# Semi-Slant Warped Product Submanifolds of a Kenmotsu Manifold

Authors: Falleh R. Al-Solamy; Meraj Ali Khan
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

We study semi-slant warped product submanifolds of a Kenmotsu manifold. We obtain a characterization for warped product submanifolds in terms of warping function and shape operator and finally proved an inequality for squared norm of second fundamental form.


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## Body

## 1. Introduction

In [1] Tanno classified the connected almost contact metric manifold whose automorphism group has maximum dimension, there are three classes:(a)
homogeneous normal contact Riemannian manifolds with constantϕ holomorphic sectional curvature if the sectional curvature of the plane section contains ξ, say C(X,ξ)>0;(b)
global Riemannian product of a line or a circle and Kaehlerian manifold with constant holomorphic sectional curvature,C(X,ξ)=0;(c)
a warped product spaceR×fCn, if C(X,ξ)<0.Manifolds of class (a) are characterized by some tensorial equations, it has a Sasakian structure and manifolds of class (b) are characterized by some tensor equations called Cosymplectic manifolds. Kenmotsu [2] obtained some tensorial equations to Characterize manifolds of class (c), these manifolds are called Kenmotsu manifolds.The notion of semi-slant submanifolds of almost Hermitian manifolds was introduced by Papaghiuc [3] after that cabrerizo et al. [4] defined and studied semi-slant submanifolds in the setting of almost contact manifolds.Bishop and O’Neill [5] introduced the notion of warped product manifolds. These manifolds are generalization of Riemannian product manifolds and occur naturally. Recently, many important physical applications of warped product manifolds have been discovered, giving motivation to study of these spaces with differential geometric point of view. For instance, it has been accomplished that warped product manifolds provide an excellent setting to model space time near black hole or bodies with large gravitational fields (c.f., [5–7]). Due to wide applications of these manifolds in physics as well as engineering this becomes a fascinating and interesting topic for research, and many articles are available in literature (c.f., [3, 8, 9]).Recently, Atçeken [10] proved that the warped product submanifolds of type Nθ×fNT and Nθ×fN⊥ of a Kenmotsu manifold M̅ do not exist where the manifolds Nθ and NT (resp. N⊥) are proper slant and invariant (resp., anti-invariant) submanifolds of Kenmotsu manifold M̅, respectively. After that Siraj-Uddin et al. [11] investigated warped product of the types NT×fNθ and N⊥×fNθ and obtained some interesting results. In this continuation we obtain a characterization and an inequality for squared norm of second fundamental form.


