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

Authors: Ibrahim Al-Dayel; Meraj Ali Khan
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

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.


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## Body

## 1. Introduction

The 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. The analysis performed in [2, 5] proved that a nonconstant function λ on a complete Riemannian manifold Un,g satisfies the differential equation(1)∇2λ+cg=0,if and only if Un,g is isometric to Euclidean space Rn, where c is constant.Moreover, Garcia-Rio et al. [4] proved that, under some restrictions, the Riemannian manifold is isometric to warped product U×Rf, 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(2)d2fdt2+μ1f=0,if and only if there exist a nonconstant function ϕ:Un⟶R with a negative eigenvalue μ1≤0, which is the solution of following differential equation:(3)Δϕ+μ1ϕ=0.The categorization of differential equations on Riemannian manifold has become a fascinating topic of research and has been investigated by numerous researchers, for instance, [6–9].Recently, Al-Dayel et al. [6] studied the impact of differential equation (2) on Riemannian manifold Ln,g by taking the concircular vector field and proved that, under certain conditions, the Riemannian manifold Ln,g is isometric to Euclidean manifold Rn. Similarly, by taking gradient conformal vector field, Chen et al. [10] identified that Riemannian manifold Nn,g is isometric to the Euclidean space Rn. However, in [11], it has been proved that the complete totally real submanifold in CPn (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 manifoldsL1,g1 and L2,g2 with corresponding Riemannian metrics g1 and g2, and let ψ:L1⟶R be a positive differentiable function. If x and y are projection maps such that x:L1×L2⟶L1 and y:L1×L2⟶L2, which are defined as xm,n=m and ym,n=n∀m,n∈L1×L2, then L=L1×L2 is called warped product manifold if the Riemannian structure on L satisfies(4)gE¯,F¯=g1x∗E¯,x∗F¯+ψ°x2g2y∗E¯,y∗F¯,for all E¯,F¯∈TL. The function ψ represents the warping function of L1×L2. 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 [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]. The 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. The 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.


