# Warped Product Submanifolds of Riemannian Product Manifolds

Authors: Falleh R. Al-Solamy; Meraj Ali Khan
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We study warped product of the typeNθ×fNT and Nθ×fN⊥, where Nθ, NT, and N⊥ are proper slant, invariant, and anti-invariant submanifolds,
respectively, and we prove some basic results and finally obtain
some inequalities for squared norm of second fundamental form.


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## Body

## 1. Introduction

Bishop and O’Neil [1] introduced the notion of warped product manifolds that occur naturally; for example, surface of revolution is a warped product manifold. With regard to physical applications of these manifolds, one may realize that space time around a massive star or a black hole can be modeled on warped product manifolds [2]. CR-warped product was introduced by Chen [3]; he studied warped product CR-submanifolds in the setting of Kaehler manifolds and showed that there does not exist warped product of the form N⊥×fNT; therefore he considered warped product CR-submanifolds of type NT×N⊥ and established a relationship between the warping function f and the squared norm of second fundamental form [3]. In [4] Atçeken studied semi-slant warped product of Riemannian product manifolds. In fact they proved that there exists no warped product if spheric submanifold of warped product submanifold is proper slant submanifold. On the other hand they proved the existence of warped product of the type Nθ×fNT and Nθ×fN⊥ via some examples. In this continuation we have studied the warped product submanifolds in which proper slant submanifolds are totally geodesic; that is, we study the warped product of the types Nθ×NT and Nθ×N⊥ and called them semi-slant warped product and hemi-slant warped product submanifolds, respectively.


