# Characterizing Inequalities for Biwarped Product Submanifolds of Sasakian Space Forms

Authors: Meraj Ali Khan; Ibrahim Al-dayel
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

The biwarped product submanifolds generalize the class of product submanifolds and are particular case of multiply warped product submanifolds. The present paper studies the biwarped product submanifolds of the typeST×ψ1S⊥×ψ2Sθ in Sasakian space forms S¯c, where ST, S⊥, and Sθ are the invariant, anti-invariant, and pointwise slant submanifolds of S¯c. Some characterizing inequalities for the existence of such type of submanifolds are proved; besides these inequalities, we also estimated the norm of the second fundamental form.


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## Body

## 1. Introduction

Because of its expected applications in material science and relativistic theory, the investigation of warped product manifolds has obtained a conspicuous subject in the field of differential geometry; for example, warped products give numerous major solutions for Einstein field equations [1]. The theory of warped product manifolds is being used to demonstrate space-time close to the black holes [2]. The warped product P×rS21 represents Schwartzschild space-time, with base P=R×R+, r>0, and fibre S21 that is sphere with radius one. However, the Schwartzschild space-time will transform into a black hole under some instances [3].In the paper [4], some of the inherent properties of warped product manifolds were investigated. Chen (see [5]) undertook the very first extrinsic study of warped product manifolds in the almost complex setting while acquiring certain existence results for CR-submanifolds to be CR-warped product submanifolds in Kaehler manifolds. Hasegawa and Mihai [6], on the other hand, analyzed contact CR-warped product submanifolds in almost contact environments. Many other people have investigated warped product manifolds in contact geometry, yielding an assortment of existence outcomes for instance (see [7–10]).Another general class of warped product semislant submanifolds and contact CR-warped product submanifolds is the warped product pointwise semislant submanifold. The analysis was then continued by I. Mihai and S. Uddin in the framework of Sasakian manifolds, and few ideal inequalities relating to the second fundamental form and warping function were obtained. In the papers (see [11–13]), warped product pointwise semislant submanifolds for almost contact and almost complex manifolds were investigated.One more generalized class of product manifolds is biwarped product manifolds, which are a subclass of multiply warped product manifolds. Chen and Dillen [14] looked at multiply warped product submanifolds immersed in Kaehler manifolds and found the remarkable optimum inequalities for them. Biwarped product submanifolds have recently been investigated (cf., see [15, 16]). Also, there is a recent paper [17], which initiates the study of inequalities for biwarped product submanifolds of nearly trans-Sasakian manifolds.In this manuscript, authors established some inequalities for the squared norm of the second fundamental forms. These inequalities generalize several results available in the literature. Since Sasakian manifolds are particular cases of the nearly trans-Sasakian manifolds, therefore, a natural question arises that the inequalities obtained in the present paper may be particular case of inequalities obtained in [17]. Although in the paper [17] authors studied whole norm of second fundamental form, in the present study, the inequalities for second fundamental form are obtained by taking the restriction on the distribution μ, which is a part of the normal distribution. Therefore, our main results are different from the results obtained in [17] except some initial results.Basically, in this manuscript, we look at biwarped product submanifolds of Sasakian space forms and determine some interesting inequalities. In terms of warping functions and slant functions, we estimate the norm of the second fundamental form. As a result, the equity case is taken into account.The article is structured as follows. Second section is contributed to fundamental concepts, formulae, and results that are essential for the paper’s next analysis. We prove our key findings in Section3 by looking into the nature of biwarped product submanifolds in Sasakian space forms.Throughout the text, we used some abbreviations like Biwarped Product≡ BW-P, Sasakian space form ≡ S-S-F, totally Geodesic ≡ T-G, and totally umbilical ≡ T-U.