## 2. Preliminaries

A2n+1 dimensional C∞ manifold M̅ is said to have an almost contact structure if there exist on M̅ a tensor field ϕ of type (1,1), a vector field ξ, and 1-form η satisfying the following properties:
(2.1)ϕ2=-I+η⊗ξ,ϕ(ξ)=0,η∘ϕ=0,η(ξ)=1.
There always exists a Riemannian metric g on an almost contact manifold M̅ satisfying the following conditions:(2.2)g(ϕX,ϕY)=g(X,Y)-η(X)η(Y),η(X)=g(X,ξ),
where X,Y are vector fields on M̅.An almost contact metric structure(ϕ,ξ,η,g) is said to be Kenmotsu manifold, if it satisfies the following tensorial equation [2]:(2.3)(∇̅Xϕ)Y=g(ϕX,Y)ξ-η(Y)ϕX,
for any X,Y∈TM̅, where TM̅ is the tangent bundle of M̅ and ∇̅ denotes the Riemannian connection of the metric g. Moreover, for a Kenmotsu manifold(2.4)∇̅Xξ=X-η(X)ξ.LetM be a submanifold of an almost contact metric manifold M̅ with induced metric g and if ∇ and ∇⊥ are the induced connection 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),∇̅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, respectively, for the immersion of M into M̅ and they are related as
(2.6)g(h(X,Y),N)=g(ANX,Y),
where g denotes the Riemannian metric on M̅ as well as on M.For anyX∈TM, we write
(2.7)ϕX=PX+FX,
where PX is the tangential component and FX is the normal component of ϕX.Similarly, for anyN∈T⊥M, we write
(2.8)ϕN=tN+fN,
where tN is the tangential component and fN is the normal component of ϕN. The covariant derivatives of the tensor field P and F are defined as
(2.9)(∇̅XP)Y=∇XPY-P∇XY,(∇̅XF)Y=∇X⊥FY-F∇XY.
From (2.3), (2.5), (2.7) and (2.8) we have
(2.10)(∇̅XP)Y=AFYX+th(X,Y)-g(X,PY)ξ-η(Y)PX,(2.11)(∇̅XF)Y=fh(X,Y)-h(X,PY)-η(Y)FX.Definition 2.1 (see [12]).
A submanifoldM of an almost contact metric manifold M̅ is said to be slant submanifold if for any x∈M and X∈TxM-〈ξ〉 is constant. The constant angle θ∈[0,π/2] is then called slant angle of M in M̅. If θ=0 the submanifold is invariant submanifold, if θ=π/2 then it is anti-invariant submanifold, if θ≠0, π/2 then it is proper slant submanifold.For slant submanifolds of contact manifolds Cabrerizo et al. [13] proved the following lemma.Lemma 2.2.
LetM be a submanifold of an almost contact manifold M̅, such that ξ∈TM then M is slant submanifold if and only if there exists a constant λ∈[0,1] such that
(2.12)P2=λ(I-η⊗ξ),
where λ=-cos2θ.
Thus, one has the following consequences of above formulae:(2.13)g(PX,PY)=cos⁡2⁡θ[g(X,Y)-η(X)η(Y)],(2.14)g(FX,FY)=sin⁡2⁡θ[g(X,Y)-η(X)η(Y)].A submanifoldM of M̅ is said to be semi-slant submanifold of an almost contact manifold M̅ if there exist two orthogonal complementary distributions DT and Dθ on M such that(i)
TM=DT⊕Dθ⊕〈ξ〉,(ii)
the distributionDT is invariant that is, ϕDT⊆DT,(iii)
the distributionDθ is slant 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 ϕ of the normal bundle T⊥M, then in the case of semi-slant submanifold, the normal bundle T⊥M can be decomposed as
(2.15)T⊥M=μ⊕FDθ.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.As a generalization of the product manifolds and in particular of a semi-slant product submanifold, one can consider warped product of manifolds which are defined asDefinition 2.3.
Let(B,gB) and (F,gF) be two Riemannian manifolds with Riemannian metric gB and gF, respectively, and f be a positive differentiable function on B. The warped product of B and F is the Riemannian manifold (B×F,g), where
(2.16)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(2.17)‖X‖2=‖dπ1X‖2+f2(p)‖dπ2X‖2.Bishop and O’Neill [5] proved the following.Theorem 2.4.
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
(2.18)g(∇f,X)=Xf,
for all X∈TM.Corollary 2.5.
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, one denotes byNT and Nθ an invariant and a slant submanifold, respectively, of an almost contact metric manifold M̅.Khan et al. [14] proved the following corollary.Corollary 2.6.
LetM̅ be a Kenmotsu manifold and N1 and N2 be any Riemannian submanifolds of M̅, then there do not exist a warped product submanifold M=N1×fN2 of M̅ such that ξ is tangential to N2.
Thus, one assumes that the structure vector fieldξ is tangential to N1 of a warped product submanifold N1×fN2 of M̅.In this paper we will consider the warped product of the typeNθ×fNT and NT×fNθ. The warped product of the type Nθ×fNT is called warped product semi-slant submanifolds; this type of warped product is studied by Atçeken [10], they proved that the warped product Nθ×fNT does not exist. The warped product of the type NT×fNθ is called semi-slant warped product; these submanifolds were studied by Siraj-Uddin et al. [11] and they proved the following LemmaLemma 2.7.
LetM=NT×fNθ be warped product semi-slant submanifold of a Kenmotsu manifold M̅ such that ξ is tangent to NT, where NT and Nθ are invariant and proper slant submanifolds of M̅. then(i)
g(h(X,Z),FPZ)=g(h(X,PZ),FZ)={Xlnf-η(X)}cos2θ∥Z∥2,(ii)
g(h(X,Z),FZ)=-PXlnf∥Z∥2,
for anyX∈TNT and Z∈TNθ.ReplacingX by PX in part (ii) of above lemma one has(2.19)g(h(PX,Z),FZ)=Xln⁡⁡f‖Z‖2.


