# A General Inequality for CR-Warped Products in Generalized Sasakian Space Form and Its Applications

Authors: Yanlin Li; Akram Ali; Rifaqat Ali
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

In the present paper, by considering the Gauss equation in place of the Codazzi equation, we derive new optimal inequality for the second fundamental form of CR-warped product submanifolds into a generalized Sasakian space form. Moreover, the inequality generalizes some inequalities for various ambient space forms.


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## Body

## 1. Introduction

The fundamental idea of warped product manifolds was first initiated in [1] with manifolds of negative curvature. Let N1 and N2 be two Riemannian manifolds endowed with Riemannian matrices g1 and g2, respectively, such that f:N1⟶0,∞ is a positive smooth function on N1. Then, the warped product M=N1×fN2 is characterized as the product manifold N1×N2 with the equipped metric g=g1+f2g2. In particular, if f=constant, then M turned to be a Riemannian product manifold; otherwise, M is called a nontrivial warped product manifold. Let M=N1×fN2 be a nontrivial warped product manifold. Then,
(1)∇XZ=∇ZX=XlnfZ,for any vector fields X,Y∈ΓTN1 and Z∈ΓTN2. If we consider a local orthonormal frame e1,e2,⋯,en such that ei1≤i≤n1∈N1 and ejn1+1≤j≤n∈N2, we have
(2)∑1≤i≤n1∑n1+1≤j≤nKei∧ej=n2Δff.In [2], Chen established the inequality for the squared norm of the mean curvature and the warping function f of a CR-warped product NT×fN⊥, where N⊥ is a totally real submanifold and NT is a holomorphic submanifold, isometrically immersed in a complex space form as follows.Theorem 1 (see [2]).
NTn1×fN⊥n2 be a CR-warped product into a complex space form M~4c with constant sectional curvature c. Then,
(3)h2≥2n2∇lnf2+Δlnf+2n1c,where Δ is the Laplacian operator of NT. Moreover, the equality holds if and only if NT is totally geodesic and N⊥ is totally umbilical in M~4c.Moreover, Theorem1 is extended to CR-warped product submanifolds in a generalized Sasakian space form by using the same technique.Theorem 2 (see [3]).
LetNTn1×fN⊥n2 be a contact CR-warped product submanifold of a generalized Sasakian space form M~λ1,λ2,λ3 such that the structure vector field ξ is tangent to base manifold. Then, the following inequality is satisfied:
(4)h2≥2n2∇lnf2−Δlnf+1+4n1n2λ1+1,where Δ denotes the Laplace operator on NTn1. The equality holds if and only if NTn1 is a totally geodesic submanifold of M~λ1,λ2,λ3; in this case, NTn1 is a generalized Sasakian space form of λ1+3λ3.Furthermore, Mustafa et al. [4] recalled some fundamental problems of CR-warped products in Kenmotsu space forms as to simple relationships between the second fundamental form and the main intrinsic invariants by using the Gauss equation. In [5–7], some sharp inequalities are established for the sectional curvature of warped product pointwise semislant submanifolds in various space forms such as a Sasakian space form, a cosymplectic space form, a Kenmotsu space form, and a complex space form in terms of the Laplacian and the squared norm of a warping function with pointwise slant immersions. Afterward, several geometers [1, 2, 4, 8–18] obtained similar inequalities for different types of warped products in different kinds of structures.Al-Ghefari et al. [3] proved the existence of CR-warped product submanifolds of type NT×fN⊥ in trans-Sasakian manifolds. They obtained an inequality for the second fundamental form with constant sectional curvature in terms of a warping function. Moreover, the nonexistence of CR-warped products of the form N⊥×fNT in a generalized Sasakian space form was proved in [19].In this paper, we shall establish a Chen-type inequality for CR-warped product submanifolds in a generalized Sasakian space form by considering the nontrivial caseNT×fN⊥. We also find some applications of the inequality in the compact Riemannian manifold by using integration theory on manifolds. Our future work then is combining the work done in this paper with the techniques of singularity theory presented in [20–23] to explore new results on manifolds.


