# Study of Differential Equations on Warped Product Semi-Invariant Submanifolds of the Generalized Sasakian Space Forms

Authors: Ibrahim Al-Dayel
Journal: Advances in Mathematical Physics
 (2021)

---

## Abstract

The purpose of the present paper is to study the applications of Ricci curvature inequalities of warped product semi-invariant product submanifolds in terms of some differential equations. More precisely, by analyzing Bochner’s formula on these inequalities, we demonstrate that, under certain conditions, the base of these submanifolds is isometric to Euclidean space. We also look at the effects of certain differential equations on warped product semi-invariant product submanifolds and show that the base is isometric to a special type of warped product under some geometric conditions.


---

## Body

## 1. Introduction

Bishop and O’Neill [1] evaluated the geometry of manifolds having negative curvature and noticed that Riemannian product manifolds do have nonnegative curvature. As a result, they came up with the recommendation of warped product manifolds, which are described as follows.Consider two Riemannian manifoldsL1,g1 and L2,g2 with corresponding Riemannian metrics g1 and g2 and ψ:L1⟶R be a positive differentiable function. If x and y are projection maps such that x:L1×L2⟶L1 and y:L1×L2⟶L2, which are defined as xm,n=m and ym,n=n∀m,n∈L1×L2, then, L¯=L1×L2 is called warped product manifold if the Riemannian structure on L satisfies
(1)gE¯,F¯=g1x∗E¯,x∗F¯+ψ∘x2g2y∗E¯,y∗F¯,for all E¯,F¯∈TL¯. The function ψ represents the warping function of LTn1×L2. The Riemannian product manifold is a special case of warped product manifold in which the warping function ψ=1. The study of Bishop and O’Neill [1] revealed that these types of manifolds have a wide range of applications in physics and theory of relativity. It is well known that the warping function is the solution of some partial differential equations; for example, the Einstein field equation can be solved by the approach of warped product [2]. The warped product is also applicable in the study of space time near black holes [3].On the other hand, the analysis of differential equation on Riemannian manifolds yields some important geometric and isometric intrinsic properties. It is well known that categorization of differential equation has a major influence on the global analysis of Riemannian manifolds. Tanno [4] explored various aspects of differential equations on Riemannian manifolds in 1978. The approach of differential equations was used by the authors [5, 6] to describe the Euclidean sphere. These calculations demonstrated that a nonconstant function λ on a complete Riemannian manifold Un,g satisfies the differential equation as follows:
(2)∇2λ+kg=0,if and only if Un,g is congruent to Euclidean space Rn, where k is constant.Furthermore, under some geometric conditions, Garcia-Rio et al. [6] proved that the Riemannian manifold is isometric to the warped product U×fR, where U is a complete Riemannian manifold, R is the Euclidean line, and f is the warping function. Moreover, warping function f is the solution of the following differential equation:
(3)d2fdt2+μ1f=0,if and only if there exists a nonconstant function ϕ:Un⟶R with an eigenvalue λ1<0, which satisfies the following differential equation:
(4)Δϕ+μ1ϕ=0.The categorization of differential equations on Riemannian manifolds turns into an attractive research subject that has been explored by various researchers, for example, [7–11].Al-Dayel et al. [7] recently investigated the effect of the differential equation (3) on the Riemannian manifold Ln,g using the concircular vector field, showing that the Riemannian manifold Ln,g is isometric to the Euclidean manifold Rn. By using the gradient conformal vector field, Chen et al. [12] discovered that the Riemannian manifold Ln,g is isometric to the Euclidean space Rn. However, it has been shown in [13] that the complete totally real submanifold in CPn (complex projective space) with bounded Ricci curvature satisfying (4) is isometric to a special form of hyperbolic space.Latterly, Ali et al. [8] characterized warped product submanifolds in Sasakian space form by the approach of differential equation. The purpose of this paper is to study the impact of differential equation on warped product semi-invariant product submanifolds in the framework of generalized Sasakian space form.


