# Geometric Mechanics on Warped Product Semi-Slant Submanifold of Generalized Complex Space Forms

Authors: Yanlin Li; Ali H. Alkhaldi; Akram Ali
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

In this study, we develop a general inequality for warped product semi-slant submanifolds of typeMn=NTn1×fNϑn2 in a nearly Kaehler manifold and generalized complex space forms using the Gauss equation instead of the Codazzi equation. There are several applications that can be developed from this. It is also described how to classify warped product semi-slant submanifolds that satisfy the equality cases of inequalities (determined using boundary conditions). Several results for connected, compact warped product semi-slant submanifolds of nearly Kaehler manifolds are obtained, and they are derived in the context of the Hamiltonian, Dirichlet energy function, gradient Ricci curvature, and nonzero eigenvalue of the Laplacian of the warping functions.


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## Body

## 1. Introduction

We can examine the energy, angles, and lengths of their second fundamental form using certain warped product manifolds. These manifolds are generalizations of Riemannian product manifolds and provide examples of manifolds with a strictly negative curvature from a mathematical standpoint. They can be usefully applied to models of spacetime around black holes and bodies with enormous gravitational fields from a mechanical standpoint. From the geometric standpoint of applied mathematics, their warping functions can solve numerous partial differential equations (see [1, 2]). Bishop and O’Neill [3] first proposed the concept of warped product manifolds in order to analyze negative curvature manifolds. Here’s how they define it.Definition 1.
LetN1,g1 and N2,g2 be two Riemannian manifolds, where f:N1⟶0,∞ and γ1:N1×N2⟶N1,γ2:N1×N2⟶N2; the orthogonal projection maps are defined as γ1t,s=t and γ2t,s=s for any t,s∈N1×N2. Then, the warped product N1×fN2 is a product manifold that is N1×N2 associated with the Riemannian structure; in other words,
(1)gX,Y=g1γ1∗X,γ1∗Y+f∘γ12g2γ2∗X,γ2∗Y,for any X,Y∈TMn, where the tangent map is denoted by ∗ and f represents a warping function of Mn.The theory of slant submanifolds is currently under investigation; it was first established by Chen in [4] for nearly Hermitian manifolds. The almost complex (holomorphic) and entirely real submanifolds are specific examples among the classes of slant submanifolds. As a result, the warped product semi-slant submanifold is the most basic generalization of a CR-warped product submanifold. al-Solamy et al. [5] recently investigated a warped product semi-slant submanifold of a nearly Kaehler manifold, proving that no such warped product semi-slant submanifold exists of the form Mn=Nϑn2×fNTn1, where Nϑn2 is a proper slant submanifold and NTn1 is a complex submanifold.The researchers next looked into warped products of the typeMn=NTn1×fNϑn2 and came up with a number of fascinating conclusions, including characterizations and an inequality. We refer to [6] for a survey of warped product submanifolds. We first remark that the utilization of the Codazzi equation in Chen’s study [7] problem atomizes attempts to extend its results from the warped product semi-slant submanifold setting, due to the slant angle’s involvement. We use a novel strategy in this work, substituting the Codazzi equation (used in [7]) with the Gauss equation. As a generalization of the contact CR-warped products, we construct a sharp general inequality for warped product semi-slant submanifolds isometrically immersed in a generalized space form. We also investigate nontrivial warped product semi-slant submanifolds of type Mn=Nϑn2×fNTn1 that are isometrically immersed in an arbitrary nearly Kaehler manifold; we obtain results (cf. Theorem 21) and consider interesting applications thereof (cf. Theorem 22).Chen developed a general inequality for the CR-warped product of complex space forms in [7]. Furthermore, in [8–11], the classifications of contact CR-warped products in spheres that satisfy the equality cases similarly were given. The classifications of the totally geodesic and totally umbilical submanifolds are examples of how these relations might be used to classify equalities in the derived inequality. For various types of inequalities, several authors (in [12–17]) have presented thorough classifications of CR-warped products in complex projective space forms and Lagrangian submanifolds in complex space forms. Motivated by previous studies, we derived necessary and sufficient conditions to determine whether a compact oriented warped product semi-slant submanifold in a generalized complex space forms is trivial (cf. Theorems 24, 25, and 26, and Corollaries 27 and 28).Calin and Chang presented a geometric approach to Riemannian manifolds in [18], identifying its applicability to partial differential equations that implement a Lagrangian formalism on Riemannian manifolds; for example, they considered its application to the energy-momentum tensor and conservation laws; the Hamiltonian formalism; Hamilton-Jacobi theory; harmonic functions, maps, and geodes; and harmonic functions, maps, and geodes. Let us note that the geometry of a Riemannian manifold can be thought of as a compact Riemannian submanifold with a boundary; in other words, ∂M≠∅. We considered the Euler–Lagrange equation, kinetic energy function, and Hamiltonian approach to warped product submanifolds for which the warping function plays an important role as a positive differential function for such identities because of the influence of the slant angle in a warped product semi-slant submanifold of a nearly Kaehler we provide (cf. Theorems 29, 30, and 32).The effect of Ricci curvature on the structure of warped products is investigated. In Riemannian geometry, one important question arises: What is the geometric meaning of Ricci curvature? Answer: Ricci-flat manifolds require us to solve the Riemannian manifold’s Einstein field equations with a vanishing cosmological constant geometrically.We study the Ricci curvature on the structure of warped products. One fundamental question arises: What is the geometric meaning of Ricci curvature in Riemannian geometry? Answer: Geometrically, Ricci-flat manifolds require us to solve the Einstein field equations of the Riemannian manifold with a vanishing cosmological constant. In general relativity, the Ricci tensor corresponds to the universe’s matter content via Einstein field equations. The degree to which matter tends to converge or diverge over time is determined by this term of spacetime curvature. As a result, in physics, Ricci curvature is more essential than Riemannian curvature, and geometric obstacles of the Ricci curvature and Ricci tensor will be found in warped product manifolds (for further details, see [12, 19] and the references therein). Our next goal is to look into the physical implications of these issues in terms of warping functions. We propose our result (cf. Theorem 33) to enable our study to uncover the useful applications of the obtained inequality in physics. The work described in this paper will be combined with the singularity theory techniques presented in [20–24].The following is a breakdown of the paper’s structure: We review some basic formulas and definitions in Section2 and give a quick overview of semi-slant submanifolds. In Section 3, we analyze warped product semi-slant submanifolds and prove an inequality for an intrinsic invariant in a nearly Kaehler manifold in terms of the second basic form, the squared norm of the warping function, and the Laplacian of the warping functions. The case of equality is also examined. In this section, we get the main result for warped product semi-slant submanifolds immersed isometrically in a nearly Kaehler manifold. In Section 4, we use boundary conditions to explain multiple classifications of such inequalities for Riemannian and compact Riemannian submanifolds. In Section 5, we strengthen the second fundamental form inequality in a virtually Kaehler manifold for warped product semi-slant submanifolds and CR-warped product submanifolds. We also show that the warped product semi-slant manifold in a nearly Kaehler manifold becomes a Riemannian product under a set of complicated requirements expressed in terms of the kinetic energy function and the Hamiltonian of the warping function. In Section 6, we prove that the compact warped product semi-slant submanifold of a virtually Kaehler manifold is either a CR-warped product manifold or a simple Riemannian product manifold in terms of the gradient Ricci curvature of warped functions.


