Characterizations of contact CR-warped product submanifolds of nearly Sasakian manifolds

submanifolds of nearly Sasakian manifolds

Akram Ali, Wan Ainun Mior Othman, Cenap Ozel

Abstract.In this paper, we study warped product contact CR-submanifolds of a nearly Sasakian manifold. We work out the characterizations in terms of tensor fields under which a contact CR-submanifold of a nearly Sasakian manifold reduces to a warped product submanifold.

M.S.C. 2010: 53C40, 53C42, 53B25.

Key words: CR-warped product; totally umbilical; totally geodesic; contact CR- submanifold; nearly Sasakian manifold.

1 Introduction

The geometry of warped product manifolds provide an excellent setting to model space time near black holes and bodies with large gravitational fields. B. Y. Chen [5]

initiated the study of CR-warped product submanifold in a Kaehler manifold. Many geometers studied geometric properties in terms canonical structure tensors T and F. B.Y.Chen [4] obtained characterization in terms T, i.e., a CR-subamnifold of a Kaehler manifold is a CR-product if and only if∇T = 0. Motivated by Chen’s paper, the result was extended for warped product submanifolds in almost contact setting by M. I. Munteanu [11]. Recently, V. A. Khan [10] studied contact CR-warped product submanifold of Kenmotsu manifold in terms tensor fields. Therefore its natural to see that how the non-triviality of covariant derivatives of T and F gives rise to a warped product submanifold. Moreover, I. Hesigawa and I. Mihai [6] worked out the necessary and sufficient conditions involving the shape operator of a contact CR- submanifold into a Sasakian manifold under which a submanifold is reduced to contact CR-warped product submanifold. In this paper, we study contact CR-submanifold in brief not in details as our aim is to discuss the warped products. We prove some existence results of contact CR-warped product of a nearly Sasakian manifold by its characterizations in terms of tensor fieldsT andF. We obtain some initial results on contact CR-submanifold of a nearly Sasakian manifold.

The paper is organized as follows: in Section 2, we review and collect some nec- essary results. In Section 3, we define a contact CR-submanifold in a nearly Sasakian

Balkan Journal of Geometry and Its Applications, Vol.21, No.2, 2016, pp. 9-20.∗

⃝c Balkan Society of Geometers, Geometry Balkan Press 2016.

manifolds and derive the integrability conditions and totally geodesic foliation of in- volving distributions. In Section 4, we obtain some results on characterizations of warped product submanifolds in terms of endomorphismsT andF.

2 Preliminaries

An odd (2n+ 1)−dimensional manifold Mfis said to be an almost contact metric manifoldif it admits an endomorphismφof its tangent bundleTM¯, a vector fieldξ (calledstructure vector field), andη(the dual 1-form), satisfying the following:

(2.1) φ²=−I+η⊗ξ, η(ξ) = 1, φ(ξ) = 0, η◦φ= 0, g(φU, φV) =g(U, V)−η(U)η(V), η(U) =g(U, ξ),

for anyU, V tangent toMf. Similarly, an almost contact metric manifoldMfis called Sasakian structureif

(∇eUφ)V =g(U, V)ξ−η(V)U and ∇eUξ=−φU.

Furthermore, an almost contact metric manifold is known to be a nearly Sasakian manifold if [16]

(2.2) (∇eUφ)V + (∇eVφ)U = 2g(U, V)ξ−η(V)U−η(U)V,

for any vector fieldsU, V onMf, where ∇e denotes the Riemannian connection with respect tog.

Now let M be a submanifold of Mf. Then we will denote by ∇ is a induced Riemannian connection on M and g is a Riemannian metric on Mf as well as the metric induced onM. LetT M andT^⊥M are Lie algebras of vector fields tangent to M and normal toM, respectively and∇^⊥ the induced connection onT^⊥M. Denote byF(M) the algebra of smooth functions onM and by Γ(T M) theF(M)-module of smooth sections ofT M overM. Then the Gauss and Weingarten formulas are given by

(2.3) ∇eUV =∇UV +h(U, V),

(2.4) ∇eUN =−A_NU+∇^⊥UN,

for any U, V ∈ Γ(T M) and N ∈ Γ(T^⊥M), where h and AN are the second fun- damental form and the shape operator (corresponding to a normal vector field N) respectively, for an immersion ofM intoMf. They are related as:

(2.5) g(h(U, V), N) =g(ANU, V).

