Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 49 (2008), No. 2, 383-397.

CR-Warped Product Submanifolds of Nearly Kaehler Manifolds

Bayram S.ahin Rıfat Günes.

Department of Mathematics, Inonu University 44280, Malatya, Turkey e-mail: bsahin@inonu.edu.tr rgunes@inonu.edu.tr

Abstract. In this paper, we study warped product CR-submanifolds of nearly Kaehler manifolds. We extend some results of B. Y. Chen [7] on warped product CR-submanifolds of Kaehler manifolds to warped product CR-submanifolds of nearly Kaehler manifolds. We also give an example for such submanifolds in six dimensional sphere. MSC 2000: 53C42, 53C40, 53C15 Keywords: CR-submanifold, warped product, nearly Kaehler manifold

1. Introduction

Let (M̄, g) be an almost Hermitian manifold. This means [22] that M̄ admits a tensor field J of type (1, 1) on M̄ such that, ∀X, Y ∈ Γ(T M̄), we have

J^{2} = −I, g(X, Y) = g(JX, JY).  (1.1)

An almost Hermitian manifold M̄ is called nearly Kaehler manifold if

( ¯∇_{X}J)X = 0, ∀X ∈ Γ(T M̄),  (1.2)

which is equivalent to

( ¯∇_{X}J)Y + ( ¯∇_{Y} J)X = 0, ∀X, Y ∈ Γ(T M̄),  (1.3)

where ¯∇ is the Levi-Civita connection on M̄. T. Fukami and S. Ishihara [12] proved that there exists a nearly Kaehlerian structure on a six dimensional sphere

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2008 Heldermann Verlag

384 B. S.ahin, R. Günes.: CR-Warped Product Submanifolds …

S^{6} by making use of the properties of the Cayley division algebra. This structure is called as the canonical nearly Kaehlerian structure on S^{6}. In fact, A. Gray [13] showed that S^{6} has some almost complex structures, which are different from the canonical almost complex structures, but nonetheless nearly Kaehlerian.

CR-submanifolds of almost Hermitian manifolds were defined by A. Bejancu [3] as a generalization of complex and totally real submanifolds. A CR-submanifold is called proper if it is neither complex nor totally real submanifold. The geometry of CR-submanifolds has been studied in several papers since then. We note that the geometry of CR-submanifolds of nearly Kaehler manifolds has been also studied by several authors, [1], [2], [10], [15], [17], [18], [21].

On the other hand, in [4], R. L. Bishop and B. O’Neill introduced a class of warped product manifolds as follows: Let (B, g_{1}) and (F, g_{2}) be two Riemannian manifolds, f : B → (0, ∞) and π : B×F → B, η : B×F → F the projection maps given by π(p, q) = p and η(p, q) = q for every (p, q) ∈ B ×F. The warped product M = B × F is the manifold B × F equipped with the Riemannian structure such that

g(X, Y) = g_{1}(π_{∗}X, π_{∗}Y) + (foπ)^{2}g_{2}(η_{∗}X, η_{∗}Y)

for every X and Y of M and ∗ is symbol for the tangent map. The function f is called the warping function of the warped product manifold. In particular, if the warping function is constant, then the manifold M is said to be trivial. Let X, Y be vector fields on B and V, W vector fields on F, then from Lemma 7.3 of [4], we have

∇_{X}V = ∇_{V} X = (\frac{Xf}{f})V  (1.4)

where ∇ is the Levi-Civita connection on M.

Recently, B. Y. Chen [7], [8] considered warped product CR-submanifolds of Kaehler manifolds and showed that there do not exist warped product CRsubmanifolds in the form M_{⊥} ×_{f} M_{T} such that M_{T} is a holomorphic (complex) submanifold and M_{⊥} is a totally real submanifold of a Kaehler manifold M̄. Then he introduced the notion of CR-warped products of Kaehler manifolds as follows: A submanifold of a Kaehler manifold is called CR-warped product if it is the warped product M_{T} ×_{f} M_{⊥} of a holomorphic submanifold M_{T} and a totally real submanifold M_{⊥}. He also established a sharp relationship between the warping function f of a warped product CR-submanifold M_{T} ×_{f} M_{⊥} of a Kaehler manifold M̄ and the squared norm of the second fundamental form ∥ h ∥^{2}. After the papers of B. Y. Chen, CR-warped product submanifolds have been studied in various manifolds [5], [14], [19], [20].

It is known that every Kaehler manifold is a nearly Kaehler but the converse is not true in general. Therefore, the aim of this paper is to extend the concept of CR-warped product submanifolds of Kaehler manifolds to warped product CRsubmanifolds of nearly Kaehler manifolds. Our main results improve some results of B. Y. Chen to CR-warped product submanifolds of nearly Kaehler manifolds. First, we prove that there exist no warped product CR-submanifolds in the form M_{⊥} ×_{f} M_{T} such that M_{⊥} is a totally real submanifold and M_{T} is a holomorphic submanifold of a nearly Kaehler manifold M̄. Then we consider warped product CR-submanifolds in the form M_{T} ×_{f} M_{⊥} in a nearly Kaehler manifold M̄ and give an example. We prove a characterization theorem and obtain necessary condition for CR-warped product submanifolds to be a CR-product. We obtain a sharp inequality for the squared norm of the second fundamental form in terms of the warping function for CR-warped product submanifolds of nearly Kaehler manifolds.

