# Warped Product Submanifolds of LP-Sasakian Manifolds

Authors: S. K. Hui; S. Uddin; C. Özel; A. A. Mustafa
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

We study of warped product submanifolds, especially warped product hemi-slant submanifolds of LP-Sasakian manifolds. We obtain the results on the nonexistance or existence of warped product hemi-slant submanifolds and give some examples of LP-Sasakian manifolds. The existence of warped
product hemi-slant submanifolds of an LP-Sasakian manifold is also ensured by an interesting example.


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## Body

## 1. Introduction

The notion of warped product manifolds was introduced by Bishop and O'Neill [1], and later it was studied by many mathematicians and physicists. These manifolds are generalization of Riemannian product manifolds. The existence or nonexistence of warped product manifolds plays some important role in differential geometry as well as in physics.On the analogy of Sasakian manifolds, in 1989, Matsumoto [2] introduced the notion of LP-Sasakian manifolds. The same notion is also introduced by Mihai and Roşca [3] and obtained many interesting results. Later on, LP-Sasakian manifolds are also studied by several authors.The notion of slant submanifolds in a complex manifold was introduced and studied by Chen [4], which is a natural generalization of both invariant and anti-invariant submanifolds. Chen [4] also found examples of slant submanifolds of complex Euclidean spaces C2 and C4. Then, Lotta [5] has defined and studied the slant immersions of a Riemannian manifold into an almost contact metric manifold and proved some properties of such immersions. Also, Cabrerizo et al. [6] studied slant immersions of K-contact and Sasakian manifolds.In 1994, Papaghuic [7] introduced the notion of semi-slant submanifolds of almost Hermitian manifolds. Then, Cabrerizo et al. [8] defined and investigated semi-slant submanifolds of Sasakian manifolds. The idea of hemi-slant submanifolds was introduced by Carriazo as a particular class of bi-slant submanifolds and he called them anti-slant submanifolds [9]. Recently, these submanifolds were studied by Sahin for their warped products of Kähler manifolds [10]. Recently, Uddin [11] studied warped product CR-submanifolds of LP-Sasakian manifolds.The purpose of the present paper is to study the warped product hemi-slant submanifolds of LP-Sasakian manifolds. The paper is organized as follows. Section2 is concerned with some preliminaries. Section 3 deals with the study of warped and doubly warped product submanifolds of LP-Sasakian manifolds. In Section 4, we define hemi-slant submanifolds of LP-contact manifolds and investigate their warped products. Section 5 consists some examples of LP-Sasakian manifolds and their warped products.