## 2. Preliminaries

An almost Hermitian manifoldM~,J,g of a 2m-dimensional space, such that J is an almost complex structure and g is a Riemannian metric, satisfies
(2)aJ2=−I,bgJX,JY=gX,Y,for any X,Y on M~2m, where the identity map is denoted by I. Let ΓTM~2m denote the set of all vector fields tangent to M~2m; the Levi-Civita connection defined on M~2m is denoted by ∇~. Then, a Kaehler manifold with an almost complex structure J satisfies
(3)∇~XJY=0,for any X,Y∈ΓTM~2m and M~2m. Moreover, if the almost complex structure J is such that
(4)∇XJY+∇YJX=0,for any vector field X,Y tangent to M~2m, then the manifold M~ represents a nearly Kaehler manifold [25–27]. The above equation is similar to the following:
(5)∇XJX=0,X∈ΓTM~2m.Assume thatM~n is a complex space form of constant holomorphic sectional curvature 4κ and it is denoted by M~n4κ. The curvature tensor R~ of M~n4κ can be expressed as
(6)R~W1,W2W3=κgW2,W3W1−gW1,W3W2+gW3,JW2JW1−gW3,JW1JW2+2gW1,JW2JW3,for all W1,W2,W3∈ΓTN~. Based on the cases κ<0,κ=0,andκ>0, ℕ~n4κ is the complex hyperbolic space ℂHn, complex Euclidean space ℂn, or the complex projective space ℂPn. Now, we consider the generalized complex space forms which are a natural generality of complex space forms and a special family of Hermitian manifolds. Actually, a generalized complex space form is a RK-manifold of constant type α with constant holomorphic sectional curvature κ. Moreover, it is denoted by M~2mκ,α. Hence, the curvature tensor R~ for generalized complex space is given by
(7)R~X,Y,Z,W=κ+3α4gY,ZgX,W−gY,WgX,Z+κ−α4gX,JZgJY,W−gY,JZgJX,W+2gX,JYgJZ,W,for any X,Y,Z,W∈ΓTM~2m. Thus, for the more classifications of generalized complex space forms, we refer to [10, 11, 28–32]. The curvature tensor R~ for a nearly Kaehler 6-sphere is given by
(8)R~X,Y,Z,W=gY,ZgX,W−gY,WgX,Z,for any X,Y,Z,WTM~S6.A submanifold is denoted by theMn of an almost Hermitian manifold M~2m with an induced Riemannian metric g. However, ∇⊥ and ∇ represent the induced Riemannian connections on the normal bundle T⊥Mn and tangent bundle TMn of Mn, respectively. Thus, the Gauss and Weingarten formulas are defined as
(9)i∇~XY=∇XY+hX,Y,ii∇~XN=−ANX+∇X⊥N,for every N∈ΓT⊥Mn and X,Y∈ΓTMn, where AN and h denote the shape operator and second fundamental form for an immersion of M into M~2m, respectively. Now, for any N∈ΓT⊥Mn and X∈ΓTMn, we have
(10)iJX=PX+FX,iiJN=tN+fN,where FXfN and PXtN are the normal and tangential components of JNJX, respectively. From (2), it can be clearly seen that, for each X,Y∈ΓTMn, we have
(11)agPX,Y=−gX,PY,bP2=∑i,j=1ng2Pei,ej,for each ei,i=1,⋯,n tangent to Mn. Assuming M~2m to be a Riemannian manifold and Mn a submanifold of M~2m, the Gauss equation can be defined as
(12)R~X,Y,Z,W=RX,Y,Z,W+ghX,Z,hY,W−ghX,W,hY,Z,for any X,Y,Z,W∈ΓTMn, where R~ and R represent the curvature tensors on M~2m and Mn, respectively. Furthermore, totally umbilical and totally geodesic submanifolds satisfy hX,Y=gX,YH and hX,Y=0, respectively, for any X,Y∈ΓTMn, where H is the mean curvature vector of Mn. If H=0, then Mn is called a minimal submanifold. The mean curvature vector H is expressed in terms of e1,e2⋯,en, which is the so-called orthonormal frame of the tangent space TMn; it is defined as
(13)H=1ntraceh=1n∑i=1nhei,ei,where n=dimM. Moreover, we have
(14)h2=∑i,j=1nghei,ej,hei,ej,hijr=ghei,ej,er,for which eii=1,⋯n and err=n+1,⋯2m are orthonormal frames tangent to Mn and normal to Mn, respectively. The scalar curvature τ for a submanifold Mn of an almost complex manifold M~2m is given by
(15)τTMn=∑1≤i≠j≤nKei∧ej.In the above equation,ei and ej represent the span of the plane section, and its sectional curvature is denoted by Kei∧ej. Let Gr be an r-plane section on TMn and e1,e2⋯er be any orthonormal basis of Gr. Then, the scalar curvature τGr of Gr is defined as
(16)τGr=∑1≤i≠j≤rKei∧ej.Letϕ be a differential function defined on Mn. Thus, the gradient ∇⟶ϕ is given as
(17)g∇⟶ϕ,X=Xf,∇⟶ϕ=∑i=1neiϕei.Thus, from the above equation, the Hamiltonian in a local orthonormal frame is defined as(18)Hdϕ,x=12∑j=1ndfej2=12∑j=1nejϕ2=12∇ϕ2.Moreover, the LaplacianΔf of f is also given by
(19)Δϕ=∑i=1n∇eieiϕ−eieiϕ=−∑i=1ng∇eigradϕ,ei.Similarly, the Hessian tensor of functionf is given by
(20)Δϕ=−TraceHϕ=−∑i=1nHessϕei,ei,where Hϕ denotes the Hessian tensor. The compact manifold Mn is considered as being without a boundary; that is, ∂Mn=∅. Thus, we have the following lemma.Lemma 2 (see [18]; Hopf’s lemma).
LetMn be a connected and compact Riemannian manifold and ϕ a smooth function on Mn such that Δϕ≥0Δϕ≤0. Then, ϕ is a constant function on Mn.Moreover, the integration of the Laplacian of the smooth function, defined on a compact-orientated Riemannian manifoldMn without boundary, vanishes with respect to the volume element of such a manifold, and we obtain the following formula:
(21)∫MnΔϕdV=0,where dV denotes the volume of Mn (see [33]).Theorem 3 (see [18]).
The Euler–Lagrange equation for the Lagrangian is(22)Δϕ=0.Hopf’s lemma becomes the uniqueness theorem for the Dirichlet problem if manifoldMn has a boundary.Theorem 4 (see [18]).
LetMn be a connected and compact manifold and f a positive differentiable function on Mn such that Δϕ=0,OnMn. Thus, ϕ=0, where ∂Mn is the boundary of M.Moreover, letMn be a compact Riemannian manifold and f be a positive differentiable function on Mn. Then, the Dirichlet energy function is defined as described in [18]; that is,
(23)Eϕ=12∫Mn∇ϕ2dV.IfMn is compact, then 0≤Eϕ<∞. We provide the following definition of a slant submanifold.Definition 5 (see [4]).
AssumeTxMn−0 to be a set containing all nonzero tangent vector fields of immersion Mn in an almost Hermitian manifold M~2m at a point x∈Mn. Then, for each vector X∈TxMn at point x∈Mn, the angle between JX and the tangent space TxM is considered to be the Wirtinger angle of X at x∈Mn; this is denoted as ϑX. In this case, a submanifold Mn of M~2m is called a slant submanifold such that ϑ is a slant angle.It is clear that the slant submanifolds include totally real and holomorphic submanifolds. However, Chen proved the following characterization theorem of slant submanifolds.Theorem 6.
LetM~2m be an almost Hermitian manifold and Mn be a submanifold of M~2m. Then, Mn is slant if and only if there exists a constant λ∈0,1 such that
(24)P2=−λI,where λ=cos2ϑ for a slant angle ϑ defined on the tangent bundle TMn of Mn.Hence, we have the following consequences of Theorem6:
(25)gPX,PY=cos2ϑgX,Y,gFX,FY=sin2ϑgX,Y,for any X,Y∈ΓTMn.In an essentially Hermitian manifold, another group of submanifolds known as semi-slant submanifolds exists as a natural generalization of slant submanifolds, CR-submanifolds, and holomorphic and antiholomorphic submanifolds. Papaghiuc researched and defined semi-slant submanifolds in [34] as a natural extension of CR-submanifolds of an almost Hermitian manifold. The following is the definition of a semi-slant submanifold.Definition 7.
A Riemannian submanifoldMn of an almost Hermitian manifold M~2m is defined as a semi-slant submanifold if there exist two complementary distributions D and Dϑ such that
(i)
TM=D⊕Dϑ(ii)
D is a holomorphic distribution; that is, JD=D(iii)
Dϑ is called a slant distribution if slant angle ϑ≠0,π/2Remark 8.
For a semi-slant submanifold, let us consider the dimensions ofD and Dϑ in d1 and d2; then, Mn is holomorphic if d2=0 and slant if d1=0. Furthermore, if ϑ=π/2 and d1=0, then Mn is represented as an antiholomorphic (totally real) submanifold. Moreover, Mn is referred to as a proper semi-slant submanifold if ϑ differs from 0 and π/2. We can also define Mn as proper if d1≠0 and d2≠0.Remark 9.
Ifν is an invariant normal subspace under an almost complex structure J of the normal bundle T⊥Mn, then this normal bundle is decomposed as
(26)T⊥Mn=FDϑ⊕ν.


