The present paper studies the applications of Obata’s differential equations on the Ricci curvature of the pointwise semislant warped product submanifolds. More precisely, by analyzing Obata’s differential equations on pointwise semislant warped product submanifolds, we demonstrate that, under certain conditions, the base of these submanifolds is isometric to a sphere. We also look at the effects of certain differential equations on pointwise semislant warped product submanifolds and show that the base is isometric to a special type of warped product under some geometric conditions.

Taif University1. Introduction

The study of Obata [1] has become a vital investigation technique for geometric analysis. Basically, Obata described the Obata equation as a characterization theorem for a regular sphere in terms of a differential equation. According to Obata, if Mn,g is a complete Riemannian manifold, then the nonconstant function f on Mn satisfies the differential equation ∇2f+cfg=0 or Hessianf+cfg=0 if and only if Mn is isometric to n-dimensional sphere of radius c. A significant number of studies have been conducted on this topic. As a result, the Euclidean space, Euclidean sphere, and complex projective space are recognized domains in the analysis of differential geometry of manifolds, for instance, [2–17]. As a special case, the differential equation ∇2f=cg signifies the Euclidean space, where c is a constant; infact, this was proved by Tashiro [17]. In [18], Lichnerowicz has proved that, under some geometric condition, there exists an isometry between Mn,g and Sn. However, Deshmukh and Al-Solamy used Obata’s differential equation and showed the connected Riemannian manifold isometric to n-dimensional sphere of radius c if the Ricci curvature of Mn,g satisfies the inequality 0<Ric≤n−12−nc/μ1c for a constant c, where μ1 is the first eigenvalue of the Laplacian. Furthermore, Al-Dayael and Khan [19] proved that, under certain conditions, the base of contact CR-warped product submanifolds NT×fN⊥ is isometric to a sphere. Recently, Mofarreh et al. [20] used Obata’s differential equation on warped product submanifolds of Sasakian space form and established some characterization.

On the contrary, Bishop and O’Neill [21] 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 manifolds L1,g1 and L2,g2 with corresponding Riemannian metrics g1 and g2 and ψ:L1⟶R as 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 is warping function of LTn1×L2. The Riemannian product manifold is a special case of warped product manifold in which the warping function is constant. The study of Bishop and O’ Neill [21] revealed that these types of manifolds have 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, Einstein field equation can be solved by the approach of warped product [22]. The warped product is also applicable in the study of space time near to black holes [23].

2. Preliminaries

Let M¯ be an almost Hermitian manifold with an almost complex structure J and almost Hermitian metric g, i.e., J2=−I and gJE,JF=gE,F, for all E,F∈TM¯. If J is parallel with respect to the Levi-Civita connection ∇¯ on M¯, i.e., ∇¯EJF=0, for all E,F∈TM¯, then M¯,g,J is called the Kaehler manifold. A Kaehler manifold M¯ is called the complex space form if and only if it has constant holomorphic sectional curvature denoted by M¯c. The curvature tensor of M¯c is given by(2)R¯E,F,G,H=c4gF,GgE,H−gE,GgF,H+gE,JGgJF,H−gF,JGgJE,H+2gE,JFJG,for all E,F,g∈TM¯.

Let M be a submanifold of dimension n isometrically immersed in a m-dimensional complex space form M¯mc. For an orthonormal basis e1,e2,…,en of the tangent space TxM, the mean curvature vector Hx and its squared norm are given by(3)Hx=1n∑i=1nσei,ei,H2=1n2∑i,j=1ngσei,ei,σej,ej,where σ is the second fundamental form of M and n is the dimension of the submanifold.

The scalar curvature of M¯ is denoted by τ¯L¯ and is defined as(4)τM¯=∑1≤α<β≤mκαβ,where καβ=κ¯eα∧eβ and m is the dimension of the complex space form M¯c.

