# Warped Product Semi-Invariant Submanifolds in Almost Paracontact Riemannian Manifolds

Authors: Mehmet Atçeken
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

We show that there exist no proper warped product semi-invariant submanifolds in almost paracontact Riemannian manifolds such that totally geodesic submanifold and totally umbilical submanifold of the warped product are invariant and anti-invariant, respectively. Therefore, we consider warped product semi-invariant submanifolds in the formN=N⊥×fNT
by reversing two factor manifolds NT
and N⊥. We prove several fundamental properties of warped product semi-invariant submanifolds in an almost paracontact Riemannian manifold and establish a general inequality for an arbitrary warped product semi-invariant submanifold. After then, we investigate warped product semi-invariant submanifolds in a general almost paracontact Riemannian manifold which satisfy the equality case of the inequality.


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## Body

## 1. Introduction

It is well known that the notion of warped products plays some important role in differential geometry as well as physics. The geometry of warped product was introduced by Bishop and O'Neill [1]. Many geometers studied different objects/structures on manifolds endowed with an warped product metric (see [2–6]).Recently, Chen has introduced the notion of CR-warped product in Kaehlerian manifolds and showed that there exist no proper warped product CR-submanifolds in the formN=N⊥×fNT in Kaehlerian manifolds. Therefore, he considered warped product CR-submanifolds in the form N=NT×fN⊥ which is called CR-warped product, where NT is an invariant submanifold, and N⊥ is an anti-invariant submanifold of Kaehlerian manifold M̅ (see [2, 7, 8]). Analogue results have been obtained for Sasakian ambient as the odd dimensional version of Kaehlerian manifold by Hasegawa and Mihai in [3] and Munteanu in [9].Almost paracontact manifolds and almost paracontact Riemannian manifolds were defined and studied byŞato̅ [10]. After then, many authors studied invariant and anti-invariant submanifolds of the almost paracontact Riemannian manifold M with the structure (F,g,ξ,η), when ξ is tangent to the submanifold, and ξ is not tangent to the submanifold [11].We note that CR-warped products in Kaehlerian manifold are corresponding warped product semi-invariant submanifolds in almost paracontact Riemannian manifolds. In this paper, we showed that there exist no warped product semi-invariant submanifolds in the formN=NT×fN⊥ in contrast to Kaehlerian manifolds (see Theorem 3.1). So, from now on we consider warped product semi-invariant submanifolds in the form N=N⊥×fNT, where N⊥ is an anti-invariant submanifold, and NT is an invariant submanifold of an almost paracontact Riemannian manifold M by reversing the two factor manifolds NT and N⊥, and it simply will be called warped product semi-invariant submanifold in the rest of this paper (see Example 3.3 and Theorem 3.4).


## 2. Preliminaries

Although there are many papers concerning the geometry of semi-invariant submanifolds of almost paracontact Riemannian manifolds (see [11–13]), there is no paper concerning the geometry of warped product semi-invariant submanifolds of almost paracontact Riemannian manifolds in literature so far. So the purpose of the present paper is to study warped product semi-invariant submanifolds in almost paracontact Riemannian manifolds. We first review basic formulas and definitions for almost paracontact Riemannian manifolds and their submanifolds, which we shall use for later.LetM be an (m+1)-dimensional differentiable manifold. If there exist on M a (1,1) type tensor field F, a vector field ξ, and 1-form η satisfying(2.1)F2=I-η⊗ξ,η(ξ)=1,