We note that although the proof of Theorem 4.1 is same as that of Theorem 5.1 in [7], the proof of preparatory lemmas for the proof of Theorem 4.1 is different.

2. Preliminaries

In this section, we will recall the canonical nearly Kaehlerian structure on a 6dimensional unit sphere S^{6}. We will also brief some formulas and definitions which will be useful later. Let C be the Cayley division algebra generated by {e_{o}, e_{i}, (1 ≤ i ≤ 7)} over real number field R and C_{+} be the subspace of C consisting of all purely imaginary Cayley numbers. We may identify C_{+} with a 7dimensional Euclidean space R^{7} with the canonical inner product <, >. A vector cross product for the vectors in C_{+} = R^{7} is defined by

x ∧ y =< x, y > e_{o} + xy, x, y ∈ C_{+}.  (2.1)

Then the multiplication table is given by the following:

1 0 e_{3} −e_{2} e_{5} −e_{4} −e_{7} −e_{6}
2 −e_{3} 0 e_{1} e_{6} −e_{7} −e_{4} e_{5}
3 e_{2} −e_{1} 0 −e_{7} −e_{6} −e_{5} e_{4}
e_{j} ∧ e_{k} = 4 −e_{5} −e_{6} e_{7} 0 e_{1} e_{2} −e_{3}
5 e_{4} e_{7} e_{6} −e_{1} 0 −e_{3} −e_{2}
6 −e_{7} e_{4} −e_{5} −e_{2} e_{3} 0 e_{1}
7 e_{6} −e_{5} −e_{4} e_{3} e_{2} −e_{1} 0

On S^{6}(1) define a (1, 1) tensor field J by putting

J_{p}X = p ∧ X

for p ∈ S^{6}(1) and X ∈ T_{p}S^{6}. The above almost complex structure J together with the induced Riemannian metric <, > on S^{6} gives a nearly Kaehlerian structure on S^{6} [12]. It is well known that S^{6} does not admit any Kaehlerian structures. Let M be a Riemannian manifold isometrically immersed in a nearly Kaehler manifold M̄ and denote by the same symbol g the Riemannian metric induced on M. Let Γ(T M) be the Lie algebra of vector fields in M and Γ(T M^{⊥}) the set of all vector fields normal to M. Denote by ∇ the Levi-Civita connection of M. Then the Gauss and Weingarten formulas are given by

¯∇_{X}Y = ∇_{X}Y + h(X, Y)  (2.2)

and

¯∇_{X}N = −A_{N}X + ∇_{X}^{⊥}N  (2.3)

for any X, Y ∈ Γ(T M) and any N ∈ Γ(T M^{⊥}), where ∇^{⊥} is the connection in the normal bundle T M^{⊥}, h is the second fundamental form of M and A_{N} is the Weingarten endomorphism associated with N. The second fundamental form and the shape operator A are related by

g(A_{N}X, Y) = g(h(X, Y), N).  (2.4)

Let M̄ be a nearly Kaehler manifold with complex structure J and M is a Riemannian manifold isometrically immersed in M̄. Then M is called holomorphic (complex) if J(T_{p}M) ⊂ T_{p}M, for every p ∈ M, where T_{p}M denotes the tangent space to M at the point p. M is called totally real if J(T_{p}M) ⊂ T_{p}M^{⊥} for every p ∈ M, where T_{p}M^{⊥} denotes the normal space to M at the point p. The submanifold M is called a CR-submanifold [3] if there exists a differentiable distribution D : p → D_{p} ⊂ T_{p}M such that D is invariant with respect to J and the complementary distribution D^{⊥} is anti-invariant with respect to J. It is clear that holomorphic and totally real submanifolds are CR-submanifolds with D^{⊥} = {0} and D = {0}, respectively. Moreover, every real hypersurface of an almost Hermitian manifold is a CR-submanifold. For a real hypersurface M of an almost Hermitian M̄ with a unit normal vector field ξ, the tangent vector Jξ on M is called a characteristic vector field of M. A unit tangent vector X on M is called a principal vector if X is an eigenvector of the shape operator A_{ξ}, the corresponding eigenvalue is called the principal curvature at X. Let M be a CR-submanifold of a nearly Kaehler manifold, then we have

T M = D ⊕ D^{⊥}.

Hence, we also have

T M^{⊥} = JD^{⊥} ⊕ ν,

where ν denotes the complementary distribution to JD^{⊥} in the normal bundle of M. We note that ν is also invariant with respect to J. For any X ∈ T M we write

JX = T X + F X,  (2.5)

where T X ∈ Γ(D) and F X ∈ Γ(JD^{⊥}). Similarly, for any vector field normal to M, we put

JN = BN + CN,  (2.6)

where BN ∈ Γ(D^{⊥}) and CN ∈ Γ(ν).

