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 structure F.

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 type M=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

Let M̅ 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]:

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:

g(ϕX,ϕY)=g(X,Y)-η(X)η(Y).
An almost contact metric manifold M̅ is called Sasakian if

(∇̅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

∇̅Xξ=-ϕX.

Let M 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

∇̅XY=∇XY+h(X,Y),∇̅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

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 any X∈TM, we write

ϕX=PX+FX,
where PX is the tangential component and FX is the normal component of ϕX.

Similarly, for any N∈T⊥M, we write

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 vector X 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 [<xref ref-type="bibr" rid="B5">3</xref>]).

Let M 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
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)

g(PX,PY)=cos2θ[g(X,Y)-η(X)η(Y)],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

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]:

∇XV=∇VX=(Xlnλ)V,
where X is tangent to N1 and V is tangent to N2.

Let M=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

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

∇XZ=(Xlnf1)Z+(Zlnf2)X
for each X in TN1 and Z in TN2 [7].

If neither f1 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 manifolds N1 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

∇Vξ=(ξlnf1)V+(Vlnf2)ξ.
Thus from (2.4), (2.5), (2.8), and (3.5), we get

∇̅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

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, if ξ is tangent to N2 and for any U∈TN1 we have

∇̅Uξ=(ξlnf2)U+(Ulnf1)ξ+h(U,ξ)=-PU-FU,
which gives

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 submanifolds M=N1×λN2 of a Sasakian manifold M̅, if the structure vector field ξ is tangent to the first factor N1 then second factor N2 is an anti-invariant submanifold. On the other hand the warped product M=N1×λN2 is trivial if the structure vector field ξ is tangent to N2.

To study the warped product submanifolds N1×λN2 with structure vector field ξ tangent to N1, we have obtained the following lemma.

Lemma 3.1 (see [<xref ref-type="bibr" rid="B13">12</xref>]).

Let M=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

ξlnλ=0,

AFZX=-th(X,Z),

g(h(X,Z),FY)=g(h(X,Y),FZ),

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

TM=𝒟⊕𝒟θ⊕{ξ}.

Similarly we say that M 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

T⊥M=FTM⊕μ,
where μ is an invariant subbundle of T⊥M.

From the above note, we see that for warped product submanifolds N1×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:

N⊥×λNθ,

Nθ×λN⊥.

The above two types of warped product pseudo-slant submanifolds are trivial if the structure vector field ξ 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 have

Theorem 4.1.

There do not exist the warped product Pseudo-slant submanifolds M=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 any X∈TNθ and Z∈TN⊥, we have
(∇̅Xϕ)Z=∇̅XϕZ-ϕ∇̅XZ.
Using (2.3), (2.5), (2.6), and the fact that ξ is tangent to N⊥, we obtain
-η(Z)X=-AFZX+∇X⊥FZ-P∇XZ-F∇XZ-th(X,Z)-fh(X,Z).
Comparing tangential and normal parts, we get
η(Z)X=AFZX+P∇XZ+th(X,Z)
Equation (4.5) takes the form on using (3.2) as
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
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
(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
(Zlnλ)cos4θ∥X∥2=-cos2θg(h(PX,Z),FX)+cos2θg(h(X,PX),FZ)
or
(Zlnλ)cos2θ∥X∥2=g(h(X,PX),FZ)-g(h(PX,Z),FX).
Adding equations (4.8) and (4.10), we get
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
(Zlnλ)cos2θ∥X∥2=0.
Since Nθ is proper slant and X is nonnull, then
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.

Let M=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.

As M=NT×λN⊥ is a warped product submanifold with ξ tangent to NT, then by (2.3),
(∇̅Xϕ)Z=0,
for any X∈TNT and Z∈TN⊥. Using this fact in the formula
(∇̅Uϕ)V=∇̅UϕV-ϕ∇̅UV
for each U,V∈TM̅, thus, we obtain
∇̅XϕZ=ϕ∇̅XZ.
Then from (2.5) and (2.6), we get
-AFZX+∇X⊥FZ=ϕ(∇XZ+h(X,Z)).
Which on using (2.8) and (2.9) yields
-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
∇X⊥FZ=(Xlnλ)FZ+fh(X,Z).
Taking the product in (4.19) with FW1 for any W1∈TN⊥, we get
g(∇X⊥FZ,FW1)=(Xlnλ)g(FZ,FW1)+g(fh(X,Z),FW1)
or
g(∇X⊥FZ,FW1)=(Xlnλ)g(ϕZ,ϕW1)+g(ϕh(X,Z),ϕW1).
Then from (2.2), we have
g(∇X⊥FZ,FW1)=(Xlnλ)g(Z,W1).
On the other hand, we have
(∇̅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
g((∇̅XF)Z,FW1)=0,
for any X∈TNT and Z,W1∈TN⊥.

Now, if W2∈TNT then using the formula (4.23), we get

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
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.

Acknowledgment

The authors are thankful to the referee for his valuable suggestion and comments which have improved this paper.

LottaA.Slant submanifolds in contact geometryCabrerizoJ. L.CarriazoA.FernándezL. M.FernándezM.Semi-slant submanifolds of a Sasakian manifoldCabrerizoJ. L.CarriazoA.FernándezL. M.FernándezM.Slant submanifolds in Sasakian manifoldsBishopR. L.O'NeillB.Manifolds of negative curvatureChenB.-Y.Geometry of warped product CR-submanifolds in Kaehler manifoldsKhanK. A.KhanV. A.Siraj-UddinWarped product submanifolds of cosymplectic manifoldsMunteanuM.-I.A note on doubly warped product contact CR-submanifolds in trans-Sasakian manifoldsSahinB.Nonexistence of warped product semi-slant submanifolds of Kaehler manifoldsHasegawaI.MihaiI.Contact CR-warped product submanifolds in Sasakian manifoldsBlairD. E.ÜnalB.Doubly warped productsUddinS.On warped product CR-submanifolds of Sasakian manifoldssubmitted