then M is said to be an almost paracontact manifold, where ⊗, the symbol, denotes the tensor product. In the almost paracontact manifold, the following relations hold good:(2.2)Fξ=0,η∘F=0,rank(F)=m.An almost paracontactmanifoldM is said to be an almost paracontact metric manifold if Riemannian metric g on M satisfies(2.3)g(FX,FY)=g(X,Y)-η(X)η(Y),η(X)=g(X,ξ)
for all X,Y∈Γ(TM) [14], where Γ(TM) denotes the differentiable vector field set on M. From (2.2) and (2.3), we can easily derive the relation(2.4)g(FX,Y)=g(X,FY).An almost paracontact metric manifold is said to be an almost paracontact Riemannian manifold with(F,g,ξ,η)-connection if ∇̅F=0 and ∇̅η=0, where ∇̅ denotes the connection on M. Since F2=I-η⊗ξ, the vector field ξ is also parallel with respect to ∇̅ [11, 13].In the rest of this paper, let us suppose thatM is an almost paracontact Riemannian manifold with structure (F,g,ξ,η)-connection.LetM be an almost paracontact Riemannian manifold, and let N be a Riemannian manifold isometrically immersed in M. For each x∈N, we denote by Dx the maximal invariant subspace of the tangent space TxN of N. If the dimension of Dx is the same for all x in N, then Dx gives an invariant distribution D on N.A submanifoldN in an almost paracontact Riemannian manifold is called semi-invariant submanifold if there exists on N a differentiable invariant distribution D whose orthogonal complementary D⊥ is an anti-invariant distribution, that is, F(D⊥)⊂TN⊥, where TN⊥ denotes the orthogonal vector bundle of TN in TM. A semi-invariant submanifold is called anti-invariant (resp., invariant) submanifold if dim(Dx)=0 (resp., dim(Dx⊥)=0). It is called proper semi-invariant submanifold if it is neither invariant nor anti-invariant submanifold.A semi-invariant submanifoldN of an almost paracontact Riemannian manifold M is called a Riemannian product if the invariant distribution D and anti-invariant distribution D⊥ are totally geodesic distributions in N. The geometry notion of the semi-invariant submanifolds has been studied by many geometers in the various type manifolds. Authors researched the fundamental properties of such submanifolds (see references).LetN1 and N2 be two Riemannian manifolds with Riemannian metrics g1 and g2, respectively, and f a differentiable function on N1. Consider the product manifold N1×N2 with its projection π1:N1×N2→N1 and π2:N1×N2→N2. The warped product manifold N=N1×fN2 is the manifold N1×N2 equipped with the Riemannian metric tensor such that(2.5)g(X,Y)=g1(π1*X,π1*Y)+f2(π1(x))g2(π2*X,π2*Y)
for any X,Y∈Γ(TN), where * is the symbol for the tangent map. Thus we have g=g1+f2g2, where f is called the warping function of the warped product. The warped product manifold N=N1×fN2 is characterized by the fact that N1 and N2 are totally geodesic and totally umbilical submanifolds of N, respectively. Hence Riemannian products are special classes of the warped products [4].In this paper, we define and study a new class of warped product semi-invariant submanifolds in an almost paracontact Riemannian manifolds, namely, we investigate the class of warped product semi-invariant submanifolds, and we establish the fundamental theory of such submanifolds.Now, letN be an isometrically immersed submanifold in an almost paracontact Riemannian manifold M. We denote by ∇ and ∇̅ the Levi-Civita connections on N and M, respectively. Then the Gauss and Weingarten formulas are, respectively, defined by(2.6)∇̅XY=∇XY+h(X,Y),∇̅XV=-AVX+∇X⊥V