## 2. Preliminaries

An almost contact metric manifoldM~,g,φ,η,ξ is an odd-dimensional manifold M~, endowed with a field φ of an endomorphism on the tangent space, the Reeb vector field ξ, a 1-form η and admits Riemannian metric g satisfying
(5)φ2=−I+η⊗ξ,ηξ=1,(6)gφU,φV=gU,V−ηXηY,(7)ηU=gU,ξ,for any U,V∈ΓTM~. An almost contact metric manifold M~,g,φ,η,ξ is said to be trans-Sasakian manifold (cf. [12, 13]) if
(8)∇~UφV=αgU,Vξ−ηUV−βgφU,V−ηVφU,for anyU,V∈ΓTM~, where ∇~ is the Riemannian connection on M~,g. If we replace U=ξ and V=ξ in (8), we find that ∇~ξφξ=0, which implies that ∇~ξξ=0. For a trans-Sasakian manifold, (8) implies
(9)∇~Xξ=−αφX+βX−ηXξ.Remark 3.
We classify a trans-Sasakian manifold in the following way:(a)
Ifα=0 and β=0 in (8), a trans-Sasakian manifold becomes a cosymplectic manifold [7](b)
Ifα=1 and β=0 in (8), it is a Sasakian manifold [5](c)
Ifα=0 and β=1 in (8), it is a Kenmotsu manifold [6](d)
α-Sasakian manifold and β-Kenmotsu manifold can be derived from the tans-Sasakian manifold when β=0 and α=0 in (8), respectivelyGiven an almost contact metric manifoldM~,φ,η,ξ, it is said to be a generalized Sasakian space form M~λ1,λ2,λ3 if there exist three functions λ1,λ2, and λ3 on M~ such that the curvature tensor R~ is
(10)R~X,Y,Z,W=λ1gX,WgY,Z−gX,ZgY,W+λ2gX,φZgφY,W−gY,φZgφX,W+2gX,φYgφZ,W+λ3ηXηZgY,W−ηYηZgX,W+gX,ZηYηW−gY,ZηXηW,for any X,Y,Z,W∈ΓTM~ [24].Remark 4.
The characteristics are as follows:(a)
Ifλ1=c+3/4 and λ2=λ3=c−1/4, then M~ is a Sasakian space form [25](b)
Ifλ1=c−3/4 and λ2=λ3=c+1/4, then M~ is a Kenmotsu space form [6](c)
Ifλ1=λ2=λ3=c/4, then M~ is a cosymplectic space form [26]Let∇ and ∇⊥ be the induced Riemannian connections on the tangent bundle TM and the normal bundle T⊥M of a submanifold M of an almost contact metric manifold M~,φ,η,ξ with the induced metric g. Then, the Gauss and Weingarten formulas are given by
(11)i∇~UV=∇UV+hU,V,ii∇~UN=−ANU+∇U⊥N,for U,V∈ΓTM and N∈ΓT⊥M, where h and AN are the second fundamental form and the shape operator on M. We have the relation:
(12)ghU,V,N=gANU,V,for U,V∈ΓTM and N∈ΓT⊥M. For any tangent vector U∈ΓTM and normal vector N∈ΓT⊥M, we have
(13)iφU=TU+FU,iiφN=tN+fN,where TUtN and FUfN are tangential and normal components of φUφN, respectively. If T is identically zero, then a submanifold M is called a totally real submanifold. The Gauss equation with curvature tensors R~ and R on M~ and M, respectively, is defined by
(14)R~U,V,Z,W=RU,V,Z,W+ghU,Z,hV,W−ghU,W,hV,Z,for any U,V,Z,W∈ΓTM. The mean curvature vector H for a local frame e1,e2,⋯,en of the tangent space TM on M is defined by
(15)iH=1ntraceh=1n∑i=1nhei,ei,iiT2=∑i,j=1ng2φei,ej.The scalar curvatureτ for a Riemannian submanifold M is given by
(16)τ=∑1≤i=j≤nKei∧ej,where Kei∧ej is the sectional curvature of section plane and spanned by ei and ej. Let Gr be an r-plane section on TM and let e1,e2,⋯,er be a orthonormal basis of Gr. Then, the scalar curvature τGr of Gr is given by
(17)τGr=∑1≤i=j≤rKei∧ej.Similarly, we classify a Riemannian submanifoldM said to be totally umbilical and totally geodesic if hU,V=gU,VH and hU,V=0, respectively, for any U,V∈ΓTM.Furthermore, ifH=0, then M is minimal in M~,φ,η,ξ. If φ preserves any tangent space of M tangent to the structure vector field ξ, i.e., φTpM⊆TpM, for each p∈M; then, M is called an invariant submanifold. Similarly, M is called an anti-invariant submanifold tangent to the Reeb vector field ξ if φTpM⊆T⊥M, for each p∈M. To generalize these definitions, we give the following definition.Definition 5.
A submanifoldM including the structure vector field ξ of an almost contact metric manifold M~,φ,η,ξ is characterized to be a contact CR-submanifold if the pair of orthogonal distributions D and D⊥ exists such that
(i)
TM=D⊕D⊥⊕<ξ>, where <ξ> is 1-dimensional distribution spanned by ξ(ii)
the distributionDis invariant, i.e., φD⊆D(iii)
the distributionD⊥ is anti-invariant, i.e., φD⊥⊆T⊥MIf the dimensions ofinvariant distribution D and anti-invariant distribution D⊥ of a contact CR-submanifold of M~,φ,η,ξ are d1 and d2, respectively, such that d2=0, then M is invariant and anti-invariant if d1=0. It is called a proper contact CR-submanifold if neither d1=0 nor d2=0. The normal bundle T⊥M of a contact CR-submanifold with an invariant subspace μ under φ can be decomposed as
(18)T⊥M=φD⊥⊕μ.M is a compact orientable Riemannian submanifold without boundary. Thus, we have
(19)∫MΔfdV=0,where dV is the volume element of M [27].


