CR-submanifold

Let be an almost Hermitian manifold (cf. also Hermitian structure), where is an almost-complex structure on and is a Riemannian metric on satisfying for any vector fields and on . A real submanifold of is said to be a complex (holomorphic) submanifold if the tangent bundle of is invariant under , i.e. for any . Let be the normal bundle of . Then is called a totally real (anti-invariant) submanifold if for any .

In 1978, A. Bejancu [a1] introduced the notion of a CR-submanifold as a natural generalization of both complex submanifolds and totally real submanifolds. More precisely, is said to be a CR-submanifold if there exists a smooth distribution on such that:

is a holomorphic distribution, that is, for any ;

the complementary orthogonal distribution of is a totally real distribution, that is, for any .

The above concept has been mainly investigated from the viewpoint of differential geometry (cf. [a2], [a3], [a5], [a6], [a7]).

Let be the second fundamental form of the CR-submanifold . Then one says that is -geodesic, -geodesic or mixed geodesic if vanishes on or , or for any in and in , respectively.

From the viewpoint of complex analysis, a CR-submanifold is an imbedded CR-manifold in a complex manifold. In this case a real hypersurface of a complex manifold is a CR-submanifold (cf. [a4]).

References

[a1]	A. Bejancu, "CR submanifolds of a Kaehler manifold I" Proc. Amer. Math. Soc. , 69 (1978) pp. 134–142
[a2]	A. Bejancu, "Geometry of CR submanifolds" , Reidel (1986)
[a3]	D.E. Blair, B.Y. Chen, "On CR submanifolds of Hermitian manifolds" Israel J. Math. , 34 (1979) pp. 353–363
[a4]	A. Boggess, "CR manifolds and tangential Cauchy–Riemann complex" , CRC (1991)
[a5]	B.Y. Chen, "Geometry of submanifolds and its applications" , Tokyo Sci. Univ. (1981)
[a6]	K. Yano, M. Kon, "CR submanifolds of Kaehlerian and Sasakian manifolds" , Birkhäuser (1983)
[a7]	K. Yano, M. Kon, "Structures on manifolds" , World Sci. (1984)