for any  X,Y∈Γ(TN), V∈Γ(TN⊥), where ∇⊥ is the connection in the normal bundle TN⊥,h is the second fundamental form of N, and AV is the shape operator. The second fundamental form h and the shape operator A are related by(2.7)g(AVX,Y)=g(h(X,Y),V).Now, letN be a differentiable manifold, and we suppose that N is an isometrically immersed submanifold in almost paracontact Riemannian manifold M. We denote by g the metric tensor of M as well as that induced on N. For any vector field X tangent to N, we put(2.8)FX=tX+nX,
where tX and nX denote the tangential and normal components of FX, respectively. For any vector field V normal to N, we also put(2.9)FV=BV+CV,
where BV and CV denote the tangential and normal components of FV, respectively. The submanifold N is said to be invariant if n is identically zero, that is, F(TN)=TN. On the other hand, N is said to be anti-invariant submanifold if t is identically zero, that is, F(TN)⊂(TN⊥).We note that for any invariant submanifoldN of an almost paracontact Riemannian manifold M, if ξ is normal to N, then the induced structure from the almost paracontact structure on N is an almost product Riemannian structure whenever t is nontrivial. If ξ is tangent to N, then the induced structure on N is an almost paracontact Riemannian structure.Furthermore, we say thatN is a semi-invariant submanifold if there exist two orthogonal distributions D1 and D2 such that(1)
TN splits into the orthogonal direct sum TN=D1⊕D2;(2)
the distributionD1 is invariant, that is, F(D1)⊆D1;(3)
the distributionD2 is anti-invariant, that is, F(D2)⊆TN⊥.Given any submanifoldN of an almost paracontact Riemannian manifold M, from (2.4) and (2.8) we have(2.10)g(tX,Y)=g(X,tY),g(nX,V)=g(X,BV)
for any X,Y∈Γ(TN), V∈Γ(TN⊥).From now on we suppose that the vector fieldξ is tangent to N.We recall the following general lemma from [1] for later use.Lemma 2.1.
LetN=N1×fN2 be a warped product manifold with warping function f, then one has (1)
∇XY∈Γ(TN1)for   eachX,Y∈Γ(TN1),(2)
∇XZ=∇ZX=X(lnf)Z,for  eachX∈Γ(TN1),Z∈Γ(TN2),(3)
∇ZW=∇ZN2W-g(Z,W)gradf/f,for  eachZ,W∈Γ(TN2), where ∇ and ∇N2 denote the Levi-Civita connections on N and N2, respectively.LetN be a semi-invariant submaniold of an almost paracontact Riemannian manifold M. We denote by the invariant distribution D and anti-invariant distribution D⊥. We also denote the orthogonal complementary of F(D⊥) in TN⊥ by ν, then we have direct sum(2.11)TN⊥=F(D⊥)⊕ν.
We can easily see that ν is an invariant subbundle with respect to F.


## 3. Warped Product Semi-Invariant Submanifolds in an Almost Paracontact Riemannian Manifold

Useful characterizations of warped product semi-invariant submanifolds in almost paracontact Riemannian manifolds are given the following theorems.Theorem 3.1.
IfN=NT×fN⊥ is a warped product semi-invariant submanifold of an almost paracontact Riemannian manifold M such that NT is an invariant submanifold and N⊥ is an anti-invariant submanifold of M, then N is a usual Riemannian product.Proof.
Letξ be normal to N. Taking into account that h is symmetric and using (2.3), (2.6), (2.7), and considering Lemma 2.1(2), for X∈Γ(TNT) and Z,W∈Γ(TN⊥), we have
(3.1)g(∇XZ,W)=g(∇ZX,W)=g(∇̅ZX,W)=g(F∇̅ZX,FW)+η(∇̅ZX)η(W),Xln(f)g(Z,W)=g(∇̅ZFX,FW)=g(h(Z,FX),FW)=g(∇̅FXZ,FW)=g(∇̅FXFZ,W)=-g(AFZFX,W)=-g(h(FX,W),FZ)=-g(∇̅WFX,FZ)=-g(∇̅WX,Z)=-Xln(f)g(W,Z),
which implies that Xln(f)=0.
Ifξ is tangent to N, then ξ can be written as follows:
(3.2)ξ=ξ1+ξ2,
where ξ1∈Γ(TNT) and ξ2∈Γ(TN⊥). Since ∇̅Xξ=0, from the Gauss formulae, we have
(3.3)h(X,ξ)=0,∇Xξ=0
for any X∈Γ(TN). Considering Lemma 2.1(2), we get
(3.4)∇Zξ1=ξ1(lnf)Z=0,∇Xξ2=X(lnf)ξ2=0,∇ξ2ξ1=∇ξ1ξ2=ξ1(lnf)ξ2=0
for any X∈Γ(TNT) and Z∈Γ(TN⊥). If ξ2 is identically zero, then from Lemma 2.1 we have
(3.5)∇Zξ1=∇ξ1Z=ξ1(lnf)Z=0,∇Xξ1∈Γ(TNT).