## 2. Preliminaries

A2n+1−dimensional C∞−manifold S¯ is said to have an almost contact structure if on S¯ there exist a tensor field ϕ of type 1,1, a vector field ξ, and a 1-form η satisfying the following properties [18]:(1)ϕ2=−I+η⊗ξ,ϕξ=0,η°ϕ=0,ηξ=1.The manifoldS¯ with the structure ϕ,ξ,η is called almost contact metric manifold. There exists a Riemannian metric g on an almost contact metric manifold S¯, satisfying the following:(2)ηE1=gE1,ξ,gϕE1,ϕE2=gE1,E2−ηE1ηE2,for all E1,E2∈TS¯ where TS¯ is the tangent bundle of S¯.An almost contact metric manifoldS¯ϕ,ξ,η,g is said to be Sasakian manifold if it satisfies the following relation [18]:(3)∇¯E1ϕF=gE1,E2ξ−ηE2E1,for any E1,E2∈TS¯, where ∇¯ denotes the Riemannian connection of the metric g. More details of almost contact metric manifold can be seen in [18]. For a Sasakian manifold, we have(4)∇¯E1ξ=−ϕE1.A Sasakian manifoldS¯ is said to be a Sasakian space form if it has constant ϕ-holomorphic sectional curvature c and is denoted by S¯c. The curvature tensor R¯ of S-S-F S¯c is given by(5)R¯E1,E2E3=c−34gE2,E3E1−gE1,E3E2+c−14gE1,ϕE3ϕE2−gE2,ϕE3ϕE1+2gE1,ϕE2ϕE3+ηE1ηE3E2−ηE2ηE3E1+gE1,E3ηE2ξ−gE2,E3ηE1ξ,for all vector fields E1,E2,andE3 on S¯.LetS be a submanifold of an A-C-M manifold S¯ with induced metric g. The Riemannian connection ∇¯ of S¯ induces canonically the connections ∇ and ∇⊥ on the tangent bundle TS and the normal bundle T⊥S of S, respectively, and then the Gauss and Weingarten formulae are governed by(6)∇¯E1E2=∇E1E2+σE1,E2,(7)∇¯E1V=−AVE1+∇E1⊥V,for each E1,E2∈TS and V∈T⊥S, where σ and AV are the second fundamental form and the shape operator, respectively, for the immersion of S into S¯. They are related as(8)gσE1,E2,V=gAVE1,E2,where g is the Riemannian metric on S¯ as well as the induced metric on S.For a submanifoldS⟶S¯, the equation of Codazzi is provided by(9)R¯E1,E2E3⊥=∇E1⊥σE2,E3−∇E2⊥σE1,E3+σ∇E3E1,E3−σ∇E1E2,E3+σE1,∇E2E3−σE2,∇E1E3,where R¯E1,E2E3⊥ is the normal component of the curvature tensor R¯E1,E2E3.IfTE1 and NE1 represent the tangential and normal part of ϕE1, respectively, for any E1∈TS, one can write(10)ϕE1=TE1+NE1.Similarly, for anyV∈T⊥S, we write(11)ϕV=tV+nV,where tV and nV are the tangential and normal parts of ϕV, respectively. Thus, T (resp. n) is 1-1 tensor field on TS (respectively, T⊥S), and t (respectively, n) is a tangential (respectively, normal) valued 1-form on T⊥S (respectively, TS). The covariant derivatives of the tensor fields ϕ, T, and N are defined as(12)∇¯E1ϕE2=∇¯E1ϕE2−ϕ∇¯E1E2,(13)∇¯E1TE2=∇E1TE2−T∇E1E2,∇¯E1NE2=∇E1⊥NE2−N∇E1E2.From equations (3), (6), (7), (10), and (11), we have(14)∇¯E1TE2=ANE2E1+tσE1,E2−gE1,E2ξ−ηE2E1,∇¯E1NE2=nσE1,E2−σE1,TE2.The mean curvature vectorΠ of S is defined as(15)Π=1k∑i=1kσui,ui,where k is the dimension of S and u1,u2,…,uk is a local orthonormal basis of S. The squared norm of the second fundamental form σ is defined as(16)σ2=∑i,j=1kgσui,uj,σui,uj.A submanifoldS of S¯ is said to be a T-G submanifold if σE1,E2=0 and T-U submanifold if σE1,E2=gE1,E2Π, for each E1,E2∈TS.The notion of slant submanifolds in contact geometry was first defined by Lotta [19]. Later, these submanifolds were studied by Cabrerizo et al. [20]. Now, we have following definition of slant submanifolds.Definition 1.
A submanifoldM of an almost contact metric manifold M¯ is said to be slant submanifold if for any x∈M and X∈TxM−ξ, the angle between X and ϕX is constant. The constant angle θ∈0,π/2 is then called slant angle of M in M¯. If θ=0, the submanifold is invariant submanifold, and if θ=π/2, then it is anti-invariant submanifold. If θ≠0,π/2, it is proper slant submanifold.