3. Warped products M_{⊥} ×_{f} M_{T} in nearly Kaehler manifolds

In this section we consider CR-submanifolds in a nearly Kaehler manifold M̄ which are warped products of the form M_{⊥} ×_{f} M_{T} such that M_{⊥} is a totally real submanifold and M_{T} is a holomorphic submanifold of M̄.

Theorem 3.1. Let M̄ be a nearly Kaehler manifold. Then there do not exist warped product CR-submanifolds in the form M = M_{⊥} ×_{f} M_{T} in M̄ such that M_{⊥} is a totally real submanifold and M_{T} is a holomorphic submanifold of M̄.

Proof. Let us suppose that M be a warped product CR-submanifold in the form M = M_{⊥}×_{f} M_{T} in M̄ such that M_{⊥} is a totally real submanifold and M_{T} is a holomorphic submanifold of M̄. Then, from (2.2), we have g(∇_{X}Z, X) = g( ¯∇_{X}Z, X) for X ∈ Γ(T M_{T}) and Z ∈ Γ(T M_{⊥}). Using (1.4), we get Z(ln f)g(X, X) = g( ¯∇_{X}Z, X). Since D^{⊥} and D are orthogonal, we obtain Z(ln f)g(X, X) = −g(Z, ¯∇_{X}X). Then, from (1.1), we derive Z(ln f)g(X, X) = −g(JZ, J ¯∇_{X}X). Hence, we get Z(ln f)g(X, X) = −g(JZ, ¯∇_{X}JX −( ¯∇_{X}J)X). Thus, using (1.2) we obtain

Z(ln f)g(X, X) = −g(JZ, ¯∇_{X}JX).

Then, from (2.2), we have

Z(ln f)g(X, X) = −g(JZ, h(X, JX))  (3.1)

for X ∈ Γ(T M_{T}) and Z ∈ Γ(T M_{⊥}).

On the other hand, from (2.3), we have

g(A_{JZ}X, JX) = −g( ¯∇_{X}JZ, JX)

for X ∈ Γ(T M_{T}) and Z ∈ Γ(T M_{⊥}). Hence, we get

g(A_{JZ}X, JX) = g(JZ, ¯∇_{X}JX).

Since ¯∇ is torsion free, we have

g(A_{JZ}X, JX) = g(JZ, [X, JX] + ¯∇_{JX}X).

Since [X, JX] ∈ Γ(T M) and JZ ∈ Γ(T M^{⊥}), we derive

g(A_{JZ}X, JX) = g(JZ, ¯∇_{JX}X).

Then, from (1.1), we can write

g(A_{JZ}X, JX) = −g(JZ, ¯∇_{JX}J^{2}X).

Hence, we have

g(A_{JZ}X, JX) = −g(JZ, ( ¯∇_{JX}J)JX + J ¯∇_{JX}JX).

Then, from (1.2), we derive

g(A_{JZ}X, JX) = −g(JZ, J ¯∇_{JX}JX).

Thus, using the second equation of (1.1), we arrive at

g(A_{JZ}X, JX) = −g(Z, ¯∇_{JX}JX).

Then, we get

g(A_{JZ}X, JX) = g( ¯∇_{JX}Z, JX).

Using (2.2) we obtain

g(A_{JZ}X, JX) = g(∇_{JX}Z, JX).

Thus, from (1.4) and (2.4, ) we have

g(h(X, JX), JZ) = Z(ln f)g(JX, JX).

Using (1.1), we arrive at

g(h(X, JX), JZ) = Z(ln f)g(X, X)  (3.2)

for X ∈ Γ(T M_{T}) and Z ∈ Γ(T M_{⊥}). Thus, from (3.1) and (3.2), we conclude that

2Z(ln f)g(X, X) = 0.

Since D is Riemannian, we get

Z(ln f) = 0.

This implies that f is constant on M_{⊥} which shows that M is a usual product. Thus proof is complete.

Remark 1. We note that Theorem 3.1 is a generalization of Theorem 3.1 in [7].

4. Warped products M_{T} ×_{f} M_{⊥} in nearly Kaehler manifolds

Theorem 3.1 shows that there exist no warped product CR-submanifolds in the form M = M_{⊥} ×_{f} M_{T} in M̄ such that M_{⊥} is a totally real submanifold and M_{T} is a holomorphic submanifold of M̄. Therefore, in this section, we consider warped product CR-submanifolds in the form M = M_{T} ×_{f} M_{⊥} such that M_{⊥} is a totally real submanifold and M_{T} is a holomorphic submanifold of a nearly Kaehler manifold M̄. Follow Chen’s definition, we call such warped product CR-submanifolds CR-warped products. Now, we give an example of CR-warped product in S^{6}.

