MATEMATIQKI VESNIK

59 (2007), 205–210

UDK 514.763.4 originalni nauqni rad

research paper

NOTES ON DOUBLY WARPED AND DOUBLY TWISTED PRODUCT CR-SUBMANIFOLDS OF KAEHLER MANIFOLDS

Bayram Şahin

Abstract. Recently, B. Y. Chen studied warped product CR-submanifolds [7] and twisted product CR-submanifolds [6] in Kaehler manifolds. In this paper, we have checked the existence of other product CR-submanifolds such as doubly warped and doubly twisted products in Kaehler manifolds.

1. Introduction

Warped product manifolds were introduced by R. L. Bishop and B. O’Neill in [2] to construct new examples of negatively curved manifolds. Later, this notion has been generalized in several ways. Let (B, g_{B}) and (F, g_{F}) be semi-Riemannian manifolds of dimensions m and n, respectively and let π : B × F → B and σ : B × F → F be the canonical projections. Also let b : B × F → (0, ∞), f : B × F → (0, ∞) be smooth functions. Then the doubly twisted product ([10], [15]) of (B, g_{B}) and (F, g_{F}) with twisting functions b and f is defined to be the product manifold M = B × F with metric tensor g = f^{2} g_{B} ⊕ b^{2} g_{F}. We denote this kind manifolds by _{f}B ×_{b} F. Denote by F (B) the algebra of smooth functions on B and by Γ(E) the F (B) module of smooth sections of a vector bundle E (same notation for any other bundle) over B. If X ∈ Γ(T B) and V ∈ Γ(T F), then from Proposition 1 of [10], we have

∇_{X}V = V (lnf)X + X(lnb)V,  (1.1)

where ∇ denotes the Levi-Civita connection of the doubly twisted product _{f}B ×_{b} F of (B, g_{B}) and (F, g_{F}). In particular, if f = 1, then B ×_{b} F is called the twisted product of (B, g_{B}) and (F, g_{F}) with twisting function b. We note that the notion of twisted products was introduced in [5]. If M = B ×_{b} F is a twisted product manifold, then (1.1) becomes

∇_{X}V = X(lnb)V.  (1.2)

AMS Subject Classification: 53C40, 53C42, 53C15.

Keywords and phrases: Doubly warped product; doubly twisted product; CR-submanifold; Kaehler manifold.

Moreover, if b only depends on the points of B, then B×_{b}F is called warped product of (B, g_{B}) and (F, g_{F}) with warping function b. In this case, for X ∈ Γ(T B) and V ∈ Γ(T F), from Lemma 7.3 of [2], we have

∇_{X}V = X(lnb)V,  (1.3)

where b depends on the points of B and X ∈ Γ(T B), V ∈ Γ(T F).

As a generalization of the warped product of two semi-Riemannian manifolds, fB ×_{b} F is called the doubly warped product of semi-Riemannian manifolds (B, g_{B}) and (F, g_{F}) with warping functions b and f if only depend on the points of B and F, respectively.

In [7], B. Y. Chen considered warped product CR-submanifolds of Kaehler manifolds and showed that there exist no warped product CR-submanifolds in the form M_{⊥} ×_{f} M_{T}, where M_{⊥} is a totally real submanifold and M_{T} is a holomorphic submanifold of a Kaehler manifold M̄. Then he introduced CR-warped products which are warped product CR-submanifolds in the form M_{T} ×_{f} M_{⊥} such that M_{T} is a holomorphic submanifold and M_{⊥} is a totally real submanifold of M̄. CR-warped products (or CR-products) have been also studied in [3], [4], [8], [9], [11], [12], [13], [14].

B. Y. Chen [6] also introduced twisted product CR-submanifolds in Kaehler manifolds and showed that a twisted product CR-submanifold in the form M_{⊥} ×_{λ} M_{T} is a CR-product. Then he considered twisted product CR-submanifolds in the form M_{T} ×_{λ} M_{⊥} and established a general sharp inequality for twisted product CR-submanifolds in Kaehler manifolds.

In this paper, we investigate the existence of other product (doubly warped and doubly twisted product) CR-submanifolds in Kaehler manifolds. In fact, we show that there do not exist doubly warped and doubly twisted product CR-submanifolds in Kaehler manifolds.

2. Preliminaries

Let (M̄, g) be a Kaehler manifold. This means 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), ( ¯∇_{X}J)Y = 0  (2.1)

where g is Riemannian metric and ¯∇ is the Levi-Civita connection on M̄.

Doubly warped and doubly twisted product CR-submanifolds … 207

Let M̄ be a 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. In 1978, A. Bejancu introduced [1] a new class of submanifolds of Kaehler manifolds as follows. A submanifold M is called a CR-submanifold if there exists on M a differentiable distribution D : p → D_{p} ⊂ T_{p}M such that D is invariant with respect to J and its orthogonal complement D^{⊥} is totally real distribution, i.e, J(D_{p}^{⊥}) ⊆ T_{p}M^{⊥}. Obviously, holomorphic and totally real submanifolds are CR-submanifolds having D_{p} = T_{p}M and D_{p} = 0, respectively. A CR-submanifold is called proper if it is neither holomorphic nor totally real.

Let M be a Riemannian manifold isometrically immersed in M̄ and denote by the same symbol g the Riemannian metric induced on M. Let T M be the Lie algebra of vector fields in M and 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 ∈ T M and any N ∈ 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)

For Kaehler manifolds and their submanifolds, see [17].