## 3. Warped Product semi-slant Submanifolds of Nearly Kaehler Manifolds

We will go through some of the findings on warped product manifolds in this section. References [5, 6, 35–40] provide more information. We derive our main inequality for the squared norm of the second fundamental form in terms of constant holomorphic sectional curvature using numerous geometric conditions for the mean curvature of a warped product semi-slant submanifold.In particular, a warped product manifold is classified to betrivial if the warping function is constant. In such cases, we refer to the warped product manifold as a Riemannian product manifold. It was proven in [3] that, for X∈ΓTN1 and Z∈ΓTN2, the following is satisfied:
(27)∇XZ=∇ZX=XlnfZ,where ∇ denotes the Levi-Civita connection on Mn. We recall the following lemma obtained in [3].Lemma 10.
A warped product manifoldMn=N1×fN2. Thus,
(i)
∇XY∈ΓTN1(ii)
∇ZW=∇Z′W−gZ,W∇lnf
for anyZ,W∈ΓTN2 and X,Y∈ΓTN1, where ∇′ is the Levi-Civita connection on N2.Remark 11.
If the warping functionf is constant, then M=N1×fN2 is a trivial warped product or a simple Riemannian product.Remark 12.
In a nontrivial warped product manifoldMn=N1×fN2, the manifold N1 is totally geodesic, and N2 is a totally umbilical submanifold in Mn.Letφ:Mn=N1×fN2⟶M~2m be an isometric immersion of a warped product manifold N1×fN2 in an arbitrary Riemannian manifold M~2m. Furthermore, let n1,n2, and n represent the real dimensions of N1,N2,, and Mn, respectively. Then, for any unit tangent vectors Y and W on N1 and N2, respectively, we have
(28)KY∧W=g∇W∇YY−∇Y∇WY,W=1f∇YYf−Y2f.If we consider the local orthonormal framee1,e2,⋯,en such that the vectors e1,e2⋯,en1 are tangential to N1 and en1+1⋯,en are tangential to N2, then in view of the Gauss equation (12), we can deduce that
(29)τTMn=τ~TMn+∑r=12m∑1≤i≠j≤nhiirhjjr−hijr2,for each j=n1+1⋯n.Hereafter, we will denote the corresponding dimensions as indices. Recall that [5] proved several results for both types of warped product semi-slant submanifolds in nearly Kaehler manifolds.Theorem 13 (see [5]).
There does not exist a proper warped product submanifold of the formMn=Nϑn2×fNTn1 in a nearly Kaehler manifold M~2m such that Nϑn2 is a proper slant submanifold and NTn1 is a holomorphic submanifold of M~2m.Lemma 14.
For a nontrivial warped product semi-slant submanifoldMn=NTn1×fNϑn2 in a nearly Kaehler manifold M~2m, we have the following equalities:
(i)
ghJX,Z,FW+ghJX,W,FZ=2XlnfgZ,W(ii)
ghX,Z,FW+ghX,W,FZ=−2JXlnfgZ,W
for anyX∈ΓTNTn1 and Z,W∈ΓTNϑn1.Proof.
For the first part of the proof, using (9) (i) and the orthogonality of vector fields, we establish that
(30)ghJX,W,FZ=−g∇~WJZ,JX−g∇~WJX,PZ.
From the covariant derivative of the almost complex structureJ and from (27), we derive that
(31)ghJX,W,FZ=XlnfgZ,W−gJX,∇~WJZ−JXlnfgW,PZ.
Using the structure equation (4) in the above equation and in (10) (i), we find that
(32)ghJX,W,FZ=XlnfgZ,W+g∇~ZFW,JX−g∇~ZJX,PW+g∇~ZX,W−JXlnfgW,PZ.
Thus, from (27) and (10) (ii), we finally obtain
(33)ghJX,W,FZ+ghJX,Z,FW=2XlnfgZ,W,which is (i). Replacing X wit JX in (i) and using (2) (i), we obtain the required result (ii). The lemma is proven completely.Lemma 15.
Assume thatMn=NTn1×fNϑn2 is a nontrivial warped product semi-slant submanifold in a nearly Kaehler manifold; thus, we have
(i)
ghJX,PZ,FPW+ghJX,PW,FPZ=2Xlnfcos2ϑgZ,W(ii)
ghX,PZ,FPW+ghX,PW,FPZ=−2JXlnfcos2ϑgZ,W
for anyX∈ΓTNTn1 and Z,W∈ΓTNϑn2.Proof.
ReplacingZ and W by PZ and PW, respectively, and using (24) in Lemma 14 (i)–(ii), we directly obtain (i) and (ii), respectively. This completes the proof of the lemma.Remark 16.
In particular, if we substituteZ=W in Lemma 10, then Lemma 10 coincides with Lemma 3.1 in [5].Lemma 17.
LetMn=NTn1×fNϑn2 be a warped product semi-slant submanifold of a nearly Kaehler manifold M~2m. Then,
(i)
ghX,Z,FPW+ghX,W,FPZ=2/3cos2ϑXlnfZ2(ii)
ghX,PZ,FW+ghX,PW,FZ=2/3cos2ϑXlnfZ2
for anyX∈ΓTNTn1 and Z,W∈ΓTNϑn2.Proof.
ReplacingZ by Z+W in Lemma 5.2 in [41], and using the linearity property of vector fields, we derive (i) and (ii). This completes the proof of the lemma.Lemma 18.
Assume thatMn=NTn1×fNϑn2 is a nontrivial warped product semi-slant submanifold in a nearly Kaehler manifold. For any X∈ΓTNTn1 and Z,W∈ΓTNϑn2, we have
(i)
ghJX,Z,FPW+ghJX,W,FPZ=−2/3cos2ϑJXlnfZ2(ii)
ghJX,PZ,FW+ghJX,PW,FZ=−2/3cos2ϑJXlnfZ2Proof.
Similarly, by replacingX with JX in Lemma 17 (i)–(ii), we arrive at our required results (i) and (ii) using (2).To prove the general inequality, we require an orthonormal frame for orthonormal vector fields, as well as some preparatory results.Lemma 19.
LetMn be a warped product semi-slant submanifold in a nearly Kaehler manifold M~2m. Thus,
(i)
ghX,Y,FZ=0(ii)
ghJX,JY,ξ=−ghX,Y,ξ
for anyX,Y∈ΓTNTn1, Z∈ΓTNϑn2, and ξ∈Γν.Proof.
The first part of the proof is trivial; the second part can be proved in a similar manner to Lemma 5.1 in [39].Lemma 20.
Letφ:Mn=NTn1×fNϑn2⟶M~2m be an isometric immersion of a warped product semi-slant submanifold in a nearly Kaehler manifold M~2m. Then, NTn1 is a minimal submanifold of M~2m, and the squared norm of the mean curvature of Mn is given by
(34)H2=1n2∑r=n+12mhn1+1n1+1r+⋯+hnnr2,where H denotes the mean curvature vector. Moreover, n1,n2,n, and 2m are dimensions of NTn1,Nϑn2,NTn1×fNϑn2, and M~2m, respectively.Proof.
The above lemma can be readily proven in a similar manner to Lemma 5.2 in [39].Main inequality for warped product semi-slant submanifolds.Theorem 21.
Letφ:Mn=NTn1×fNϑn2⟶M~2m be an isometric immersion of a warped product semi-slant submanifold Mn in a nearly Kaehler manifold M~2m. Thus,
(i)
The squared norm of the second fundamental form ofMn satisfies(35)h2≥2n2∇lnf2+δ−n2Δlnf,where δ=τ~TMn−τ~TNTn1−τ~TNϑn2,n2 is the dimension of the slant submanifold Nϑn2, and Δ is the Laplacian operator of NTn1.
(ii)
The equality holds in (35) if and only if NTn1 is totally geodesic and Nϑn2 is totally umbilical in M~2m and if Mn is a minimal submanifold of M~2mProof.
The proof proceeds in a similar manner to the proof of Theorem29 [31] if we consider the nearly Kaehler manifold, instead of nearly trans-Sasakian manifold.