Example 1. Consider Sekigawa’s example [21] of CR-submanifold in S^{6}, which is the image of the mapping of S^{2} × S^{1} into S^{6}:

Ψ(y, t) = Ψ((y_{2}, y_{4}, y_{6}), e^{\sqrt{−1}t})
= (y_{2} cos t)e_{2} − (y_{2} sin t)e_{3} + (y_{4} cos 2t)e_{4}
+(y_{4} sin 2t)e_{5} + (y_{6} cos t)e_{6} + (y_{6} sin t)e_{7}

where (y_{2}, y_{4}, y_{6}) ∈ S^{2} and e^{\sqrt{−1}t} ∈ S^{1}. Then the tangent bundle T M of submanifold M is given by

Z_{1} = −y_{2} sin t e_{2} − y_{2} cos te_{3} − 2y_{4} sin 2te_{4} + 2y_{4} cos 2te_{5} − y_{6} sin te_{6} − y_{6} cos te_{7}
Z_{2} = y_{6} cos t e_{2} − y_{6} sin te_{3} − y_{2} cos t e_{6} − y_{2} sin t e_{7}
Z_{3} = y_{6} cos 2t e_{4} + y_{6} sin 2t e_{5} − y_{4} cos t e_{6} − y_{4} sin t e_{7}.

It is known that M is a CR-submanifold of S^{6} such that D = span{Z_{2}, Z_{3}} and D^{⊥} = span{Z_{1}}. Moreover, we can derive that D is integrable. Denoting the integral manifolds of D and D^{⊥}by M_{T} and M_{⊥}, respectively, then the induced metric tensor is

ds^{2} = (y_{6}^{2} + y_{2}^{2})dy_{2}^{2} + y_{2} y_{4}dy_{2} dy_{4} + (y_{6}^{2} + y_{4}^{2})dy_{4}^{2} + (1 + 3y_{4}^{2})dt^{2}
= g_{M_{T}} + (1 + 3y_{4}^{2})g_{M_{⊥}}.

Thus it follows that M is a CR-warped product submanifold of S^{6} with warping function f = \sqrt{((1 + 3y_{4}^{2}))}.

Remark 2. We note that H. Hashimoto and K. Mashimo [15] gave a generalization of Sekigawa’s example. They also showed that the induced metric of their generalization is a warped product metric (see: [15], Lemma 8). Thus, there are many three dimensional CR-warped product submanifolds in S^{6}.

Next we give a characterization for CR-warped products in a nearly Kaehler manifolds. We first need the following lemmas.

Lemma 4.1. Let M be a CR-submanifold of a nearly Kaehler manifold M̄. Then, the anti-invariant distribution is integrable if and only if

g(∇_{Z}X, V) = \frac{1}{2}g(A_{JZ}V + A_{JV} Z, JX)  (4.1)

for Z, V ∈ Γ(D^{⊥}) and X ∈ Γ(D).

Proof. From (2.2), we have

g([Z, V], X) = g( ¯∇_{Z}V, X) − g(∇_{V} Z, X)

for Z, V ∈ Γ(D^{⊥}) and X ∈ Γ(D). Then, using (1.1), we get

g([Z, V], X) = g(J ¯∇_{Z}V, JX) − g(∇_{V} Z, X).

Hence, we derive

g([Z, V], X) = g(−( ¯∇_{Z}J)V + ¯∇_{Z}JV, JX) − g(∇_{V} Z, X).

Thus, using (1.3), we obtain

g([Z, V], X) = g( ¯∇_{V} J)Z, JX) + g( ¯∇_{Z}JV, JX) − g(∇_{V} Z, X).

Hence, we have

g([Z, V], X) = g(( ¯∇_{V} JZ − J ¯∇_{V} Z, JX) + g( ¯∇_{Z}JV, JX) − g(∇_{V} Z, X).

Then, using (2.2) and (2.3), we obtain

g([Z, V], X) = −g(A_{JZ}V + A_{JV} Z, JX) − 2g(∇_{V} Z, X).

Then, we have

g([Z, V], X) = −g(A_{JZ}V + A_{JV} Z, JX) − 2g([V, Z], X) − 2g(∇_{Z}V, X).

Hence we get

g([V, Z], X) = 2g(∇_{Z}X, V) − g(A_{JZ}V + A_{JV} Z, JX),

which proves our assertion.

Lemma 4.2. Let M be a CR-submanifold of a nearly Kaehler manifold M̄. Then, we have

g(∇_{X}Y, Z) = g(JY, A_{JZ}X) − g(JY, ∇_{Z}JX) + g(Y, ∇_{Z}X)  (4.2)

for X, Y ∈ Γ(D) and Z ∈ Γ(D^{⊥}).

Proof. Using (2.2) and (1.1), we have g(∇_{X}Y, Z) = −g(JY, J ¯∇_{X}Z) for X, Y ∈ Γ(D) and Z ∈ Γ(D^{⊥}). Hence, we get g(∇_{X}Y, Z) = −g(JY, ¯∇_{X}JZ − ( ¯∇_{X}J)Z). Then, from (1.3), we obtain

g(∇_{X}Y, Z) = −g(JY, ¯∇_{X}JZ) − g(JY, ( ¯∇_{Z}J)X).

Hence, we get

g(∇_{X}Y, Z) = −g(JY, ¯∇_{X}JZ) − g(JY, ¯∇_{Z}JX − J ¯∇_{Z}X).

