# CR-manifold

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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]).

How to Cite This Entry:
CR-manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=CR-manifold&oldid=11362
This article was adapted from an original article by A. Bejancu (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article