### 3.1. Applications of Theorem21 to Generalized Complex Space Forms

In this section, we prove our main theorem using Theorem21 for a generalized complex space form. Then, we give the following result.Theorem 22.
Assume thatφ:Mn=NTn1×fNϑn2⟶M~2mκ,α be an isometric immersion of a warped product semi-slant NTn1×fNϑn2 into generalized complex space form M~2mκ,α admitting a nearly Kaehler structure with constant holomorphic sectional curvature κ of constant type α. Then,
(i)
The squared norm of the second fundamental form ofMn satisfies(36)h2≥2n2κ+3α4n1−Δlnf+∇lnf2,where n1=dimNT,n2=dimRNϑ,n=dimNT×fNϑ and 2m=dimM~2mκ,α.
(ii)
The equality holds in (36), if and only if Nϑn2 and NTn2 are totally umbilical and totally geodesic submanifolds in M~, respectively. Moreover, Mn is a minimal submanifold of M~2mκ,αProof.
Letting us substituteX=W=ei and Y=Z=ej in (8), we get
(37)R~ei,ej,ej,ei=κ+3α4gei,eigej,ej−gei,ejgei,ej+κ−α4gei,JejgJej,ei−gei,Jeigej,Jej+2g2Jej,ei.
Taking the summation over the basis vector fields ofTMn such that 1≤i≠j≤n, one shows that
(38)2τ~TMn=c+3α4nn−1+3c−α4∑1≤i≠j≤ng2Pei,ej.
Mn is a warped product of holomorphic and proper slant submanifolds in a generalized complex space form M~2mc,α. Thus, we set the following frame of orthonormal vector fields as
(39)e1,e2=Je1,⋯e2d1−1,e2d1=Je2d1−1,e2d1+1,e2d1+2=secϑPe2d1+1,⋯e2d1+2d2−1e2d1+2d2=secϑPed1−1.
From using the above orthonormal frame, we obtain(40)g2Jei,ei+1=1,fori∈1⋯n1−1=cos2ϑfori∈n1+1,⋯n1+n2−1.
Thus, it is easily seen that(41)∑i,j=1ng2Pei,ej=n1+n2cos2ϑ.
From (38) and (40), it follows that
(42)2τ~TMn=κ+3α4nn−1+3κ−α4n1+n2cos2ϑ.
Similarly, forTNTn1, we derive
(43)2τ~TNTn1=κ+3α4n1n1−1+3κ−α4n1.
Now using fact thatP2=n2cos2ϑ, for slant bundle TNϑn2, one derives
(44)2τ~TNϑn2=κ+3α4n2n2−1+3κ−α4n2cos2ϑ.
Therefore, substituting (41), (42), and (40) in Theorem 21, we get the required result (29). The equality case follows from Theorem 21 (ii). Thus, the proof is complete.Notably, the following corollary can be readily obtained in terms of the Hessian tensor of the warping functionlnf for a warped product submanifold.Corollary 23.
Letφ:Mn=NTn1×fNϑn2⟶M~2mκ,α be an isometric immersion of a warped product NTn1×fNϑn2 into generalized complex space form M~2mκ,α admitting a nearly Kaehler structure. Then,
(45)h2≥2n2κ+3α4n1+∇lnf2+TraceHlnf,where Hlnf is the Hessian tensor of the warping function lnf.