## 3. Main Inequalities of CR-Warped Products

We are mentioning that in the following study, we shall consider the structure fieldξ tangent to the base manifold of warped product manifold. In this main section, we classify the contact CR-warped product submanifolds in a trans-Sasakian manifold.Lemma 6.
LetM=NT×fN⊥ be a CR-warped product submanifold in a trans-Sasakian manifold. Then,
(20)ghφX,Y,φZ=ghX,Y,φZ=0,ghX,X,β=−ghφX,φX,β,for X,Y∈ΓTNT, Z,W∈ΓTN⊥, and β∈Γμ.Proof.
From (11)(i), (8), and (5), we obtain
(21)ghφX,Y,φZ=g∇¯YφX,φZ=g∇¯YX,Z−η∇¯YXηZ.
SinceNT is totally geodesic in M with ξ∈ΓTNT, (9) implies the results.☐Lemma 7.
Letℓ:M=NT×fN⊥⟶M~,φ,η,ξ be an isometric immersion from an n-dimensional contact CR-warped product submanifold into a trans-Sasakian manifold M~,φ,η,ξ such that NT is invariant submanifold of dimension n1=2d1+1 tangent to ξ. Then, NT is always ℓ-minimal submanifold of M~.Proof.
We skip the proof of the above lemma due to the similar proof of Theorem 4.2 in [4].☐By helping the above lemma, the following result can be obtained as follows.Proposition 8.
Assume thatℓ:M=NT×fN⊥⟶M~ is an isometric immersion of an n-dimensional contact CR-warped product submanifold M into a trans-Sasakian manifold M~. Thus,
(i)
the squared norm of the second fundamental form ofM is satisfied:(22)h2≥2n2∇lnf2+τ~TM−τ~TNT−τ~TN⊥−n2Δlnf,where n2 is the dimension of anti-invariant submanifold N⊥ and Δ is the Laplacian operator of NT(ii)
the equality holds in (22) if and only if NT is totally geodesic and N⊥ is totally umbilical in M~. Moreover, M is minimal submanifold of M~Proof.
It can be easily proven as the proof of Theorem 4.4 in [4] if we consider a Riemannian submanifold as a CR-warped product submanifold, and the base manifold is a trans-Sasakian manifold instead of a Kenmotsu manifold.
Now, we prove our main theorem using Proposition8 for a generalized Sasakian space form.☐Theorem 9.
Letℓ:M=NT×fN⊥⟶M~λ1,λ2,λ3 be an isometric immersion from an n-dimensional contact CR-warped product submanifold of a generalized Sasakian space form M~λ1,λ2,λ3. Then, the second fundamental form is given by
(23)h2≥2n2∇lnf2+λ1n1+32λ2−λ3−Δlnf,where n1=dimNT, n2=dimN⊥, and Δ is the Laplacian operator on NT. The equality holds in (23) if and only if NT and N⊥ are totally geodesic and totally umbilical submanifolds in M~λ1,λ2,λ3, respectively, and hence, M is a minimal submanifold of M~λ1,λ2,λ3.Proof.
SubstitutingX=W=ei and Y=Z=ej in (10), we get
(24)R~ei,ej,ej,ei=λ1gei,eigej,ej−gei,ejgei,ej+λ2gei,φejgφej,ei−gei,φeigej,φej+2g2φej,ei+λ3gei,eiηejηej−gei,ejηeiηei+gej,ejηeiηeigej,eieiηej.
Summing up along the orthonormal vector fields ofM, it can be derived from the above as
(25)2τ~TM=λ1nn−1+3λ2∑1≤i=j≤ng2φei,ej−2λ3n−1.
As for ann-dimensional CR-warped product submanifold tangent ξ, one can derive T2=n−1 from (15)(ii); we obtain
(26)2τ~TM=λ1nn−1+3n−1λ2−2λ3n−1.
On the other hand, by helping the frame field ofTN⊥, we have
(27)2τ~TN⊥=λ1n2n2−1.
Similarly, we considered thatξ is tangent to invariant submanifold NT. Then, using the frame vector fields of TNT, we get from (24)
(28)2τ~TNT=λ1n1n1−1+3λ2n1−1−2λ3n1−1.
Therefore, using (26), (27), and (28) in Proposition 8, we get the required result. The equality case follows from Proposition 8. Thus, the proof is completed.☐