Now for anyU ∈Γ(T M), we write

(2.6) φU=T U+F U,

whereT U andF Uare the tangential and the normal components ofφU, respectively.

Similarly, for anyN ∈ Γ(T^⊥M), we have φN =tN +f N, where tN (resp. f N)

are the tangential (resp. the normal) component of φN, respectively. Its easy to observe that for eachU, V ∈Γ(T M),we have g(T U, V) =−g(U, T V). The covariant derivatives of the endomorphismsφ, T andF are defined respectively as:

(2.7) (∇eUφ)V =∇eUφV −φ∇¯UV, (2.8) (∇eUT)V =∇UT V −T∇UV, (2.9) (∇eUF)V =∇^⊥UF V −F∇UV,

for all U, V ∈ Γ(TM¯) and N ∈ Γ(T^⊥M). A submanifold M of an almost contact metric manifoldMfis said to betotally umbilicalif

(2.10) h(U, V) =g(U, V)H,

whereH is the mean curvature vector ofM. Furthermore, if h(U, V) = 0, ∀U, V ∈ Γ(T M), thenM is said to betotally geodesicand ifH = 0, themM is calledminimal in Mf. Let us denote tangential and normal parts of (∇eUφ)V by PUV and QUV, respectively. Then it can be decomposed as:

(∇eUφ)V =PUV +QUV.

(2.11) PUV = (∇eUT)V −AF VU−th(U, V).

(2.12) QUV = (∇eUF)V +h(U, T V)−f h(U, V).

Similarly, let us denote the tangential and normal parts of (∇eUϕ)N by PUN and QUN, respectively. Thus we have decomposed as:

(∇eUφ)N =PUN+QUN, for anyN∈Γ(T^⊥M), wherePUN andQUN are defined as:

PUN = (∇eUt)N+T ANX−Af NU, QUN = (∇eUf)N+h(tN, U) +F ANU.

For anearly Sasakianstructure we have

(2.13) (a)PUV +PVU = 2g(U, V)ξ−η(V)U−η(U)V, (b)QUV +QVU = 0, for eachU, V ∈Γ(T M). It is straightforward to verify the following properties of P andQ, which will be further used:

(2.14)

(i) PU+VW =PUW+PVW, (ii) QU+VW =QUW+QVW, (iii) PU(W +Z) =PUW +PUZ, (iv) QU(W +Z) =QUW+QUZ, (v) g(PUV, W) =−g(V,PUW), (vi) g(QUV, N) =−g(V,PUN),

(vii) PUφV +QUφV =−φ(PUV +QUV).

































3 Contact CR-submanifolds

A submanifold M tangent to the structure vector field ξ of an almost contact met- ric manifold Mfis said to be invariant, if φ(T_xM) ⊆ (T_xM), and anti-invariant if φ(T_xM)⊂(T_x^⊥M) for eachx∈M.

Definition 3.1. A submanifold M tangent to structure vector field ξ of an almost contact metric manifoldMf is said to be acontact CR-submanifold, if there exist a pair of orthogonal distributionsDandD^⊥ such that:

(i) T M =D ⊕ D^⊥⊕< ξ >, where< ξ >is an 1-dimensional distribution spanned byξ,

(ii) Dis invariant, i.e.,φ(D)⊆ D,

(iii) D^⊥ is anti-invariant, i.e., φ(D^⊥)⊂(T^⊥M).

Letd₁ and d₂ be, respectively, the dimensions of the invariant distributionD and of the anti-invariant distribution D^⊥ of a contact CR-submanifold of a given almost contact metric manifoldMf. ThenM isinvariantifd₂= 0, andanti-invariant ifd1 = 0. It is called proper contact CR-subamnifold if neither d1 = 0 nor d2 = 0.