Let e1,…,en be an orthonormal basis of the tangent space TxM, and if eγ belongs to the orthonormal basis en+1,…em of the normal space T⊥M, then we have(5)σαβγ=gheα,eβ,eγ,σ2=∑α,β=1ngheα,eβ,heα,eβ.

The global tensor field for orthonormal frame of vector field e1,…,en on M¯ is defined as(6)SE,F=∑i=1ngRei,EF,ei,for all E,F∈TxM, where R is the Riemannian curvature tensor. The above tensor is called the Ricci tensor. If we fix a distinct vector eu from e1,…,en on Mn, which is governed by χ, then the Ricci curvature is defined by(7)Rχ=∑α=1α≠unκeα∧eu.

The submanifold M of an almost Hermitian manifold M¯ is called a pointwise slant submanifold if, at each point x∈M, the Wirtinger angle θX between JX and TxM is independent of the choice of the nonzero vector X∈TxM. In this case, the angle θ is treated as a function on M, which is called the slant function of the pointwise slant submanifold [24].

A submanifold M of an almost Hermitian manifold M¯ is called a pointwise semislant submanifold if there exist two orthogonal complementary distributions D and D⊥ such that TM=D⊕D⊥, where D is a holomorphic distribution, i.e., JD=D and D⊥ is a pointwise slant distribution with slant function θ [24].

Biwarped product submanifolds of the type M=NT×f1N⊥×f2Nθ of a Kaehler manifold M¯ have been studied by Tastan [25], where NT, N⊥, and Nθ are invariant, anti-invariant, and slant submanifolds. Furthermore, Khan and Khan [26] extended the study of biwarped product submanifold in the complex space form; more precisely, they studied the warped product of the type M=NT×f1N⊥×f2Nθ, where NT, N⊥, and Nθ are invariant, anti-invariant, and slant submanifolds of the complex space form M¯c, respectively. Recently, Ishan and Khan [27] used biwarped product submanifolds and calculated the Ricci curvature inequalities of biwarped product submanifold. Simultaneously, as a special case, they also obtained the Ricci curvature for pointwise semislant warped product submanifold of the form M=NT×fNθ, where NT is the invariant submanifold and Nθ is the pointwise slant submanifold. More details of these types of submanifolds are available in [24, 28]. Basically, Ishan and Khan [27] proved the following result.

Theorem 1.

(see Corollary 4.2 in [27]). Let Mn=NTn1×fNθn2 be a pointwise semislant warped product submanifold isometrically immersed in a complex space form M¯c. Then, for each orthogonal unit vector field χ∈TxM, either tangent to MTn1 or Mθn2, the Ricci curvature satisfies the following inequalities:

If χ is tangent to MTn1, then(8)14n2H2≥Rχ+n2Δff+c4n−n1n2−12.

If χ is tangent to Mθn2, then(9)14n2H2≥Rχ+n2Δff+c4n−n1n2+1−32cos2θ,

where n1 and n2 are the dimensions of the invariant submanifold NTn1 and the pointwise slant submanifold, respectively.

The equality case can be seen in [27]. Moreover, for the warped product submanifold M=MTn1×fN⊥n2, we have n2Δf/f=n2Δlnf−∇lnf2 [8]. Using this fact in the Theorem 1, we obtain the following theorem.

Theorem 2.

Let Mn=NTn1×fNθn2 be a pointwise semislant warped product submanifold isometrically immersed in a complex space form M¯c; then, for each orthogonal unit vector field χ∈TxM, either tangent to NTn1 or Nθn2, the Ricci curvature satisfies the following inequalities:

If χ is tangent to NTn1, then(10)Rχ+n2Δlnf≤14n2H2+n2∇lnf2−c4n−n1n2−12.

If χ is tangent to Nθn2, then(11)Rχ+n2Δlnf≤14n2H2+n2∇lnf2−c4n−n1n2+1−32cos2θ,

where n1 and n2 are the dimensions of the invariant submanifold NTn1 and the pointwise slant submanifold Nθn2, respectively.

