PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 92(106) (2012), 157–163 DOI: 10.2298/PIM1206157U

WARPED PRODUCT CR-SUBMANIFOLDS OF LORENTZIAN β-KENMOTSU MANIFOLDS

Siraj Uddin, Cenap Ozel, and Viqar Azam Khan

Communicated by Vladimir Dragović

Abstract. We study the geometry of warped product submanifolds of Lorentzian β-Kenmotsu manifolds. We obtain a characterization result for CR-warped products.

1. Introduction

The notion of warped product manifolds was introduced by Bishop and O’Neill [2] and then it was studied by many mathematicians and physicists. These manifolds are the generalization of the Riemannian product manifolds. Chen [4] has studied the geometry of warped product submanifolds in Kaehler manifold and showed that the warped product submanifold of the type N_{⊥} ×_{f} N_{⊤} is trivial. Later, many research articles have recently appeared exploring the existence or nonexistence of warped product submanifolds in known spaces [1, 6, 9].

Matsumoto [8] introduced the notion of Lorentzian almost paracontact metric manifolds. Later on, many geometers studied submanifolds of Lorentzian almost paracontact manifolds [5, 10].

As Kenmotsu manifolds are themselves warped product spaces, it is interesting to study warped product submanifolds in Kenmotsu manifolds. In this paper we consider the warped product submanifolds of the types M = N_{⊥} ×_{f} N_{⊤} and M = N_{⊤} ×_{f} N_{⊥} in an arbitrary Lorentzian β-Kenmotsu manifold M̄, where N_{⊤} and N_{⊥} are the invariant and anti-invariant submanifolds of M̄, respectively.

2. Preliminaries

Let M̄ be a (2n + 1)-dimensional manifold of class C^{∞} endowed with an endomorphism ϕ of its tangent bundle T M̄, a vector field ξ, which is called the structure vector field, a 1-form η and a Lorentzian metric g with signature (−, +, ⋯, +) satisfying: ϕ^{2}X = X + η(X)ξ, η(ξ) = −1, ϕ(ξ) = 0, η ∘ ϕ = 0, (2.1)

2010 Mathematics Subject Classification: 53C25, 53C40, 53B25.

Key words and phrases: Warped product, CR-submanifold, Lorentzian β-Kenmotsu manifold, canonical structure.

g(ϕX, ϕY) = g(X, Y) + η(X)η(Y), η(X) = g(X, ξ). (2.2)

for any vector fields X, Y on M̄. Then such a structure (ϕ, ξ, η, g) is termed as Lorentzian paracontact structure and the manifold M̄ with a Lorentzian paracontact structure is called a Lorentzian paracontact manifold [8].

Our purpose is to define the warped product submanifolds of a Lorentzian β-Kenmotsu manifold, that is a manifold with a paracontact structure and a compatible Lorentzian metric g, (M̄^{2n+1} , ϕ, ξ, η, g) satisfying (2.1) and (2.2) with the following additional condition:

(2.3)  ( ¯∇_{X} ϕ)Y = β{g(ϕX, Y)ξ + η(Y)ϕX},

for any X, Y ∈ T M̄, where ¯∇ is the Levi–Civita connection with respect to the Lorentzian metric g. Thus, a Lorentzian paracontact manifold satisfying (2.3) is called a Lorentzian β-Kenmotsu manifold [10]. From (2.3), it is easy to obtain that

(2.4)  ¯∇_{X} ξ = β{X + η(X)ξ}

Now, let M be a submanifold of M̄. Let T M be the Lie algebra of vector fields in M and T^{⊥} M the set of all vector fields normal to M. If ∇ is the Levi–Civita connection on M, then Gauss–Weingarten formulas are given by

¯∇_{X} Y = ∇_{X} Y + h(X, Y) (2.5)

¯∇_{X} N = −A_{N} X + ∇_{X}^{⊥} N (2.6)

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

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

where g denotes the metric on M̄ as well as the induced metric on M [11]. For any X ∈ T M, we write

(2.8)  ϕX = P X + F X

where P X is the tangential component of ϕX and F X is the normal component of ϕX, respectively. Similarly, for any vector field N normal to M, we put

(2.9)  ϕN = BN + CN

where BN and CN are tangential and normal components of ϕN, respectively. The covariant derivatives of the tensor fields P and F are defined as

( ¯∇_{X} P)Y = ∇_{X} P Y − P ∇_{X} Y, (2.10)

( ¯∇_{X} F)Y = ∇_{X}^{⊥} F Y − F ∇_{X} Y (2.11)

for all X, Y ∈ T M.