Moreover, if ν is an invariant subspace under φ of the normal bundle T^⊥M, then in case of contact CR-submanifold, the normal bundleT^⊥M can be decomposed as T^⊥M =FD^⊥⊕ν. Let us denote the orthogonal projections onDandD^⊥byP1and P2, respectively. Then, for anyU ∈Γ(T M), we define

(3.1) U =P1U+P2U+η(U)ξ,

whereP1U ∈Γ(D) andP2U ∈Γ(D^⊥.From (2.5), (2.6) and (3.1), we have T U =φP1U, F U =φP2U.

It is straightforward to observe that we obtain:

(i)T P₂= 0, (ii)F P₁= 0, (iii)t(T^⊥M)⊆ D^⊥, (iv)f(T^⊥M)⊂ν.

Lemma 3.1. Let M be a contact CR-submanifold of a nearly Sasakian manifoldMf. ThenD ⊕ξis integrable if and only if,

2g(∇XY, Z) =g(AφZX, φY) +g(AφZY, φX), for anyX, Y ∈Γ(D⊕< ξ >)andZ,∈Γ(D^⊥).

Proof. Let X, Y ∈ Γ(D⊕ < ξ >) and Z ∈ Γ(D^⊥). Then, by using (2.1) and (2.3), we have g(∇XY, Z) = g(∇eXY, Z) = g(φ∇eXY, φZ) + η(e∇XY)η(Z). Since η(Z) = 0, we getg(∇XY, Z) =g(φ∇eXY, φZ). Thus by (2.7), we obtaing(∇XY, Z) = g(∇eXφY, φZ)−g((∇eXφ)Y, φZ). From theGauss formulaand the structure equations (2.2), we getg(∇XY, Z) =g(h(X, φY), φZ) +g((∇eYφ)X, φZ)−2g(X, Y)g(ξ, φZ) + η(X)g(Y, φZ) +η(Y)g(X, φZ). Again, by using (2.3), (2.7) and (2.1), we obtain g(∇XY, Z) =g(h(X, φY), φZ) +g(h(Y, φX), φZ)−g(∇eYX, Z). Then, by a property of the Lie bracket and by the relation between the second fundamental form and the

shape operator, we get the desired result.

Lemma 3.2. The anti-invariant distribution D^⊥ of the contact CR-submanifoldM of a nearly Sasakian manifoldMfdefines a totally geodesic foliation if and only if

g(h(Z, φX), φW) +g(h(W, φX), φZ) = 0, for anyX ∈Γ(D⊕< ξ >)andZ, W ∈Γ(D^⊥).

Proof. LetZ, W ∈Γ(D^⊥) andX ∈Γ(D⊕< ξ >). By using (2.1) and (2.3), we have g(∇ZW, X) = g(φ∇¯ZW, φX). From (2.7), we get g(∇ZW, X) = g(∇eZφW, φX)− g((∇eZφ)W, φX). Thus (2.4) and (2.2) imply thatg(∇ZW, X) =−g(AφWZ, φX) + g((∇eWφ)Z, φX). Again, by using (2.7) and (2.4), we arrive at 2g(∇ZW, X) =

−g(A_φWZ, φX)−g(A_φZW, φX) +g([Z, W], X), which proves our assertion.

4 Warped product CR-submanifods

The warped products manifolds were introduced by Bishop and O’Neill [2]. They defined the warped products as follows: letM₁andM₂be two Riemannian manifolds with the corresponding Riemannian metrics g₁ and g₂ respectively, and let f be a positive differentiable function onM₁. Consider the product manifoldM₁×M₂with its projectionsπ₁ : M₁×M₂ → M₁ and π₂ : M₁×M₂ → M₂. Then their warped product manifoldM =M₁×fM₂is the Riemannian manifoldM₁×M₂= (M₁×M₂, g) equipped with the Riemannian structureggiven by

g(X, Y) =g1(π1∗X, π2∗Y) + (f◦π1)²g2(π2∗X, π2∗Y),

for any vector fieldsX, Y tangent toM, where∗is the symbol for the tangent maps.