Etayo [21] presented the idea of pointwise slant submanifolds as a generalization of slant submanifolds in the context of almost Hermitian manifolds. Further, Chen and Garay [22] looked into pointwise slant submanifolds for almost Hermitian manifolds and came up with some important results. However, Park [23] expanded the definition of pointwise slant submanifolds in almost contact metric manifolds, which was an important development in this direction. Nevertheless, for almost contact metric manifolds, Uddin and Alkhalidi [24] revised the concept of pointwise slant submanifolds. More specifically, a submanifolds S of an almost contact metric manifold S¯ are claimed to be pointwise slant submanifold if for any E∈TpS in the sense that ξ is tangential to S, and the angle θE between ϕE and TxS−0 is independent of the choice of nonzero vector field E∈TpS−0. In this case, θ is viewed as the slant function of the pointwise slant submanifold, which is a function on S. We now have the following descriptive theorem.Theorem 1 (see [24]).
LetS be a submanifold of an A-C-M manifold S¯ such that ξ∈TS. Then, S is pointwise slant iff(17)T2=cos2θ−I+η⊗ξ,where θ is the real valued function on TS.As a result, the above formula has the following implications:(18)gTE1,TE2=cos2θgE1,E2−ηE1ηE2,(19)gNE1,NE2=sin2θgE1,E2−ηE1ηE2,∀E1,E2∈TS.One can conceive the warped product of manifolds as a generalization of the product manifolds, which are explained as follows.Consider two Riemannian manifoldsS1,g1 and S2,g2 with corresponding Riemannian metrics g1 and g2 and ψ:S1⟶R be a positive differentiable function. If x and y are projection maps such that x:S1×S2⟶S1 and y:S1×S2⟶S2, which are defined as xm,n=m and ym,n=n∀m,n∈S1×S2, then S=S1×S2 is called warped product manifold if the Riemannian structure on S satisfies(20)gE1,E2=g1x∗E¯1,x∗E¯2+ψ°x2g2y∗E¯1,y∗E¯2,for all E¯1,E¯2∈TS. The function ψ represents the warping function of S1×S2. We can generalize this definition to multiply W-P manifolds as follows.LetSii=1,2,…,k be Riemannian manifolds with respective Riemannian metrics gii=1,2,…,k, and let ψii=2,3,…,k be positive valued functions on S1. Then, the product manifold S=S1×S2×⋯Sk equipped with Riemannian metric g given by(21)g=π1∗g1+∑i=2kψi°π12πi∗gi,is said to multiply W-P manifold denoted by S=S1×ψ2S2×⋯×ψkSk where πi are the projection maps of S onto i, respectively, and πi∗ are their respective tangent maps for i=1,2,…,k. The functions ψi are known as the warping functions [14]. If the warping functions are constants, the warped product is simply a Riemannian product, known as a trivial multiply warped product.The analysis of multiply warped product manifolds has recently gained attention in both complex and almost contact settings [14, 25]. We may define biwarped product manifolds as a special case of multiply warped product manifolds by using i=3 in the above description. For the BW-P manifold S¯=S0×ψ1S1×ψ2S2 with the Levi-Civita connection, ∇¯ and ∇i denote the Levi-Civita connection of Si for i=0,1,2. Some formulae relating to covariant derivatives for a BW-P manifold are given in the following lemma.Lemma 1 (see [26]).
LetS¯=S0×ψ1S1×ψ2S2 be a BW-P manifold. Then, we have(22)∇¯EF=∇E0F,(23)∇¯EG=∇¯GE=ElnψiG,for E,F∈TS0 and G∈TSi, i=1,2.∇ψ is the gradient of ψ and is defined as(24)g∇ψ,E=Eψ,∀E∈TS. Let S be an m−dimensional Riemannian manifold with the Riemannian metric g, and let u1,u2,…um be an orthogonal basis of TS. As a consequence of (24), we have(25)∇ψ2=∑i=1muiψ2.The Laplacian ofψ is defined by(26)Δψ=∑i=1m∇uiuiψ−uiuiψ.Hopf’s lemma is now described.Lemma 2 (see [27]).
LetS be an n-dimensional connected compact Riemannian manifold. If ψ is a differentiable function on S such that Δψ≥0 everywhere on S (or Δψ≤0 everywhere on S), then ψ is a constant function.


