Characterization of Skew CR-Warped Product Submanifolds in Complex Space Forms via Differential Equations

Recently, we have obtained Ricci curvature inequalities for skew CR-warped product submanifolds in the framework of complex space form. By the application of Bochner’s formula on these inequalities, we show that, under certain conditions, the base of these submanifolds is isometric to the Euclidean space. Furthermore, we study the impact of some differential equations on skew CR-warped product submanifolds and prove that, under some geometric conditions, the base is isometric to a special type of warped product.


Introduction
e studies [1,2] provide both important intrinsic geometric as well as isometric properties of Riemannian manifolds via differential equations. It is well known that the classification of differential equations has a significant effect on the global study of Riemannian manifolds. In 1978, Tanno [2] studied various aspects of differential equations on Riemannian manifolds. In particular, the authors of [3,4] characterized Euclidean sphere by the approach of differential equations. e analysis performed in [2,5] proved that a nonconstant function λ on a complete Riemannian manifold (U n , g) satisfies the differential equation if and only if (U n , g) is isometric to Euclidean space R n , where c is constant. Moreover, Garcia-Rio et al. [4] proved that, under some restrictions, the Riemannian manifold is isometric to 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 satisfies the second-order differential equation if and only if there exist a nonconstant function ϕ: U n ⟶ R with a negative eigenvalue μ 1 ≤ 0, which is the solution of following differential equation: e categorization of differential equations on Riemannian manifold has become a fascinating topic of research and has been investigated by numerous researchers, for instance, [6][7][8][9].
Recently, Al-Dayel et al. [6] studied the impact of differential equation (2) on Riemannian manifold (L n , g) by taking the concircular vector field and proved that, under certain conditions, the Riemannian manifold (L n , g) is isometric to Euclidean manifold R n . Similarly, by taking gradient conformal vector field, Chen et al. [10] identified that Riemannian manifold (N n , g) is isometric to the Euclidean space R n . However, in [11], it has been proved that the complete totally real submanifold in CP n (complex projective space) with bounded Ricci curvature satisfying (3) is isometric to a special class of hyperbolic space.
On the contrary, Bishop and O'Neill [12] studied the geometry of manifolds having negative curvature and confirmed that Riemannian product manifolds always have nonnegative curvature. As a result, they proposed the idea of warped product manifolds, and these manifolds are defined 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 let 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 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. e function ψ represents the warping function of L 1 × L 2 . e Riemannian product manifold is a special case of warped product manifold in which the warping function is constant. e study of Bishop and O'Neill [12] 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 the warped product [13]. e warped product is also applicable in the study of space time near to black holes [14].
Latterly, Ali et al. [7] characterized warped product submanifolds in Sasakian space form by the approach of the differential equation. e purpose of this paper is to study the impact of differential equation on skew CR-warped product submanifolds in the framework of the complex space form.