It follows that the warping function f is a constant and N is usual Riemannian product. Hence the proof is complete.
If the warping function is constant, then the metric on the “second” factor could be modified by an homothety, and hence, the warped product becomes a direct product.Now, we give two examples for almost paracontact Riemannian manifold and their submanifolds in the formN=N⊥×fNT to illustrate our results. Firstly, we construct an almost paracontact metric structure on ℝ2n+1 (see Example 3.2) and after give an example which is concerning its submanifold (see Example 3.3).Example 3.2.
Let(3.6)ℝ2n+1={(x1,x2,…,xn,y1,y2,…,yn,t)∣xi,yi,t∈ℝ,i=1,2,…,n}.
The almost paracontact Riemannian structure (F,g,ξ,η) is defined on ℝ2n+1 in the following way:
(3.7)F(∂∂xi)=∂∂yi,F(∂∂yi)=∂∂xi,F(∂∂t)=0,ξ=∂∂t,η=dt.
If Z=λi(∂/∂xi)+μi(∂/∂yi)+ν(∂/∂t)∈T(ℝ2n+1), then we have
(3.8)g(Z,Z)=∑i=1nλi2+∑i=1nμi2+ν2.
From this definition, it follows that
(3.9)g(Z,ξ)=η(Z)=ν,g(FZ,FZ)=g(Z,Z)-η2(Z),Fξ=0,η(ξ)=1
for an arbitrary vector field Z. Thus (ℝ2n+1,F,g,ξ,η) becomes an almost paracontact Riemannian manifold, where g and {∂/∂xi,∂/∂yi,∂/∂t} denote usual inner product and standard basis of T(ℝ2n+1), respectively.Example 3.3.
LetN be a submanifold in ℝ5 with coordinates (x1,x2,y1,y2,t) given by
(3.10)x1=vcosθ,x2=vsinθ,y1=vcosβ,y2=vsinβ,t=2u.
It is easy to check that the tangent bundle of N is spanned by the vectors
(3.11)Z1=cosθ∂∂x1+sinθ∂∂x2+cosβ∂∂y1+sinβ∂∂y2,Z2=-vsinθ∂∂x1+vcosθ∂∂x2,Z3=-vsinβ∂∂y1+vcosβ∂∂y2,Z4=2∂∂t.
We define the almost paracontact Riemannian structure of ℝ5 by
(3.12)F(∂∂xi)=-∂∂xi,F(∂∂yi)=∂∂yi,F(∂∂t)=0,η=12dt.
Then with respect to the almost paracontact Riemannian structure F of ℝ5, the space F(TN) becomes
(3.13)FZ1=-cosθ∂∂x1-sinθ∂∂x2+cosβ∂∂y1+sinβ∂∂y2,FZ2=vsinθ∂∂x1-vcosθ∂∂x2,FZ3=-vsinβ∂∂y1+vcosβ∂∂y2,FZ4=0.
Since FZ1 and FZ4 are orthogonal to N  and FZ2, FZ3 are tangent to N, D and D⊥ can be taken subspace sp{Z1,Z4} and subspace sp{Z2,Z3}, respectively, where ξ can be taken as Z4 for FZ4=0 and η(Z4)=1. Furthermore, the metric tensor of N is given by
(3.14)g=2(du2+dv2)+v2(dθ2+dβ2)=2gN⊥+v2gNT.