## 3. Main Results

In the present section, first we trace the existence of BW-P submanifoldsS=S1×ψ1S2×ψ2S3 for any Riemannian submanifolds S1, S2, and S3 in Sasakian manifolds with warping functions ψ1 and ψ2 and then we demonstrate our key findings. Hasegawa and Mihai [6] set up the following result.Theorem 2.
LetS¯ be a 2m+1−dimensional Sasakian manifold. Then, there do not exist W-P submanifolds S=S⊥×ψST such that S⊥ is an anti-invariant submanifold tangent to ξ and ST an invariant submanifold of S¯.We draw the conclusion based on the above result; that is, ifST, S⊥, and Sθ are invariant, anti-invariant, and pointwise proper slant submanifolds, then BW-P submanifolds of the forms S⊥×ψ1ST×ψ2Sθ and Sθ×ψ1S⊥×ψ2ST in a Sasakian manifold do not exist. From [23], we have the following observation.Theorem 3.
LetS¯ be a 2m+1− dimensional Sasakian manifold. Then, there do not exist W-P submanifolds S=Sθ×ψST tangential to S such that Sθ is pointwise proper slant submanifold and ST is invariant submanifold of S¯, respectively.It can be deduced by Theorem2 that BW-P submanifolds of the types Sθ×ψ1ST×ψ2S⊥ and S⊥×ψ1Sθ×ψ2ST in a Sasakian manifold are trivial.Park identified the existence of the warped product pointwise semislant submanifolds of Sasakian manifolds of the formST×ψSθ in his paper [23], with warping function ψ, where ST and Sθ are the holomorphic and pointwise slant submanifolds of S¯, as well as proving the next lemma.Lemma 3.
LetS=ST×ψSθ be a W-P pointwise semislant submanifold of a Sasakian manifold S¯ such that ξ∈TST, where ST and Sθ are invariant and pointwise slant submanifolds of S¯, respectively. Then,(27)gσE,G,NTH=cos2θElnψgG,H−ϕElnψgG,TH−ηEgG,TH,for any E∈TST and G,H∈TSθ.Consider the biwarped product submanifolds of the typeST×ψ1S⊥×ψ2Sθ of a Sasakian manifold S¯,ϕ,ξ,η with warping functions ψ1 and ψ2 such that ST, S⊥, and Sθ are the invariant, anti-invariant, and pointwise slant submanifolds of S¯ correspondingly. To address the question of which factor of the biwarped product submanifold is parallel to ξ, we have the following.for all E∈TST and G∈TSθ.where u0=ξ,u1,u2,…,up,ϕu1,ϕu2,…,ϕup and u1,u2,…,uq,secθTu1,…,secθTuq are the basis of the orthonormal vector fields on TST and TSθ, respectively.Proof. Choosing unit vector fields E∈TST, F∈TS⊥, and G∈TSθ and using (5) and (19), we have(49)R¯E,ϕE,G,NG=−c−12sin2θE2G2,(50)R¯E,ϕE,F,ϕF=−c−12E2F2.Theorem 4.
LetS¯ be a Sasakian manifold. If ST×ψ1S⊥×ψ2Sθ is a biwarped product submanifold of S¯ such that ST, S⊥, and Sθ are the invariant, anti-invariant, and pointwise slant submanifolds of S¯, respectively, then we have the following:(i)
Ifξ∈TS⊥, ψ1 is constant(ii)
Ifξ∈TSθ, ψ2 is constantProof.
The proof of this theorem can be deduced directly from Proposition 1 in [17] for β=0.Remark 1.
Proposition 1 in [17] was proved for slant distributions, and the same proof is also valid for pointwise slant distributions.
As a consequence of the above, we can point out that there are no any nontrivial BW-P submanifolds of the typeST×ψ1S⊥×ψ2Nθ of a Sasakian manifold if the vector field ξ is tangential to S⊥ or Sθ.
Now, letS=ST×ψ1S⊥×ψ2Sθ be a BW-P submanifold of a Sasakian manifold S¯ and consider the vector field ξ tangent to be ST. If D is invariant distribution, D⊥ is anti-invariant and Dθ is pointwise slant distribution with the slant function θ. The following decomposition refers to the tangent bundle TM:(28)TS=D⊕D⊥⊕Dθ⊕ξ.
The normal bundleT⊥S is decompounded as(29)T⊥S=ϕD⊥⊕NDθ⊕μ,where μ is the invariant orthogonal complementary distribution of ϕD⊥⊕NDθ in T⊥S.