Preliminaries
Let L be an almost Hermitian manifold with an almost complex structure J and a Hermitian metric g, i.e., J 2 � − I and g(JX, JY) � g(X, Y), for all vector fields X, Y on L. If the almost complex structure J satisfies for all X, Y ∈ TL, where D is Levi-Civita connection on L, then (L, J) is called a Kaehler manifold. A Kaehler manifold L is called a complex space form if it has constant holomorphic sectional curvature and denoted by L(c). e curvature tensor of the complex space form L(c) is given by for any X, Y, Z, W ∈ TL.
Let L be an n− dimensional Riemannian manifold isometrically immersed in a m− dimensional Riemannian manifold L. en, the Gauss and Weingarten formulas are X ξ, respectively, for all X, Y ∈ TL and ξ ∈ T ⊥ L, where D is the induced Levi-Civita connection on L, ξ is a vector field normal to L, h is the second fundamental form of L, D ⊥ is the normal connection in the normal bundle T ⊥ L, and A ξ is the shape operator of the second fundamental form. For any X ∈ TL and N ∈ T ⊥ L, JX and JN can be decomposed as follows: where PX (respectively, tN) is the tangential and FX (respectively, fN) is the normal component of JE (respectively, JN). It is evident that g(ϕX, Y) � g(PX, Y), for any X, Y ∈ T x L; this implies that g(PX, Y) + g(X, PY) � 0. us, P 2 is a symmetric operator on the tangent space T x L, for all x ∈ L. e eigenvalues of P 2 are real and diagonalizable. Moreover, for each x ∈ L, one can observe where I denotes the identity transformation on T x L and Furthermore, it is easy to observe that KerF � S 1 x and KerP � S 0 x , where S 1 x is the maximal holomorphic subspace of T x L and S 0 x is the maximal totally real subspace of T x L, and these distributions are denoted by S T and S ⊥ , respectively. If − μ 2 1 (x), . . . , − μ 2 k (x) are the eigenvalues of P 2 at x, then T x L can be decomposed as for any x ∈ M.
If, in addition, each μ i is constant on L, then L is called a skew CR-submanifolds [15]. It is significant to account that 2 Mathematical Problems in Engineering CR-submanifolds are particular class of skew CR-submanifold with k � 1, Definition 1. A submanifold L of a Kaehler manifold L is said to be a skew CR-submanifold of order 1 if L is a skew CR-submanifold with k � 1 and μ 1 is constant. For any orthonormal basis e 1 , e 2 , . . . , e n of the tangent space T x L, the mean curvature vector Ω(x) and its squared norm are defined as follows: g h e i , e i , h e j , e j , where n is the dimension of L. If h � 0, then the submanifold is said to be totally geodesic and minimal if Ω � 0. If h(E 1 , E 2 ) � g(E 1 , E 2 )Ω, for all E 1 , E 2 ∈ TL, then L is called totally umbilical. e scalar curvature of L is denoted by τ(L) and is defined as where κ αβ � κ(e α ∧ e β ) and m is the dimension of the Riemannian manifold L.
e global tensor field for orthonormal frame of vector field e 1 , . . . , e n on L is defined as for all E 1 , E 2 ∈ T x L, 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 L n , which is governed by χ, then the Ricci curvature is defined by A submanifold L of a Kaehler manifold L is said to be skew CR-warped product submanifolds if it is warped product of the type L � L 1 × f L ⊥ , where L 1 is a semislant submanifold which was defined by N. Papaghiuc [16] and L ⊥ is a totally real submanifold. Sahin [17] proved the existence skew CR-warped product submanifolds. Recently, Ali Khan and Al-Dayel [18] studied Skew CR-warped product submanifolds of the form L d � L ⊥ is totally real submanifold. More precisely, they obtained Ricci curvature inequalities for these submanifolds as follows.  Let f be a real-valued differential function on a Riemannian manifold L n ; then, the Bochner formula [19] is stated as where R L denotes Ricci tensor and H(f) is the Hessian of the function f.

Main Results
In this section, we obtain some characterization by the application of Bochner formula.
If χ ∈ S and satisfying the equality, then the base submanifold L ⊥ and λ 1 is the eigenvalue corresponding to the eigenfunction lnf.
Proof. Since χ ∈ S, by equation (16), By the assumption that R L (χ) ≥ K, we have Since R L (χ) ≥ K, on applying the theorem of Myers [1], according to this, if Ricci curvature is greater than by a positive constant, then base manifold L d 3 1 is compact. On integrating (22) and using Green's theorem, we obtain Let H(lnf) be the Hessian of the warping function lnf; then, we have |H(lnf) − nI| 2 � |H(lnf)| 2 + n 2 |I| 2 − 2ng(I, H(lnf)), where n is real number. e above formula provides Putting n � (λ 1 /(d 1 + d 2 )) and integrating the last equation with respect to dV (volume element), we obtain Using (19), with the fact Δlnf � λ 1 lnf, we have Combining (27) and (28), we derive By the assumption R L (∇f, ∇f) ≥ K, the above equation changes to Using (24), the last inequality leads to If (20) holds, then the above inequality produces erefore, we have H(lnf)(X, X) � (λ 1 /(d 1 + d 2 )). Hence, by the application of Tashiro's result [5], the fibre L 1 is isometric to Euclidean space R d 1 +d 2 .

□
Let lnf be an eigenfunction corresponding to the eigenvalue λ 1 satisfying Δlnf � λ 1 lnf, and we have\scale 90% (38) Again, using Δlnf � λ 1 lnf, it is easy to see that which on integrating provides Putting K � (λ 1 /(d 1 + d 2 )) in (40), we have Furthermore, integrating (16) and applying Green's Lemma, we find □ From the above two expressions, we have On using the assumption that R L (χ) ≥ K, for K > 0, Equivalently, 6 Mathematical Problems in Engineering