## 2. Preliminaries

Ann-dimensional smooth manifold M is said to be an LP-Sasakian manifold [3] if it admits a (1,1) tensor field ϕ, a unit timelike contravariant vector field ξ, an 1-form η, and a Lorentzian metric g, which satisfy
(2.1)η(ξ)=-1,g(X,ξ)=η(X),ϕ2X=X+η(X)ξ,(2.2)g(ϕX,ϕY)=g(X,Y)+η(X)η(Y),∇¯Xξ=ϕX,(2.3)(∇¯Xϕ)(Y)=g(X,Y)ξ+η(Y)X+2η(X)η(Y)ξ,
where ∇¯ denotes the operator of covariant differentiation with respect to the Lorentzian metric g. It can be easily seen that, in an LP-Sasakian manifold, the following relations hold:
(2.4)ϕξ=0,η(ϕX)=0,rank⁡ϕ=n-1.
Again, we put
(2.5)Ω(X,Y)=g(X,ϕY)
for any vector fields X, Y tangent to M. The tensor field Ω(X,Y) is a symmetric (0,2) tensor field [2]. Also, since the vector field η is closed in an LP-Sasakian manifold, we have [2]
(2.6)(∇¯Xη)(Y)=Ω(X,Y),Ω(X,ξ)=0,
for any vector fields X and Ytangent to M.LetN be a submanifold of an LP-Sasakian manifold M with induced metric g and let ∇ and ∇⊥ be the induced connections on the tangent bundle TN and the normal bundle T⊥N of N, respectively. Then, the Gauss and Weingarten formulae are given by
(2.7)∇¯XY=∇XY+h(X,Y),(2.8)∇¯XV=-AVX+∇X⊥V,
for all X, Y∈TN and V∈T⊥N, where h and AV are second fundamental form and the shape operator (corresponding to the normal vector field V), respectively, for the immersion of N into M. The second fundamental form h and the shape operator AV are related by [12]
(2.9)g(h(X,Y),V)=g(AVX,Y)
for any X,Y∈TN and V∈T⊥NFor anyX∈TN, we may write
(2.10)ϕX=EX+FX,
where EX is the tangential component and FX is the normal component of ϕX.Also, for anyV∈T⊥N, we have
(2.11)ϕV=BV+CV,
where BV and CV are the tangential and normal components of ϕV, respectively. The covariant derivatives of the tensor fields E and F are defined as
(2.12)(∇¯XE)Y=∇XEY-E∇XY,(2.13)(∇¯XF)Y=∇X⊥FY-F∇XY
for any X,Y∈TN.Throughout the paper, we considerξ to be tangent to N. The submanifold N is said to be invariant if F is identically zero, that is, ϕX∈TN for any X∈TN. On the other hand, N is said to anti-invariant if E is identically zero, that is, ϕX∈T⊥N for any X∈TN.Furthermore, for a submanifold tangent to the structure vector fieldξ, there is another class of submanifolds which is called a slant submanifold. For each nonzero vector X tangent to N at x∈N, the angle θ(X), 0≤θ(X)(π/2) between ϕX and EX is called the slant angle or wirtinger angle. If the slant angle is constant then the submanifold is called aslant submanifold. Invariant and anti-invariant submanifolds are particular classes of slant submanifolds with slant angle θ=0 and θ=π/2, respectively. A slant submanifold is said to be proper slant if the slant angle θ lies strictly between 0 and π/2, that is, 0<θ<π/2 [6].Theorem 2.1 (see [13]).
LetN be a submanifold of a Lorentzian almost paracontact manifold M such that ξ is tangent to N. Then, N is slant submanifold if and only if there exists a constant λ∈[0,1] such that
(2.14)E2=λ(I+η⊗ξ).Furthermore, ifθ is the slant angle of N, then λ=cos2θ. Also from (2.14), we have
(2.15)g(EX,EY)=cos⁡2θ[g(X,Y)+η(X)η(Y)],(2.16)g(FX,FY)=sin⁡2θ[g(X,Y)+η(X)η(Y)]
for any X, Y tangent to N.The study of semi-slant submanifolds of almost Hermitian manifolds was introduced by Papaghuic [7], which was extended to almost contact manifold by Cabrerizo et al. [8]. The submanifold N is called semi-slant submanifold of M if there exist an orthogonal direct decomposition of TN as
(2.17)TN=D1⊕D2⊕{ξ},
where D1 is an invariant distribution, that is, ϕ(D1)=D1 and D2 is slant with slant angle θ≠0. The orthogonal complement of FD2 in the normal bundle T⊥N is an invariant subbundle of T⊥N and is denoted by μ. Thus, we have for a semi-slant submanifold
(2.18)T⊥N=FD2⊕μ.
For an LP-contact manifold this study is extended by Yüksel et al. [13].


