APPLICATION OF AN INTEGRAL FORMULA TO CR-SUBMANIFOLDS OF COMPLEX HYPERBOLIC SPACE

The purpose of this paper is to study n-dimensional compact CR-submanifolds of complex hyperbolic space CH(n


Introduction
Let M be a complex space form of constant holomorphic sectional curvature c and let M be an n-dimensional CR-submanifold of (n − 1) CR-dimension in M. Then M has an almost contact metric structure (F,U,u,g) (see Section 2) induced from the canonical complex structure of M. Hence on an n-dimensional CR-submanifold of (n − 1) CRdimension, we can consider two structures, namely, almost contact structure F and a submanifold structure represented by second fundamental form A. In this point of view, many differential geometers have classified M under the conditions concerning those structures (cf.[3,5,8,9,10,11,12,14,15,16]).In particular, Montiel and Romero [12] have classified real hypersurfaces M of complex hyperbolic space CH (n+1)/2 which satisfy the commutativity condition (C) by using the S 1 -fibration π : H n+2 1 → CH (n+1)/2 of the anti-de Sitter space H n+2 1 over CH (n+1)/2 , and obtained Theorem 4.1 stated in Section 2. We notice that among the model spaces in Theorem 4.1, the geodesic hypersphere is only compact.
In this paper, we will investigate n-dimensional compact CR-submanifold of (n − 1) CR-dimension in complex hyperbolic space and provide a characterization of the geodesic hypersphere, which is equivalent to condition (C), by using the following integral formula established by Yano [17,18]: 988 Application of an integral formula to CR-submanifolds where X is an arbitrary vector field tangent to M. Our results of the paper are complex hyperbolic versions of those in [6,15].

Preliminaries
Let M be an n-dimensional CR-submanifold of (n − 1) CR-dimension isometrically immersed in a complex space form M(n+p)/2 (c).Denoting by (J, ḡ) the Kähler structure of M(n+p)/2 (c), it follows by definition (cf.[5,6,8,9,13,16]) that the maximal J-invariant subspace of the tangent space T x M of M at each point x in M has constant dimension (n − 1).So there exists a unit vector field U 1 tangent to M such that where Ᏸ ⊥ x denotes the subspace of T x M complementary orthogonal to Ᏸ x .Moreover, the vector field ξ 1 defined by is normal to M and satisfies (2.4) Hence we have, for any tangent vector field X and for a local orthonormal basis {ξ 1 , ξ α } α=2,...,p of normal vectors to M, the following decomposition in tangential and normal components: ) Since the structure (J, ḡ) is Hermitian and J 2 = −I, we can easily see from (2.5) and (2.6) that F and P are skew-symmetric linear endomorphisms acting on T x M and T x M ⊥ , respectively, and that where T x M ⊥ denotes the normal space of M at x and g the metric on M induced from ḡ. Furthermore, we also have and consequently, (2.10) J. S. Pak and H. S. Kim 989 Next, applying J to (2.5) and using (2.6) and (2.10), we have from which, taking account of the skew-symmetry of P and (2.7), (2.12) Thus (2.6) may be written in the form These equations tell us that (F,g,U 1 ,u 1 ) defines an almost contact metric structure on M (cf.[5,6,8,9,16]), and consequently, n = 2m + 1 for some integer m.
We denote by ∇ and ∇ the Levi-Civita connection on M(n+p)/2 (c) and M, respectively.Then the Gauss and Weingarten formulas are given by for any vector fields X, Y tangent to M.Here ∇ ⊥ denotes the normal connection induced from ∇ in the normal bundle TM ⊥ of M, and h and A α the second fundamental form and the shape operator corresponding to ξ α , respectively.It is clear that h and A α are related by (2.16) We put (2.17) Then (s αβ ) is the skew-symmetric matrix of connection forms of ∇ ⊥ .Now, using (2.14), (2.15), and (2.17), and taking account of the Kähler condition ∇J = 0, we differentiate (2.5) and (2.6) covariantly and compare the tangential and normal parts.Then we can easily find that (2.18) for any X, Y tangent to M.
990 Application of an integral formula to CR-submanifolds In the rest of this paper, we suppose that the distinguished normal vector field ξ 1 is parallel with respect to the normal connection ∇ ⊥ .Hence (2.17) gives s 1α = 0, α = 2,..., p, (2.22) which, together with (2.21), yields On the other hand, the ambient manifold M(n+p)/2 (c) is of constant holomorphic sectional curvature c and consequently, its Riemannian curvature tensor R satisfies for any X, Ȳ , Z tangent to M(n+p)/2 (c) (cf.[1,2,4,19]).So, the equations of Gauss and Codazzi imply that for any X, Y , Z tangent to M with the aid of (2.22), where R denotes the Riemannian curvature tensor of M.Moreover, (2.11) and (2.25) yield ) where Ric and ρ denote the Ricci tensor and the scalar curvature, respectively, and is the mean curvature vector (cf.[1,2,4,19]).

Codimension reduction of CR-submanifolds of CH (n+p)/2
Let M be an n-dimensional CR-submanifold of (n − 1) CR-dimension in a complex hyperbolic space CH (n+p)/2 with constant holomorphic sectional curvature c = −4.
Applying the integral formula (1.2) to the vector field U 1 , we have and consequently,