Thus, using (1.1), (2.2) and (2.3) we obtain (4.2).

Lemma 4.3. Let M be a CR-warped product of a nearly Kaehler manifold M̄. Then, we have

g(h(X, JY), JZ) = 0  (4.3)

and

g(h(X, Z), JV)) = −JX(ln f)g(Z, V)  (4.4)

for X, Y ∈ Γ(T M_{T}) and Z, V ∈ Γ(T M_{⊥}).

Proof. From (1.4), we get g(∇_{X}X, Z) = X(ln f)g(X, Z) = 0 for X ∈ Γ(T M_{T}) and Z ∈ Γ(T M_{⊥}). Then, using (2.2) and (1.1), we have g(J ¯∇_{X}X, JZ) = 0. Hence

g(−( ¯∇_{X}J)X + ¯∇_{X}JX, JZ) = 0.

Thus, (1.2) implies

g( ¯∇_{X}JX, JZ) = 0.

Using (2.2), we get

g(h(X, JX), JZ) = 0.

Then, by polarization for h, we derive

g(h(X, JY), JZ) + g(h(Y, JX), Z) = 0.

Now, since M = M_{T} ×_{f} M_{⊥} is a warped product manifold, D is integrable. Then, from a result of [10], it follows that h(X, JY) = h(JX, Y) for X, Y ∈ Γ(D). Hence, we obtain (4.3).

On the other hand, from (2.3), we get g(A_{JZ}X, V) = −g( ¯∇_{X}JZ, V) for X ∈ Γ(D) and Z, V ∈ Γ(D^{⊥}). Hence, we have

g(A_{JZ}X, V) = −g(( ¯∇_{X}J)Z + J ¯∇_{X}Z, V).

Using (1.2), we obtain

g(A_{JZ}X, V) = g(( ¯∇_{Z}J)X, V) − g(J ¯∇_{X}Z, V).

By direct computations, from (1.1), we derive

g(A_{JZ}X, V) = g( ¯∇_{Z}JX − J ¯∇_{Z}X, V) + g( ¯∇_{X}Z, JV).

Using, again (1.1), we have

g(A_{JZ}X, V) = g( ¯∇_{Z}JX, V) + g( ¯∇_{Z}X, JV) + g( ¯∇_{X}Z, JV).

Thus, from (2.2), we get

g(A_{JZ}X, V) = g(∇_{Z}JX, V) + g(h(Z, X), JV) + g(h(X, Z), JV).

Since h is symmetric, (1.4) and (2.4) imply that

g(h(V, X), JZ) = JX(ln f)g(Z, V) + 2g(h(X, Z), JV)  (4.5)

for X ∈ Γ(D) and Z, V ∈ Γ(D^{⊥}). Interchanging the role of Z and V, we obtain

g(h(Z, X), JV) = JX(ln f)g(Z, V) + 2g(h(X, V), JZ)  (4.6)

Thus from (4.5) and (4.6) we get

g(h(X, Z), JV) = −JX(ln f)g(Z, V).

Now, we are ready to prove a characterization theorem for CR-warped products in nearly Kaehler manifolds. But we first recall that we have the following result of S. Hiepko [16], (cf. [11], Remark 2.1):

Let D_{1} be a vector subbundle in the tangent bundle of a Riemannian manifold M and D_{2} be its normal bundle. Suppose that the two distributions are involutive. We denote the integral manifolds of D_{1} and D_{2} by M_{1} and M_{2}, respectively. Then M is locally isometric to warped product M_{1} ×_{f} M_{2} if the integral manifold M_{1} is totally geodesic and the integral manifold M_{2} is an extrinsic sphere, i. e, M_{2} is a totally umbilical submanifold with parallel mean curvature vector.

Theorem 4.1. A proper CR-submanifold of a nearly Kaehler manifold M̄ is locally CR-warped product if and only if

A_{JZ}X = −(JX)(μ)Z, g(∇_{Z}X, Y) = 0  (4.7)

for some function μ such that W (μ) = 0, W ∈ Γ(D^{⊥}) and

2g(∇_{Z}X, V) = g(A_{JZ}V + A_{JV} Z, JX),  (4.8)

where X, Y ∈ Γ(D) and Z, V ∈ Γ(D^{⊥}).

Proof. If M is a CR-warped product, then, M_{T} is totally geodesic in M. Thus, g(∇_{X}Y, Z) = 0 for X, Y ∈ Γ(D) and Z ∈ Γ(D^{⊥}). Hence, we obtain g(Y, ∇_{X}Z) = 0. Then from (1.4) we get g(Y, ∇_{Z}X) = 0 which is the second equation of (4.7). On the other hand, from (4.3), we have g(h(X, JY), JZ) = 0 for X, Y ∈ Γ(D) and Z, V ∈ Γ(D^{⊥}). Then, using (2.4), we get g(A_{JZ}X, JY) = 0. This implies that A_{JZ}X ∈ Γ(D^{⊥}). On the other hand, from (2.4) and (4.4) we have

g(A_{JZ}X, V) = −JX(ln f)g(Z, V)

for X ∈ Γ(D) and Z, V ∈ Γ(D^{⊥}). Moreover, f is a function on M_{T}, we have W (ln f) = 0 for W ∈ Γ(D^{⊥}). This proves the first equation of (4.7). Furthermore, since A is self-adjoint, from (4.4), we have

g(A_{JZ}V, JX) = X(ln f)g(Z, V).