The second fundamental formσ can be written as a consequence of the above direct decomposition.(30)σE1,E2=σϕD⊥E1,E2+σNDθE1,E2+σμE1,E2,for E1,E2∈TS, where σϕD⊥E1,E2, σNDθE1,E2, and σμE1,E2 are the components of σE1,E2 in the normal sub-bundles ϕD⊥, NDθ and μ, respectively. Moreover if F1,F2,…,Fq be a local orthonormal frame of vector fields of Dθ, then(31)σNDθE1,E2=∑r=1qσrE1,E2NFr,where(32)σrE1,E2=csc2θgσE1,E2,NFr.
We create an example of a BW-P submanifold of the formS=ST×ψ1S⊥×ψ2Sθ in Sasakian manifold with ξ∈ST.Example 1.
It is well known thatR2m+1,ϕ0,ξ,η,g denotes a Sasakian manifold with its standard Sasakian structure given by(33)η=12dz−∑i=1myidxi,ξ=2∂∂z,g=η⊗η+14∑i=1mdxi⊗dxi+dyi⊗dyi,ϕ0∑i=1mXi∂∂xi+Yi∂∂yi+Z∂∂z=∑i=1mYi∂∂xi−Xi∂∂yi+∑i=1mYiyi∂∂z.
Consider the submanifold as follows:(34)S=2u,0,wet,0,0,v,0,setcosθ,setsinθ,sint,f∈R11.
And consider a frameu1,u2,u3,u4,u5,u6 of orthogonal vector fields tangent to M as(35)u1=2∂∂x1+y1∂∂z,u2=2∂∂y1,u3=2et∂∂x3,u4=2etcosθ∂∂y3+2etsinθ∂∂y4,u5=2sint∂y5,u6=2∂∂z=ξ.
It is then simple to note thatD= span u1,u2,u6, D⊥= span u5, and Dθ=span u3,u4 defined as the invariant, anti-invariant, and pointwise slant distributions with the slant function θ∈0,π/2 on the Sasakian manifold R11. If we denote the integral manifold of D, D⊥, and Dθ by ST, S⊥, and Sθ correspondingly, then the metric g on S is given by(36)g=gST+sin2tgS⊥+e2tgSθ.
Then,S=ST×ψ1S⊥×ψ2Sθ is a BW-P submanifold with the warping functions ψ1=sint, ψ2t=et.
First, we demonstrate some preliminary findings.Lemma 4.
LetST×ψ1S⊥×ψ2Sθ be a BW-P submanifold of a Sasakian manifold S¯. Then,(i)
ξlnψ1=0 and ξlnψ2=0(ii)
gσϕE,G,NG=ElnψG2(iii)
gσϕE,H,JH=ElnψH2(iv)
gσϕE,G,ϕσE,G=σμE,G2+cos2θElnψ2G2(v)
gσϕE,H,ϕσE,H=σμE,H2for all E∈TNT, H∈TN⊥, and G∈TNθ, where σμ is the μ component of the second fundamental form σ.Proof.
The part (i) can be deduced from Proposition 2 of [17]. Moreover, the parts (ii) and (iii) can be concluded from equations (30) and (20) in [17], respectively.
Using (6) and (3), we can prove part (iv) as(37)σϕE,G=−ηEG+ϕσE,G+ϕ∇GE−∇GϕE.
By applying (23), the above equation can now be written as(38)σϕE,G=−ηEG+ϕσE,G+Elnψ2ϕG−ϕElnψ2G.
Comparing the normal parts,(39)σϕE,G=ϕσμE,G+Elnψ2NG.
On taking Riemannian product withϕσE,G, we find(40)gσϕE,G,ϕσE,G=σμE,G2+Elnψ2gϕσE,G,NG.
Calculating the last term of (40) by using (3) and (6) and (23),(41)gϕσE,G,NG=gσϕE,G,NG−sin2θElnψ2G2.
Utilizing part (ii), we get(42)gϕσE,G,NG=cos2θElnψ2G2.
Using the above equation in (40), we get the required result. Part (v) of the lemma can also be verified in a similar way.Lemma 5.
LetS=ST×ψ1S⊥×ψ2Sθ be a BW-P submanifold of a Sasakian manifold S¯. Then,(43)gσE,TG,NG=−gσE,G,NTG=−cos2θElnψG2,Proof.
