CR-submanifold

Let $ ( M,J,g ) $ be an almost Hermitian manifold (cf. also Hermitian structure), where $ J $ is an almost-complex structure on $ M $ and $ g $ is a Riemannian metric on $ M $ satisfying $ g ( JX,JY ) = g ( X,Y ) $ for any vector fields $ X $ and $ Y $ on $ M $. A real submanifold $ N $ of $ M $ is said to be a complex (holomorphic) submanifold if the tangent bundle $ TN $ of $ N $ is invariant under $ J $, i.e. $ J ( T _ {x} N ) = T _ {x} N $ for any $ x \in N $. Let $ TN ^ \perp $ be the normal bundle of $ N $. Then $ N $ is called a totally real (anti-invariant) submanifold if $ J ( T _ {x} N ) \subset T _ {x} N ^ \perp $ for any $ x \in N $.

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, $ N $ is said to be a CR-submanifold if there exists a smooth distribution $ D $ on $ N $ such that:

$ D $ is a holomorphic distribution, that is, $ J ( D _ {x} ) = D _ {x} $ for any $ x \in N $;

the complementary orthogonal distribution $ D ^ \perp $ of $ D $ is a totally real distribution, that is, $ J ( D _ {x} ^ \perp ) \subset T _ {x} N ^ \perp $ for any $ x \in N $.

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

Let $ h $ be the second fundamental form of the CR-submanifold $ N $. Then one says that $ N $ is $ D $- geodesic, $ D ^ \perp $- geodesic or mixed geodesic if $ h $ vanishes on $ D $ or $ D ^ \perp $, or $ h ( X,Y ) = 0 $ for any $ X $ in $ D $ and $ Y $ in $ D ^ \perp $, 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 $ N $ of a complex manifold $ ( M,J ) $ 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)