Interchanging the role of Z and V, we get

g(A_{JV} Z, JX) = X(ln f)g(Z, V).

Thus, from above two equations, we obtain

g(A_{JZ}V + A_{JV} Z, JX) = 2X(ln f)g(Z, V).

Then, using (1.4), we arrive at

g(A_{JZ}V + A_{JV} Z, JX) = 2(∇_{Z}X, V)

which implies (4.8). Conversely, suppose that M is a CR-submanifold of a nearly Kaehler manifold M̄ satisfying (4.7) and (4.8). Then, from (4.2) and (4.7), it follows that D is totally geodesic. (4.8) implies that D^{⊥} is integrable. Let M_{⊥} be a leaf of D^{⊥}. We now denote the second fundamental form of M_{⊥} in M by h_{2}. From (4.8), we also have

−2g(∇_{Z}V, X) = g(A_{JZ}V + A_{JV} Z, JX)

for X ∈ Γ(D) and Z, V ∈ Γ(D^{⊥}). Then, since A is self-adjoint, (4.7) implies that

−2g(∇_{Z}V, X) = X(μ)g(Z, V) + X(μ)g(V, Z).

Thus, we obtain

g(h_{2}(Z, V), X) = g(∇_{Z}V, X) = −X(μ)g(Z, V).

This shows that M_{⊥} is totally umbilical in M. Moreover, by direct computations, we get

g(∇_{Z} grad μ, X) = g(∇_{Z} grad μ, X)
= [Zg(grad μ, X) − g(grad μ, ∇_{Z}X)]
= [Z(X(μ)) − [Z, X]μ − g(grad μ, ∇_{X}Z)]
= [[Z, X]μ + X(Z(μ)) − [Z, X]μ − g(grad μ, ∇_{X}Z)]
= [X(Z(μ)) − g(grad μ, ∇_{X}Z)].

Since Z(μ) = 0, we obtain

g(∇_{Z} grad μ, X) = −g(grad μ, ∇_{X}Z).

On the other hand, since grad μ ∈ Γ(TM_{T}) and M_{T} is totally geodesic in M, it follows that ∇_{X}Z ∈Γ(T M_{⊥}) for X ∈Γ(D) and Z ∈Γ(D^{⊥}). Hence g(∇_{Z} grad μ, X)= 0. Then the spherical condition is also fulfilled, that is M_{⊥} is an extrinsic sphere in M. Thus we conclude that M is a warped product and proof is complete.

We recall that a CR-submanifold is called mixed geodesic [3] if h(X, Z) = 0 for X ∈ Γ(D) and Z ∈ Γ(D^{⊥}). Using (4.4) we have the following:

Corollary 4.1. A mixed totally geodesic CR-warped product M = M_{T} ×_{f} M_{⊥} in a nearly Kaehler manifold M̄ is a CR-product.

In particular, if JD^{⊥} = T M^{⊥}, i. e., M is an anti-holomorphic submanifold of M̄, we have:

Corollary 4.2. An anti-holomorphic warped product submanifold M = M_{T} ×_{f} M_{⊥} in a nearly Kaehler manifold M̄ is a CR-product if and only if M is mixed totally geodesic.

In the rest of this section, we obtain an inequality for the squared norm of the second fundamental form in terms of the warping function for CR-warped products in a nearly Kaehler manifold. We note that this inequality was proved in [7] for Kaehler case.

Lemma 4.4. Let M be a CR-submanifold of a nearly Kaehler manifold M̄. Then, we have:

g(h(Z, Z), JW) = g(h(Z, W), JZ)  (4.9)

for Z, W ∈ Γ(D^{⊥}).

Proof. From (2.2), we have g(h(Z, Z), JW) = g( ¯∇_{Z}Z, JW) for Z, W ∈ Γ(D^{⊥}). Using (1.1), we get g(h(Z, Z), JW) = −g(J ¯∇_{Z}Z, W). Hence, we derive

g(h(Z, Z), JW) = −g(−( ¯∇_{Z}J)Z + ¯∇_{Z}JZ, W).

Thus, using (1.2) and (2.3), we obtain

g(h(Z, Z), JW) = g(A_{JZ}Z, W).

Then, from (2.4), we have (4.9).

Theorem 4.2. Let M = M_{T} ×_{f} M_{⊥} be a CR-warped product in a nearly Kaehler manifold M̄. Then we have

(1) The squared norm of the second fundamental form of M satisfies

∥ h ∥^{2}≥ 2p ∥ ∇(ln f) ∥^{2},  (4.10)

where ∇ ln f is the gradient of ln f and p is the dimension of M_{⊥}.