## 2. Preliminaries

A2k+1-dimensional C∞-manifold L¯ is said to have an almost contact structure if there exists on L¯ a tensor field ϕ of the type 1,1, a vector field χ, and a 1-form η satisfying
(5)ϕ2=−I+η⊗χ,ϕχ=0,η∘ϕ=0,ηχ=1.On an almost contact metric manifoldL¯, there is always a Riemannian metric g that meets the following requirements:
(6)ηE=gE,χ,gϕE,ϕF=gE,F−ηEηF,for all E,F∈TL¯.An almost contact metric manifold is said to be nearly Sasakian manifold, if(7)∇¯EϕF+∇¯FϕE=−2gE,Fχ+ηFE+ηEF,for all E,F∈TL¯.In [14], Alegre et al. gave the concept of generalized Sasakian space form as that an almost contact metric manifold L¯,f,χ,η,g whose curvature tensor R¯ satisfies
(8)R¯E,F,G,W=ϕ1gF,GgE,W−gE,GgF,W−ϕ2gϕE,GgϕF,W−gϕE,WgϕF,G+2gϕE,FgϕG,W−ϕ3ηGηFgE,W−ηEgG,W+ηWηEgF,G−ηFgE,G,for any vector fields E,F,G,W and certain differentiable functions ϕ1,ϕ2,ϕ3 on L¯. A generalized Sasakian space form with functions ϕ1,ϕ2,ϕ3 is denoted by L¯ϕ1,ϕ2,ϕ3. If ϕ1=c+3/4, ϕ2=ϕ3=c−1/4, then M¯ϕ1,ϕ2,ϕ3 is a Sasakian space form M¯c [14]. If ϕ=c−3/4,ϕ2=ϕ3=c+1/4, then M¯ϕ1,ϕ2,ϕ3 is a Kenmotsu space form M¯c [14], and if ϕ1=ϕ2=ϕ3=c/4, then M¯ϕ1,ϕ2,ϕ3 is a cosymplectic space form M¯c [14].A submanifoldL of an almost contact metric manifold L¯ is called semi-invariant submanifolds (contact CR-submanifolds) if there exist two orthogonal complementary distributions D and D⊥ satisfying the following conditions:
(i)
TL=D⊕D⊥⊕ξ, where ξ is the distribution spanned by the structure vector field ξ(ii)
D is invariant distribution, i.e., ϕD⊂TL(iii)
D⊥ is anti-invariant, i.e., ϕD⊥⊆T⊥LRecently, we [15] studied warped product semi-invariant product submanifolds of the type Ln=LTn1×fL⊥n2 isometrically immersed in the generalized Sasakian space form admitting a nearly Sasakian structure, where LTn1 is an invariant submanifold of dimension n1 and L⊥n2 is a totally real submanifold of dimension n2. More precisely, the computed Ricci curvature inequalities for these submanifolds are as follows:Theorem 1.
LetLn=LTn1×fL⊥n2 be a warped product semi-invariant submanifold isometrically immersed in a generalized Sasakian space form L¯mϕ1,ϕ2,ϕ3 with nearly Sasakian structure. Then, for each orthogonal unit vector field ξ∈TxM orthogonal to χ, either tangent to NTn1 or N⊥n2, the Ricci curvature satisfies the following inequalities:
(i)
Ifξ∈TLTn1, then(9)Ricξ≤14n2H2−n2Δlnf+n2∇lnf2+3ϕ22+n+n1n2−1ϕ1−n2+1ϕ3(ii)
Ifξ∈TL⊥n2, then(10)Ricξ≤14n2H2−n2Δlnf+n2∇lnf2+n+n1n2−1ϕ1−n2+1ϕ3
The equality cases can be seen in [15].
Letf be a real-valued differential function on a Riemannian manifold Ln, then the Bochner formula [16] is stated as
(11)12Δ∇f2=RL∇f,∇f+Hf2+g∇Δf,∇f,where RL denotes Ricci tensor and Hf is the Hessian of the function f.