The proof of the present lemma can be concluded from equation (33) in [17] for β=0.Lemma 6.
On a BW-P submanifoldS=ST×ψ1S⊥×ψ2Sθ of a Sasakian manifold S¯, we have(44)∑i=1p∑j,k=12qgσϕui,uk,Nujgσui,Tuk,Nuj−gσui,uk,Nujgσϕui,Tuk,Nuj=−4qcos2θ∇lnψ22,Proof.
First, we will modify the left-hand term as follows:(45)∑i=1p∑j,k=12qgσϕui,uk,Nujgσui,Tuk,Nuj=∑i=1p∑j=12qgσϕui,uj,Nujgσui,Tuj,Nuj+∑j≠k=12qgσϕui,uk,Nujgσui,Tuk,Nuj=∑i=1p∑j=12qgσϕui,uj,Nujgσui,Tuj,Nuj+∑j=1qgσϕui,uj,Nuj+qgσui,Tuj,Nuj+q+∑j=1qgσϕui,uj+q,Nujgσui,Tuj+q,Nuj=∑i=1p∑j=12qgσϕui,uj,Fujgσui,Puj,Fuj+sec2θ∑j=1qgσϕui,uj,NTujgσui,Tuj,NTuj−∑j=1qgσϕui,Tuj,Nujgσui,uj,Nuj..
On using part (ii) of Lemmas4 and 5 and combining (25), we find(46)∑i=1p∑j,k=12qgσϕui,uk,Nujgσui,Tuk,Nuj=∑i=1p−2qcos2θuilnψ22−2qcos2θϕuilnψ22−2qϕuilnψ2ηui=−2qcos2θ∇lnψ22.
Replacingui by ϕui in the above equation, we get(47)∑i=1p∑j,k=12qgσui,uk,Fujgσϕui,Tuk,Nuj=2qcos2θ∇lnψ22.
We get the required result while subtracting the above two findings.
The following characterization is now established.Theorem 5.
LetS=ST×ψ1S⊥×ψ2Sθ be a BW-P submanifold of a Sasakian space form S¯c such that ST is a compact submanifold. The following characterization is now established. If the following inequalities hold, S is a Riemannian product submanifold.(48)∑i=12p∑j=12qσμui,uj2+∑k=1rσμui,fk2≤pc−1qsin2θ+r2−−2qcos2θ+2cot2θ∇lnψ22∑i=1p∑j=12qgσμϕui,uj,σμui,Tuj≥0,where σμ represents the projection of σ in μ and 2p+1, 2q, and r are the dimensions of ST, Sθ, and S⊥ subsequently while f1,f2,…,fr is a local orthonormal basis of TS⊥.Then, again by the Codazzi equation,(51)R¯E,ϕE,G,NG=g∇E⊥σϕE,G,NG−g∇ϕX⊥σE,G,NG+gσE,∇ϕEG,NG−gσϕE,∇EG,NG−gσ∇EϕN,G,NG+gσ∇ϕEE,G,NG.The estimations of the terms engaged with (51) are presently processed. Foremost, we have(52)g∇E⊥σϕE,G,NG=EgσϕE,G,NG−gσϕE,G,∇E⊥NG.Applying the part (ii) of Lemma4 in the last equation, we find(53)g∇E⊥σϕE,G,NG=E2lnψ2G2+2Elnψ22G2−gσϕE,G,∇E⊥NG.Calculating the last term of (53) and using (10), we have(54)gσϕE,G,∇E⊥NG=gσϕE,G,∇¯EϕG−TG.By the use of (6) and (12), the previous equation changes to(55)gσϕE,G,∇E⊥NG=gσϕE,G,∇¯EϕG+ϕ∇¯EG−gσϕE,G,σE,TG.By the application of (3), (6), and (23) and part (ii) and (iii) of Lemma 4, we get(56)gσϕE,G,∇E⊥NG=Elnψ221+cos2θG2+σμE,G2−gσϕE,G,σE,TG.Making use of (56) in (53), we find(57)g∇E⊥σϕE,G,NG=E2lnψ2G2+Elnψ22sin2θG2−σμE,G2+gσϕE,G,σE,TG.In similar fashion, we are able to write(58)g∇ϕE⊥σE,G,NG=−ϕE2lnψ2G2−ϕElnψ22sin2θG2+σμϕE,G2+gσE,G,σϕE,TG.We