(2) If the equality sign of (4.10) holds identically, then M_{T} is a totally geodesic submanifold and M_{⊥} is a totally umbilical submanifold of M̄. Moreover, M is a minimal submanifold in M̄

(3) When M is anti-holomorphic and p > 1, the equality sign of (4.10) holds identically if and only if M_{⊥} is totally umbilical submanifold of M̄.

(4) If M is real hypersurface of M̄ (that is p = 1), then the equality sign of (4.10) holds identically if and only if the characteristic vector field Jξ of M is a principal vector field with zero as its principal curvature. Also in this case, the equality sign of (4.10) holds identically if and only if M is a minimal hypersurface in M̄.

Proof. We take an orthonormal frame {e_{1}, e_{2q}, e_{2q+1}, …, e_{m}, e_{1}^{∗} = J(e_{2q+1}), …, e_{p}^{∗}, e_{1}, …, e_{2r}} of M̄ along M such that {e_{1}, …, e_{2q}}, {e_{2q+1}, …, e_{m}} and {e_{1}, …, e_{2r}} are bases of D, D^{⊥} and ν, respectively. Since

∥ h ∥^{2}=∥ h(D, D) ∥^{2} + ∥ h(D^{⊥}, D^{⊥}) ∥^{2} +2 ∥ h(D^{⊥}, D) ∥^{2} .

We have

∥ h ∥^{2} = ∑_{k=m+1}^{n} ∑_{i,j=1}^{2q} g(h(e_{i}, e_{j}), ē_{k})^{2} + ∑_{k=m+1}^{n} ∑_{s,l=1}^{p} g(h(e_{s}, e_{l}), ē_{k})^{2}

+2 ∑_{k=m+1}^{n} ∑_{i=1}^{2q} ∑_{s=1}^{p} g(h(e_{i}, e_{s}), ē_{k})^{2}

where {ē_{k}}_{k=1}^{n} is a basis of T M^{⊥}. Then, from the decomposition of normal bundle of a CR-submanifold of a nearly Kaehler manifold, we get

∥ h ∥^{2} = ∑_{k=m+1}^{n} ∑_{i,j=1}^{2q} g(h(e_{i}, e_{j}), ē_{k})^{2} + ∑_{k=m+1}^{n} ∑_{s,l=1}^{p} g(h(e_{s}, e_{l}), ē_{k})^{2}

+2 ∑_{t,s=1}^{p} ∑_{i=1}^{2q} g(h(e_{i}, e_{s}), Je_{t})^{2} + 2 ∑_{l=1}^{2r} ∑_{i=1}^{2q} ∑_{s=1}^{p} g(h(e_{i}, e_{s}), Je_{l})^{2}.

Then, from (4.4) we have

∥ h ∥^{2} = ∑_{k=m+1}^{n} ∑_{i,j=1}^{2q} g(h(e_{i}, e_{j}), ē_{k})^{2} + ∑_{k=m+1}^{n} ∑_{s,l=1}^{p} g(h(e_{s}, e_{l}), ē_{k})^{2}

+2p ∥ ∇(ln f) ∥^{2} +2 ∑_{l=1}^{2r} ∑_{i=1}^{2q} ∑_{s=1}^{p} g(h(e_{i}, e_{s}), Je_{l})^{2}.

This proves (4.10). If the equality case of (4.10) holds identically, we obtain

h(D, D) = 0, h(D^{⊥}, D^{⊥}) = 0, h(D, D^{⊥}) ⊂ JD^{⊥}.  (4.11)

Since M_{T} is totally geodesic in M, the first equation in (4.11) implies that M_{T} is totally geodesic in M̄. On the other hand, using (2.2) and (1.4) we get

h_{2}(Z, V) = −(∇ ln f)g(Z, V)  (4.12)

for Z, V ∈ Γ(M_{⊥}). Then, (4.12) and the second equation (4.11) imply that M_{⊥} is totally umbilical in M̄. (4.11) also implies that M is minimal in M̄.

Now suppose that M is an anti-holomorphic warped product in M̄. Then from (4.3) we get

h(D, D) = 0.  (4.13)

Thus, if M_{⊥} is totally umbilical in M̄, we can write

h̄(Z, V) = g(Z, V)H̄,  (4.14)

for Z, V ∈ Γ(D^{⊥}), where H̄ is a normal vector field to M_{⊥}. Since

h̄(Z, V) = h_{2}(Z, V) + h(Z, V),

(4.14) implies that there is a normal vector field N such that

h(Z, V) = g(Z, V)N.

Hence, for each unit vector field V ∈ Γ(D^{⊥}) and each unit vector field Z ∈ Γ(D^{⊥}), perpendicular to V, we derive

g(N, JV) = g(h(Z, Z), JV).

Then using (4.9) we get

g(N, JV) = g(h(Z, V), JZ) = g(Z, V)g(N, JZ) = 0.  (4.15)

Since M is anti-holomorphic, (4.15) implies that either p = 1 or

h(D^{⊥}, D^{⊥}) = 0.  (4.16)

Then, from (4.4), (4.13) and (4.16), it follows that the equality case holds, when p > 1.