3. Main Results

In this section, we study the application of Obata’s differential equation on pointwise semislant submanifolds Mn=MTn1×fMθn2 in the complex space form M¯c by using the Ricci curvature. Now, we have the following result.

Theorem 3.

Let Mn=NTn1×fNθn2 be a compact orientable pointwise semislant warped product submanifold isometrically immersed in a complex space form Mmc with positive Ricci curvature Rχ≥0,χ∈TNTn1, satisfying the following relation:(12)∇2λ2=3μ1cn1n2n−n1n2−12−3μ1n24n1n2H2,where μ1>0 is an eigenvalue of the warping function λ=lnf. Then, the base manifold NTn1 is isometric to the sphere Sn1μ1/n1 with constant sectional curvature μ1/n1.

Proof.

Let χ∈TNTn1, and consider that λ=lnf and define the following relation as(13)∇2λ−tλI2=∇2λ2+t2λ2I2−2tλg∇2λ,I,where I is the identity operator on the submanifold TNTn1, and we know that I2=traceII∗=n1 and(14)g∇2λ,I∗=trace∇2λ,I∗=trace∇2λ.

Assuming μ1 is an eigenvalue of the eigen function λ, then Δλ=μ1λ. Thus, we obtain(16)∇2λ−tλI2=∇2λ2+n1t2−2tλλ2.

On the contrary, we obtain Δλ2=2λΔλ+∇λ2 or μ1λ2=2μ1λ2+∇λ2 which implies that λ2=−1/μ1∇λ2; using this in equation (16), we have(17)∇2λ−tλI2=∇2λ2+2t−n1t2μ1∇λ2.

In particular, t=−μ1/n1 in equation (17), and integrating with respect to volume element dV,(18)∫Mn∇2λ+μ1n1λI2dV=∫Mn∇2λ2dV−3μ1n1∫Mn∇λ2dV.

Integrating inequality (10) and using the fact ∫MnΔϕdV=0, we have(19)∫MnRicχdV≤n24∫MnH2dV+n2∫Mn∇λ2dV+−c4n−n1n2−12VolMn.

From equations (18) and (19), we derive(20)1n2∫MnRicχdV≤n24n2∫MnH2dV−n13μ1∫Mn∇2λ+μ1n1λI2dV+n13μ1∫Mn∇2λ2dV−c4n−n1n2−12VolMn.

According to assumption Ricχ≥0, the above inequality gives(21)∫Mn∇2λ+μ1n1ϕI2dV≤3n2μ14n1n2∫MnH2dV+∫Mn∇2λ2dV−3μ1cn1n2n−n1n2−12VolMn.

From equation (12), we obtain(22)∫Mn∇2λ+μ1n1λI2dV≤0,but we know that(23)∫Mn∇2λ+μ1n1λI2dV≥0.

Combining the last two statements, we obtain(24)∫Mn∇2λ+μ1n1λI2dV=0⟹∇2λ=−μ1n1λI.

Since the warping function λ=lnf is not constant function on Mn, so equation (24) is Obata’s [1] differential equation with constant c=μ1/n1>0. As μ1>0, therefore, the base submanifold NTn1 is isometric to the sphere Sn1μ1/n1 with constant sectional curvature μ1/n1. This proves the theorem.

If we consider that the unit vector field χ∈TNθn2, then, by adopting the similar steps as in the proof of Theorem 3, we have the following theorem.

Theorem 4.

Let Mn=NTn1×fNθn2 be a compact orientable pointwise semislant warped product submanifold isometrically immersed in a complex space form Mmc with positive Ricci curvature Rχ≥0,χ∈TNθn2, satisfying the following relation:(25)∇2λ2=3μ1cn1n2n−n1n2+1−32cos2θ−3μ1n24n1n2H2,where μ1>0 is an eigenvalue of the warping function λ=lnf. Then, the base manifold NTn1 is isometric to the sphere Sn1μ1/n1 with constant sectional curvature μ1/n1.