have the following from part (ii) of Lemma4:(59)gANGG,ϕE=Elnψ2G2.Changing outE by ∇EE (applying the totally geodesicness of ST, ∇EE∈TST) in the previous equation, we obtain(60)gANGG,ϕ∇EE=∇EElnψ2G2.By (6), the equation above has the following form:(61)gANGG,ϕ∇¯EE−ϕσE,E=∇EElnψ2G2.It is simple to verify thatσE,F∈μ, for all E,F in TST by using the fact that the first factor ST is totally geodesic in S. Substituting this and (12) in the last equation, we obtain(62)gσ∇EϕE,G,NG=∇EElnψ2G2.Adopting similar steps, we can put(63)gσ∇ϕEE,G,NG=−∇ϕEϕElnψ2G2.By part (ii) of Lemma4 and (23), we get(64)gσϕX,∇EG,NG=Elnψ22G2,(65)gσE,∇ϕEG,NG=−ϕElnψ22G2.Substituting values of (49) and (57)–(65) in (51), we obtain(66)−c−12sin2θE2G2=E2lnψ2G2+ϕE2lnψ2G2−Elnψ22cos2θG2−ϕElnψ22cos2θG2−σμE,G2−σμϕE,G2−∇EElnψ2G2−∇ϕEϕElnψ2G2+gσϕE,G,σE,TG−gσE,G,σϕE,TG.On using (30), (32), (6), and (3), the previous equation becomes(67)−c−12sin2θE2G2=E2lnψ2G2+ϕE2lnψ2G2−Elnψ22cos2θG2−ϕElnψ22cos2θG2−σμE,G2−σμϕE,G2−∇EElnψ2G2−∇ϕEϕElnψ2G2+csc2θ∑j=12qgσϕE,G,NFjgσE,TG,NFj−gσϕE,G,NFjgσϕE,TG,NFjFj2+2gσμϕE,G,σμE,TG.Letu0=ξ,u1,u2,…,up,up+1=ϕu1,up+2=ϕu2,…,u2p=ϕup be the orthonormal frame on TST and u1,u2,…,uq,secθTu1,secθTu2,…,secθTuq be an orthonormal frame on TSθ. Taking sum of the above equation with the indices i=1,2,…,p and j=1,2,…2q and making use of (25) and (26) and part (iii) of Lemma 4, we get(68)2qΔlnψ2=pqc−1sin2θ−2qcos2θ∇lnψ22−∑i=12p∑j=12qσμui,uj2−4qcot2θ∇lnψ22+2∑i=1p∑j=12qgσμϕui,uj,σμui,Tuj.In the similar way, forE∈TST and F∈TS⊥ again using the Codazzi equation, we can prove the following:(69)−c−12E2F2=E2lnψ1F2+ϕE2lnψ1F2−σμE,F2−σμϕE,F2−∇EElnψ1F2−∇ϕEϕElnψ1F2.Letf1,f2,…,fr be an orthonormal frame of TS⊥. Taking sum by using i=1,2,…,p and l=1,2,…,r and simultaneously using (24) and (25), the above equation yields the following:(70)rΔlnψ1=c−1.p.r2−∑i=12p∑l=1rσμui,fl2.From (69),(71)∑i=12p∑j=12qσμui,uj2≤pqc−1sin2θ−2qcos2θ+2cot2θ∇lnψ22∑i=1p∑j=12qgσμϕui,uj,σμui,Puj≥0.This meansΔlnψ2≥0, so by application of Hopf’s Lemma, lnψ2 is constant that indicates ψ2 is constant. Moreover, in (70), if(72)∑i=12p∑l=1rσμui,fl2≤c−1.p.r2,then Δlnψ1≥0, so by Hopf’s lemma, lnψ1 is constant that implies that the W–F ψ1 is constant. We get the necessary result when these two statements are combined.The squared norm of the second fundamental form is obtained using the warping functions and the slant function in the following theorem.Theorem 6.