A warped product manifoldM =M₁×fM₂is said to betrivialor simplyRiemannian product manifold, if the warping functionfis constant. We recall below several results for warped product manifolds.

Lemma 4.1. [2] LetM =M1×M2 be a warped product manifold with the warping functionf. Then

(i) ∇XY ∈Γ(T M₁)is the lift of ∇XY on M₁, (ii) ∇XZ=∇ZX = (Xlnf)Z,

(iii) ∇ZW =∇^′ZW −g(Z, W)∇lnf,

for eachX, Y ∈ Γ(T M₁) andZ, W ∈Γ(T M₂), where ∇lnf is the gradient of lnf, defined byg(∇lnf, X) =Xlnf, where∇ and∇^′ denote the Levi-Civita connections onM andM₂, respectively.

The non-existence of warped product contact CR-submanifolds of the typeM = M_⊥×fMT andM =MT ×fM_⊥ was proved in [16], when the structure vector field ξis tangent toMT and to M_⊥, respectively.

Lemma 4.2. [16] Let a CR-warped product submanifold MT ×f M_⊥ of a nearly Sasakian manifold Mf be, such that M_T and M_⊥ are invariant and anti-invariant submanifolds ofMf, respectively. Then we have

(i) ξlnf = 0,

(ii) g(h(X, Y), φZ) = 0,

(iii) g(h(X, Z), φZ) =−(φXlnf) +η(X)||Z||², (iv) g(h(ξ, Z), φZ) =−||Z||²,

for eachX, Y ∈Γ(T MT)andZ∈Γ(T M_⊥).

Before stating the characterization theorem, we first discuss several properties of contact CR-warped product submanifolds of a nearly Sasakian manifold. For any X, Y ∈Γ(T M_T), the properties (2.11) and (2.14)(i) imply that

(4.1) (∇eXT)Y + (∇eYT)X = 2th(X, Y) + 2g(X, Y)ξ−η(X)Y −η(Y)X.

Since, MT is totally geodesic in M, (∇eXT)Y completely lies onMT and its second fundamental identically vanishes. Thus, by comparing the component which is tan- gent toMT in formula (4.1), we obtainth(X, Y) = 0, which implies thath(X, Y)∈ν, and

(4.2) (∇eXT)Y + (∇eYT)X = 2g(X, Y)ξ−η(X)Y −η(Y)X.

If we setY =ξin the above equation, we derive

(∇eXT)ξ+ (∇eξT)X = 2g(X, ξ)ξ−η(X)ξ−η(ξ)X

= 2η(X)ξ−η(X)ξ−X.

From (2.8), it can be easily seen that

(∇eξT)X =−(∇eXT)ξ+η(X)ξ−X

(4.3) =T∇Xξ+η(X)ξ−X.

Lemma 4.3. Let M =MT ×fM_⊥ be a CR-warped product submanifold of a nearly Sasakian manifoldMf. In this case, we have:

(∇eZT)X = (T Xlnf)Z, (∇eUT)Z =g(P2U, Z)T∇lnf, for eachU ∈Γ(T M),X ∈Γ(T M_T)andZ∈Γ(T M_⊥).

Proof. From Lemma 4.1(ii), we have ∇XZ = ∇ZX = (Xlnf)Z, for any X ∈ Γ(T MT) andZ ∈Γ(T M_⊥). Using the above equation in (2.8), we obtain

(∇eZT)X=∇ZT X−T∇ZX= (T Xlnf)Z−(Xlnf)T Z= (T Xlnf)Z, which is the first part of the Lemma. On the other hand, from (2.8), for each Z ∈ Γ(T M_⊥) andU ∈Γ(T M), we derive

(∇eUT)Z =∇UT Z−T∇UZ=−T∇UZ.

From property (3.1), the above equation can be expressed as:

(∇eUT)Z=−T∇P₁UZ−T∇P₂UZ−η(U)T∇ξZ.

