# Some Results on Warped Product Submanifolds of a Sasakian Manifold

Authors: Siraj Uddin; V. A. Khan; Huzoor H. Khan
Journal: International Journal of Mathematics and Mathematical Sciences
 (2010)

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## Abstract

We study warped product Pseudo-slant submanifolds of Sasakian manifolds. We prove a theorem for the existence of warped product submanifolds of a Sasakian manifold in terms of the canonical structureF.


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## Body

## 1. Introduction

The notion of slant submanifold of almost contact metric manifold was introduced by Lotta [1]. Latter, Cabrerizo et al. investigated slant and semislant submanifolds of a Sasakian manifold and obtained many interesting results [2, 3].The notion of warped product manifolds was introduced by Bishop and O'Neill in [4]. Latter on, many research articles appeared exploring the existence or nonexistence of warped product submanifolds in different spaces (cf. [5–7]). The study of warped product semislant submanifolds of Kaehler manifolds was introduced by Sahin [8]. Recently, Hasegawa and Mihai proved that warped product of the type N⊥×λNT in Sasakian manifolds is trivial where NT and N⊥ are ϕ-invariant and anti-invariant submanifolds of a Sasakian manifold, respectively [9].In this paper we study warped product submanifolds of a Sasakian manifold. We will see in this paper that for a warped product of the typeM=N1×λN2, if N1 is any Riemannian submanifold tangent to the structure vector field ξ of a Sasakian manifold M̅ then N2 is an anti-invariant submanifold and if ξ is tangent to N2 then there is no warped product. Also, we will show that the warped product of the type M=N⊥×λNθ of a Sasakian manifold M̅ is trivial and that the warped product of the type NT×λN⊥ exists and obtains a result in terms of canonical structure.


## 2. Preliminaries

LetM̅ be a (2m+1)-dimensional manifold with almost contact structure (ϕ,ξ,η) defined by a (1,1) tensor field ϕ, a vector field ξ, and the dual 1-form η of ξ, satisfying the following properties [10]:(2.1)ϕ2=-I+η⊗ξ,ϕξ=0,η∘ϕ=0,η(ξ)=1.
There always exists a Riemannian metric g on an almost contact manifold M̅ satisfying the following compatibility condition:(2.2)g(ϕX,ϕY)=g(X,Y)-η(X)η(Y).
An almost contact metric manifold M̅ is called Sasakian if(2.3)(∇̅Xϕ)Y=g(X,Y)ξ-η(Y)X
for all X,Y in TM̅, where ∇̅ is the Levi-Civita connection of g on M̅. From (2.3), it follows that(2.4)∇̅Xξ=-ϕX.LetM be submanifold of an almost contact metric manifold M̅ with induced metric g and if ∇ and ∇⊥ are the induced connections on the tangent bundle TM and the normal bundle T⊥M of M, respectively, then Gauss and Weingarten formulae are given by(2.5)∇̅XY=∇XY+h(X,Y),(2.6)∇̅XN=-ANX+∇X⊥N,
for each X,Y∈TM and N∈T⊥M, where h and AN are the second fundamental form and the shape operator (corresponding to the normal vector field N), respectively, for the immersion of M into M̅. They are related as(2.7)g(h(X,Y),N)=g(ANX,Y),
where g denotes the Riemannian metric on M̅ as well as the one induced on M.For anyX∈TM, we write(2.8)ϕX=PX+FX,
where PX is the tangential component and FX is the normal component of ϕX.Similarly, for anyN∈T⊥M, we write(2.9)ϕN=tN+fN,
where tN is the tangential component and fN is the normal component of ϕN. We shall always consider ξ to be tangent to M. The submanifold M is said to be invariant if F is identically zero, that is, ϕX∈TM for any X∈TM. On the other hand, M is said to be anti-invariant if P is identically zero, that is, ϕX∈T⊥M, for any X∈TM.For each nonzero vectorX tangent to M at x, such that X is not proportional to ξ, we denote by θ(X) the angle between ϕX and PX.M is said to be slant [3] if the angle θ(X) is constant for all X∈TM-{ξ} and x∈M. The angle θ is called slant angle or Wirtinger angle. Obviously, if θ=0,M is invariant and if θ=π/2,M is an anti-invariant submanifold. If the slant angle of M is different from 0 and π/2 then it is called proper slant.A characterization of slant submanifolds is given by the following.Theorem 2.1 (see [3]).
LetM be a submanifold of an almost contact metric manifold  M̅, such that  ξ∈TM. Then  M is slant if and only if there exists a constant  δ∈[0,1] such that
(2.10)P2=δ(-I+η⊗ξ).
Furthermore, in such case, if θ is slant angle, then δ=cos2θ.Following relations are straightforward consequences of (2.10)(2.11)g(PX,PY)=cos2θ[g(X,Y)-η(X)η(Y)],(2.12)g(FX,FY)=sin2θ[g(X,Y)-η(X)η(Y)]
for any X,Y tangent to M.