## 3. Warped and Doubly Warped Products

The notion of warped product manifolds was introduced by Bishop and O’Neill [1]. They defined the warped product manifolds as follows.Definition 3.1.
Let(N1,g1) and (N2,g2) be two semi-Riemannian manifolds and f be a positive differentiable function on N1. Then, the warped product of N1 and N2 is a manifold, denoted by N1×fN2=(N1×N2,g), where
(3.1)g=g1+f2g2.
A warped product manifold N1×fN2 is said to be trivial if the warping function f is constant.
More explicitely, if the vector fieldsX and Y are tangent to N1×fN2 at (x,y), then
(3.2)g(X,Y)=g1(π1*X,π1*Y)+f2(x)g2(π2*X,π2*Y),
where πi(i=1,2) are the canonical projections of N1×N2 onto N1 and N2, respectively, and * stands for the derivative map.
LetN=N1×fN2 be a warped product manifold, which means that N1 and N2 are totally geodesic and totally umbilical submanifolds of N, respectively.For the warped product manifolds, we have the following result for later use [1].Proposition 3.2.
LetN=N1×fN2 be a warped product manifold. Then, (I)
∇XY∈TN1 is  the  lift  of ∇XY on N1,(II)
∇UX=∇XU=(Xlnf)U,(III)
∇UV=∇U′V-g(U,V)∇lnf,for any X,Y∈TN1 and U,V∈TN2, where ∇ and ∇′ denote the Levi-Civita connections on N and N2, respectively.Doubly warped product manifolds were introduced as a generalization of warped product manifolds by Ünal [14]. A doubly warped product manifold of N1 and N2, denoted as f2N1×f1N2 is endowed with a metric g defined as
(3.3)g=f22g1+f12g2,
where f1 and f2 are positive differentiable functions on N1 and N2, respectively.In this case formula (II) of Proposition3.2 is generalized as
(3.4)∇XZ=(Xln⁡f1)Z+(Zln⁡f2)X
for each X in TN1 and Z in TN2 [15].One has the following theorem for doubly warped product submanifolds of an LP-Sasakian manifold [11].Theorem 3.3.
LetN=f2N1×f1N2 be a doubly warped product submanifold of an LP-Sasakian manifold M where N1 and N2 are submanifolds of M. Then, f2 is constant and N2 is anti-invariant if the structure vector field ξ is tangent to N1, and f1 is constant and N1 is anti-invariant if ξ is tangent to N2.The following corollaries are immediate consequences of the above theorem.Corollary 3.4.
There does not exist a proper doubly warped product submanifold in LP-Sasakian manifolds.Corollary 3.5.
There does not exist a warped product submanifoldN1×fN2 of an LP-Sasakian manifold M such that ξ is tangent to N2.From the above theorem and Corollary3.5, we have only the remaining case is to study the warped product submanifold N1×fN2 with structure vector field ξ is tangent to N1.