## 2. Preliminaries

LetL¯ be an almost Hermitian manifold with an almost complex structure J and a Hermitian metric g, i.e., J2=−I and gJX,JY=gX,Y, for all vector fields X,Y on L¯. If the almost complex structure J satisfies(5)D¯XJY=0,for all X,Y∈TL¯, where D¯ is Levi-Civita connection on L¯, then L¯,J is called a Kaehler manifold.A Kaehler manifoldL¯ is called a complex space form if it has constant holomorphic sectional curvature and denoted by L¯c. The curvature tensor of the complex space form L¯c is given by(6)R¯X,Y,Z,W=c4gY,ZgX,W−gX,ZgY,W+gX,JZgJY,W−gY,JZgJX,W+2gX,JYgJZ,W,for any X,Y,Z,W∈TL¯.LetL be an n−dimensional Riemannian manifold isometrically immersed in a m−dimensional Riemannian manifold L¯. Then, the Gauss and Weingarten formulas are D¯XY=DXY+hX,Y and D¯Xξ=−AξX+DX⊥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 Aξ 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:(7)JX=PX+FX,andJN=tN+fN,where PX (respectively, tN) is the tangential and FX (respectively, fN) is the normal component of JE (respectively, JN).It is evident thatgϕX,Y=gPX,Y, for any X,Y∈TxL; this implies that gPX,Y+gX,PY=0. Thus, P2 is a symmetric operator on the tangent space TxL, for all x∈L. The eigenvalues of P2 are real and diagonalizable. Moreover, for each x∈L, one can observe(8)Sxλ=KerP2+μ2xIx,where I denotes the identity transformation on TxL and μx∈0,1 such that −μ2x is an eigenvalue of P2x. Furthermore, it is easy to observe that KerF=Sx1 and KerP=Sx0, where Sx1 is the maximal holomorphic subspace of TxL and Sx0 is the maximal totally real subspace of TxL, and these distributions are denoted by ST and S⊥, respectively. If −μ12x,…,−μk2x are the eigenvalues of P2 at x, then TxL can be decomposed as(9)TxL=Sxμ1⊕Sxμ2⊕⋯⊕Sxμk.EverySxλi, 1≤i≤k, is a P−invariant subspace of TxL. Moreover, if μi≠0, then Sxλi is even dimensional; the submanifold L of a Kaehler manifold L¯ is a generic submanifold if there exists an integer k and functions μi1≤i≤k defined on L with μi∈0,1 such that(i)
Each−μi2x,1≤i≤k, is a distinct eigenvalue of P2 with(10)TxL=SxT⊕Sx⊥⊕Sxλ1⊕⋯⊕Sxλk,for any x∈M.(ii)
The distributions ofSxT,Sx⊥ and Sxμi,1≤i≤k, are independent of x∈L.If, in addition, eachμi is constant on L, then L is called a skew CR-submanifolds [15]. It is significant to account that CR-submanifolds are particular class of skew CR-submanifold with k=1,ST=0,S⊥=0, and μ1 is constant. If S⊥=0, ST1≠0, and k=1, then L is semislant submanifold, whereas if ST=0,S⊥≠0, and k=1, then L is a hemi-slant submanifold.Definition 1.
A submanifoldL 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 μ1 is constant. For any orthonormal basis e1,e2,…,en of the tangent space TxL, the mean curvature vector Ωx and its squared norm are defined as follows:(11)Ωx=1n∑i=1nhei,ei,Ω2=1n2∑i,j=1nghei,ei,hej,ej,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 hE1,E2=gE1,E2Ω, for all E1,E2∈TL, then L is called totally umbilical.
The scalar curvature ofL¯ is denoted by τ¯L and is defined as(12)τL=∑1≤α<β≤mκαβ,where καβ=κ¯eα∧eβ and m is the dimension of the Riemannian manifold L¯.
Lete1,…,en be an orthonormal basis of the tangent space TxL, and if eγ belongs to the orthonormal basis en+1,…,em of the normal space T⊥L, then we have(13)hαβγ=gheα,eβ,eγ,h2=∑α,β=1ngheα,eβ,heα,eβ.
The global tensor field for orthonormal frame of vector fielde1,…,en on L is defined as(14)SE1,E2=∑i=1ngRei,E1E2,ei,for all E1,E2∈TxL, where R is the Riemannian curvature tensor. The above tensor is called the Ricci tensor. If we fix a distinct vector eu from e1,…,en on Ln, which is governed by χ, then the Ricci curvature is defined by(15)Ricχ=∑α=1α≠unκeα∧eu.
A submanifoldL of a Kaehler manifold L¯ is said to be skew CR-warped product submanifolds if it is warped product of the type L=L1×fL⊥, where L1 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 Ld=L1d1+d2×fL⊥d3, where L1 is semislant submanifold with d1−dimensional holomorphic distribution S and d2− dimensional slant distribution Sθ and L⊥d3 is totally real submanifold. More precisely, they obtained Ricci curvature inequalities for these submanifolds as follows.Theorem 1.
LetLd=L1d1+d2×fL⊥d3 be a S−minimal complete skew CR-warped product submanifold isometrically immersed in a Complex space form L¯mc. If the holomorphic and slant distributions S and Sθ are integrable with integral submanifolds LTd1 and Lθd2, respectively, then, for each orthogonal unit vector field χ∈TxLd, either tangent to LTd1 or Lθd2 or L⊥d3, the Ricci curvature satisfies the following expressions.(i)
Ifχ∈TLTd1, then(16)RLχ+d3Δlnf≤14d2Ω2+d3∇lnf2−c4d−d1d2−d2d3−d1d3−12.(ii)
Ifχ∈TLθd2, then(17)RLχ+d3Δlnf≤14d2Ω2+d3∇lnf2−c4d−d1d2−d2d3−d1d3+1−32cos2θ.(iii)
Ifχ∈TL⊥d2, then(18)RLχ+d3Δlnf≤14d2Ω2+d3∇lnf2−c4d−d1d2−d2d3−d1d3+1,where d1, d2, and d3 are the dimensions of STd1, Sθd2, and S⊥d3, respectively, and Ω is the mean curvature vector.Letf be a real-valued differential function on a Riemannian manifold Ln; then, the Bochner formula [19] is stated as(19)12Δ∇f2=RL∇f,∇f+Hf2+g∇Δf,∇f,where RL denotes Ricci tensor and Hf is the Hessian of the function f.