## 3. Warped and Doubly Warped Product Manifolds

Let(N1,g1) and (N2,g2) be two Riemannian manifolds and λ a positive differentiable function on N1. The warped product of N1 and N2 is the Riemannian manifold N1×λN2=(N1×N2,g), where(3.1)g=g1+λ2g2.
A warped product manifold N1×λN2 is said to be trivial if the warping function λ is constant. We recall the following general formula on a warped product [4]:(3.2)∇XV=∇VX=(Xlnλ)V,
where X is tangent to N1 and V is tangent to N2.LetM=N1×λN2 be a warped product manifold then N1 is totally geodesic and N2 is totally umbilical submanifold of M, respectively.Doubly warped product manifolds were introduced as a generalization of warped product manifolds by Ünal [11]. A doubly warped product manifold of N1 and N2, denoted as f2N1×f1N2 is the manifold N1×N2 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 (3.2) is generalized as(3.4)∇XZ=(Xlnf1)Z+(Zlnf2)X
for each X in TN1 and Z in TN2 [7].If neitherf1 nor f2 is constant we have a nontrivial doubly warped product M=f2N1×f1N2. Obviously in this case both N1 and N2 are totally umbilical submanifolds of M.Now, we consider a doubly warped product of two Riemannian manifoldsN1 and N2 embedded into a Sasakian manifold M̅ such that the structure vector field ξ is tangent to the submanifold M=f2N1×f1N2. Consider ξ is tangent to N1, then for any V∈TN2 we have(3.5)∇Vξ=(ξlnf1)V+(Vlnf2)ξ.
Thus from (2.4), (2.5), (2.8), and (3.5), we get(3.6)∇̅Vξ=(ξlnf1)V+(Vlnf2)ξ+h(V,ξ)=-PV-FV.
On comparing tangential and normal parts and using the fact that ξ,V, and PV are mutually orthogonal vector fields, (3.6) implies that(3.7)Vlnf2=0,ξlnf1=0,h(V,ξ)=-FV,PV=0.
This shows that f2 is constant and N2 is an anti-invariant submanifold of M̅, if the structure vector field ξ is tangent to N1.Similarly, ifξ is tangent to N2 and for any U∈TN1 we have(3.8)∇̅Uξ=(ξlnf2)U+(Ulnf1)ξ+h(U,ξ)=-PU-FU,
which gives(3.9)Ulnf1=0,ξlnf2=0,PU=0,h(U,ξ)=-FU.
That is, f1 is constant and N1 is an anti-invariant submanifold of M̅.Note.
From the above conclusion we see that for warped product submanifoldsM=N1×λN2of a Sasakian manifoldM̅, if the structure vector fieldξis tangent to the first factorN1then second factorN2is an anti-invariant submanifold. On the other hand the warped productM=N1×λN2is trivial if the structure vector fieldξis tangent toN2.To study the warped product submanifoldsN1×λN2 with structure vector field ξ tangent to N1, we have obtained the following lemma.Lemma 3.1 (see [12]).
LetM=N1×λN2 be a proper warped product submanifold of a Sasakian manifold  M̅, with  ξ∈TN1, where  N1 and  N2 are any Riemannian submanifolds of  M̅. Then   (i)
ξlnλ=0,(ii)
AFZX=-th(X,Z),(iii)
g(h(X,Z),FY)=g(h(X,Y),FZ),(iv)
g(h(X,Z),FW)=g(h(X,W),FZ) for any X,Y∈TN1 and Z,W∈TN2.