## 3.1. Applications of Theorem21 to Generalized Complex Space Forms

In this section, we prove our main theorem using Theorem21 for a generalized complex space form. Then, we give the following result.Theorem 22.
Assume thatφ:Mn=NTn1×fNϑn2⟶M~2mκ,α be an isometric immersion of a warped product semi-slant NTn1×fNϑn2 into generalized complex space form M~2mκ,α admitting a nearly Kaehler structure with constant holomorphic sectional curvature κ of constant type α. Then,
(i)
The squared norm of the second fundamental form ofMn satisfies(36)h2≥2n2κ+3α4n1−Δlnf+∇lnf2,where n1=dimNT,n2=dimRNϑ,n=dimNT×fNϑ and 2m=dimM~2mκ,α.
(ii)
The equality holds in (36), if and only if Nϑn2 and NTn2 are totally umbilical and totally geodesic submanifolds in M~, respectively. Moreover, Mn is a minimal submanifold of M~2mκ,αProof.
Letting us substituteX=W=ei and Y=Z=ej in (8), we get
(37)R~ei,ej,ej,ei=κ+3α4gei,eigej,ej−gei,ejgei,ej+κ−α4gei,JejgJej,ei−gei,Jeigej,Jej+2g2Jej,ei.
Taking the summation over the basis vector fields ofTMn such that 1≤i≠j≤n, one shows that
(38)2τ~TMn=c+3α4nn−1+3c−α4∑1≤i≠j≤ng2Pei,ej.
Mn is a warped product of holomorphic and proper slant submanifolds in a generalized complex space form M~2mc,α. Thus, we set the following frame of orthonormal vector fields as
(39)e1,e2=Je1,⋯e2d1−1,e2d1=Je2d1−1,e2d1+1,e2d1+2=secϑPe2d1+1,⋯e2d1+2d2−1e2d1+2d2=secϑPed1−1.
From using the above orthonormal frame, we obtain(40)g2Jei,ei+1=1,fori∈1⋯n1−1=cos2ϑfori∈n1+1,⋯n1+n2−1.
Thus, it is easily seen that(41)∑i,j=1ng2Pei,ej=n1+n2cos2ϑ.
From (38) and (40), it follows that
(42)2τ~TMn=κ+3α4nn−1+3κ−α4n1+n2cos2ϑ.
Similarly, forTNTn1, we derive
(43)2τ~TNTn1=κ+3α4n1n1−1+3κ−α4n1.
Now using fact thatP2=n2cos2ϑ, for slant bundle TNϑn2, one derives
(44)2τ~TNϑn2=κ+3α4n2n2−1+3κ−α4n2cos2ϑ.
Therefore, substituting (41), (42), and (40) in Theorem 21, we get the required result (29). The equality case follows from Theorem 21 (ii). Thus, the proof is complete.Notably, the following corollary can be readily obtained in terms of the Hessian tensor of the warping functionlnf for a warped product submanifold.Corollary 23.
Letφ:Mn=NTn1×fNϑn2⟶M~2mκ,α be an isometric immersion of a warped product NTn1×fNϑn2 into generalized complex space form M~2mκ,α admitting a nearly Kaehler structure. Then,
(45)h2≥2n2κ+3α4n1+∇lnf2+TraceHlnf,where Hlnf is the Hessian tensor of the warping function lnf.