## 3. Main Results

In this section, we obtain some characterization by the application of Bochner formula.Theorem 2.
LetLd=L1d1+d2×fL⊥d3 be a S−minimald−dimensional complete skew CR-warped product submanifold in a complex space form L¯mc, where L1 is a semislant submanifold with d1− dimensional invariant distribution S and d2−dimensional slant distribution Sθ, such that Ricci curvature RLχ≥K,K>0. If χ∈S and satisfying the equality,(20)λ1+d3K=λ1d3d+d24Ω2−c4d−d1d2−d2d3−d1d3−12,then the base submanifold L1d1+d2 is isometric to Euclidean space Rd1+d2, where d3 is the dimension of the anti-invariant submanifold L⊥d3 and λ1 is the eigenvalue corresponding to the eigenfunction lnf.Proof.
Sinceχ∈S, by equation (16),(21)RLχ+d3Δlnf≤14d2Ω2+d3∇lnf2−c4d−d1d2−d2d3−d1d3−12.
By the assumption thatRLχ≥K, we have(22)K+d3Δlnf≤14d2Ω2+d3∇lnf2−c4d−d1d2−d2d3−d1d3−12.
SinceRLχ≥K, on applying the theorem of Myers [1], according to this, if Ricci curvature is greater than by a positive constant, then base manifold L1d3 is compact. On integrating (22) and using Green’s theorem, we obtain(23)VolL1K≤d24∫LtΩ2dV+d3∫Ld∇lnf2dV−∫Ldc4d−d1d2−d2d3−d1d3−12dV,or(24)∫Ld∇lnf2dV≥Kd3VolL1−d24d3∫LdΩ2dV+1d3∫Ldc4d−d1d2−d2d3−d1d3−12dV.
LetHlnf be the Hessian of the warping function lnf; then, we have(25)Hlnf−nI2=Hlnf2+n2I2−2ngI,Hlnf,where n is real number. The above formula provides(26)Hlnf−nI2=2nΔlnf+n2d1+d2+Hlnf2.
Puttingn=λ1/d1+d2 and integrating the last equation with respect to dV (volume element), we obtain(27)∫LdHlnf−λ1d1+d2I2dV=∫LdHlnf2dV+∫Ldλ12d1+d2dV.
Using (19), with the fact Δlnf=λ1lnf, we have(28)∫LdHlnf2dV=−λ1∫Ld∇lnf2dV−∫LdRL∇lnf,∇lnf.
Combining (27) and (28), we derive(29)∫LdHlnf−λ1d1+d2I2dV=∫Ldλ12d1+d2dV−λ1∫Ld∇lnf2dV−∫LdRL∇f,∇fdV.
By the assumptionRL∇f,∇f≥K, the above equation changes to(30)∫LdHlnf−λ1d1+d2I2dV≤∫Ldλ12d1+d2dV−λ1∫Ld∇lnf2dV−KVolL1.
Using (24), the last inequality leads to(31)∫LdHlnf−λ1d1+d2I2dV≤∫Ldλ12d1+d2dV−∫Ldλ1Kd3+KdV+λ1d3∫Ldc4d−d1d2−d2d3−d1d3−12dV−λ1d24d3∫LdΩ2dV.
If (20) holds, then the above inequality produces(32)Hlnf−λ1d1+d2I2=0.
Therefore, we haveHlnfX,X=λ1/d1+d2. Hence, by the application of Tashiro’s result [5], the fibre L1 is isometric to Euclidean space Rd1+d2.If we consider the unit vector fieldχ∈TLθd2 or χ∈TL⊥d3, then we have the following results which can be proved by adopting the similar steps in Theorem 2.Theorem 3.