## 2. Preliminaries

Let(M1,g1) and (M2,g2) be the Riemannian manifolds with dimensions m1 and m2, respectively, and let M1×M2 be Riemannian product manifold of M1 and M2. We denote projection mapping of T(M1×M2) onto TM1 and TM2 by σ★ and π★, respectively. Then we have σ★+π★=I,σ★2=σ★,π★2=π★, and σ★∘π★=π★∘σ★=0, where ★ denotes the differential.Riemannian metric of the Riemannian product manifoldM=M1×M2 is defined by
(2.1)g(X,Y)=g1(σ★X,σ★Y)+g2(π★X,π★Y),
for any X,Y∈TM-. If we set F=σ★-π★, then F2=I,  F≠I, and g satisfies the condition
(2.2)g(FX,Y)=g(X,FY),
for any X,Y∈TM-; thus F defines an almost Riemannian product structure on M-. We denote Levi-Civita connection on M- by ∇-; then the covariant derivative of F is defined as
(2.3)(∇-XF)Y=∇-XFY-F∇-XY,
for any X,Y∈TM-. We say that F is parallel with respect to the connection ∇- if we have (∇-XF)Y=0. Here from [5], we know that F is parallel; that is, F is Riemannian product structure.LetM- be a Riemannian product manifold with Riemannian product structure F and M an immersed submanifold of M-; we also denote by g the induced metric tensor on M as well as on M-. If ∇- is the Levi-Civita connection on M-, then the Gauss and Weingarten formulas are given, respectively, as
(2.4)∇-XY=∇XY+h(X,Y),(2.5)∇-XV=-AVX+∇X⊥V,
for any X,Y∈TM and V∈T⊥M, where ∇ is the connection on M and ∇⊥ is the connection in the normal bundle, h is the second fundamental form of M, and AV is the shape operator of M. The second fundamental form h and the shape operator AV are related by
(2.6)g(AVX,Y)=g(h(X,Y),V).
For any X∈TM, we can write
(2.7)FX=TX+NX,
where TX and NX are the tangential and normal components of FX, respectively, and for V∈T⊥M,
(2.8)FV=tV+nV,
where tV and nV are the tangential and normal components of FV, respectively, and the submanifold M is said to be invariant if N is identically zero. On the other hand M is said to be an anti-invariant submanifold if T is identically zero.The covariant derivatives ofT, N, t, and n are defined as
(2.9)(∇-XT)Y=∇XTY-T∇XY,(2.10)(∇-XN)Y=∇X⊥NY-N∇XY,(2.11)(∇-Xt)V=∇XtV-t∇XV,(2.12)(∇-Xn)V=∇X⊥nV-n∇X⊥Y.
Using (2.4)–(2.9) we get
(2.13)(∇-XT)Y=ANYX+th(X,Y),(2.14)(∇-XN)Y=h(X,TY)-nh(X,Y).LetM be an immersed submanifold of a Riemannian product manifold M-, for each nonzero vector X tangent to M at a point x, and we denote by θ(x) the angle between FX and TxM. The angle θ(x) is called the slant angle of immersion.LetM be an immersed submanifold of a Riemannian product manifold M-. M is said to be slant submanifold of Riemannian product manifold M- if the slant angle θ(x) is constant which is independent of choice of x∈M and X∈TM.Invariant and anti-invariant submanifolds are particular cases of slant submanifolds with anglesθ=0 and θ=π/2, respectively. A slant submanifold which is neither invariant nor anti-invariant is called proper slant submanifold. The following characterization of slant submanifolds of Riemannian product manifolds is proved by Atçeken [6].Theorem 2.1.
LetM be an immersed submanifold of a Riemannian product manifold M-. Then M is a slant submanifold if and only if there exists a constant λ∈[0,1] such that T2=λI.Moreover, ifθ is the slant angle of M, then it satisfies λ=cos2θ.Hence, for a slant submanifold we have the following relations which are consequences of the above theorem:(2.15)g(TX,TY)=cos2θg(X,Y),(2.16)g(NX,NY)=sin2θg(X,Y)