## 4. Compact-Orientated Warped Product semi-slant Submanifolds

In this section, we consider compact Riemannian manifolds without boundaries; that is,∂Mn=∅. Applying these to warped product semi-slant submanifolds, and using integration theory on the manifold, we obtain several characterizations.Theorem 24.
LetMn=NTn1×fNϑn2 be a compact-orientated warped product semi-slant submanifold generalized complex space form M~2mκ,α admitting a nearly Kaehler structure. Then, Mn is a trivial warped product if and only if
(46)h2≥2κ+3α4n2n1.Proof.
From Theorem21, we have
(47)h2≥2κ+3α4n2n1−n2Δlnf+n2∇lnf2.
This implies that(48)n2∇lnf2+2κ+3α4n2n1−h2≤n2Δlnf.
Applying integration theory on the compact-orientated Riemannian manifoldMn without boundary, and then using (21), we obtain
(49)∫Mn2κ+3α4n2n1+n2∇lnf2−h2dV≤∫MnΔlnfdV=0.
Now, if the inequality (44) holds, then from (47), we find that
(50)∫Mn∇lnf2dV≤0,which is impossible for a positive integrable function; hence, ∇lnf=0; that is, f is a constant function on Mn. Thus, by Remark 11 on warped product manifolds, Mn is trivial. The converse proof is straightforward.To prove the equality case, we must prove the following theorem for later use.Theorem 25.
Letφ be a Dϑ-minimal isometric immersion of a warped product semi-slant submanifold NTn1×fNϑn2 in a nearly Kaehler manifold M~2m. If Nϑn2 is totally umbilical in M~2m, then ϕ is Nϑn2-totally geodesic.Proof.
Let us assume that the second fundamental forms ofMn and M~2m are denoted by h∗ and h~, respectively; we define
(51)hZ,W+h∗Z,W=h~Z,W.for any vector fields Z and W that are tangential to Nϑn2. Thus, the above hypothesis and Remark 12 show that Nϑn2 is totally umbilical in M~2m, owing to its being totally umbilical in Mn. Then, following Lemma 10 (ii), equation (47) can be written as
(52)hZ,W=gZ,Wξ+∇lnf,where the vector field ξ is normal to ΓTNϑ and is such that ξ∈ΓTMn. Assuming that e1∗,⋯en2∗ is an orthonormal frame of the slant submanifold Nϑn2, then by taking a summation over the vector fields of Nϑn2 in equation (49), we obtain
(53)∑i,j=1n2hei∗,ej∗=ξ+∇lnf∑i,j=1n2gei∗,ej∗.
The left-hand side of the above equation identically vanishes due to theDϑ-minimality of φ, such that ∑i,j=1n2hei∗,ej∗=0. Then, equation (50) takes the following form:
(54)n2ξ+∇lnf=0.
This implies thatNϑn2 is nonempty, such that
(55)ξ=−∇lnf.
Thus, from (50) and (53), it follows that hZ,W=0, for every Z,W∈ΓTNϑn2. This means that φ is Nϑn2-totally geodesic. This completes the proof of the theorem.Theorem 26.
LetMn=NTn1×fNϑn2 be a compact-orientated warped product semi-slant submanifold in a generalized complex space form M~2mκ,α admitting a nearly Kaehler structure. Then, Mn is a trivial warped product if and only if
(56)∑i=1n1∑j=1n2hνei,ej2=κ+3α4n2n1,where n1=dimNT and n2=dimRNϑ. Moreover, hν is a component of h in Γν.Proof.
We assume that the equality sign holds in (36); then, we have
(57)κ+3α4n2n1+2n2∇lnf2=hD,D2+hDϑ,Dϑ2+2hD,Dϑ2+2n2Δlnf.
However, the equality case of inequality (36) implies that NTn1 is totally geodesic in a nearly Kaehler manifold; this means that hei,ej=0, for any 1≤i,j≤2d1. Moreover, Nϑn2 is totally umbilical and can be written as
(58)het∗,es∗=get∗,es∗H,for any 1≤t,s≤2d2. Furthermore, Mn is a minimal submanifold of a nearly Kaehler manifold; thus, its mean curvature vector H should be zero; that is, H=0; hence, het∗,es∗=0,forevery1≤t,s≤2d2 through Theorem 25(59)κ+3α4n2n1=n2Δlnf+hD,Dϑ2−n2∇lnf2.
We assume thatMn is a compact submanifold; thus, Mn is closed and bounded; hence, by integrating the above equation over the volume element dV of Mn and using (21), we find that
(60)∫MnhD,Dϑ2−n2∇lnf2dV=∫Mnκ+3α4n2n1dV.