LetLd=L1d1+d2×fL⊥d3 be a S−minimald−dimensional complete skew CR-warped product submanifold in a complex space form L¯mc, where L1 is a semislant submanifold with d1− dimensional invariant distribution S and d2−dimensional slant distribution Sθ, such that Ricci curvature RLχ≥K,K>0. If χ∈Sθ and satisfying the following equality,(33)λ1+d3K=λ1d3d+d24Ω2−c4d−d1d2−d2d3−d1d3+1−32cos2θ.Then, the base submanifoldL1d1+d2 is isometric to Euclidean space Rd1+d2, where d3 is the dimension of the anti-invariant submanifold L⊥d3 and λ1 is the eigenvalue corresponding to the eigenfunction lnf.Theorem 4.
LetLd=L1d1+d2×fL⊥d3 be a S−minimald−dimensional compete skew CR-warped product submanifold in a complex space form L¯mc, where L1 is a semislant submanifold with d1−dimensional invariant distribution S and d2-dimensional slant distribution Sθ, such that Ricci curvature RLχ≥K,K>0. If χ∈S⊥ and satisfying the following equality,(34)λ1+d3K=λ1d3d+d24Ω2−c4d−d1d2−d2d3−d1d3+1.Then, the base submanifoldL1d1+d2 is isometric to Euclidean space Rd1+d2, where d3 is the dimension of the anti-invariant submanifold L⊥d3 and λ1 is the eigenvalue corresponding to the eigenfunction lnf.Now, we have next result which is based on the study of Garcia-Rio et al. [4].Theorem 5.
LetLd=L1d1+d2×fL⊥d3 be a S−minimal complete skew CR-warped product submanifold in a complex space form L¯mc, where L1 is a semislant submanifold with d1-dimensional invariant distribution S and d2-dimensional slant distribution Sθ, such that Ricci curvature RLχ>K,K>0. If χ∈S and satisfying the following relation,(35)d2Ω2+4d1+d2d3λ1Hlnf2=4d1+d2d3λ1c4d−d1d2−d2d3−d1d3−12+K,for λ1<0. Then, L1 is isometric to warped product of the type R×θU, where R is the Euclidean line and U is a complete Riemannian manifold with the warping function θ, which satisfies the differential equation dθ2/dt2+λ1θ=0.Proof.
For the warping functionlnf, defining the following equation on L1d1+d2,(36)KlnfI+Hlnf2=K2lnf2I2+Hlnf2+2KlnfgI,Hlnf.
However, we know thatI2=trII∗=d1+d2 and gHlnf,I∗=trI∗Hlnf=trHlnf; using these facts, the above equation leads to(37)KlnfI+Hlnf2=Hlnf2+d1+d2K2lnf2−2KlnfΔlnf.
Letlnf be an eigenfunction corresponding to the eigenvalue λ1 satisfying Δlnf=λ1lnf, and we have(38)KlnfI+Hlnf2=Hlnf2+d1+d2K2−2Kλ1lnf2.
Again, usingΔlnf=λ1lnf, it is easy to see that(39)∇lnf22=lnfλ1lnf−∇lnf2,which on integrating provides(40)∫Ldlnf2dV=1λ1∫Ld∇lnf2,∫LdHlnf+KlnfI2dV=∫LdHlnf2dV+d1+d2K2λ1−2K∫Ld∇lnf2dV.
PuttingK=λ1/d1+d2 in (40), we have(41)∫LdHlnf+λ1d1+d2lnfI2dV=∫LdHlnf2dV−λ1d1+d2∫Ld∇lnf2dV.