for any X,Y∈TM.Papaghuic [7] introduced a class of submanifolds in almost Hermitian manifolds called semi-slant submanifolds; this class includes the class of proper CR-submanifolds and slant submanifolds. Cabrerizo et al. [8] initiated the study of contact version of semi-slant submanifolds and also gave the notion of Bi-slant submanifolds. A step forward Carriazo [9] defined and studied Bi-slant submanifolds and simultaneously gave the notion of anti-slant submanifolds; after that V. A. Khan and M. A. Khan [10] have studied anti-slant submanifolds with the name pseudo-slant submanifolds. Recently, Sahin [11] renamed these submanifolds and studied these submanifolds with the name hemi-slant submanifolds for their warped product.Definition 2.2.
A submanifoldM of a Riemannian product manifold is said to be semi-slant submanifold if there exist two orthogonal complementary distributions DT and Dθ such that DT is invariant and Dθ is slant distribution with slant angle θ≠0.It is straight forward to see that semi-invariant submanifolds and slant submanifolds are semi-slant submanifolds withθ=π/2 and DT={0}, respectively.Ifμ is invariant subspace under F of the normal bundle T⊥M, then in the case of semi-slant submanifold, the normal bundle T⊥M can be decomposed as
(2.17)T⊥M=μ⊕NDθ.A semi-slant submanifoldM is called a semi-slant product if the distributions DT and Dθ are parallel on M. In this case M is foliated by the leaves of these distributions.Definition 2.3.
A submanifoldM of a Riemannian product manifold is called hemi-slant submanifold if it is endowed with two orthogonal complementary distributions D⊥ and Dθ such that D⊥ is totally real and Dθ is slant distribution with slant angle θ≠π/2.It is easy to see that semi-invariant submanifolds and slant submanifolds are semi-slant submanifolds withθ=0 and D⊥={0}, respectively. The normal bundle T⊥M can be decomposed as follows:
(2.18)T⊥M=μ⊕ND⊥⊕NDθ.AsD⊥ and Dθ are orthogonal distributions on M, then it is easy to see that the distributions ND⊥ and NDθ are mutually perpendicular. In fact, the decomposition (2.18) is an orthogonal direct decomposition. A hemi-slant submanifold M is called a hemi-slant product if the distributions D⊥ and Dθ are parallel on M.As a generalization of product manifold and in particular of semi-slant product submanifolds (hemi-slant product submanifolds) one can consider warped product of manifolds which are defined as.Definition 2.4.
Let(B,gB) and (F,gF) be two Riemannian manifolds with Riemannian metric gB and gF, respectively, and f a positive differentiable function on B. The warped product of B and F is the Riemannian manifold (B×F,g), where
(2.19)g=gB+f2gF.
For a warped product manifoldN1×fN2, we denote by D1 and D2 the distributions defined by the vectors tangent to the leaves and fibers, respectively. In other words, D1 is obtained by the tangent vectors of N1 via the horizontal lift and D2 is obtained by the tangent vectors of N2 via vertical lift. In case of semi-slant warped product submanifolds D1 and D2 are replaced by DT and Dθ, respectively.
The warped product manifold(B×F,g) is denoted by B×fF. If X is the tangent vector field to M=B×fF at (p,q), then
(2.20)∥X∥2=∥dπ1X∥2+f2(p)∥dπ2X∥2.
Bishop and O’Neill [1] proved the following.Theorem 2.5.
LetM=B×fF be warped product manifolds. If X,Y∈TB and V,W∈TF, then(i)
∇XY∈TB,(ii)
∇XV=∇VX=(Xf/f)V,(iii)
∇VW=(-g(V,W)/f)∇f.∇f is the gradient of f and is defined as
(2.21)g(∇f,X)=Xf,
for all X∈TM.Corollary 2.6.
On a warped product manifoldM=N1×fN2, the following statements hold: (i)
N1 is totally geodesic in M;(ii)
N2 is totally umbilical in M.Throughout, we denote byNT, N⊥, and Nθ invariant, anti-invariant, and slant submanifolds, respectively, of a Riemannian product manifold M-.