Now, letX=ei and Z=ej for 1≤i≤n1 and 1≤j≤n2, respectively; then, using (14) and expressing in terms of the orthonormal frame, we have
(61)hD,Dϑ2=∑k=12m∑i=1n1∑j=1n1ghei,ej∗,ek2.
In the above equation, the first term on the right-hand side is theFDϑ-component and the second term is the ν-component. Let us suppose that Mn=NTn1×fNϑn2 is a warped product semi-slant submanifold of an n-dimension in a nearly Kaehler manifold M~2m of 2m dimensions, such that dimNT=n1=2d1 and dimRNϑ=n2=2d2. We assume that the tangent spaces of NTn1 and Nϑn2 are D and Dϑ, respectively. We further assume that e1,e2,⋯,ed1,ed1+1=Je1,⋯,e2d1=Jed1 is a local orthonormal frame of TNTn1 and that e2d1+1=e1∗,⋯,e2d1+d2=ed2∗,e2d1+d2+1=ed2+1∗=secϑPe1∗,⋯,en1+n2=en2∗=secϑPed2∗ is a local orthonormal frame of TNϑn2. Thus, the orthonormal frames of the normal subbundles FDϑ and ν are en+1=e~1=cscϑFe1∗,⋯,en+d2=e~d2=cscϑFe1∗,en+d2+1=e~d2+1=cscϑsecϑFPe1∗,⋯,en+2d2=e~2d2=cscϑsecϑFPed2∗ and en+2d2+1,⋯,e2m, respectively. Taking a summation over the vector fields on NTn1 and Nϑn2 and using adapted frame fields, we obtain
(62)hD,Dϑ2=csc2ϑ∑i=1d1∑j,k=1d2ghei,ej∗,Fek∗2+csc2ϑsec2ϑ∑i=1d1∑j,k=1d2ghei,Pej∗,Fek∗2+csc2ϑsec2ϑ∑i=1d1∑j,k=1d2ghJei,ej∗,FPek∗2+csc2ϑsec2ϑ∑i=1d1∑j,k=1d2ghJei,ej∗,FPek∗2+csc2ϑsec4ϑ∑i=1d1∑j,k=1d2ghJei,Pej∗,FPek∗2+csc2ϑsec2ϑ∑i=1d1∑j,k=1d2ghJei,Pej∗,Fek∗2+csc2ϑ∑i=1d1∑j,k=1d2ghJei,ej∗,Fek∗2+csc2ϑsec4ϑ∑i=1d1∑j,k=1d2ghei,Pej∗,FPer∗2+∑r=n+n2+12m∑i=1n1∑j=1n2ghei,ej,er2.
Then, using Lemmas14 to 18 in the above equations, we derive
(63)hD,Dϑ2=2csc2ϑ+19cot2ϑ∑i=1d1∑j=1d2eilnf2gej∗,ej∗2+2csc2ϑ+19cot2ϑ∑i=1d1∑j=1d2Jeilnf2gej∗,ej∗2+∑r=n+n2+12m∑i=1n1∑j=1n2ghei,ej,er2.
From (17), the last equation can be expressed as
(64)hD,Dϑ2=2csc2ϑ+19cot2ϑ∇lnf2∑j=1d2gej∗,ej∗2+∑r=n+n2+12m∑i=1n1∑j=1n2ghei,ej,er2,which implies that
(65)hD,Dϑ2=n21+109cot2ϑ∇lnf2+∑i=1n1∑j=1n2hνei,ej2.
Then, from (58) and (63), it follows that
(66)∫Mn∑i=1n1∑j=1n2hμei,ej2+109n2cot2ϑ∇lnf2dV=∫Mnκ+3α4n2n1dV.
If (54) holds identically, then from (66), we find that either f is constant on Mn or cotϑ=0. However, Mn is a proper semi-slant submanifold; thus, Mn is a simple Riemannian product. The converse proof follows immediately from (66). Hence, the theorem is proven completely.Corollary 27.
Assume thatMn=NTn1×fNϑn2 is a warped product semi-slant submanifold in a generalized complex space form M~2mκ,α admitting a nearly Kaehler structure. Suppose that NTn1 is a compact invariant submanifold and λT is a nonzero eigenvalue of the Laplacian on NTn1. Then,
(67)∫NTn1h2dVT≥∫NTn1κ+3α4n2n1dV+2n2λT∫NTn1lnf2dVT,where dVT is the volume element on NTn1. The equality sign holds if and only if the following are satisfied:
(i)
∇lnf=λTlnf(ii)
A warped product semi-slant submanifoldMn=NTn1×fNϑn2 is both NTn1- and Nϑn2-totally geodesicProof.
Thus, using the minimum principle property, we have(68)∫NTn1∇lnf2dVT≥λT∫NTn1lnf2dVT,
The equality holds if and only if one has∇lnf=λTlnf. Thus, from (44) and (66), we require the result (65). Similarly, it is clear that the equality sign of (65) holds identically if and only if the warped product is both NTn1- and Nϑn2-totally geodesic. This completes the proof of the corollary.Corollary 28.
LetMn=NTn1×fNϑn2 be a warped product semi-slant submanifold in a generalized complex space form M~2mκ,α admitting a nearly Kaehler structure such that NTn1 is compact, and let λT be a nonzero eigenvalue of the Laplacian on NTn1. Then,
(69)∫NTn1∑i=1n1∑j=1n2hνei,ej2dVT≥∫NTn12κ+3α4n2n1dV+109n2cot2ϑλT∫NTn1lnf2dVT.
The equality sign holds if and only if the following are satisfied:(a)
∇lnf=λTlnf(b)
A warped product semi-slant submanifoldMn=NTn1×fNϑn2 is both NTn1- and Nϑn2-totally geodesicProof.
The proof follows from (67) and (66). This completes the proof of the theorem.