If p = 1, then M is a real hypersurface in M̄. Then, the characteristic vector field Jξ is a principal vector field with zero as its principal curvature if and only if (4.16) holds. Thus, in this case, we also have equality case. If the characteristic vector field Jξ is a principal vector field with zero as its principal curvature, from (4.11) we also know that the condition (4.16) holds if and only if M is minimal in M. The converse is clear. Thus proof is complete.

Remark 3. Thus we show that Theorem 5.1 of [7] is also valid for warped product CR-submanifolds in the form M_{T} ×_{f} M_{⊥} in a nearly Kaehler manifold M̄ such that M_{T} is a holomorphic submanifold and M_{⊥} is a totally real submanifold of M̄. It is also clear that Theorem 4.2 is a generalization of Theorem 5.1 in [7].

Acknowledgements. We express our sincere thanks to the referee for many helpful suggestions.

References

[1] Bashir, M. A.: On the three-dimensional CR-submanifolds of the six-

dimensional sphere. Internat. J. Math. Math. Sci. 14(4) (1991), 675–678.

Zbl 0748.53032

[2] Bashir, M. A.: On CR-submanifolds of the six-dimensional sphere. Internat. J. Math. Math. Sci. 18(1) (1995), 201–203. Zbl 0828.53049

[3] Bejancu, A.: Geometry of CR-Submanifolds. Kluwer Academic Publishers, Dortrecht 1986. Zbl 0605.53001

[4] Bishop, R. L.; O’ Neill, B.: Manifolds of Negative Curvature. Trans. Amer. Math. Soc. 145 (1969), 1–49. Zbl 0191.52002

[5] Bonanzinga, V.; Matsumoto, K.: Warped product CR-submanifold in locally conformal Kaehler Manifold. Periodica Math. Hungar. Mat. 48 (2004), 207– 221. Zbl 1104.53049

[6] Chen, B. Y.: Geometry of Slant Submanifolds. Katholieke Universiteit Leuven, Leuven 1990. Zbl 0716.53006

[7] Chen, B. Y.: Geometry of warped product CR-submanifolds in Kaehler manifold. Monatsh. Math. 133 (2001), 177–195. Zbl 0996.53044

[8] Chen, B. Y.: Geometry of warped product CR-submanifolds in Kaehler manifolds II. Monatsh. Math. 134 (2001), 103–119. Zbl 0996.53045

[9] Chen, B. Y.: CR-warped products in complex projective spaces with compact holomorphic factor. Monatsh. Math. 141 (2004), 177–186. Zbl 1063.53056

[10] Deshmukh, S.; Shahid, M. H.; Ali, S.: CR-submanifolds of a nearly Kaehler manifold. II. Tamkang J. Math. 17(4) (1986), 17–27. Zbl 0674.53056

[11] Dillen, F.; Nölker, S.: Semi-paralellity, multi rotation surfaces and the helix property. J. Reine Angew Math. 435 (1993), 33–63. Zbl 0770.53039

[12] Fukami, T.; Ishihara, S.: Almost Hermitian structures on S^{6}. Tōhoku Math. J. 7 (1955), 151–156. Zbl 0068.36001 [13] Gray, A.: Vector cross products on manifolds. Trans. Amer. Math. Soc. 141

(1969), 465–504. Zbl 0182.24603 [14] Hasegawa, I.; Mihai, I.: Contact CR-warped product submanifolds in Sasakian

Manifolds. Geometriae Dedicata 102 (2003), 143–150. Zbl 1066.53103 [15] Hashimoto, H.; Mashimo, K.: On some 3-dimensional CR submanifolds in

S^{6}. Nagoya Math. J. 156 (1999), 171–185. Zbl 0957.53023 [16] Hiepko, S.: Eine innere Kennzeichnung der verzerrten Produkte. Math. Ann.

241 (1979), 209–215. Zbl 0387.53014 [17] Husain, S. I.; Deshmukh, S.: CR-submanifolds of a nearly Kaehler manifold.

Indian J. Pure Appl. Math. 18(11) (1987), 979–990. Zbl 0632.53053 [18] Khan, K. A.; Khan, V. A.; Husain, S. I.: Totally umbilical CR-submanifolds

of nearly Kähler manifolds. Geometriae Dedicata 50(1) (1994), 47–51.

Zbl 0805.53053

[19] Mihai, I.: Contact CR-warped product submanifolds in Sasakian space forms. Geometriae Dedicata 109 (2004), 165–173. Zbl 1114.53052

[20] Munteanu, M. I.: Warped product contact CR-submanifolds of Sasakian space forms. Publ. Math. Debrecen, 66(1–2) (2005), 75–120. Zbl 1063.53052

[21] Sekigawa, K.: Some CR-submanifolds in a 6-dimensional sphere. Tensor (N.S.) 41(1) (1984), 13–20. Zbl 0547.53027

[22] Yano, K.; Kon, M.: Structures on Manifolds. World Scientific, Singapore 1984. Zbl 0557.53001

Received March 15, 2007