Furthermore, integrating (16) and applying Green’s Lemma, we find(42)∫LdRLχdV≤d24∫LdΩ2dV+d3∫Ld∇lnf2dV−∫Ldc4d−d1d2−d2d3−d1d3−12dV.From the above two expressions, we have(43)1d3∫LdRLχdV≤d24d3∫LdΩ2dV+d1+d2λ1∫LdHlnf2dV−d1+d2λ1∫LdHlnf+λ1d1+d2lnfI2dV−∫Ldc4d−d1d2−d2d3−d1d3−12dV.On using the assumption thatRLχ≥K, for K>0,(44)∫LdHlnf+λ1d1+d2lnfI2dV≤d2λ14d1+d2d3∫LdΩ2dV+∫LdHlnf2dV−λ1d1+d2d3∫LdKdV−λ1d1+d2∫Ldc4d−d1d2−d2d3−d1d3−12dV.Equivalently,(45)∫LdHlnf+λ1d1+d2lnfI2dV≤∫Ldλ1d1+d2d24d3Ω2−c4d−d1d2−d2d3−d1d3−12+Kd3+Hlnf2dV.By assumption (35), we obtain(46)d2Ω2+4d1+d2d3λ1Hlnf2=4d1+d2d3λ1c4d−d1d2−d2d3−d1d3−12+Kd3.In view of (45) and (46), we find(47)Hlnf+λ1d1+d2lnfI2≤0,orHlnf+λ1d1+d2lnfI=0.By taking trace of the above equation, we obtain(48)Δlnf+λ1lnf=0.Now, by the application of the result obtained in [4], together with the fact that Ld=L1d1+d2×fL⊥d3 is nontrivial, we deduced that L1 is isometric to a warped product of the form R×θU, where U is complete Riemannian manifold. In addition, the warping function θ is the solution of the differential equation dθ2/dt2+λ1θ=0. Hence, the proof is completed.Similarly, we can prove the following theorems by taking the unit vector fieldχ tangent to Lθd2 and L⊥d3, respectively.Theorem 6.
LetLd=L1d1+d2×fL⊥d3 be a S−minimal complete skew CR-warped product submanifold in a complex space form L¯mc, where L1 is a semislant submanifold with d1-dimensional invariant distribution S and d2-dimensional slant distribution Sθ, such that Ricci curvature RLχ>K,K>0. If χ∈Sθ and satisfying the following relation,(49)d2Ω2+4d1+d2d3λ1Hlnf2=4d1+d2d3λ1c4d−d1d2−d2d3−d1d3+1−32cos2θ+K,for λ1<0. Then, L1 is isometric to warped product of the type R×θU, where R is the Euclidean line and U is a complete Riemannian manifold with the warping function θ, which satisfies the differential equation dθ2/dt2+λ1θ=0.Theorem 7.
LetLd=L1d1+d2×fL⊥d3 be a S−minimal complete Skew CR-warped product submanifold in a complex space form L¯mc, where L1 is a semislant submanifold with d1-dimensional invariant distribution S and d2-dimensional slant distribution Sθ, such that Ricci curvature RLχ>K,K>0. If χ∈S⊥ and satisfying the following relation,(50)d2Ω2+4d1+d2d3λ1Hlnf2=4d1+d2d3λ1c4d−d1d2−d2d3−d1d3+1+K,for λ1<0. Then, L1 is isometric to warped product of the type R×θU, where R is the Euclidean line and U is a complete Riemannian manifold with the warping function θ, which satisfies the differential equation dθ2/dt2+λ1θ=0.


## 4. Conclusions

This paper studies the geometric behavior of ordinary differential equations on the skew CR-warped product submanifolds. More precisely, we obtain characterizing theorems for skew CR-warped product submanifolds of complex space forms via differential and integral theory on Riemannian manifolds. Therefore, the present article provides a wonderful correlation of theory of differential equations with the warped product submanifolds.



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