Then Lemma 4.1(ii) andT Z= 0 yield

(∇eUT)Z= (P1Ulnf)T Z−T∇P₂UZ−η(U)(ξlnf)T Z=−T∇P₂UZ.

Thus by the property(ii) of Lemma 4.1, we finally get

(∇eUT)Z=−T∇^′P₂UZ+g(P₂U, Z)T∇lnf =g(P₂U, Z)T∇lnf,

which is the second part of the Lemma, which completes the proof.

Theorem 4.4. Let M be a contact CR-submanifold of a nearly Sasakian manifold Mf, with both the distributions integrable. ThenM is locally isometric to a CR-warped product if and only if

(4.4) (∇eUT)U = (T P1U µ)P2U+||P1U||²ξ+||P2U||²T∇µ−η(U)P1U, or, equivalently,

(∇eUT)V + (∇eVT)U = (T P1Vlnf)P2U+ (T P1Ulnf)P2V + 2g(P1U, P1V)ξ

(4.5) + 2g(P₂U, P₂V)T∇lnf −η(U)P₁V −η(V)P₁U,

for each U, V ∈ Γ(T M) and for any C^∞-function µ on M, with Zµ = 0 for each Z∈Γ(D^⊥).

Proof. Let M be a contact CR-warped product submanifold of Mf. Then for any U ∈Γ(T M) with the property (3.1), we can write:

(∇eUT)U = (∇eP₁UT)P1U+ (∇eP₂UT)P1U +η(U)(∇eξT)P1U +(∇eUT)P₂U+η(U)(e∇UT)ξ.

From (4.2) and (4.3), for the contact CR-warped product case, from Lemma 4.3 we obtain

(∇eUT)U =||P₁U||²ξ−η(P₁U)P₁U + (T P₁Ulnf)P₂U+||P₂U||²T∇lnf +η(U)T∇P₁Uξ−η(U)T∇P₁Uξ.

Since lnf =µ, we get

(∇eUT)U = (T P1U µ)P2U+||P1U||²ξ+||P2U||²T∇µ−η(U)P1U,

which is the desired result (4.4). Furthermore, by replacingU byU+V in the above equation and using the linearity of vector fields, we get (4.5). Conversely, suppose thatM is a contact CR-submanifold of Mf, with both distributions integrable onM such that (4.5) holds for aC^∞-functionµonM, withZµ = 0 for eachZ ∈Γ(D^⊥).

Then for anyX, Y ∈Γ(D⊕< ξ >) and the fact thatP2X = 0, by using in (4.5) we obtain

(4.6) (e∇XT)Y + (∇eYT)X = 2g(X, Y)ξ−η(X)Y −η(Y)X.

SinceMfis a nearly Sasakian manifold, then from (2.11) and (2.13)(a) we derive (4.7) (∇eXT)Y + (∇eYT)X = 2th(X, Y) + 2g(X, Y)ξ−η(X)Y −η(Y)X.

Thus from (4.6) and (4.7), we obtain th(X, Y) = 0. This means that h(X, Y) ∈ ν. However, D⊕ < ξ > is integrable, and hence from Lemma 3.1, we infer that g(∇XY, Z) = 0 for eachZ ∈Γ(D^⊥), which means that∇XY ∈Γ(D⊕< ξ >), i.e., the leaves of the distribution D⊕ < ξ > are totally geodesic in M. On the other hand, for anyZ, W ∈Γ(D^⊥) and from (4.5), we obtain

(4.8) (∇eZT)W+ (∇eWT)Z= 2g(Z, W)T∇µ.

From (2.11) and (2.13)(a), we get

(4.9) (∇eZT)W + (∇eWT)Z =AF WZ+AF ZW + 2th(Z, W) + 2g(Z, W)ξ.

Thus from (4.8) and (4.9), it follows that

A_{F W}Z+A_{F Z}W = 2g(Z, W)T∇µ−2th(Z, W)−2g(Z, W)ξ.