LORENTZIAN β-KENMOTSU MANIFOLDS 159

A submanifold M, of a Lorentzian β-Kenmotsu manifold M̄ is called CRsubmanifold if it admits a differentiable invariant distribution D whose orthogonal complementary distribution D^{⊥} is anti-invariant i.e., T M = D ⊕ D^{⊥} ⊕ ⟨ξ⟩ with ϕ(D_{p}) ⊆ D_{p} and ϕ(D_{p}^{⊥} ) ⊂ T_{p}^{⊥} M, for every p ∈ M. A CR-submanifold is known to be invariant, anti-invariant and proper if D^{⊥} = 0, D = 0 and D ≠ 0 ≠ D^{⊥} , respectively.

Note that ξ is time like vector field and all vector field in D ⊕ D^{⊥} are space like.

Let M be an m-dimensional CR-submanifold of (2n+1)-dimensional Lorentzian β-Kenmotsu manifold (M̄^{2n+1} , ϕ, ξ, η, g). Then, F (T_{p}M) is a subspace of T_{p}^{⊥} M. Then for every p ∈ M, there exists an invariant subspace μ_{p} of T_{p}M̄ such that

T_{p}M̄ = T_{p}M ⊕ F (T_{p}M) ⊕ μ_{p}.

3. Warped Product Submanifolds

The study of warped product manifolds was initiated by Bishop and O’Neill [2]. They defined these manifolds as follows:

Definition 3.1. Let (B, g_{1}) and (F, g_{2}) be two semi-Riemannian manifolds with metric g_{1} and g_{2} respectively and f a positive differentiable function on B. The warped product of B and F is the manifold B ×_{f} F = (B × F, g), where

g = g_{1} + f^{2}g_{2}.

More explicitly, if U is tangent to M = B ×_{f} F at (p, q), then

∥U∥^{2} = ∥dπ_{1}U∥^{2} + f^{2}(p)∥dπ_{2}U∥^{2}

where π_{i}(i = 1, 2) are the canonical projections of B×F onto B and F, respectively.

A warped product manifold B ×_{f} F is said to be trivial if the warping function f is constant. We recall that on a warped product manifold, one has

(3.1)  ∇_{U}V = ∇_{V}U = (U lnf)V

for any vector fields U tangent to B and V tangent to F [2].

Throughout the paper, we denote by N_{⊤} and N_{⊥} , the invariant and antiinvariant submanifolds of a Lorentzian β-Kenmotsu manifold M̄, respectively. Then their warped product CR-submanifolds are one of the following forms:

(i) M = N_{⊥} ×_{f} N_{⊤} , (ii) M = N_{⊤} ×_{f} N_{⊥} .

For case (i), when ξ ∈ T N_{⊤} , we have the following theorem.

Theorem 3.1. There does not exist a warped product CR-submanifold M = N_{⊥} ×_{f} N_{⊤} of a Lorentzian β-Kenmotsu manifold M̄ such that N_{⊤} is an invariant submanifold tangent to ξ and N_{⊥} is an anti-invariant submanifold of M̄.

Proof. Let M = N_{⊥} ×_{f} N_{⊤} be a warped product CR-submanifold of a Lorentzian β-Kenmotsu manifold M̄ such that N_{⊤} is an invariant submanifold tangent to ξ and N_{⊥} is an anti-invariant submanifold of M̄. Then by equation (3.1), we get

∇_{X} Z = ∇_{Z} X = (Z lnf)X

for any vector fields Z and X tangent to N_{⊥} and N_{⊤} , respectively.

In particular,

(3.2)  ∇_{Z} ξ = (Z lnf)ξ,

whereas by (2.4), (2.5) and the fact that ξ is tangent to N_{⊤} , we have

(3.3)  ∇_{Z} ξ = βZ, h(Z, ξ) = 0.