## 3. Semi-Slant Warped Product Submanifolds

In this section we will consider the warped product of the typeNθ×fNT.For the warped product of the typeNθ×fNT by Theorem 2.5 we have
(3.1)∇XZ=∇ZX=ZlnfX,
for any Z∈TNθ and X∈TNT.Lemma 3.1.
LetM=Nθ×fNT be a semi-slant warped product submanifold of a Riemannian product manifold; then(i)
g(h(X,Z),NW)=0,(ii)
g(h(X,X),NZ)=TZlnf∥X∥2,
for anyX∈TNT and Z,W∈TNθ.Proof.
For anyZ,W∈TNθ, (∇ZT)W∈TNθ; then from (2.13)
(3.2)(∇ZT)W=ANWZ+th(Z,W).
Taking inner product with X∈TNT we have
(3.3)g(h(X,Z),NW)=0.
This is part (i) of the lemma.
Now for anyX,Y∈TNT, from (2.13) and (2.9),
(3.4)∇XTY-T∇XY=th(X,Y).
Taking inner product with Z∈TNθ, the above equation yields
(3.5)-g(∇XZ,TY)-g(∇XTZ,Y)=g(h(X,Y),NZ).
Using (3.1), the above equation gives
(3.6)-Zlnfg(X,TY)+TZlnfg(X,Y)=g(h(X,Y),NZ).
In particular
(3.7)TZlnfg(X,X)=g(h(X,X),NZ).
This proves part (ii) of the lemma. Now we have the following corollary.Corollary 3.2.
For the warped product of the typeNθ×fNT following statements are equivalent: (i)
H∈μ,(ii)
θ=π/2 or the warping function f is constant; that is, there does not exist warped product.Proof.
SinceNT is totally umbilical, then from (3.7)
(3.8)g(H,NZ)=TZlnf.
Replacing Z by TZ and using Theorem 2.1, we get
(3.9)g(H,NTZ)=cos2θZlnf.
The proof follows from (3.9).Now we have the following characterization for semi-slant warped product submanifolds.Theorem 3.3.
A semi-slant submanifoldM of Riemannian product manifolds M- with integrable invariant distribution DT and the slant distribution Dθ is locally a semi-slant warped product if and only if ∇XTZ∈Dθ and there exist a C∞-function α on M with Xα=0 for all X∈DT such that
(3.10)ANZX=TZlnfX,
for all X∈DT and Z∈Dθ.Proof.
IfM is a semi-slant warped product of the type Nθ×fNT, then for any X∈TNT and Z∈TNθ from (2.9), (2.13), and (3.1), we have
(3.11)TZlnfX-ZlnfTX=ANZX+th(X,Z).
Taking inner product with X, the above equation gives
(3.12)g(ANZX,X)=TZlnfg(X,X).
By part (i) of Lemma 3.1, we also have
(3.13)g(ANZX,W)=0.
From (3.12) and (3.13) we have the following equation:
(3.14)ANZX=TZlnfX.
Conversely, let M be a semi-slant submanifold of M- satisfying the hypothesis of the theorem; then for any Z∈TNθ and Y∈TNT we have
(3.15)g(h(Z,Y),NZ)=g(ANZY,Z)=0.
This mean h(Z,Y)∈μ.
From (2.14), we have
(3.16)-N∇ZY=fh(Z,Y)-h(Z,TY).
Comparing components of μ and NDθ, we get
(3.17)N∇ZY=0.
It is evident from the above equation that ∇ZY∈DT; this means ∇ZW∈Dθ for any Z,W∈Dθ and hence Dθ is totally geodesic. Further, let NT be a leaf of DT and hT a second fundamental form of the immersion NT in M; then for any X,Y∈DT and Z,W∈Dθ(3.18)g(hT(X,Y),FW)=g(∇XY,FW).
or
(3.19)g(hT(X,Y),FW)=-g(∇XTW-ANWX,Y).
Using the hypothesis, we get
(3.20)g(hT(X,Y),FW)=TWlnfg(X,Y),
Finally, the above equation yields
(3.21)hT(X,Y)=g(X,Y)∇α.
That is, NT is totally umbilical and as Xα=0, for all X∈DT, this means that mean curvature vector of NT is parallel; that is, the leaves of DT are extrinsic spheres in M. Hence by virtue of result of [12] which says that if the tangent bundle of a Riemannian manifold M splits into an orthogonal sum TM=E0⊕E1 of nontrivial vector subbundles such that E1 is spherical and its orthogonal complement E0 is auto parallel, then the manifold M is locally isometric to a warped product M0×fM1, we can say M is locally semi-slant warped product submanifold Nθ×fNT, where warping function f=eα.
Let us denote byDT and Dθ the tangent bundles on NT and Nθ, respectively, and let {X1,…,Xp,Xp+1=FX1,…,X2p=FXp} and {Z1,…,Zq,Zq+1=TZ1,…,Z2q=TZq} be local orthonormal frames of vector fields on NT and Nθ, respectively, with 2p and 2q being real dimensions:
(3.22)∥h∥2=∑i,j=12p‍g(h(Xi,Xj),h(Xi,Xj))+∑i=12p‍∑r=12q‍g(h(Xi,Zr),h(Xi,Zr))+∑r,s=12q‍g(h(Zr,Zs),h(Zr,Zs)).
Now, on a semi-slant warped product submanifold of a Riemannian product manifold, we prove the following.Theorem 3.4.
LetM=Nθ×fNT be a semi-slant warped product submanifold of a Riemannian product manifold M- with NT and Nθ invariant and slant submanifolds, respectively, of M-. Then the squared norm of the second fundamental form h satisfies
(3.23)∥h∥2≥4p(1+cos2θ)csc2θ∥∇lnf∥2.Proof.
ForNθ×fNT, in view of decomposition (2.17), we may write
(3.24)h(X,Y)=hNDθ(X,Y)+hμ(X,Y),
for each X,Y∈TM, where hNDθ(X,Y)∈NDθ and hμ(X,Y)∈μ with
(3.25)hNDθ(X,Y)=∑r=12q‍hr(X,Y)NZr,
where
(3.26)hr(X,Y)=csc2θg(h(X,Y),NZr),
for each Z∈TNθ. For any X∈TNT and Z∈TNθ, by (3.25) we have
(3.27)g(hNDθ(Xi,Xi),hNDθ(Xi,Xi))=g(hr(Xi,Xi)NZr,hr(Xi,Xi)NZr)+∑s≠rg(hs(Xi,Xi)NZr,hs(Xi,Xi)NZr).
Now, using (3.26), (3.7), and (2.16), the above equation takes the form
(3.28)g(hNDθ(Xi,Xi),hNDθ(Xi,Xi))=csc2θ(TZrlnf)2+sin2θ∑s≠r‍(hs(Xi,Xi))2.
Now summing over i=1,…,2p, r,s=1,…,2q and again using (3.7) and (3.26), we have
(3.29)g(hNDθ(Xi,Xi),hNDθ(Xi,Xi))=2p(1+cos2θ)csc2θ∥∇lnf∥2(1+sin2θcsc2θ),
or
(3.30)g(hNDθ(Xi,Xi),hNDθ(Xi,Xi))=4p(1+cos2θ)csc2θ∥∇lnf∥2,
By similar calculation, from (3.25), (3.26), (3.3), and (2.16) it is easy to see that
(3.31)g(hNDθ(Xi,Zr),hNDθ(Xi,Zr))=0.
The result follows from (3.22), (3.30), and (3.31).