## 4. Geometric Applications

Remark 10.
Considerλ1=c−3/4 and λ2=λ3=c+1/4 in Theorem 9. It is the generalization of Theorem 4.6 in [4] for the result of contact CR-warped products in Kenmotsu space forms.Remark 11.
If we putλ1=c+3/4 and λ2=λ3=c−1/4 in Theorem 9, then it generalizes Corollary 4.6 in [5].Remark 12.
Ifλ1=λ2=λ3=c/4 in Theorem 9, then Theorem 9 coincides with Theorem 1.2 in [26].Corollary 13.
Letlnf be a harmonic function on NT. Then there does not exist any CR-warped product submanifold NT×fN⊥ into a generalized Sasakian space form M~λ1,λ2,λ3withc≤−λ1.Corollary 14.
Assume thatlnf is a nonnegative eigenfunction on NT with the corresponding nonzero positive eigenvalue. Then, there does not exist any CR-warped product submanifold NT×fN⊥ into a generalized Sasakian space form M~λ1,λ2,λ3 with c≤−λ1.Theorem 15.
LetM=NT×fN⊥ be a compact orientated CR-warped product into a generalized Sasakian space form M~λ1,λ2,λ3. Then, M is a simply Riemannian product if
(29)h2≥2λ1n1n2+3λ2n2−2λ3n2,where n1=dimNT and n2=dimN⊥.Proof.
From Theorem9, we get
(30)h2≥2λ1n1n2+3λ2n2−2λ3n3−n2Δlnf+n2∇lnf2,n2∇lnf2+2λ1n1n2+3λ2n2−2λ3n3−h2≤n2Δlnf.
We obtain(31)∫NT×q2λ1n1n2+3λ2n2−2λ3n2+n2∇lnf2−h2dV≤n2∫NT×qΔlnfdV=0.
Now, if(32)h2≥2λ1n1n2+3λ2n2−2λ3n2.
Then, from (31), we find
(33)∫NT×q∇lnf2dV≤0,which is impossible for a positive integral function, and hence, ∇lnf=0, i.e., f is a constant function on NT. Thus, by the definition of a warped product manifold, M is trivial. The converse part is straightforward.☐Corollary 16.
Assume thatM=NT×fN⊥ is a CR-warped product submanifold in a generalized Sasakian space form M~λ1,λ2,λ3. Let NT be a compact invariant submanifold and γ be nonzero eigenvalue of the Laplacian on NT. Then,
(34)∫NT×qh2dVT≥2λ1n1n2+3λ2n2−2λ3n2VolNT+2n2γ∫NT×qlnf2dVT.Proof.
From the minimum principle property, we obtain(35)∫NT∇lnf2dVT≥γ∫NTlnf2dVT.
From (23) and (35), we get the required result (34).☐



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