LetS¯c be a 2n+1−dimensional S-C-F and ST×ψ1S⊥×ψ2Sθ be an m−dimensional BW-P submanifold such that ST is a 2p−dimensional invariant submanifold, S⊥ is a r− dimensional anti-invariant submanifold, and Sθ be a 2q−dimensional proper pointwise slant submanifold of S¯c. If(73)∑i=1p∑j=12qgσϕui,uj,σui,Puj≥0,then(i)
The squared norm of the second fundamental formσ satisfies(74)σ2≥pc−1qsin2θ+r2+2qsin2θ∇lnψ22+r∇lnψ12−2qΔlnψ2−rΔlnψ1.(ii)
The equality sign of (74) satisfies identically if and only if(i)
ST is T-G invariant submanifold of S¯c. Hence, ST is a S–C-F.(ii)
S⊥ and Sθ are T-U submanifolds of S¯c.(iii)
∑i=1p∑j=12qgσϕui,uj,σui,Puj=0.Proof.
From (69), we have(75)∑i=12p∑j=12qσμui,uj2≥pqc−1sin2θ−2qcos2θ+2cot2θ∇lnψ2−2qΔlnψ.
For the orthonormal framesu0=ξ,u1,u2,…,up,up+1=ϕu1,up+2=ϕu2,…,u2p=ϕup and u1,u2,…,uq,secθTu1,secθTu2,…,secθTuq, in view of formulae (31) and (32) and part (ii) of Lemma 4, we get(76)∑i=02p∑j=12qσNDθui,uj2=∑i=02p∑j,k=12qcsc2θgσui,uj,Nuk2=csc2θ∑i=02p∑j=12qgσui,uj,Nuj2+∑j≠k=12qgσui,uj,Nuk2=csc2θ∑i=02p2quilnψ2+sec2θ∑j=1qgσui,uj,NTuj2+gσui,Tuj,Nuj2.
Further, using Lemma5 and (25), the above equation is reduced to(77)∑i=02p∑j=12qσFDθui,uj2=2qcsc2θ∇lnψ2+2qcot2θ∇lnψ2.
Now, for anyE∈TNT and F∈TN⊥, from part (iii) of Lemma 1, we have(78)gσϕE,F,ϕF=Elnψ1F2,gσξ,F,ϕF=0.
By the above equations for the frameu0=ξ,u1,u2,…,up,up+1=ϕu1,up+2=ϕu2,…,u2p=ϕup and f1,f2,…,fr, it is simple to conclude that(79)∑i=12p∑l=1rhϕD⊥ui,fl2=r∇lnψ12.
Moreover, from (3) and (23), we get(80)gσE,G,ϕF=0,gσE,F,NG=0,for all E∈TST, G∈TSθ, and F∈TS⊥. We can deduce the following from these two findings:(81)∑i=12p∑j=12qσϕD⊥ui,uj2=0,(82)∑i=12p∑k=1rσϕDθui,fk2=0.
From (75), (77), (79), (81), and (82), we get the required inequality.
To prove the part (ii), letσ′ and σ″ be the second fundamental forms for the immersion of Sθ and S⊥ in S, respectively. Then, for any G,K∈TSθ and E∈TST, using the Gauss formula, we have(83)gσ′G,K,E=g∇GK,E=−Elnψ2gG,K.
By (24), we obtain(84)gσ′G,K,E=−gG,Kg∇lnψ2,E,or(85)σ′G,K=−gG,K∇lnψ2.
Accordingly, for anyF1,F2∈TS⊥ and E∈TST, we have(86)σ″F1,F2=−gF1,F2∇lnψ1.
If the equality sign of (74) holds identically, then we have(87)σD,D=0,σD⊥,D⊥=0,σDθ,Dθ=0,(88)gσμϕD,Dθ,σμD,TDθ=0.
The first condition of (87) suggests that ST is T-G submanifold in S. Then, again it is not difficult to see that gσE1,ϕE2,NG=0 andgσE,ϕE2,ϕF=0, for all E1,E2∈TST,G∈TSθ,andF∈TS⊥ It follows that ST is T-G in S¯c and hence is a S-C-F. The second condition of (87) with (86) implies that S⊥ is T-U. Besides, the third condition of (87) along with (85) suggests that Sθ is a T-U submanifold. This demonstrates the proof.


## 4. Conclusion

In this paper, by utilizing Hopf’s Lemma, we acquired the describing inequalities for the existence of biwarped product submanifolds of Sasakian space forms. Besides, we additionally worked out an assessment for the squared norm of the second fundamental form in terms of the warping function and slant function. To fortify our study, we gave a nontrivial example of a biwarped product submanifold in a Sasakian manifold.



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*Source: 9966248-2021-08-20.xml*