## 4.  Warped Product Hemi-Slant Submanifolds

In this section, first we define hemi-slant submanifolds of an LP-contact manifold and then we will discuss their warped products.Definition 4.1.
A submanifoldN of an LP-contact manifold M is said to be a hemi-slant submanifold if there exist two orthogonal complementary distributions D1 and D2 satisfying: (i)
TN=D1⊕D2⊕〈ξ〉,(ii)
D1 is a slant distribution with slant angle θ≠π/2,(iii)
D2 is anti-invariant, that is, ϕD2⊆T⊥N.If μ is ϕ-invariant subspace of the normal bundle T⊥N, then in case of hemi-slant submanifold, the normal bundle T⊥N can be decomposed as
(4.1)T⊥N=FD1⊕FD2⊕μ.
Now, we discuss the warped product hemi-slant submanifolds of an LP-Sasakian manifoldM. If N=N1×fN2 be a warped product hemi-slant submanifold of an LP-Sasakian manifold M and Nθ and N⊥ are slant and anti-invariant submanifolds of an LP-Sasakian manifold M, respectively then their warped product hemi-slant submanifolds may be given by one of the following forms: (i)
N⊥×fNθ,(ii)
Nθ×fN⊥. In the following theorem, we start with the case (i).Theorem 4.2.
There does not exist a proper warped product hemi-slant submanifoldN=N⊥×fNθ of an LP-Sasakian manifold M such that ξ is tangent to Nθ, where N⊥ and Nθ are anti-invariant and proper slant submanifolds of M, respectively.Proof.
LetN=N⊥×fNθ be a proper warped product hemi-slant submanifold of an LP-Sasakian manifold M such that ξ is tangent to Nθ. Then, for any X∈TNθ and U∈TN⊥, we have
(4.2)(∇¯Xϕ)U=∇¯XϕU-ϕ∇¯XU.
By virtue of (2.3) and (2.7)–(2.11), it follows from (4.2) that
(4.3)η(U)X=-AFUX+∇X⊥FU-E∇XU-F∇XU-Bh(X,U)-Ch(X,U).
Using Proposition 3.2(II) in (4.3) and then equating the tangential components, we get
(4.4)η(U)X=AFUX+(Uln⁡f)EX+Bh(X,U).
Taking the inner product with EX in (4.4) and using the fact that X and EX are mutually orthogonal vector fields, then we have
(4.5)g(AFUX,EX)+(Uln⁡f)g(EX,EX)+g(Bh(X,U),EX)=0.
Using (2.9) and (2.15), we get
(4.6)-(Uln⁡f)cos⁡2θ‖X‖2=g(h(X,EX),FU)-g(h(X,U),FEX).
Replacing X by EX in (4.6) and using (2.14), we obtain
(4.7)-(Uln⁡f)cos⁡2θ‖X‖2=-g(h(X,EX),FU)+g(h(EX,U),EX).
Adding (4.6) and (4.7), we get
(4.8)(Uln⁡f)cos⁡2θ‖X‖2=0.
Since Nθ is proper slant and X is nonnull, (4.8) yields Ulnf=0, which shows that f is constant and consequently the theorem is proved.The second case is dealt with the following theorem.Theorem 4.3.
LetN=Nθ×fN⊥ be a warped product hemi-slant submanifold of an LP-Sasakian manifold M such that Nθ is a proper slant submanifold tangent to ξ and N⊥ is an anti-invariant submanifold of M. Then, (∇¯XF)(U) lies in the invariant normal subbundle μ, for each X∈TNθ and U∈TN⊥.Proof.
ConsiderN=Nθ×fN⊥ be a warped product hemi-slant submanifold of an LP-Sasakian manifold M such that Nθ is a proper slant submanifold tangent to ξ and N⊥ is an anti-invariant submanifold of M. Then, for any X∈TNθ and U∈TN⊥, we have
(4.9)∇¯XϕU=ϕ∇¯XU.
Using (2.7) and (2.8), we obtain
(4.10)-AFUX+∇X⊥FU=ϕ(∇XU+h(X,U)).
By virtue of (2.10), (2.11) and Proposition 3.2(II), it follows from (4.10) that
(4.11)-AFUX+∇X⊥FU=(Xln⁡f)EU+(Xln⁡f)FU+Bh(X,U)+Ch(X,U).
Equating the normal components, we obtain
(4.12)∇X⊥FU=(Xln⁡f)FU+Ch(X,U).
Taking the inner product of with FW1, for any W1∈TN⊥ in (4.13), we get
(4.13)g(∇X⊥FU,FW1)=(Xln⁡f)g(FU,FW1)+g(Ch(X,U),FW1)=(Xln⁡f)g(ϕU,ϕW1)+g(ϕh(X,U),ϕW1)=(Xln⁡f)g(U,W1).
Also for any X∈TNθ and U∈TN⊥, we have
(4.14)(∇¯XF)U=∇X⊥FU-(Xln⁡f)FU.
Taking the inner product FW1 for any W1∈TN⊥ in (4.14) and using (2.1) and (2.2), we derive
(4.15)g((∇¯XF)U,FW1)=g(∇X⊥FU,FW1)-(Xln⁡f)g(U,W1).
By virtue of (4.13), the above equation yields
(4.16)g((∇¯XF)U,FW1)=0,foranyX∈TNθ,U,W1∈TN⊥.
Similarly, if any W2∈TNθ, then from (2.13), we obtain
(4.17)g((∇¯XF)U,ϕW2)=g(∇X⊥FU,ϕW2)-g(F∇XU,ϕW2).
Since the product of tangential component with normal is zero and Nθ is a proper slant submanifold, we may conclude from (4.17) that
(4.18)g((∇¯XF)U,ϕW2)=0foranyX,W2∈TNθ,U∈TN⊥.
From (4.16) and (4.18), it follows that (∇¯XF)(U)∈μ and hence the proof is complete.


