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

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.


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 ðL 1 , g 1 Þ and ðL 2 , g 2 Þ with corresponding Riemannian metrics g 1 and g 2 and ψ : L 1 ⟶ R be a positive differentiable function. If x and y are projection maps such that x : L 1 × L 2 ⟶ L 1 and y : L 1 × L 2 ⟶ L 2 , which are defined as xðm, nÞ = m and yðm, nÞ = n∀ðm, nÞ ∈ L 1 × L 2 , then, L = L 1 × L 2 is called warped product manifold if the Riemannian structure on L satisfies for all E, F ∈ T L: The function ψ represents the warping function of L n 1 T × L 2 : 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 ðU n , gÞ satisfies the differential equation as follows: if and only if ðU n , gÞ is congruent to Euclidean space R n , 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 × f R, 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: if and only if there exists a nonconstant function ϕ : U n ⟶ R with an eigenvalue λ 1 < 0, 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][8][9][10][11].
Al-Dayel et al. [7] recently investigated the effect of the differential equation (3) on the Riemannian manifold ðL n , gÞ using the concircular vector field, showing that the Riemannian manifold ðL n , gÞ is isometric to the Euclidean manifold R n . By using the gradient conformal vector field, Chen et al. [12] discovered that the Riemannian manifold ðL n , gÞ is isometric to the Euclidean space R n . However, it has been shown in [13] that the complete totally real submanifold in CP n (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.

Preliminaries
A ð2k + 1Þ-dimensional C ∞ -manifold L is said to have an almost contact structure if there exists on L a tensor field ϕ of the type ð1, 1Þ, a vector field χ, and a 1-form η satisfying On an almost contact metric manifold L, there is always a Riemannian metric g that meets the following requirements: for all E, F ∈ T L: An almost contact metric manifold is said to be nearly Sasakian manifold, if for all E, F ∈ T L: In [14], Alegre et al. gave the concept of generalized Sasakian space form as that an almost contact metric manifold ð L, f , χ, η, gÞ whose curvature tensor for any vector fields E, F, G, W and certain differentiable functions MðcÞ [14]. If ϕ = ðc − 3Þ/4,ϕ 2 = ϕ 3 = ðc + 1Þ/4, then MðcÞ [14], and if MðcÞ [14]. A submanifold L 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: Recently, we [15] studied warped product semi-invariant product submanifolds of the type L n = L ⊥ isometrically immersed in the generalized Sasakian space form admitting a nearly Sasakian structure, where L n 1 T is an invariant submanifold of dimension n 1 and L n 2 ⊥ is a totally real submanifold of dimension n 2 . More precisely, the computed Ricci curvature inequalities for these submanifolds are as follows: ⊥ be a warped product semiinvariant submanifold isometrically immersed in a generalized Sasakian space form L m ðϕ 1 , ϕ 2 , ϕ 3 Þ with nearly Sasakian structure. Then, for each orthogonal unit vector field ξ ∈ T x M orthogonal to χ, either tangent to N n 1 T or N n 2 ⊥ , the Ricci curvature satisfies the following inequalities: Advances in Mathematical Physics The equality cases can be seen in [15]. Let f be a real-valued differential function on a Riemannian manifold L n , then the Bochner formula [16] is stated as where R L denotes Ricci tensor and Hð f Þ is the Hessian of the function f .

Main Results
In this section, we obtain some characterization by the application of Bochner's formula.
⊥ be a n-dimensional warped product semi-invariant product submanifold in a generalized Sasakian space form L m ðcÞ, where L n 1 T is a n -dimensional invariant submanifold and L n 2 ⊥ is an anti-invariant submanifold. Such that Ricci curvature R L ðξÞ ≥ b, b > 0: If ξ ∈ TL n 1 T and satisfying the following equality: then, the base submanifold L n 1 T is isometric to R n 1 (Euclidean space).
Proof. Since ξ ∈ TN n 1 T , by equation (9) By the assumption that R L ðξÞ ≥ b, we have Since the Ricci curvature R L ðξÞ is bounded below by b ≥ 0, then by virtue of theorem of Myers [17], the base manifold L n 1 T is compact. On integrating (9) and using Green's theorem, we have ð16Þ Suppose Hðln f Þ denotes the Hessian of the warping function ln f , then we have after some calculations, the above formula turns to Putting t = λ 1 /n 1 and integrating the last equation with respect to dV (volume element), we get using (11), with the fact Δlnf = λ 1 lnf , we have Merging (19) and (20), we derive By the assumption R L ð∇f ,∇f Þ ≥ b, the above equation yields 3 Advances in Mathematical Physics Using (16), the last inequality leads to If (12) holds, then the above inequality produces Therefore, we have Hðlnf ÞðX, XÞ = λ 1 /n 1 : Hence, by the application of the result of Tashiro [18], the fibre L n 1 T is isometric to R n 1 (Euclidean space).
If we consider the unit vector field ξ ∈ TL n 2 ⊥ , then we have the following results which can be proved by adopting similar steps in Theorem 2. T is a n -dimensional invariant submanifold and L n 2 ⊥ is an anti-invariant submanifold. Such that Ricci curvature R L ðξÞ ≥ b, b > 0: If ξ ∈ TL n 1 T and satisfying the following equality: then, the base submanifold L n 1 T is isometric to R n 1 (Euclidean space). Now, we have the next result which is based on the study of Garcia-Rio et al. [6]. Such that Ricci curvature R L ðξÞ > b, b > 0. If ξ ∈ TN n 1 T and satisfying the following relation: for λ 1 < 0, then L n 1 T is isometric to warped product of the type R × θ U with the warping function θ, which satisfies the differential equation dθ 2 /dt 2 + λ 1 θ = 0: Proof. For the warping function lnf , defining the following equation on L n 1 T : But we know that jIj 2 = trðII * Þ = n 1 and gðHðlnf Þ, I * Þ = trðI * Hðlnf ÞÞ = trðHðlnf ÞÞ ; using these facts, the above equation leads to Let lnf is an eigenfunction corresponding to the eigenvalue λ 1 satisfying Δlnf = λ 1 lnf , we have Further, using Δlnf = λ 1 lnf , it is easy to see that which on integrating provides Thus, we have Advances in Mathematical Physics Further, integrating (9) and applying Green's lemma, we find From the above two expressions, we have On using the assumption that R L ðξÞ ≥ b, for b > 0, 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 L t = L n 1 T is isometric to a warped product of the form R × θ U, where U is complete Riemannian manifold. Moreover, the warping function θ is the solution of the differential equation dθ 2 /d t 2 + λ 1 θ = 0: Hence, the proof is completed. ☐ Similarly, we can prove the following theorems by taking the unit vector field ξ tangent to L Such that Ricci curvature R L ðξÞ > b, b > 0. If ξ ∈ TN n 2 ⊥ and satisfying the following relation: n 2 H k k 2 + 4 n 1 n 2 ð Þ λ 1 H lnf ð Þ j j 2 = 4n 1 n 2 λ 1 Á b − n + n 1 n 2 − 1 ð Þ ϕ 1 + n 2 + 1 ð for λ 1 < 0, then, L n 1 T is isometric to warped product of the type ℝ × θ U with the warping function θ, which satisfies the differential equation dθ 2 /dt 2 + λ 1 θ = 0:

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.