Thus N is a warped product semi-invariant submanifold with dimensional 5 of almost paracontact manifold ℝ5 with warping function f=v2.Now, letN=N⊥×fNT be a warped product semi-invariant submanifold of an almost paracontact Riemannian manifold M, where N⊥ is an anti-invariant submanifold, and NT is an invariant submanifold of M. If we denote the Levi-Civita connections on M and N by ∇̅ and ∇, respectively, by using (2.6) and (2.8), we have(3.15)∇̅XFY=F∇̅XY,∇XtY+h(X,tY)-AnYX+∇X⊥nY=t(∇XY)+n(∇XY)+Bh(X,Y)+Ch(X,Y)
for any X,Y∈Γ(TN). Taking into account the tangential and normal components of (3.15), respectively, we have(3.16)(∇Xt)Y=AnYX+Bh(X,Y),(3.17)(∇Xn)Y=Ch(X,Y)-h(X,tY),
where the derivatives of t and n are, respectively, defined by(3.18)(∇Xt)Y=∇XtY-t(∇XY),(∇Xn)Y=∇X⊥nY-n(∇XY).
Next, we are going to investigate the geometric properties of the leaves of the warped product semi-invariant submanifolds in an almost paracontact Riemannian manifold.Theorem 3.4.
LetN be a warped product semi-invariant submanifold of an almost paracontact Riemannian manifold M. Then the invariant distribution D and the anti invariant distribution D⊥ are always integrable.Proof.
From (3.16) and considering lemma 2.1(1), we have
(3.19)∇̅XFU=F∇̅XU,Xln(f)tU+h(X,tU)=Xln(f)tU+Bh(X,U)+Ch(X,U)
for any X∈Γ(D⊥)  and U∈Γ(D). From the tangential and normal components of (3.19), respectively, we arrive at
(3.20)Bh(X,U)=0,(3.21)h(X,tU)=Ch(X,U).
By using (3.16) and (3.20) we get
(3.22)AnXU=-X(lnf)tU.
Furthermore, by using the Gauss-Weingarten formulas and taking into account that D⊥ is totally geodesic in N and it is anti-invariant in M, by direct calculations, it is easily to see that
(3.23)AnYX=-Bh(X,Y),
which is also equivalent to
(3.24)AnYX=AnXY
for any X,Y∈Γ(D⊥). Moreover, using (2.4) and (2.7) and making use of A being self-adjoint, we obtain
(3.25)g(AnXY,Z)=g(h(Y,Z),nX)=g(∇̅ZY,FX)=g(∇̅ZFY,X)=-g(AnYZ,X)=-g(AnYX,Z),
which gives us
(3.26)AnXY=-AnYX
for any X,Y∈Γ(D⊥) and Z∈Γ(TN). Thus from (3.24) and (3.26), we arrive at
(3.27)AnXY=0,Bh(X,Y)=0
for any  X,Y∈Γ(D⊥). Furthermore, by using (2.6), (2.8), and (2.9) and considering Lemma 2.1(3), we have
(3.28)h(U,tV)+∇UtV=F(∇UV)+Fh(V,U)=F(∇U′V-g(V,U)gradff)+Bh(V,U)+Ch(V,U)=t(∇U′V)-g(V,U)n(gradff)+Bh(V,U)+Ch(V,U)
for any  V,U∈Γ(D), where ∇′ denote the Levi-Civita connection on D. Taking into account  the normal and tangential components of (3.28), respectively, we have
(3.29)h(U,tV)=-g(U,V)n(gradff)+Ch(U,V),(3.30)∇U′tV-g(tV,U)gradff=t(∇U′V)+Bh(V,U).
From (3.29), we can easily see that
(3.31)h(U,tV)=h(V,tU)
for any U,V∈Γ(D). Finally, by using (3.17) and (3.31), we have
(3.32)n([V,U])=n(∇VU-∇UV)=∇V⊥nU-(∇VnU)-∇U⊥nV+(∇Un)V=(∇Un)V-(∇Vn)U=Ch(U,V)-h(U,tV)-Ch(V,U)+h(V,tU)=0
for any V,U∈Γ(D), that is, [V,U]∈Γ(D).