It follows from (3.2) and (3.3) that Z lnf = 0, for all Z ∈ T N_{⊥} i.e., f is constant for all Z ∈ T N_{⊥} . This completes the proof. □

Now, the other case, when ξ tangent to N_{⊥} is dealt in the following two results.

Lemma 3.1. Let M = N_{⊥} ×_{f} N_{⊤} be a warped product CR-submanifold of a Lorentzian β-Kenmotsu manifold M̄ such that ξ is tangent to N_{⊥} . Then

(i) ξ lnf = β, (ii) g(h(X, ϕX), F Z) = −{βη(Z) + Z lnf}∥X∥^{2},

for any X ∈ T N_{⊤} and Z ∈ T N_{⊥} .

Proof. If ξ ∈ T N_{⊥} then for any X ∈ T N_{⊤} , we have
(3.4)  ∇_{X} ξ = (ξ lnf)X.

On the other hand, from (2.4) and the fact that ξ is tangent to N_{⊥} , we have ¯∇_{X} ξ = βX. Using (2.5), we obtain

(3.5)  ∇_{X} ξ = βX, h(X, ξ) = 0.

By equations (3.4) and (3.5), it follows that ξ lnf = β. Now, for any X ∈ T N_{⊤} and Z ∈ T N_{⊥} , we have ( ¯∇_{X} ϕ)Z = ¯∇_{X} ϕZ − ϕ ¯∇_{X} Z. Using (2.3), (2.6), (2.8), (2.9) and by the orthogonality of two distributions, we derive

βη(Z)ϕX = −A_{F Z} X + ∇_{X}^{⊥} F Z − P ∇_{X} Z − F ∇_{X} Z − Bh(X, Z) − Ch(X, Z).

Equating the tangential components, we get

−βη(Z)ϕX = A_{F Z} X + P ∇_{X} Z + Bh(X, Z).

Taking the product with ϕX and using (2.2) and (3.1), we derive

−βη(Z)∥X∥^{2} = g(A_{F Z} X, ϕX) + (Z lnf)g(P X, ϕX) + g(Bh(X, Z), ϕX)
= g(h(X, ϕX), F Z) + (Z lnf)g(ϕX, ϕX) + g(ϕh(X, Z), ϕX).

Then from (2.2), we obtain

{□ \atop (3.6)} g(h(X, ϕX), F Z) = −\{βη(Z) + Z lnf\}∥X∥^{2}.

Theorem 3.2. Let M = N_{⊥} ×_{f} N_{⊤} be a warped product CR-submanifold of a Lorentzian β-Kenmotsu manifold M̄ such that ξ is tangent to N_{⊥} . If h(X, ϕX) ∈ μ the invariant normal subbundle of M, then Z lnf = −βη(Z) for all X ∈ T N_{⊤} and Z ∈ T N_{⊥} .

Proof. The assertion follows from formula (3.6) by using the given fact. □

For the warped product of the type N_{⊤} ×_{f} N_{⊥} , we have the following theorem.

Theorem 3.3. There does not exist a warped product CR-submanifold M = N_{⊤} ×_{f} N_{⊥} of a Lorentzian β-Kenmotsu manifold M̄ such that ξ is tangent to N_{⊥} .

Proof. As ξ ∈ T N_{⊥} , then by formula (3.1), we have

(3.7)  ∇_{X} ξ = (X lnf)ξ,

for any X ∈ T N_{⊤} . Whereas from (2.4), (2.5) and the fact that ξ ∈ T N_{⊥} , we have

(3.8)  ∇_{X} ξ = βX, h(X, ξ) = 0

From (3.7) and (3.8), it follows that X lnf = 0, for all X ∈ T N_{⊤} , and this means that f is constant on N_{⊤} . This proves the theorem. □

The remaing case, when ξ ∈ T N_{⊤} is dealt with the following two theorems.

Theorem 3.4. Let M = N_{⊤} ×_{f} N_{⊥} be a warped product CR-submanifold of a Lorentzian β-Kenmotsu manifold M̄ such that ξ is tangent to N_{⊤} . Then ( ¯∇_{X} F)Z lies in the invariant normal subbundle for each X ∈ T N_{⊤} and Z ∈ T N_{⊥} .

Proof. For any X ∈ T N_{⊤} and Z ∈ T N_{⊥} , we have
g(ϕ ¯∇_{X} Z, ϕZ) = g( ¯∇_{X} Z, Z) = g(∇_{X} Z, Z).