Taking the inner product withφX and using (2.5) and (2.6), we obtain g(h(Z, φX), φW) +g(h(W, φX), φZ) = 2g(Z, W)g(T∇µ, φX).

From (2.3), we find that

g(∇eZφX, φW) +g(∇eWϕX, φZ) = 2g(Z, W)g(T∇µ, φX).

Using the orthogonality of vector fields, we derive

g(∇eZφW, φX) +g(∇eWϕZ, φX) =−2g(Z, W)g(T∇µ, φX).

Then by using the property of covariant derivative (2.7), we find that

−2g(Z, W)g(φ∇µ, φX) =g((e∇Zφ)W+ (∇eWφ)Z, φX) +g(φ∇eZW, φX) +g(φ∇eWZ, φX).

Applying the property of nearly Sasakian structure in the first term of the right hand side of the last equation and (2.1), we get

g(∇eZW, X) +g(∇eWZ, X) =−2g(Z, W)g(∇µ, X)−2g(Z, W)η(∇µ)η(X), which implies that

g(∇ZW+∇WZ, X) =−2g(Z, W)g(∇µ, X)−2g(Z, W)(ξlnf)η(X).

SinceD^⊥ is assumed to be integrable andξlnf = 0, we obtain (4.10) g(∇ZW, X) =−g(Z, W)g(∇µ, X).

LetM_⊥ be a leaf ofD^⊥ and leth^⊥ be the second fundamental form of the immersion ofM_⊥ intoM. Then for anyX ∈Γ(D⊕< ξ >), we find that

(4.11) g(h^⊥(Z, W), X) =g(∇ZW, X).

From (4.10) and (4.11), it can be easily seen that

g(h^⊥(Z, W), X) =−g(Z, W)g(∇µ, X), which means that

h^⊥(Z, W) =−g(Z, W)∇µ.

From the above equation, we conclude that M_⊥ is totally umbilical in M with the mean curvature vector satisfies H = −∇µ. Now we can prove that H is parallel corresponding to the normal connection D of M_⊥ in M. To this aim, we consider Y ∈Γ(D⊕< ξ >) andZ ∈Γ(D^⊥),

g(DZ∇λ, Y) =g(∇Z∇λ, Y) =Zg(∇λ, Y)−g(∇λ,∇ZY)

=Z(Y(λ))−g(∇λ,[Z, Y])−g(∇λ,∇YZ) =Y(Zλ) +g(∇Y∇λ, Z) = 0.

SinceZ(λ) = 0 for allZ∈Γ(D^⊥), we get that∇Y∇λ∈Γ(D⊕< ξ >). This means that the leaves ofD^⊥ areextrinsicspheres inM. Hence, by a result of Hiepko[7], we conclude thatM is a warped product submanifold, which completes the proof of the

theorem.

Lemma 4.5. Let M =MT ×f M_⊥ be a contact CR-warped product submanifold of a nearly Sasakian manifoldMf. Then, for all X, Y ∈Γ(T M_T)andZ, W ∈Γ(T M_⊥), the following hold true:

(i)g((∇eXF)Y, φW) = 0,

(ii)g((∇eZF)X, φW) =−Xlnf g(Z, W), (iii)g((e∇ξF)Z, φW) = 0.

Proof. Let M be a contact CR-warped product submanifold of a nearly Sasakian manifoldMf. Then for anyX, Y ∈Γ(T MT) andW ∈Γ(T M_⊥), we have

g((e∇XF)Y, φW) =−g(F∇XY, φW) =−g(∇XY, W).

The first result directly follows, by using the property given by the above equation, and henceMT istotally geodesicinM. Again, for anyX ∈Γ(T MT) andZ, W ∈Γ(T M_⊥), we infer

((e∇ZF)X, φW) =−g(F∇ZX, φW).

Using Lemma 4.1(ii), we obtain the second result. Now from (2.12), we have g((∇eξF)Z, φW) =g(QξZ+f h(ξ, Z), φW) =g(QξZ, φW), for anyZ, W ∈Γ(T M_⊥). Then from property (2.14) (v)-(vii), we yield

g((∇eξF)Z, φW) =g(φξ,PZW) = 0,

which is the last result of the Lemma, which concludes the proof.

