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

$e 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.


Introduction
e 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 (M n , g) is a complete Riemannian manifold, then the nonconstant function f on M n satisfies the differential equation ∇ 2 f + cfg � 0 or Hessian(f) + cfg � 0 if and only if M n 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][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. As a special case, the differential equation ∇ 2 f � 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 (M n , g) and S n . 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 (M n , g) satisfies the inequality 0 < Ric ≤ (n − 1)(2 − (nc/μ 1 )c) for a constant c, where μ 1 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 N T × f N ⊥ 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 (L 1 , g 1 ) and (L 2 , g 2 ) with corresponding Riemannian metrics g 1 and g 2 and ψ: L 1 ⟶ R as 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 x(m, n) � m and y(m, n) � n∀(m, n) ∈ 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 ∈ TL, the function ψ is warping function of L  [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]. e warped product is also applicable in the study of space time near to black holes [23].

Preliminaries
Let M be an almost Hermitian manifold with an almost complex structure J and almost Hermitian metric g, i.e., J 2 � − I and g(JE, JF) � g(E, F), for all E, F ∈ TM. If J is parallel with respect to the Levi-Civita connection ∇ on M, i.e., (∇ E J)F � 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). e curvature tensor of M(c) is given by for all E, F, g ∈ TM.
Let M be a submanifold of dimension n isometrically immersed in a m-dimensional complex space form M m (c).
For an orthonormal basis e 1 , e 2 , . . . , e n of the tangent space T x M, the mean curvature vector H(x) and its squared norm are given by where σ is the second fundamental form of M and n is the dimension of the submanifold. e scalar curvature of M is denoted by τ(L) and is defined as where κ αβ � κ(e α ∧e β ) and m is the dimension of the complex space form M(c). Let e 1 , . . . , e n be an orthonormal basis of the tangent space T x M, and if e c belongs to the orthonormal basis e n+1 , . . . e m of the normal space T ⊥ M, then we have e global tensor field for orthonormal frame of vector field e 1 , . . . , e n on M is defined as for all E, F ∈ T x M, where R is the Riemannian curvature tensor. e above tensor is called the Ricci tensor. If we fix a distinct vector e u from e 1 , . . . , e n on M n , which is governed by χ, then the Ricci curvature is defined by e submanifold M of an almost Hermitian manifold M is called a pointwise slant submanifold if, at each point x ∈ M, the Wirtinger angle θ(X) between JX and T x M is independent of the choice of the nonzero vector X ∈ T x M. 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., J D � D and D ⊥ is a pointwise slant distribution with slant function θ [24].
Biwarped product submanifolds of the type M � N T × f 1 N ⊥ × f 2 N θ of a Kaehler manifold M have been studied by Tastan [25], where N T , N ⊥ , and N θ 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 � 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 � N T × f N θ , where N T is the invariant submanifold and N θ 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.
where n 1 and n 2 are the dimensions of the invariant submanifold N n 1 T and the pointwise slant submanifold, respectively.
The equality case can be seen in [27]. Moreover, for the warped product submanifold M � M [8]. Using this fact in the eorem 1, we obtain the following theorem.
(ii) If χ is tangent to N n 2 θ , then where n 1 and n 2 are the dimensions of the invariant submanifold N n 1 T and the pointwise slant submanifold N n 2 θ , respectively.

Main Results
In this section, we study the application of Obata's differential equation on pointwise semislant submanifolds M n � M  T , satisfying the following relation: where μ 1 > 0 is an eigenvalue of the warping function λ � ln f. en, the base manifold N n 1 T is isometric to the sphere S n 1 (μ 1 /n 1 ) with constant sectional curvature (μ 1 /n 1 ).

Mathematical Problems in Engineering
According to assumption Ric(χ) ≥ 0, the above inequality gives From equation (12), we obtain but we know that Combining the last two statements, we obtain Since the warping function λ � ln f is not constant function on M n , so equation (24) is Obata's [1] differential equation with constant c � (μ 1 /n 1 ) > 0. As μ 1 > 0, therefore, the base submanifold N n 1 T is isometric to the sphere S n 1 (μ 1 /n 1 ) with constant sectional curvature (μ 1 /n 1 ). is proves the theorem.
If we consider that the unit vector field χ ∈ TN n 2 θ , then, by adopting the similar steps as in the proof of eorem 3, we have the following theorem.
where μ 1 > 0 is an eigenvalue of the warping function λ � ln f. en, the base manifold N n 1 T is isometric to the sphere S n 1 (μ 1 /n 1 ) with constant sectional curvature (μ 1 /n 1 ).
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 realvalued nonconstant function on a Riemannian manifold satisfying Δλ + μ 1 λ � 0 such that λ < 0, then M n 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: Motivated by the study of García-Rio et al. [16] and Ali et al. [2], we obtain the following characterization. T , satisfying one of the following relation: where μ 1 < 0 is a negative eigenvalue of the eigenfunction λ � ln ψ. en, N n 1 T is isometric to a warped product of the Euclidean line and a complete Riemannian manifold whose warping function λ � ln ψ 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 M n is compact contact CR-warped product submanifold with free boundary [29]. en, by equation (20), According to hypothesis Ricci curvature which is positive Ric(χ) > 0, then we have If equation (27) holds, then from last inequality, we get ‖∇ 2 λ + (μ 1 /n 1 )λI‖ 2 < 0, which is not possible; hence, ‖∇ 2 λ + (μ 1 /n 1 )λI‖ 2 � 0. Since μ 1 < 0, then by result of [16], the submanifold N n 1 T 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 ∈ TN n 2 θ , then we have the following result, which can be verified as eorem 5.
where μ 1 < 0 is a negative eigenvalue of the eigen function λ � ln ψ. en, N n 1 T is isometric to a warped product of the Euclidean line and a complete Riemannian manifold whose warping function λ � ln ψ satisfies the differential equation

Conclusions
is 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. erefore, 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
e author declares that she has no conflicts of interest.