3. Doubly warped and doubly twisted product CR-submanifolds

In this section, we consider CR-submanifolds which are doubly warped or doubly twisted products in the form _{f}M_{T} ×_{b} M_{⊥}, where M_{T} is a holomorphic submanifold and M_{⊥} is a totally real submanifold of M̄.

Theorem 3.1. Let M̄ be a Kaehler manifold. Then there do not exist doubly warped product CR-submanifolds which are not (singly) warped product CRsubmanifolds in the form _{f}M_{T} ×_{b} M_{⊥} such that M_{T} is a holomorphic submanifold and M_{⊥} is a totally real submanifold of M̄.

Proof. Let us suppose that M be a doubly warped product CR-submanifold of a Kaehler manifold M̄. Then from (1.1) we have

g(∇_{JX}V, X) = V (lnf)g(JX, X) = 0

for X ∈ Γ(D) and V ∈ Γ(D^{⊥}). Using (2.2) we get g( ¯∇_{JX}V, X) = 0. Since D and D^{⊥} are orthogonal, we obtain g( ¯∇_{JX}X, V) = 0. Then from (2.1) we have g( ¯∇_{JX}JX, JV) = 0. Thus from (2.2), we have

g(h(JX, JX), JV) = 0  (3.1)

for X ∈ Γ(D) and V ∈ Γ(D^{⊥}). Now, substituting X by Y + Z, Z, Y ∈ Γ(D), in (3.1) and taking into account that h is symmetric, we get

g(h(JY, JZ), JV) = 0  (3.2)

for Y, Z ∈ Γ(D) and V ∈ Γ(D^{⊥}). On the other hand, from (2.2) we have g(h(JY, JZ), JV) = g( ¯∇_{JY} JZ, JV). Using (2.1) we obtain

g(h(JY, JZ), JV) = −g(Z, ∇_{JY} V)

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

g(h(JY, JZ), JV) = −V (lnf)g(JY, Z)  (3.3)

for Y, Z ∈ Γ(D) and V ∈ Γ(D^{⊥}). Thus using (3.2) in (3.3), we obtain

V (lnf)g(JY, Z) = 0.

Hence, for Z = JY, we get

V (lnf)g(Y, Y) = 0.

Then, we derive V (lnf) = 0 due to g(Y, Y) ≠ 0. V (lnf) = 0 implies that f is constant. Thus M is a warped product CR-submanifold in the form M_{T} ×_{b} M_{⊥} which is called CR-warped product (see [7]). Thus, proof is complete.

Theorem 3.1 tells us that there exist no doubly warped product CR-submani-folds in Kaehler manifolds other than warped product CR-submanifolds. In the rest of this section, we investigate the existence of doubly twisted product CR-sub-manifolds in Kaehler manifolds.

Theorem 3.2. Let M̄ be a Kaehler manifold. Then there do not exist doubly twisted product CR-submanifolds of M̄ which are not (singly) twisted product CRsubmanifolds in the form _{f}M_{T} ×_{b} M_{⊥} such that M_{T} is a holomorphic submanifold and M_{⊥} is a totally real submanifold of M̄.

Proof. From (1.1), we have g(∇_{X}V, Y) = V (lnf)g(X, Y) for X, Y ∈ Γ(D) and V ∈ Γ(D^{⊥}). Since D and D^{⊥} are orthogonal, we get −g(∇_{X}Y, V) = V (lnf)g(X, Y). Using (2.2) and (2.1), we obtain

−g( ¯∇_{X}JY, JV) = V (lnf)g(X, Y).

Thus, from (2.2), we derive

−g(h(X, JY), JV) = V (lnf)g(X, Y)  (3.4)

for X, Y ∈ Γ(D) and V ∈ Γ(D^{⊥}). On the other hand, from (2.2), we have

g(h(X, JY), JV) = g( ¯∇_{JY} X, JV)

for X, Y ∈ Γ(D) and V ∈ Γ(D^{⊥}). Using(2.1), we get

g(h(X, JY), JV) = −g( ¯∇_{JY} JX, V).

Hence, (2.2) implies that

g(h(X, JY), JV) = g(JX, ∇_{JY} V).

Thus from (1.1), we obtain

g(h(X, JY), JV) = V (lnf)g(JX, JY).

Using (2.1) we arrive at

g(h(X, JY), JV) = V (lnf)g(X, Y)  (3.5)

for X, Y ∈ Γ(D) and V ∈ Γ(D^{⊥}). Then from (3.4) and (3.5) we obtain

V (lnf)g(X, Y) = 0.

Since D is Riemannian, we get V (lnf) = 0. This implies that f only depends the points of M_{T}. Thus we can write

g = g̃_{M_{T}} ⊕ b^{2}g_{M_{⊥}}, where g̃_{M_{T}} = f^{2} g_{M_{T}} .

Thus it follows that M is a twisted product CR-submanifold in the form M = M_{T} ×_{b} M_{⊥}, (see [6] for twisted product CR-submanifolds). Hence, we conclude that there are no doubly twisted product CR-submanifolds in Kaehler manifolds, other than twisted product CR-submanifolds.

Acknowledgment. The author is grateful to the referee for his/her very valuable suggestions.

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(received 01.05.2007, in revised form 18.06.2007)

Department of Mathematics, Inonu University, 44280 Malatya, TURKEY

E-mail: bsahin@inonu.edu.tr