Theorem 4.6. Let M be a contact CR-submanifold of a nearly Sasakian manifold Mfwith its invariant and anti-invariant distributions integrable. ThenM is a contact CR-warped product if and only if

g((∇eUF)V + (∇eVF)U, φW) =g(QP₁UP2V, φW) +g(QP₁VP2U, φW)

−(P1U µ)g(P2V, W)−(P1V µ)g(P2U, W),

for eachU, V ∈Γ(T M)andW ∈Γ(T M_⊥), whereµis aC^∞-function onM satisfying Zµ= 0 for eachZ∈Γ(T M_⊥).

Proof. LetM =MT×fM_⊥be a contact CR-warped product submanifold of a nearly Sasakian manifold Mf. Then for any U, V ∈ Γ(T M) and W ∈ Γ(T M_⊥), from the property (3.1), we obtain

(4.12)

g((∇eUF)V, φW) =g((∇eP1UF)P1V, φW) +g((∇eP2UF)P1V, φW) +η(U)g((e∇ξF)P₁V, φW) +g((∇eP1UF)P₂V, φW)

+g((e∇P2UF)P₂V, φW) +η(U)g((∇eξF)P₂V, φW) +η(V)g((∇eUF)ξ, φW).

Using (2.12) and Lemma 4.5, we get

(4.13) g((∇eUF)V, φW) =g(QP2UP₂V, φW) +g(QP1UP₂V, φW)

−(P₁V µ)g(P₂U, W).

By the polarization identity, we infer

(4.14) g((∇eVF)U, φW) =g(QP₂VP2U, φW) +g(QP₁VP2U, φW)

−(P₁U µ)g(P₂V, W).

Thus from (4.13), (4.14) and (2.14)(ii) we get the desired result. Now for the converse, suppose thatM be a CR-submanifold of a nearly Sasakian manifoldMfwith integrable distributionsDandD^⊥. Then, for anyX, Y ∈Γ(D⊕< ξ >),we obtain

g((∇eXF)Y + (∇eYF)X, φW) = 0, Using the fact thatP2X =P2Y = 0 in (4.12), from (2.10), we get

2g(f h(X, Y), φW)−g(h(X, T Y) +h(Y, T X), φW) = 0, which implies that

g(h(X, T Y), φW) +g(h(Y, T X), φW) = 0.

By applying (2.3) and (2.6), we obtain

g(∇eXϕY, φW) +g(∇eYφX, φW) = 0.

From the covariant derivative property (2.7) and (2.3), it is easily seen that g((e∇Xφ)Y + (∇eYφ)X, φW) +g(∇XY +∇YX, W) = 0.

Taking account of this observation, from the nearly Sasakian manifold property, we obtain g(∇XY +∇YX, W) = 0, which implies that∇XY +∇YX ∈Γ(D⊕< ξ >).

Since D⊕ < ξ > is assumed to be a integrable, then ∇XY ∈ Γ(D⊕ < ξ >), i.e., D⊕< ξ >is parallel. In other words, the leaves ofD⊕< ξ >are totally geodesic in M. Now, for anyX∈Γ(D⊕< ξ >) andZ, W ∈Γ(D^⊥), it follows from (4.12) that

g((∇eXF)Z, φW) +g((∇eZF)X, φW) =g(QXZ, φW)−(Xµ)g(Z, W).

From (2.12) and (2.9), we obtain

g(QXZ+f h(X, Z), φW)−g(F∇ZX, φW) =g(QXZ, φW)−(Xµ)g(Z, W), which implies that g(F∇ZX, φW) = (Xµ)g(Z, W). After simplifications, we get g(∇ZW, X) =−(Xµ)g(Z, W).Since D^⊥ is assumed to be integrable, consider a leaf M_⊥ ofD^⊥. If∇^′ denotes the induced Riemannian connection onM_⊥, andh^⊥ is the second fundamental form of the immersionM_⊥intoM, then in view of (2.3), the last equation can be written as:

g(h^⊥(Z, W), X) =−(Xµ)g(Z, W).