## 4. Hemi-Slant Warped Product Submanifolds

In this section we will study the warped product of the typeNθ×fN⊥. For warped product of type Nθ×fN⊥ from Theorem 2.5 we have
(4.1)∇XZ=∇ZX=XlnfZ,
for any X∈TNθ and Z∈TN⊥.Now we have the following lemma.Lemma 4.1.
LetM=Nθ×fN⊥ be a hemi-slant warped product submanifold of a Riemannian product manifold; then(i)
g(h(X,Z),NY)+g(h(X,Y),NZ)=0,(ii)
g(h(X,Z),NZ)=0,(iii)
g(h(Z,Z),NX)=TXlnf∥Z∥2, for any X,Y∈TNθ and Z∈TN⊥.Proof.
For anyX,Y∈TNθ, (∇-XT)Y∈TNθ; then from (2.13) we have
(4.2)g((∇-XT)Y,Z)=g(ANYX,Z)+g(th(X,Y),Z),
or equivalently the above equation gives
(4.3)g(h(X,Z),NY)+g(h(X,Y),NZ)=0,
which proves part (i).
From (2.9), (2.13), we have
(4.4)ANZX+th(X,Z)=0.
Taking inner product with Z∈TN⊥ the above equation is reduced to
(4.5)g(h(X,Z),NZ)=0.
Using (2.9), (2.13), and (4.1), we derive
(4.6)TXlnfZ=ANXZ+th(X,Z).
Taking inner product with Z∈TN⊥ and using (4.5),
(4.7)g(h(Z,Z),NX)=TXlnf∥Z∥2.
Now we have the following corollary.Corollary 4.2.
For the warped product of the typeNθ×fN⊥ following statements are equivalent: (i)
H∈μ⊕ND⊥,(ii)
the warping functionf is constant; that is, there does not exist warped product.Proof.
AsN⊥ is totally umbilical, then from (4.7) and from Theorem 2.5(4.8)g(H,NTX)=cos2θXlnf.
Since θ≠π/2, hence from the previous equation it is easy to see that statements (i) and (ii) are equivalent.
Let us denote byDθ and D⊥ the tangent bundles on Nθ and N⊥, respectively, and let {X1,X2,…,Xq,Xq+1=TX1,…,X2q=TXq} and {Z1,Z2,…,Zp} be local orthonormal frames of vector fields on Nθ and N⊥, respectively, with 2p and q being their real dimensions; then
(4.9)∥h∥2=∑r,s=12q‍g(h(Xr,Xs),h(Xr,Xs))+∑r=12q‍∑i=1pg(h(Xr,Zi),h(Xr,Zi))+∑i,j=1p‍g(h(Zi,Zj),h(Zi,Zj)).
Now, on a hemi-slant warped product submanifold of a Riemannian product manifold, we prove the following inequality.Theorem 4.3.
LetM=Nθ×fN⊥ be a hemi-slant warped product submanifold of a Riemannian product manifold M- with N⊥ and Nθ anti-invariant and slant submanifolds, respectively, of M-. Then the squared norm of the second fundamental form h satisfies
(4.10)∥h∥2≥2p(1+cos2θ)csc2θ∥∇lnf∥2.Proof.
In view of decomposition (2.18), the second fundamental form can be decomposed as follows:
(4.11)h(X,Y)=hNDθ(X,Y)+hND⊥(X,Y)+hμ(X,Y),
for each X,Y∈TM, where hNDθ(X,Y)∈NDθ, hND⊥(X,Y)∈ND⊥, and hμ(X,Y)∈μ with
(4.12)hNDθ(X,Y)=∑r=12q‍hr(X,Y)NXr′,
where
(4.13)hr(X,Y)=csc2θg(h(X,Y),NXr′),
for each Xr′∈TNθ.
Now making use of (2.18), (4.12), (4.13), and (4.7) we have
(4.14)g(hNDθ(Zi,Zi),hNDθ(Zi,Zi))=csc2θ(TXrlnf)2+sin2θ∑s≠r‍(hs(Zi,Zi))2,
for any X∈TNθ and Z∈TN⊥.
Again using (4.12) and (4.13), the previous equation gives
(4.15)g(hNDθ(Zi,Zi),hNDθ(Zi,Zi))=csc2θ(TXrlnf)2+sin2θcsc4θ(TXslnf)2.
Summing over i=1,…,p and r,s=1,…,2q, we have
(4.16)g(hNDθ(Zi,Zi),hNDθ(Zi,Zi))=2p(1+cos2θ)csc2θ∥∇lnf∥2.
Similarly, for any X∈TNθ and Z∈TN⊥ by (4.12), (4.13), and (4.5) it is easy to see that
(4.17)g(hNDθ(Xr,Zi),hNDθ(Xr,Zi))=0.
The result follows from (4.9), (4.16), and (4.17).



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