## 4. Warped Product Pseudoslant Submanifolds

The study of semislant submanifolds of almost contact metric manifolds was introduced by Cabrerizo et.al. [2]. A semislant submanifold M of an almost contact metric manifold M̅ is a submanifold which admits two orthogonal complementary distributions 𝒟 and 𝒟θ such that 𝒟 is invariant under ϕ and 𝒟θ is slant with slant angle θ≠0, that is, ϕ𝒟=𝒟 and ϕZ makes a constant angle θ with TM for each Z∈𝒟θ. In particular, if θ=π/2, then a semislant submanifold reduces to a contact CR-submanifold. For a semislant submanifold M of an almost contact metric manifold, we have(4.1)TM=𝒟⊕𝒟θ⊕{ξ}.Similarly we say thatM is an pseudo-slant submanifold of M̅ if 𝒟 is an anti-invariant distribution of M, that is, ϕ𝒟⊆T⊥M and 𝒟θ is slant with slant angle θ≠0. The normal bundle T⊥M of an pseudo-slant submanifold is decomposed as(4.2)T⊥M=FTM⊕μ,
where μ is an invariant subbundle of T⊥M.From the above note, we see that for warped product submanifoldsN1×λN2 of a Sasakian manifold M̅, one of the factors is an anti-invariant submanifold of M̅. Thus, if the manifolds Nθ and N⊥ are slant and anti-invariant submanifolds of Sasakian manifold M̅, then their possible warped product pseudo-slant submanifolds may be given by one of the following forms:(a)
N⊥×λNθ,(b)
Nθ×λN⊥.The above two types of warped product pseudo-slant submanifolds are trivial if the structure vector fieldξ is tangent to Nθ and N⊥, respectively. Here, we are concerned with the other two cases for the above two types of warped product pseudo-slant submanifolds N⊥×λNθ and Nθ×λN⊥ when ξ is in TN⊥ and in TNθ, respectively.For the warped product of the type(a), we haveTheorem 4.1.
There do not exist the warped product Pseudo-slant submanifoldsM=N⊥×Nλθ where N⊥ is an anti-invariant and  Nθ is a proper slant submanifold of a Sasakian manifold  M̅ such that  ξ is tangent to  N⊥.Proof.
For anyX∈TNθ and Z∈TN⊥, we have
(4.3)(∇̅Xϕ)Z=∇̅XϕZ-ϕ∇̅XZ.
Using (2.3), (2.5), (2.6), and the fact that ξ is tangent to N⊥, we obtain
(4.4)-η(Z)X=-AFZX+∇X⊥FZ-P∇XZ-F∇XZ-th(X,Z)-fh(X,Z).
Comparing tangential and normal parts, we get
(4.5)η(Z)X=AFZX+P∇XZ+th(X,Z)
Equation (4.5) takes the form on using (3.2) as
(4.6)η(Z)X=AFZX+(Zlnλ)PX+th(X,Z).
Taking product with PX, the left hand side of the above equation is zero using the fact that X and PX are mutually orthogonal vector fields. Then
(4.7)0=g(AFZX,PX)+(Zlnλ)g(PX,PX)+g(th(X,Z),PX).
