CR-manifold

In 1907, H. Poincaré wrote a seminal paper, [a6], in which he showed that two real hypersurfaces in 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 , .

Let be a real differentiable manifold and the tangent bundle of . One says that is a CR-manifold if there exists a complex subbundle of the complexified tangent bundle satisfying the conditions:

;

is involutive, i.e., for any complex vector fields and in the Lie bracket is also in .

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

lies in ;

for any real vector fields , in .

Thus the CR-structure on is determined either by the complex vector bundle or by the almost-complex distribution . The abbreviation CR refers to A.L. Cauchy and B. Riemann, because, for in , consists of the induced Cauchy–Riemann operators (cf. Cauchy–Riemann conditions).

A -function is called a CR-function if for all complex vector fields in . A -mapping is said to be a CR-mapping if , where is the tangent mapping of . In particular, if is a diffeomorphism, one says that is a pseudo-conformal mapping and that and are CR-diffeomorphic or, briefly, that they are equivalent. A CR-structure on is said to be realizable if is equivalent to some real hypersurface of a complex Euclidean space.

Let be the natural projection mapping. Then the Levi form for is the mapping

for any complex vector field in . If is the real hypersurface in given by the equation , where is smooth, then the Levi form for is identified with the restriction of the complex Hessian of to (cf. also Hessian matrix). When is positive- or negative-definite on , one says that 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