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.

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 manifolds and with corresponding Riemannian metrics and and be 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 satisfies for all The function represents the warping function of The Riemannian product manifold is a special case of warped product manifold in which the warping function . 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 on a complete Riemannian manifold satisfies the differential equation as follows: if and only if is congruent to Euclidean space where is constant.

Furthermore, under some geometric conditions, Garcia-Rio et al. [6] proved that the Riemannian manifold is isometric to the warped product where is a complete Riemannian manifold, is the Euclidean line, and is the warping function. Moreover, warping function is the solution of the following differential equation: if and only if there exists a nonconstant function with an eigenvalue which satisfies the following differential equation:

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 using the concircular vector field, showing that the Riemannian manifold is isometric to the Euclidean manifold . By using the gradient conformal vector field, Chen et al. [12] discovered that the Riemannian manifold is isometric to the Euclidean space . However, it has been shown in [13] that the complete totally real submanifold in (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

A -dimensional -manifold is said to have an almost contact structure if there exists on a tensor field of the type a vector field , and a 1-form satisfying

On an almost contact metric manifold , there is always a Riemannian metric that meets the following requirements: for all

An almost contact metric manifold is said to be nearly Sasakian manifold, if for all

In [14], Alegre et al. gave the concept of generalized Sasakian space form as that an almost contact metric manifold whose curvature tensor satisfies for any vector fields and certain differentiable functions on A generalized Sasakian space form with functions is denoted by . If , then is a Sasakian space form [14]. If , then is a Kenmotsu space form [14], and if then is a cosymplectic space form [14].

A submanifold of an almost contact metric manifold is called semi-invariant submanifolds (contact CR-submanifolds) if there exist two orthogonal complementary distributions and satisfying the following conditions: (i), where is the distribution spanned by the structure vector field (ii) is invariant distribution, i.e., (iii) is anti-invariant, i.e.,

Recently, we [15] studied warped product semi-invariant product submanifolds of the type isometrically immersed in the generalized Sasakian space form admitting a nearly Sasakian structure, where is an invariant submanifold of dimension and is a totally real submanifold of dimension . More precisely, the computed Ricci curvature inequalities for these submanifolds are as follows:

Theorem 1. Let be a warped product semi-invariant submanifold isometrically immersed in a generalized Sasakian space form with nearly Sasakian structure. Then, for each orthogonal unit vector field orthogonal to , either tangent to or , the Ricci curvature satisfies the following inequalities: (i)If , then(ii)If , thenThe equality cases can be seen in [15].
Let be a real-valued differential function on a Riemannian manifold , then the Bochner formula [16] is stated as where denotes Ricci tensor and is the Hessian of the function .

3. Main Results

In this section, we obtain some characterization by the application of Bochner’s formula.

Theorem 2. Let be a -dimensional warped product semi-invariant product submanifold in a generalized Sasakian space form where is a -dimensional invariant submanifold and is an anti-invariant submanifold. Such that Ricci curvature If and satisfying the following equality: then, the base submanifold is isometric to (Euclidean space).

Proof. Since , by equation (9) By the assumption that we have Since the Ricci curvature is bounded below by , then by virtue of theorem of Myers [17], the base manifold is compact. On integrating (9) and using Green’s theorem, we have or Suppose denotes the Hessian of the warping function , then we have after some calculations, the above formula turns to Putting and integrating the last equation with respect to (volume element), we get using (11), with the fact we have Merging (19) and (20), we derive By the assumption the above equation yields Using (16), the last inequality leads to If (12) holds, then the above inequality produces Therefore, we have Hence, by the application of the result of Tashiro [18], the fibre is isometric to (Euclidean space).
If we consider the unit vector field , then we have the following results which can be proved by adopting similar steps in Theorem 2.☐

Theorem 3. Let be a -dimensional warped product semi-invariant product submanifold in a generalized Sasakian space form where is a -dimensional invariant submanifold and is an anti-invariant submanifold. Such that Ricci curvature If and satisfying the following equality: then, the base submanifold is isometric to (Euclidean space).

Now, we have the next result which is based on the study of Garcia-Rio et al. [6].

Theorem 4. Let be a warped product semi-invariant product submanifold in a generalized Sasakian space form admitting the nearly Sasakian structure . Such that Ricci curvature . If and satisfying the following relation: for , then is isometric to warped product of the type with the warping function which satisfies the differential equation

Proof. For the warping function , defining the following equation on : But we know that and using these facts, the above equation leads to Let is an eigenfunction corresponding to the eigenvalue satisfying we have Further, using it is easy to see that which on integrating provides Thus, we have Choosing in (32), we have Further, integrating (9) and applying Green’s lemma, we find From the above two expressions, we have On using the assumption that for equivalently, By assumption (26), we have which is not possible; therefore, By taking trace of the above equation, we get Now, applying the result proved in [6], together with the fact that is nontrivial, we deduced that is isometric to a warped product of the form , where is complete Riemannian manifold. Moreover, the warping function is the solution of the differential equation Hence, the proof is completed.☐

Similarly, we can prove the following theorems by taking the unit vector field tangent to .

Theorem 5. Let be a warped product semi-invariant product submanifold in a generalized Sasakian space form admitting the nearly Sasakian structure . Such that Ricci curvature . If and satisfying the following relation: for , then, is isometric to warped product of the type with the warping function which satisfies the differential equation

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.

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there is no conflict of interest regarding the publication of this paper.