CR-manifold

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 fonctions 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 $n$ 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 variables complexes II." Ann. Scuola Norm. Sup. Pisa , 1 (1932) pp. 333–354