## 5. Examples on LP-Sasakian Manifolds

Example 5.1.
We consider a 3-dimensional manifoldM={(x,y,z)∈ℝ3:z>0}, where (x,y,z) are the standard coordinates in ℝ3. Let {E1,E2,E3} be a linearly independent global frame on M given by
(5.1)E1=ez∂∂x,E2=ez-ax∂∂y,E3=-∂∂z,
where a is a nonzero constant such that a≠1. Let g be the Lorentzian metric defined by g(E1,E3)=g(E2,E3=g(E1,E2)=0, g(E1,E1)=g(E2,E2)=1, g(E3,E3)=-1. Let η be the 1-form defined by η(U)=g(U,E3) for any U∈TM. Let θ be the (1,1) tensor field defined by ηE1=-E1, ϕE2=-E2, and ϕE3=0. Then, using the linearity of ϕ and g we have η(E3)=-1, ϕ2U=U+η(U)E3, and g(ϕU,ϕW)=g(U,W)+η(U)η(W) for any U,W∈TM. Thus for E3=ξ,(ϕ,ξ,η,g) defines a Lorentzian paracontact structure on M.
Let∇ be the Levi-Civita connection with respect to the Lorentzian metric g. Then, we have
(5.2)[E1,E2]=-aezE2,[E1,E3]=-E1,[E2,E3]=-E2.
Using Koszul formula for the Lorentzian metric g, we can easily calculate
(5.3)∇E1E1=-E3,∇E1E2=0,∇E1E3=-E1,∇E2E1=aezE2,∇E2E2=-aezE1-E3,∇E2E3=-E2,∇E3E1=0,∇E3E2=0,∇E3E3=0.
From the above computations, it can be easily seen that for E3=ξ, (ϕ,ξ,η,g) is an LP-Sasakian structure on M. Consequently, M3(ϕ,ξ,η,g) is an LP-Sasakian manifold.Example 5.2 (see [16]).
Letℝ5 be the 5-dimensional real number space with a coordinate system (x,y,z,t,s). Define
(5.4)η=ds-ydx-tdz,ξ=∂∂s,g=η⊗η-(dx)2-(dy)2-(dz)2-(dt)2,ϕ(∂∂x)=-∂∂x-y∂∂s,ϕ(∂∂y)=-∂∂y,ϕ(∂∂z)=-∂∂z-t∂∂s,ϕ(∂∂t)=-∂∂t,ϕ(∂∂s)=0,
the structure (ϕ,η,ξ,g) becomes an LP-Sasakian structure in ℝ5.Example 5.3.
Consider a 4-dimensional submanifoldN of ℝ7 with the cordinate system (x1,x2,…,x6,t) and the structure is defined as
(5.5)ϕ(∂∂xi)=∂∂xi,(i=1,2,3),ϕ(∂∂xj)=∂∂xj,(j=4,5,6),η=dt,ξ=-∂∂t,ϕ(∂∂t)=0,g=dxi2+dxj2+η⊗η.
Hence, the structure (ϕ,ξ,η,g) is an LP-contact structure on ℝ7. Now, for any α∈(0,π/2) and nonzero u and v, we define the submanifold N as follows:
(5.6)ω(u,v,α,t)=2(u,v,ucos⁡α,-vsin⁡α,usin⁡α,vcos⁡α,t).
Then, the tangent space TN is spanned by the vectors:
(5.7)e1=∂∂x1+cos⁡α∂∂x3+sin⁡α∂∂x5,e2=∂∂x2-sinα∂∂x4+cosα∂∂x6,e3=-usinα∂∂x3-vcosα∂∂x4+ucosα∂∂x5-vsinα∂∂x6,e4=-∂∂t.
Then the distributions Dθ=span{e1,e2,e4} is a slant distribution tangent to ξ=e4 and D⊥=span{e3} is an anti-invariant distribution, respectively. Let us denote by Nθ and N⊥ their integral submanifolds, then the metric g on N is given by
(5.8)g=2(du2+dv2)+(u2+v2)dα2.
Hence, the submanifold N=Nθ×fN⊥ is a hemi-slant-warped product submanifold of ℝ7 with the warping function f=(u2+v2).



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