In the same way, making use of (3.16) and (3.27) for any X,Y∈Γ(D⊥), we conclude that
(3.33)t([X,Y])=t(∇XY-∇YX)=∇XtY-(∇Xt)Y-∇YtX+(∇Yt)X=(∇Yt)X-(∇Xt)Y=AnXY-AnYX=0
that is, [X,Y]∈Γ(D⊥). So we obtain the desired result.Since the distributionsD and D⊥ are integrable, we denote the integral manifolds of D and D⊥ by NT and N⊥, respectively.Now, the following theorem characterizes (warped product or Riemannian product) semi-invariant submanifolds in almost paracontact manifolds.Theorem 3.5.
LetN be a submanifold of an almost paracontact Riemannian manifold M. Then N is a semi-invariant submanifold if and only nt=0.Proof.
Let us assume thatN is a semi-invariant submanifold of an almost paracontact Riemannian manifold M and by Q and P; we denote the projection operators on subspaces Γ(D⊥) and Γ(D), respectively, then we have
(3.34)P+Q=I,P2=P,Q2=Q,PQ=QP=0.
Moreover, by using (2.1), (2.8), and (2.9), if ξ is tangent to N, then we get
(3.35)X-η(X)ξ=t2X+BnX,ntX+CnX=0,(3.36)tBV+BCV=0,nBV+C2V=V
for any X∈Γ(TN) and V∈Γ(TN⊥). On the other hand, if ξ is normal to N, then (3.35) and (3.36) become, respectively,
(3.37)X=t2X+BnX,ntX+CnX=0,V-η(V)ξ=nBV+C2V,tBV+BCV=0.
From (2.8), we have
(3.38)FX=FPX+FQX,tX+nX=tPX+tQX+nPX+nQX
for any  X∈Γ(TN). From the tangential and normal components, we have
(3.39)tX=tPX+tQX,nX=nPX+nQX.
Since D is invariant and D⊥ is anti-invariant, we get
(3.40)nP=0,Qt=0.
We have
(3.41)tP=t
by virtue of Q=I-P. Now by using the right-hand side to the second equation of (3.35) and using (3.40) and (3.41), we conclude that
(3.42)nt=0,
which is also equivalent to
(3.43)Cn=0.
Conversely, for a submanifoldN of an almost paracontact Riemannian manifold M, we suppose that nt=0. For any vector fields tangent X to N and V normal to N, by using (2.4) and (3.43), we have
(3.44)g(X,FV)=g(FX,V),g(X,BV)=g(nX,V),g(X,FBV)=g(FnX,V),g(X,tBV)=g(CnX,V)=0
for all X∈Γ(TN). So we have g(tBV,X)=0. Since X,tBV∈Γ(TN), it implies tB=0 which is also equivalent to BC=0 from (3.36). Since Fξ=0, we get  tξ=nξ=0. So, from (3.35) and (3.36), we conclude
(3.45)t3=t,C3=C.
Now, if we put
(3.46)P=t2,Q=I-P,
then we can derive that P+Q=I,P2=P,Q2=Q,andPQ=QP=0 which show that Q and P are orthogonal complementary projection operators and define complementary distributions D⊥ and D, respectively, where D and D⊥ denote the distributions which are belong to subspaces TNT and TN⊥, respectively. From (3.42), (3.45), and (3.46) we can derive
(3.47)tP=t,tQ=0,QtP=0,nP=0.
These equations show that the distribution D is an invariant and the distribution D⊥ is an anti-invariant. The proof is complete.Theorem 3.6.
LetN be a semi-invariant submanifold of an almost paracontact Riemannian manifold M. Then N is a warped product semi-invariant submanifold if and only if the shape operator of N satisfies
(3.48)AFXZ=-X(μ)FZ,X∈Γ(D⊥),Z∈Γ(D)
for some function μ on N satisfying W(μ)=0, W∈Γ(D).Proof.