## 3. Semi-Slant Warped Product Submanifolds

Throughout this section we will study the warped product of the typeNT×fNθ, for these submanifolds by Theorem 2.4 we have
(3.1)∇XZ=∇ZX=Xln⁡⁡fZ,
for any X∈TNT and Z∈TNθ.Lemma 3.1.
LetM=NT×fNθ be a semi-slant warped product submanifolds of a Kenmotsu manifold M̅, then
(3.2)g(h(X,Y),FZ)=0,
for any X,Y∈TNT and Z∈TNθ.Proof.
AsNT is totally geodesic in M then (∇̅XP)Y∈TNT and therefore by formula (2.10):
(3.3)(∇̅XP)Y=th(X,Y)-g(X,PY)ξ-η(Y)PX,
taking inner product with Z∈TNθ we get (3.2).Now we have the following Characterization.Theorem 3.2.
A semi-slant submanifoldM of a Kenmotsu manifold M̅ with integrable invariant distribution DT⊕〈ξ〉 and integrable slant distribution Dθ is locally a semi-slant warped product if and only if ∇ZPZ∈Dθ and there exists a C∞- function α on M with Zα=0,
(3.4)AFZX=(Xα-η(X))PZ-PXαZ,
for all X∈DT⊕〈ξ〉 and Z∈Dθ.Proof.
From (2.10) and (3.1) we have
(3.5)AFZX+th(X,Z)=0.
Similarly,
(3.6)PXln⁡⁡fZ-Xln⁡⁡fPZ=th(X,Z)-η(X)PZ,
from (3.5) and (3.6), we get
(3.7)AFZX=Xln⁡⁡fPZ-PXln⁡⁡fZ-η(X)PZ,
taking inner product with W∈TNθ, we have
(3.8)g(AFZX,W)=(Xln⁡⁡f-η(X))g(PZ,W)-PXln⁡⁡fg(Z,W).
From Lemma 3.1 and (3.8) we get the desired result.
Conversely, letM be a semi-slant submanifold of M̅ satisfying the hypothesis of the theorem, then for any X,Y∈DT⊕〈ξ〉 and Z∈Dθ(3.9)g(h(X,Y),FZ)=0,
that means h(X,Y)∈μ. Then from (2.11)
(3.10)-F∇XY=fh(X,Y)-h(X,PY).
Since h(X,Y)∈μ, then we have F∇XY=0, that is, ∇XY∈DT⊕〈ξ〉. Hence, each leaf of DT⊕〈ξ〉 is totally geodesic in M.
Further, supposeNθ be a leaf of Dθ and hθ be second fundamental form of the immersion of Nθ in M, then for any X∈DT⊕〈ξ〉 and Z∈Dθ, we have
(3.11)g(hθ(Z,Z),ϕX)=g(∇ZZ,ϕX),
using (2.7) and (2.5), the above equation yields
(3.12)g(hθ(Z,Z),ϕX)=g(∇ZPZ,X)+g(AFZZ,X),
applying (3.4), we get
(3.13)g(hθ(Z,Z),ϕX)=-PXln⁡⁡fg(Z,Z).
Replacing X by PX, the above equation gives
(3.14)hθ(Z,Z)=∇αg(Z,Z).
From above equation it is easy to derive
(3.15)hθ(Z,W)=∇αg(Z,W),
that is, Nθ is totally umbilical and as Zα=0, for all Z∈Dθ,∇μ is defined on NT, this mean that mean curvature vector of Nθ is parallel, that is, the leaves of Dθ are extrinsic spheres in M. Hence by virtue of result of [15] 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 autoparallel, then the manifold M is locally isometric to a warped product M0×fM1, we can say M is locally semi-slant warped product submanifold NT×fNθ, where the warping function f=eα.
Let us denote byDT and Dθ the tangent bundles on NT and Nθ, respectively, and let {X0=ξ,X1,…,Xp,Xp+1=ϕX1,…,X2p=ϕXp} and {Z1,…,Zq,Zq+1=PZ1,…,Z2q=PZq} be local orthonormal frames of vector fields on NT and Nθ, respectively, with 2p and 2q being real dimension. Since h(X,ξ)=0 for all X∈TM, then the second fundamental form can be written as
