Pisa , 1 (1932) pp. 333–354
How to Cite This Entry:
CR-manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=CR-manifold&oldid=46183
This article was adapted from an original article by A. Bejancu (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article