Using the property of the gradient function, we get

g(h^⊥(Z, W), X) =−g(Z, W)g(∇µ, X),

which implies that h^⊥(Z, W) =−g(Z, W)∇µ.This means thatM_⊥istotally umbilical in M with mean curvature vector H = −∇µ. Now we can easily prove that H is parallel corresponding to the normal connectionD ofM_⊥ in M similar to Theorem 4.4, which means that the leaves of D^⊥ are extrinsic spheres in M. Then, by the result of Hiepko [7],M is a warped product submanifold, which completes the proof

of the Theorem.

Acknowledgements. The authors would like express their appreciation to the referees for their comments and valuable suggestions. This research was partially supported by grants UMRG 278-14AFR and FP 074-2015A (University of Malaya).

References

[1] N. S. Al-Luhaibi, F. R. Al-Solamy, V. A. Khan,CR-warped product submanifolds of nearly Kaehler manifolds, J. Korean. Math. Soc. 46, 5 (2009), 979–995.

[2] R. L .Bishop, B. O’Neill, Manifold of negative curvature, Trans. Amer. Math.

Soc. 145 (1969), 1–49.

[3] D. E. Blair,Contact manifolds in Riemannian geometry, Lecture Notes in Math- ematics 509, Springer-Verlag, New York 1976.

[4] B. Y. Chen,CR-subamnifold of a Kaehler manifold, I. J. Differential Geom. 16, 1-2 (1981), 305–322.

[5] B. Y. Chen,Geometry of warped product CR-submanifold in Kaehler manifolds, Monatsh. Math. 133, 3 (2001), 177–195.

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

[7] S. Hiepko,Eine innere Kennnzeichung der verzerrten Produkte, Math. Ann. 244, 3 (1979), 209–215.

[8] S. T. Hong, Warped product and black holes, Niuvo Cimento Soc. Ital. Fis. B 120, 10-11 (2005), 1221–1234.

[9] V. A. Kham, K. A. Khan and S. Uddin, CR-warped product submanifolds in a Kaehler manifolds, Southeast. Asian. Bull. Math. 33, 5 (2009), 865–874.

[10] V. A. Khan and M. Shuaib, Some warped product submanifolds of a Kenmotsu manifold, Bull. Korean Math. Soc. 51, 3 (2014), 863–881.

[11] M. I. Munteanu, Warped product contact CR-submanifold of Sasakian space forms, Publ. Math. Debrecen, 66 (6) (2005), 75–120.

[12] M. I. Munteanu, A note on doubly warped product contact CR-submanifold in trans-Sasakian manifolds, Acta. Mtah. Hungar. 16, 1-2 (2007), 121–126.

[13] S. Nolker,Isometric immersions of warped products, Diff. Geom. Appl. 6 (1996), 1–30.

[14] B. O’Neil, Semi-Riemannian geometry with applications to relativity, Pure and Applied Mathematics 103, Academic. Press, Inc., New York 1983.

[15] S. Uddin, S. H. Kon, M. A. Khan, K. Singh, Warped product semi-invariant submanifolds of nearly cosymplectic manifold, Math. Prob. Engg, Article ID- 230374 (2011).

[16] S. Uddin, A. Mustafa, C. Ozel, On warped product submanifolds of nearly Sasakian manifolds, To appear in Bull. Iranian Math. Soc. (2014).

Author’s address:

Akram Ali, Wain Ainun Mior Othman

Institute of Mathematical Sciences, Faculty of Science, University of Malaya, 50603 Kuala Lumpur, Malaysia.

E-mail: akramali133@gmail.com, wanainun@um.edu.my Cenap Ozel

Department of Mathematics, Dokuz Eylul University, Izmir, Turkey;

Department of Mathematics, Faculty of Science, King Abdul Aziz University, Jeddah, Saudi Arabia.

E-mail: cenap.ozel@gmail.com