We suppose thatN is a warped product semi-invariant submanifold in an almost paracontact Riemannian manifold M. Then from (3.22), we have
(3.49)AFXZ=-X(lnf)FZ
for any X∈Γ(D⊥) and Z∈Γ(D). Since f is the only function on N⊥, we can easily see that W(lnf)=0 for all W∈Γ(D).
Conversely, let us assume thatN is a semi-invariant submanifold in an almost paracontact Riemannian manifold M satisfying
(3.50)AFXZ=X(μ)FZ,X∈Γ(D⊥),Z∈Γ(D)
for some function μ on N satisying  W(μ)=0  for all  W∈Γ(D). Since the ambient space M is an almost paracontact Riemannian manifold and making use of (2.4) and (3.27), we arrive at
(3.51)g(∇XY,FZ)=g(∇̅XY,FZ)=g(∇̅XFY,Z)=-g(AFYX,Z)=0
for any X,Y∈Γ(D⊥) and Z∈(D). Thus the anti-invariant distribution D⊥ is totally geodesic in N. In the same way, making use of ∇̅ being Levi-Civita connection and (3.22), we have
(3.52)g(∇ZW,X)=g(∇̅ZW,X)=-g(∇̅ZX,W)=-g(∇̅ZFX,FW)=g(AFXZ,FW)=X(μ)g(Z,W)
for any Z,W∈Γ(D) and X∈Γ(D⊥), where μ=ln(1/f). Since the invariant distribution D of semi-invariant submanifold N is always integrable (Theorem 3.4) and W(μ)=0, for each W∈Γ(TNT), which implies that the integral manifold of D is an extrinsic sphere in N, that is, it is a totally umbilical submanifold and its mean curvature vector field is non-zero and parallel, thus we know that N is a Riemannian warped product N⊥×fNT, where N⊥ and NT denote the integral manifolds of the distributions of D⊥ and D, respectively, and f is the warping function. So we obtain the desired result.In the rest of this section, we are going to obtain an inequality for the squared norm of the second fundamental form by means of the warping function for warped product semi-invariant submanifolds of an almost paracontact Riemannian manifold. Now, we recall that semi-invariantN is said to be mixed geodesic (resp., D-geodesic and D⊥-geodesic) submanifold if the second fundamental form h of N satisfies h(X,Z)=0, X∈Γ(D) and Z∈Γ(D⊥) (resp., h(X,Y)=0, X,Y∈Γ(D) and h(Z,W)=0, Z,W∈Γ(D⊥)).Now, we are going to give the following lemma for later use.Lemma 3.7.
LetN=N⊥×fNT be a warped product semi-invariant submanifold of an almost paracontact Riemannian manifold M. Then one has(1)
g(h(D⊥,D⊥),FD⊥)=0,(2)
g(h(Z,W),FX)=-X(lnf)g(tZ,W), Z,W∈Γ(D), X∈Γ(D⊥),(3)
g(h(X,Z),FY)=0, for any X,Y∈Γ(D⊥) and Z∈Γ(D),(4)
g(h(D,FD),FD⊥)=0 if and only if N=N⊥×fNT is a usual Riemannian product, where D and D⊥ denote the leaves of NT and N⊥, respectively.Proof.
(1) For any X,Y,Z∈Γ(D⊥), by using (2.4) and (3.27) and considering that the ambient space is an almost paracontact Riemannian manifold, we have
(3.53)g(h(X,Y),FZ)=g(∇̅XY,FZ)=g(∇̅XFY,Z)=-g(AnYX,Z)=0.
(2) Making use of ∇̅ being Levi-Civita connection and Lemma 2.1(2.2), we get
(3.54)g(h(Z,W),FX)=g(∇̅WFZ,X)=-g(∇̅WX,tZ)=-Xlng·g(W,tZ)
for any Z,W∈Γ(D), X∈Γ(D⊥).
