Analysis of Obata’s Differential Equations on Pointwise Semislant Warped Product Submanifolds of Complex Space Forms via Ricci Curvature

Abstract

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.

1. 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 is a complete Riemannian manifold, then the nonconstant function on satisfies the differential equation or if and only if is isometric to -dimensional sphere of radius . 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 signifies the Euclidean space, where 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 and . However, Deshmukh and Al-Solamy used Obata’s differential equation and showed the connected Riemannian manifold isometric to -dimensional sphere of radius if the Ricci curvature of satisfies the inequality for a constant , where 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 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 and with corresponding Riemannian metrics and and as a positive differentiable function. If and are projection maps such that and , which are defined as and , then is called warped product manifold if the Riemannian structure on satisfiesfor all , the function is warping function of . 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 be an almost Hermitian manifold with an almost complex structure and almost Hermitian metric , i.e., and , for all . If is parallel with respect to the Levi-Civita connection on , i.e., for all , then is called the Kaehler manifold. A Kaehler manifold is called the complex space form if and only if it has constant holomorphic sectional curvature denoted by . The curvature tensor of is given byfor all .

Let be a submanifold of dimension isometrically immersed in a -dimensional complex space form . For an orthonormal basis of the tangent space , the mean curvature vector and its squared norm are given bywhere is the second fundamental form of and is the dimension of the submanifold.

The scalar curvature of is denoted by and is defined aswhere and is the dimension of the complex space form .

Let be an orthonormal basis of the tangent space , and if belongs to the orthonormal basis of the normal space , then we have

The global tensor field for orthonormal frame of vector field on is defined asfor all , where is the Riemannian curvature tensor. The above tensor is called the Ricci tensor. If we fix a distinct vector from on , which is governed by , then the Ricci curvature is defined by

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

A submanifold of an almost Hermitian manifold is called a pointwise semislant submanifold if there exist two orthogonal complementary distributions and such that , where is a holomorphic distribution, i.e., and is a pointwise slant distribution with slant function [24].

Biwarped product submanifolds of the type of a Kaehler manifold have been studied by Tastan [25], where , , and 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 , where , , and are invariant, anti-invariant, and slant submanifolds of the complex space form , 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 , where is the invariant submanifold and 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 be a pointwise semislant warped product submanifold isometrically immersed in a complex space form . Then, for each orthogonal unit vector field , either tangent to or , the Ricci curvature satisfies the following inequalities:(i)If is tangent to , then(ii)If is tangent to , thenwhere and are the dimensions of the invariant submanifold and the pointwise slant submanifold, respectively.
The equality case can be seen in [27]. Moreover, for the warped product submanifold , we have [8]. Using this fact in the Theorem 1, we obtain the following theorem.

Theorem 2. Let be a pointwise semislant warped product submanifold isometrically immersed in a complex space form ; then, for each orthogonal unit vector field , either tangent to or , the Ricci curvature satisfies the following inequalities:(i)If is tangent to , then(ii)If is tangent to , thenwhere and are the dimensions of the invariant submanifold and the pointwise slant submanifold , respectively.

3. Main Results

In this section, we study the application of Obata’s differential equation on pointwise semislant submanifolds in the complex space form by using the Ricci curvature. Now, we have the following result.

Theorem 3. Let be a compact orientable pointwise semislant warped product submanifold isometrically immersed in a complex space form with positive Ricci curvature , satisfying the following relation:where is an eigenvalue of the warping function . Then, the base manifold is isometric to the sphere with constant sectional curvature .

Proof. Let , and consider that and define the following relation aswhere is the identity operator on the submanifold , and we know that andThen, equation (13) transforms toAssuming is an eigenvalue of the eigen function , then . Thus, we obtainOn the contrary, we obtain or which implies that ; using this in equation (16), we haveIn particular, in equation (17), and integrating with respect to volume element ,Integrating inequality (10) and using the fact , we haveFrom equations (18) and (19), we deriveAccording to assumption , the above inequality givesFrom equation (12), we obtainbut we know thatCombining the last two statements, we obtainSince the warping function is not constant function on , so equation (24) is Obata’s [1] differential equation with constant . As , therefore, the base submanifold is isometric to the sphere with constant sectional curvature . This proves the theorem.
If we consider that the unit vector field , then, by adopting the similar steps as in the proof of Theorem 3, we have the following theorem.

Theorem 4. Let be a compact orientable pointwise semislant warped product submanifold isometrically immersed in a complex space form with positive Ricci curvature , satisfying the following relation:where is an eigenvalue of the warping function . Then, the base manifold is isometric to the sphere with constant sectional curvature .
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 be a real-valued nonconstant function on a Riemannian manifold satisfying such that , then is isometric to a warped product of the Euclidean line and a complete Riemannian manifold whose warping function is the solution of the following differential equation:Motivated by the study of García-Rio et al. [16] and Ali et al. [2], we obtain the following characterization.

Theorem 5. Let be a compact orientable pointwise semislant warped product submanifold isometrically immersed in a complex space form with positive Ricci curvature , satisfying one of the following relation:where is a negative eigenvalue of the eigenfunction . Then, is isometric to a warped product of the Euclidean line and a complete Riemannian manifold whose warping function satisfies the differential equation

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 is compact contact CR-warped product submanifold with free boundary [29]. Then, by equation (20),According to hypothesis Ricci curvature which is positive , then we haveIf equation (27) holds, then from last inequality, we get , which is not possible; hence, . Since , then by result of [16], the submanifold is isometric to a warped product of the Euclidean line and a complete Riemannian manifold, where the warping function on is the solution of the differential equation (28), and this proves the theorem.
Similarly, if we consider the unit vector field , then we have the following result, which can be verified as Theorem 5.

Theorem 6. Suppose be a compact orientable pointwise semislant warped product submanifold isometrically immersed in a complex space form with positive Ricci curvature and satisfying one of the following relation:where is a negative eigenvalue of the eigen function . Then, is isometric to a warped product of the Euclidean line and a complete Riemannian manifold whose warping function satisfies the differential equation

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.