## 3. Main Results

In this section, we obtain some characterization by the application of Bochner’s formula.Theorem 2.
LetLn=LTn1×fL⊥n2 be a n-dimensional warped product semi-invariant product submanifold in a generalized Sasakian space form L¯mc, where LTn1 is a n-dimensional invariant submanifold and L⊥n2 is an anti-invariant submanifold. Such that Ricci curvature RLξ≥b,b>0. If ξ∈TLTn1 and satisfying the following equality:
(12)λ1+n2b=λ1n2n+n24H2−n+n1n2−1ϕ1−3ϕ22+n2+1ϕ3,
then, the base submanifoldLTn1 is isometric to Rn1 (Euclidean space).Proof.
Sinceξ∈TNTn1, by equation (9)
(13)RLξ+n2Δlnf≤14n2H2+n2∇lnf2+n+n1n2−1ϕ1+3ϕ22−n2+1ϕ3.
By the assumption thatRLξ≥b, we have
(14)b+n2Δlnf≤14n2H2+n2∇lnf2+n+n1n2−1ϕ1+3ϕ22−n2+1ϕ3.
Since the Ricci curvatureRLξ is bounded below by b≥0, then by virtue of theorem of Myers [17], the base manifold LTn1 is compact. On integrating (9) and using Green’s theorem, we have
(15)VolLTn1b≤n24∫LTn1×qH2dV+n2∫LTn1×q∇lnf2dV+∫LTn1×qn+n1n2−1ϕ1+3ϕ22−n2+1ϕ3dV,or
(16)∫LTn1×q∇lnf2dV≥bn2VolLTn1−n24n2∫LTn1×qH2dV−1n2∫LTn1×qn+n1n2−1ϕ1+3ϕ22−n2+1ϕ3dV.
SupposeHlnf denotes the Hessian of the warping function lnf, then we have
(17)Hlnf−nI2=Hlnf2+n2I2−2ngI,Hlnf,after some calculations, the above formula turns to
(18)Hlnf−tI2=2tΔlnf+t2n1+Hlnf2.
Puttingt=λ1/n1 and integrating the last equation with respect to dV (volume element), we get
(19)∫LTn1×qHlnf−λ1n1I2dV=∫LTn1×qHlnf2dV+∫LTn1×qλ12n1dV,using (11), with the fact Δlnf=λ1lnf, we have
(20)∫LTn1×qHlnf2dV=−λ1∫LTn1×q∇lnf2dV−∫LTn1×qRL∇lnf,∇lnf.
Merging (19) and (20), we derive
(21)∫LTn1×qHlnf−λ1n1I2dV=∫LTn1×qλ12n1dV−λ1∫LTn1×q∇lnf2dV−∫LTn1×qRL∇f,∇fdV.
By the assumptionRL∇f,∇f≥b, the above equation yields
(22)∫LTn1×qHlnf−λ1n1I2dV≤∫LTn1×qλ12n1dV−bVolLTn1−λ1∫LTn1×q∇lnf2dV.
Using (16), the last inequality leads to
(23)∫LTn1×qHlnf−λ1n1I2dV≤∫LTn1×qλ12n1dV−∫LTn1×qλ1bn2+bdV−λ1n2∫LTn1×qn+n1n2−1ϕ1+3ϕ22−n2+1ϕ3dV−λ1n24n2∫LTn1×qH2dV.
If (12) holds, then the above inequality produces
(24)Hlnf−λ1n1I2=0.
Therefore, we haveHlnfX,X=λ1/n1. Hence, by the application of the result of Tashiro [18], the fibre LTn1 is isometric to Rn1 (Euclidean space).
If we consider the unit vector fieldξ∈TL⊥n2, then we have the following results which can be proved by adopting similar steps in Theorem 2.☐Theorem 3.
LetLn=LTn1×fL⊥n2 be a n-dimensional warped product semi-invariant product submanifold in a generalized Sasakian space form L¯mc, where LTn1 is a n-dimensional invariant submanifold and L⊥n2 is an anti-invariant submanifold. Such that Ricci curvature RLξ≥b,b>0. If ξ∈TLTn1 and satisfying the following equality:
(25)λ1+n2b=λ1n2n+n24H2−n+n1n2−1ϕ1+n2+1ϕ3,
then, the base submanifoldLTn1 is isometric to Rn1 (Euclidean space).Now, we have the next result which is based on the study of Garcia-Rio et al. [6].Theorem 4.
LetLn=LTn1×fL⊥n2 be a warped product semi-invariant product submanifold in a generalized Sasakian space form admitting the nearly Sasakian structure L¯mc. Such that Ricci curvature RLξ>b,b>0. If ξ∈TNTn1 and satisfying the following relation:
(26)n2H2+4n1n2λ1Hlnf2=4n1n2λ1b−n+n1n2−1ϕ1−3ϕ22+n2+1ϕ3,for λ1<0, then LTn1 is isometric to warped product of the type R×θU with the warping function θ, which satisfies the differential equation dθ2/dt2+λ1θ=0.Proof.
For the warping functionlnf, defining the following equation on LTn1:
(27)blnfI+Hlnf2=b2lnf2I2+Hlnf2+2blnfgI,Hlnf.
But we know thatI2=trII∗=n1 and gHlnf,I∗=trI∗Hlnf=trHlnf; using these facts, the above equation leads to
(28)blnfI+Hlnf2=Hlnf2+n1b2lnf2−2blnfΔlnf.
Letlnf is an eigenfunction corresponding to the eigenvalue λ1 satisfying Δlnf=λ1lnf, we have
(29)blnfI+Hlnf2=Hlnf2+n1b2−2bλ1lnf2.
Further, usingΔlnf=λ1lnf, it is easy to see that
(30)∇lnf22=lnfλ1lnf−∇lnf2,which on integrating provides
(31)∫LTn1×qlnf2dV=1λ1∫LTn1×q∇lnf2.Thus, we have
(32)∫LTn1×qHlnf+blnfI2dV=∫LTn1×qHlnf2dV+n1b2λ1−2b∫LTn1×q∇lnf2dV.
Choosingb=λ1/n1 in (32), we have
(33)∫LTn1×qHlnf+λ1n1lnfI2dV=∫LTn1×qHlnf2dV−λ1n1∫LTn1×q∇lnf2dV.
Further, integrating (9) and applying Green’s lemma, we find
(34)∫LTn1×qRLξdV≤n24∫LTn1×qH2dV+n2∫LTn1×q∇lnf2dV+∫LTn1×qn+n1n2−1ϕ1+3ϕ22−n2+1ϕ3dV.
From the above two expressions, we have(35)1n2∫LTn1×qRLξdV≤n24n2∫LTn1×qH2dV+n1λ1∫LTn1×qHlnf2dV−n1λ1∫LTn1×qHlnf+λ1n1lnfI2dV+∫LTn1×qn+n1n2−1ϕ1+3ϕ22−n2+1ϕ3dV.
On using the assumption thatRLξ≥b, for b>0,(36)∫LTn1×qHlnf+λ1n1lnfI2dV≤n2λ14n1n2∫LTn1×qH2dV+∫LTn1×qHlnf2dV−λ1n1n2∫LTn1×qbdV+λ1n1∫LTn1×qn+n1n2−1ϕ1+3ϕ22−n2+1ϕ3dV,equivalently,
(37)∫LTn1×qHlnf+λ1n1lnfI2dV≤∫LTn1×qλ1n1n24n2H2+n+n1n2−1ϕ1+3ϕ22−n2+1ϕ3+bn2+Hlnf2dV.
By assumption (26), we have
(38)Hlnf+λ1n1lnfI2≤0,which is not possible; therefore,
(39)Hlnf+λ1n1lnfI=0.
By taking trace of the above equation, we get(40)Δlnf+λ1lnf=0.
Now, applying the result proved in [6], together with the fact that Lt=LTn1×fL⊥n2 is nontrivial, we deduced that LTn1 is isometric to a warped product of the form R×θU, where U is complete Riemannian manifold. Moreover, the warping function θ is the solution of the differential equation dθ2/dt2+λ1θ=0. Hence, the proof is completed.☐Similarly, we can prove the following theorems by taking the unit vector fieldξ tangent to L⊥n2.Theorem 5.
LetLn=LTn1×fL⊥n2 be a warped product semi-invariant product submanifold in a generalized Sasakian space form admitting the nearly Sasakian structure L¯mc. Such that Ricci curvature RLξ>b,b>0. If ξ∈TN⊥n2 and satisfying the following relation:
(41)n2H2+4n1n2λ1Hlnf2=4n1n2λ1b−n+n1n2−1ϕ1+n2+1ϕ3,for λ1<0, then, LTn1 is isometric to warped product of the type ℝ×θU with the warping function θ, which satisfies the differential equation dθ2/dt2+λ1θ=0.


## 4. Conclusions

This paper studies the geometric behavior of ordinary differential equations on the warped product semi-invariant product submanifolds. More precisely, we obtain characterizing theorems for warped product semi-invariant product submanifolds of generalized Sasakian space forms via differential and integral theory on Riemannian manifolds. Therefore, the present article provides a wonderful correlation of the theory of differential equations with the warped product submanifolds.



---

*Source: 7042949-2021-08-02.xml*