(3) In the same way, we have
(3.55)g(h(X,Z),FY)=g(∇̅XZ,FY)=g(∇XtZ,Y)=Xlnfg(tZ,Y)=0
for any X,Y∈Γ(D⊥) and Z∈Γ(D).
(4) Considering Lemma 2.1(3) we derive
(3.56)g(h(W,FZ),FX)=g(∇̅FZW,FX)=g(∇̅FZFW,X)=-g(tZ,tW)X(lnf)
for any Z,W∈Γ(D) and X∈Γ(D⊥).Theorem 3.8.
LetN=N⊥×fNT be a warped product semi-invariant submanifold of an almost paracontact Riemannian manifold M. Then one has the following.(1)
The squared norm of the second fundamental form ofN in M satisfies
(3.57)∥h∥2≥1f2∥gradf∥2(Tr(t))2,
where Tr(t) denote the trace of mapping t.(2)
If the equality sign of (3.57) holds identically, then N⊥ is a totally geodesic, NT is a totally umbilical submanifolds of M, and N is a mixed geodesic submanifold in M. Furthermore, N is a minimal submanifold M if and only if Tr(t)=0 or N=N⊥×fNT is a usual Riemannian product.Proof.
Let{e1,e2,…,ep,e1,e2,…,eq,N1,N2,…,Ns,ξ} be an orthonormal basis of an almost paracontact Riemannian manifold M such that {e1,e2,…,ep} is tangent to Γ(TN⊥),{e1,e2,…,eq} is tangent to Γ(TNT), and {N1,N2,…,Ns} is tangent to Γ(ν). Taking into account  Lemma 3.7 and the basic linear algebra rules, by direct calculations, we have
(3.58)h(X,Y)=g(h(X,Y),Nj)Nj,1≤j≤s,h(Z,W)=-eilnf(tZ,W)Fei+g(h(Z,W),Nj)Nj,1≤i≤p,h(X,Z)=g(h(X,Z),Nj)Nj
for all X,Y∈Γ(TN⊥) and Z,W∈Γ(TNT). Since
(3.59)∥h∥2=∑i,j=1p∑ℓ=1sg(h(ei,ej),Nℓ)2+2∑i=1p‍∑k=1q‍∑ℓ=1s‍g(h(ei,ek),Nℓ)2+∑r,k=1q‍∑i=1p(eilnf)2g(tek,er)2+∑r,k=1q‍∑ℓ=1sg(h(ek,er),Nℓ)2,
here by direct calculations, we get
(3.60)(ei(lnf))2=1f2∥gradf∥2,Tr(t)=∑k=1qg(tek,ek).
So we conclude that
(3.61)∥h∥2≥1f2∥gradf∥2(Tr(t))2,
which proves our assertion.
Now we assume that the equality case of (3.57) holds identically, then from (3.58), respectively, we obtain
(3.62)h(D⊥,D⊥)=0,h(D,D)∈Γ(F(D⊥)),(3.63)h(D⊥,D)=0.
Since N⊥ is totally geodesic submanifold in N, the first condition in (3.62) implies that N⊥ is totally geodesic submanifold in M. Moreover, Lemma 2.1(3) shows that NT is totally umbilical submanifold in N. Therefore, the second condition in (3.62) implies that NT is also totally umbilical submanifold in M. On the other hand, (3.20) and (3.63) imply that N is mixed geodesic submanifold in M.Conclusion 3.9.
The geometry of the warped products in Riemannian manifolds is totally different from the geometry of the warped products in complex manifolds. Namely, in the complex manifolds, there exists no proper warped product CR-submanifold in the formN=N⊥×fNT (see [2, 8]) while there exists no proper warped product semi-invariant submanifold in the form N=NT×fN⊥ in Riemannian manifolds (see Theorem 3.1). The first condition in (3.62) implies that warped product CR-submanifold is minimal in complex manifolds while it does not imply that warped product semi-invariant submanifold is minimal in Riemannian product manifolds.



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*Source: 621625-2009-09-06.xml*