Using (2.7), (2.11) and the fact that ξ is tangent to N⊥, we get
(4.8)(Zlnλ)cos2θ∥X∥2=g(h(X,Z),FPX)-g(h(X,PX),FZ).
As θ≠π/2, then interchanging X by PX and taking account of (2.10), we obtain
(4.9)(Zlnλ)cos4θ∥X∥2=-cos2θg(h(PX,Z),FX)+cos2θg(h(X,PX),FZ)
or
(4.10)(Zlnλ)cos2θ∥X∥2=g(h(X,PX),FZ)-g(h(PX,Z),FX).
Adding equations (4.8) and (4.10), we get
(4.11)2(Zlnλ)cos2θ∥X∥2=g(h(X,Z),FPX)-g(h(PX,Z),FX).
The right hand side of the above equation is zero by Lemma 3.1(iv); then
(4.12)(Zlnλ)cos2θ∥X∥2=0.
Since Nθ is proper slant and X is nonnull, then
(4.13)Zlnλ=0.
In particular, for Z=ξ∈TN⊥, Lemma 3.1 (i) implies that ξlnλ=0. This means that λ is constant on N⊥. Hence the theorem is proved.Now, the other case is dealt with in the following theorem.Theorem 4.2.
LetM=NT×λN⊥ be a warped product submanifold of a Sasakian manifold  M̅ such that  NT is an invariant submanifold tangent to  ξ and  N⊥ is an anti-invariant submanifold of  M̅. Then  (∇̅XF)Z lies in the invariant normal subbundle for each  X∈TNT and  Z∈TN⊥.Proof.
AsM=NT×λN⊥ is a warped product submanifold with ξ tangent to NT, then by (2.3),
(4.14)(∇̅Xϕ)Z=0,
for any X∈TNT and Z∈TN⊥. Using this fact in the formula
(4.15)(∇̅Uϕ)V=∇̅UϕV-ϕ∇̅UV
for each U,V∈TM̅, thus, we obtain
(4.16)∇̅XϕZ=ϕ∇̅XZ.
Then from (2.5) and (2.6), we get
(4.17)-AFZX+∇X⊥FZ=ϕ(∇XZ+h(X,Z)).
Which on using (2.8) and (2.9) yields
(4.18)-AFZX+∇X⊥FZ=P∇XZ+F∇XZ+th(X,Z)+fh(X,Z).
From the normal components of the above equation, formula (3.2) gives
(4.19)∇X⊥FZ=(Xlnλ)FZ+fh(X,Z).
Taking the product in (4.19) with FW1 for any W1∈TN⊥, we get
(4.20)g(∇X⊥FZ,FW1)=(Xlnλ)g(FZ,FW1)+g(fh(X,Z),FW1)
or
(4.21)g(∇X⊥FZ,FW1)=(Xlnλ)g(ϕZ,ϕW1)+g(ϕh(X,Z),ϕW1).
Then from (2.2), we have
(4.22)g(∇X⊥FZ,FW1)=(Xlnλ)g(Z,W1).
On the other hand, we have
(4.23)(∇̅XF)Z=∇X⊥FZ-F∇XZ.
Taking the product in (4.23) with FW1 for any W1∈TN⊥ and using (4.22), (2.2), (3.2), and the fact that ξ is tangential to NT, we obtain that
(4.24)g((∇̅XF)Z,FW1)=0,
for any X∈TNT and Z,W1∈TN⊥.
Now, ifW2∈TNT then using the formula (4.23), we get
(4.25)g((∇̅XF)Z,ϕW2)=g(∇X⊥FZ,ϕW2)-g(F∇XZ,ϕW2).
As NT is an invariant submanifold, then ϕW2∈TNT for any W2∈TNT, thus using the fact that the product of tangential component with normal is zero, we obtain that
(4.26)g((∇̅XF)Z,ϕW2)=0,
for any X,W2∈TNT and Z∈TN⊥. Thus from (4.24) and (4.26), it follows that (∇̅XF)Z∈μ. Thus the proof is complete.



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*Source: 743074-2010-03-10.xml*