(3.16)‖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 Kenmotsu manifold, we prove the following.Theorem 3.3.
LetM=NT×fNθ be a semi-slant warped product submanifold of a Kenmotsu manifold M̅ with NT and Nθ invariant and slant submanifolds, respectively, of M̅. If η(X)≥2Xlnf for all X∈TNT, then the squared norm of the second fundamental form h satisfies
(3.17)‖h‖2≥4qcsc2⁡θ{1+cos⁡4θ}‖∇ln⁡⁡f‖2,
where ∇lnf is the gradient of lnf and 2q is the dimension Nθ.Proof.
In view of the decomposition (2.15), we may write
(3.18)h(U,V)=hFDθ(U,V)+hμ(U,V),
for each U,V∈TM, where hFDθ(U,V)∈FDθ and hμ(U,V)∈μ with
(3.19)hFDθ(U,V)=∑r=12qhr(U,V)FZr,(3.20)hr(U,V)=csc2⁡θg(h(U,V),FZr),
for each U,V∈TM. In view of above formulae we have
(3.21)g(hFDθ(PXi,Zr),hFDθ(PXi,Zr))=g(hr(PXi,Zr)FZr,hr(PXi,Zr)FZr)+g(hs(PXi,Zr)FZr,hs(PXi,Zr)FZr).
Now using (2.14) and (2.19)
(3.22)g(hFDθ(PXi,Zr),hFDθ(PXi,Zr))=hr(PXi,Zr)Xiln⁡⁡f+sin⁡2⁡θ∑s≠r(hs(PXi,Zr))2.
In view of (3.20) and (2.19), we get
(3.23)g(hFDθ(PXi,Zr),hFDθ(PXi,Zr))=csc2⁡θ(Xiln⁡⁡f)2+sin⁡2⁡θ∑s≠r(hs(PXi,Zr))2.
Summing over i=1,…,2p and r=1,…,2q the above equation yields
(3.24)∑i=12p∑r=12qg(hFDθ(PXi,Zr),hFDθ(PXi,Zr))=4qcsc2⁡θ‖∇ln⁡⁡f‖2+sin⁡2⁡θ∑i=12p∑r,s=1,r≠s2q(hs(PXi,Zr))2.
Since we have choose the orthonormal frame of vector fields on Dθ as {Z1,…,Zq,Zq+1=PZ1,…,Z2q=PZq}, then the second term in the right-hand side of (3.24) is written as
(3.25)csc2⁡θ∑i=12p[∑r=1q∑s=1,s≠rq{(g(h(PXi,Zr),FPZs))2+(g(h(PXi,PZr),FZs))2}∑r=1q{(g(h(PXi,Zr),FPZr))2+(g(h(PXi,PZr),FZr))2}+∑r=1q∑s=1,s≠rq{(g(h(PXi,Zr),FPZs))2+(g(h(PXi,PZr),FZs))2}].
From part (i) of Lemma 2.7, the first two terms of above equation can be written as
(3.26)csc2⁡θ∑i=12p[2q(Xiln⁡⁡f-η(Xi))2cos⁡4⁡θ].
In account of to hypothesis η(Xi)≥2Xilnf the above expression is greater than equal to the following term:
(3.27)4qcsc2⁡θ‖∇ln⁡⁡f‖2cos⁡4⁡θ.
Using above inequality into (3.24), we have
(3.28)g(hFDθ(PXi,Zr),hFDθ(PXi,Zr))≥4qcsc2⁡θ{1+cos⁡4⁡θ}‖∇ln⁡⁡f‖2.
The inequality (3.17) follows from (3.16) and (3.28).
The equality holds ifh(DT,DT)=0, h(Dθ,Dθ)=0, h(PX,Z) is orthogonal to FZ and FPZ for all X∈DT and Z∈Dθ and η(X)=2Xlnf.


## 4. Conclusion

In this paper we study nontrivial warped product submanifolds of a Kenmotsu manifold and in this study there emerge natural problems of finding the estimates of the squared norm of second fundamental form and to find the relation between shape operator and warping function. This study predict the geometric behavior of underlying warped product submanifolds. Further, as it is known that the warping function of a warped product manifold is a solution of some partial differential equations (c.f., [8]) and most of physical phenomenon is described by partial differential equations. We hope that our study may find applications in physics as well as in engineering.



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