Using (3.1), we get

(3.9)  g(ϕ ¯∇_{X} Z, ϕZ) = (X lnf)∥Z∥^{2}.

On the other hand, for any X ∈ T N_{⊤} and Z ∈ T N_{⊥} , we have

( ¯∇_{X} ϕ)Z = ¯∇_{X} ϕZ − ϕ ¯∇_{X} Z.

Using (2.3) and the fact that ξ is tangent to N_{⊤} , the left-hand side of the above equation is identically zero, then we get

(3.10)  ϕ ¯∇_{X} Z = ¯∇_{X} ϕZ.

Taking the product with ϕZ in (3.10) and making use of formula (2.6), we obtain

g(ϕ ¯∇_{X} Z, ϕZ) = g(∇_{X}^{⊥} F Z, F Z).

Then from (2.11), we derive g(ϕ ¯∇_{X} Z, ϕZ) = g(( ¯∇_{X} F)Z, F Z) + g(F ∇_{X} Z, F Z). Using (3.1), we get

g(ϕ ¯∇_{X} Z, ϕZ) = (X lnf)g(F Z, F Z) + g(( ¯∇_{X} F)Z, F Z)
= (X lnf)g(ϕZ, ϕZ) + g(( ¯∇_{X} F)Z, F Z).

Therefore by (2.2), we obtain

(3.11)  g(ϕ ¯∇_{X} Z, ϕZ) = (X lnf)∥Z∥^{2} + g(( ¯∇_{X} F)Z, F Z).

Thus (3.9) and (3.11) imply

(3.12)  g(( ¯∇_{X} F)Z, F Z) = 0.

Also, as N_{⊤} is an invariant submanifold then ϕW ∈ T N_{⊤} for any W ∈ T N_{⊤} , thus on using (2.11) and the fact that the product of tangential component with normal is zero, we obtain

(3.13)  g(( ¯∇_{X} F)Z, ϕW) = 0.

Hence from (3.12) and (3.13), it follows that ( ¯∇_{X} F)Z ∈ μ, for all X ∈ T N_{⊤} and Z ∈ T N_{⊥} . Thus, the proof is complete. □

Theorem 3.5. A proper CR-submanifold M of a Lorentzian β-Kenmotsu manifold M̄ is locally a CR-warped product if and only if the shape operator of M satisfies

(3.14)  A_{ϕZ} X = (ϕXμ)Z, X ∈ D ⊕ ⟨ξ⟩, Z ∈ D^{⊥}

for some function μ on M satisfying V (μ) = 0, for each V ∈ D^{⊥} .

Proof. Let M = N_{⊤} ×_{f} N_{⊥} be a CR-warped product submanifold of a Lorentzian β-Kenmotsu manifold M̄ with ξ ∈ T N_{⊤} , then for any X ∈ T N_{⊤} and Z, W ∈ T N_{⊥} , we have

g(A_{ϕZ} X, W) = g(h(X, W), ϕZ) = g( ¯∇_{W} X, ϕZ) = g(ϕ ¯∇_{W} X, Z)
= g( ¯∇_{W} ϕX, Z) − g(( ¯∇_{W} ϕ)X, Z).

Using (2.3), (3.1) and the fact that ξ is tangent to N_{⊤} , the above equation yields

(3.15)  g(A_{ϕZ} X, W) = (ϕX lnf)g(Z, W).

Further, we have g(h(X, Y), F Z) = g( ¯∇_{X} Y, ϕZ) = g(ϕ ¯∇_{X} Y, Z) = −g(ϕY, ¯∇_{X} Z), for each X, Y ∈ T N_{⊤} and Z ∈ T N_{⊥} . Using (3.1), we obtain g(h(X, Y), F Z) = 0 . Taking into account this fact in (3.15), we obtain (3.14).

Conversely, suppose that M is a proper CR-submanifold of a Lorentzian β-Kenmotsu manifold M̄ satisfying (3.14), then for any X, Y ∈ D ⊕ ⟨ξ⟩,

g(h(X, Y), ϕZ) = g(A_{ϕZ} X, Y) = 0.

