# 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)
How to Cite This Entry:
CR-submanifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=CR-submanifold&oldid=46184
This article was adapted from an original article by A. Bejancu (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article