In [16], García-Rio et al. studied another version of Obata’s differential equation in the characterization of Euclidean sphere. Basically, they proved that if λ be a real-valued nonconstant function on a Riemannian manifold satisfying Δλ+μ1λ=0 such that λ<0, then Mn is isometric to a warped product of the Euclidean line and a complete Riemannian manifold whose warping function λ is the solution of the following differential equation:(26)d2λdt2+μ1λ=0.

Motivated by the study of García-Rio et al. [16] and Ali et al. [2], we obtain the following characterization.

Theorem 5.

Let Mn=NTn1×fNθn2 be a compact orientable pointwise semislant warped product submanifold isometrically immersed in a complex space form Mmc with positive Ricci curvature Rχ>0,χ∈TNTn1, satisfying one of the following relation:(27)∇2λ2=3μ1cn1n2n−n1n2−12−3μ1n24n1n2H2,where μ1<0 is a negative eigenvalue of the eigenfunction λ=lnψ. Then, NTn1 is isometric to a warped product of the Euclidean line and a complete Riemannian manifold whose warping function λ=lnψ satisfies the differential equation(28)d2λdt2+μ1λ=0.

Proof.

Since we assumed that the Ricci curvature is positive, then, by Myers’s theorem, a complete Riemannian manifold with positive Ricci curvature is compact which means Mn is compact contact CR-warped product submanifold with free boundary [29]. Then, by equation (20),(29)1n2∫MnRicχdV≤n24n2∫MnH2dV−n13μ1∫Mn∇2λ+μ1n1λI2dV+n13μ1∫Mn∇2λ2dV−c4n2n−n1n2−12VolMn.

According to hypothesis Ricci curvature which is positive Ricχ>0, then we have(30)∫Mn∇2λ+μ1n1λI2dV<3n2μ14n1n2∫MnH2dV+∫Mn∇2ϕ2dV−3μ1c4n1n2n−n1n2−12VolMn.

If equation (27) holds, then from last inequality, we get ∇2λ+μ1/n1λI2<0, which is not possible; hence, ∇2λ+μ1/n1λI2=0. Since μ1<0, then by result of [16], the submanifold NTn1 is isometric to a warped product of the Euclidean line and a complete Riemannian manifold, where the warping function on R is the solution of the differential equation (28), and this proves the theorem.

Similarly, if we consider the unit vector field ξ∈TNθn2, then we have the following result, which can be verified as Theorem 5.

Theorem 6.

Suppose Mn=NTn1×fNθn2 be a compact orientable pointwise semislant warped product submanifold isometrically immersed in a complex space form Mmc with positive Ricci curvature Rχ>0,χ∈TNθn2 and satisfying one of the following relation:(31)∇2λ2=3μ1cn1n2n−n1n2+1−32cos2θ−3μ1n24n1n2H2,where μ1<0 is a negative eigenvalue of the eigen function λ=lnψ. Then, NTn1 is isometric to a warped product of the Euclidean line and a complete Riemannian manifold whose warping function λ=lnψ satisfies the differential equation(32)d2λdt2+μ1λ=0.

4. Conclusions

This paper studies the geometric behavior of ordinary differential equations on the pointwise semislant warped product submanifolds. More precisely, we obtain characterizing theorems for pointwise semislant warped product submanifolds of complex space forms via differential and integral theory on Riemannian manifolds. Therefore, the present study provides a wonderful correlation of theory of differential equations with the warped product submanifolds.

Data Availability

No data were used to support the findings of the study.

Conflicts of Interest

The author declares that she has no conflicts of interest.

Acknowledgments

This work was supported by Taif University Researchers Supporting Project (No. TURSP-2020/223), Taif University, Taif, Saudi Arabia.

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