This implies that g( ¯∇_{X} ϕY, Z) = 0, that is, g(∇_{X} Y, Z) = 0. This means D ⊕ ⟨ξ⟩ is integrable and its leaves are totally geodesic in M. Now, for any Z, W ∈ D^{⊥} and X ∈ D ⊕ ⟨ξ⟩, we have

g(∇_{Z} W, ϕX) = g( ¯∇_{Z} W, ϕX) = g(ϕ ¯∇_{Z} W, X)
= g( ¯∇_{Z} ϕW, X) − g(( ¯∇_{Z} ϕ)W, X).

Then, using (2.3) and (2.6), we obtain g(∇_{Z} W, ϕX) = −g(A_{ϕW} Z, X). Thus from (2.7), we arrive at g(∇_{Z} W, ϕX) = −g(h(Z, X), ϕW). Again using (2.7) and (3.14), we obtain

(3.16)  g(∇_{Z} W, ϕX) = −g(A_{ϕW} X, Z) = −(ϕXμ)g(Z, W).

Let N_{⊥} be a leaf of D^{⊥} and h^{⊥} be the second fundamental form of the immersion of N_{⊥} into M. Then for any Z, W ∈ D^{⊥} , we have

(3.17)  g(h^{⊥} (Z, W), ϕX) = g(∇_{Z} W, ϕX).

Hence, from (3.16) and (3.17), we conclude that

g(h^{⊥} (Z, W), ϕX) = −(ϕXμ)g(Z, W).

This means that integral manifold N_{⊥} of D^{⊥} is totally umbilical in M. Since V (μ) = 0 for each V ∈ D^{⊥} , which implies that the integral manifold of D^{⊥} is an extrinsic sphere in M, this means that the curvature vector field is nonzero and parallel along N_{⊥} . Hence by virtue of a result in [7], M is locally a warped product N_{⊤} ×_{f} N_{⊥} , where N_{⊤} and N_{⊥} denote the integral manifolds of the distributions D ⊕ ⟨ξ⟩ and D^{⊥} , respectively and f is the warping function. □

Acknowledgement. The authors are thankful to the referee for providing constructive comments and valuable suggestions.

References

1. K. Arslan, R. Ezentas, I. Mihai and C. Murathan, Contact CR-warped product submanifolds in Kenmotsu space forms, J. Korean Math. Soc. 42 (2005), 1101–1110.

2. R. L. Bishop and B. O’Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1–49.

3. D. E. Blair, Contact Manifolds in Riemannian Geometry, Lect. Notes Math. 509, Springer-Verlag, New York, 1976.

4. B. Y. Chen, Geometry of warped product CR-submanifolds in Kaehler manifold, Monatsh. Math. 133 (2001), 177–195.

5. U. C. De and K. Arslan, Certain curvature conditions on an LP-Sasakian manifold with a coefficient α, Bull. Korean Math. Soc. 46 (2009), 401–408.

6. I. Hasegawa and I. Mihai, Contact CR-warped product submanifolds in Sasakian manifolds, Geom. Dedicata 102 (2003), 143–150.

7. S. Hiepko, Eine innere Kennzeichnung der verzerrten Produkte, Math. Ann. 241 (1979), 209– 215.

8. K. Matsumoto, On Lorentzian paracontact manifolds, Bull. Yamagata Univ. Nat. Sci. 12 (1989), 151–156.

9. M. I. Munteanu, A note on doubly warped contact CR-submanifolds in trans-Sasakian manifolds, Acta Math. Hungar. 16 (2007), 121–126.

10. A. F. Yaliniz, A. Yildiz and M. Turan, On three-dimensional Lorentzian β-Kenmotsu manifolds, Kuwait J. Sci. Eng. 36 (2009), 1–14.

11. K. Yano and M. Kon, Structures on Manifolds, Series in Pure Mathematics, World Scientific, Singapore, 1984.

Institute of Mathematical Sciences (Received 17 06 2010) Faculty of Science, University of Malaya (Revised 13 03 2011 and 11 11 2011) 50603 Kuala Lumpur, Malaysia

siraj.ch@gmail.com

Department of Mathematics Abant Izzet Baysal University Bolu, Turkey

cenap.ozel@gmail.com

Department of Mathematics Aligarh Muslim University Aligarh-202002, India

viqarster@gmail.com