## 5. Applications to Dirichlet Energy Functions and Hamiltonian

We discuss connected, compact Riemannian manifolds with borders in this section; that is,∂M≠∅. Using Hopf’s lemma, we apply these to warped product submanifolds. To determine whether nontrivial warped products become trivial warped product submanifolds of nearly Kaehler manifolds, we obtain necessary and sufficient conditions in terms of Dirichlet energy (analogous to kinetic energy) and Hamiltonian of warping functions.Theorem 29.
Assume thatφ:Mn=NTn1×fNϑn2 is an isometric immersion of a warped product semi-slant in a generalized complex space form M~2mκ,α admitting a nearly Kaehler structure. A connected and compact warped product NTn1×fNϑn2 is trivial if and only if the Dirichlet energy function satisfies
(70)Elnf=920n2tan2ϑ∫Mnκ+3α4n2n1−∑i=1n1∑j=1n2hνei,ej2dV,where Elnf represents the Dirichlet energy of the warping function lnf and dV is the volume element on Mn.Proof.
Combining equations (57) and (63), we obtain
(71)κ+3α4n2n1=n2Δlnf+∑i=1n1∑j=1n2hνei,ej2+n2109cot2ϑ∇lnf2.
Taking an integration onMn over the volume element dV with a nonempty boundary in the above equation, we find that
(72)∫Mnκ+3α4n2n1dV=n2∫MnΔlnfdV+∫Mn∑i=1n1∑j=1n2hνei,ej2dV+109n2cot2ϑ∫Mn∇lnf2dV.
Then, from (23) and (70), it follows that
(73)1n2∫Mnκ+3α4n2n1dV=∫MnΔlnfdV+1n2∫Mn∑i=1n1∑j=1n2hνei,ej2dV+209cot2ϑElnf.
Equality (68) is satisfied if and only if we obtain from (71) the condition that ∫MnΔlnfdV=0, which implies that Δlnf=0. The theorem hypothesizes Mn as a connected, compact warped product semi-slant submanifold; thus, Theorem 4 implies that lnf=0⟹f=1, which means that f is constant on Mn. Thus, the theorem is proven completely.In a similar manner, we derive several characterizations in terms of the Hamiltonian.Theorem 30.
LetMn=NTn1×fNϑn2 be a connected and compact warped product semi-slant submanifold in a generalized complex space form M~2mκ,α admitting a nearly Kaehler structure. Then, Mn is a trivial warped product submanifold of NTn1 and Nϑn2 if and only if the Hamiltonian of the warping function satisfies the following equality:
(74)Hdlnf,x=920n2tan2ϑκ+3α4n2n1−1n2∑i=1n1∑j=1n2hνei,ej2.Proof.
Using (15) in (69), we derive
(75)209cot2ϑHdlnf,x+Δlnf=κ+3α4n1−1n2∑j=1n2hνei,ej2.
Equation (72) is obtained if and only if Δlnf=0 on Mn; thus, by Theorem 4, the warped product submanifold Mn is trivial. This completes the proof of the theorem.The analogy of Theorem3 for this case is classified below.Theorem 31.
Assume thatφ:Mn=NTn1×fNϑn2 is an isometric immersion of a compact warped product semi-slant submanifold in a generalized complex space form M~2mκ,α admitting a nearly Kaehler structure. Let the warping function be the solution of the Euler–Lagrange equation; then, Mn is necessarily a trivial warped product if
(76)h2≥2κ+3α4n2n1.Proof.
If the warping function satisfies the conditions of the Euler–Lagrange equation, then from Theorem3, we obtain
(77)Δlnf=0.
Thus, from (44) and (75), we derive
(78)h2≥2κ+3α4n2n1+n2∇lnf2.
Suppose that inequality (74) holds; then, (76) implies that the warping function must be constant on Mn. This completes the proof of the theorem.Theorem 32.
Assume thatφ:Mn=NTn1×fNϑn2 is an isometric immersion of a compact warped product semi-slant submanifold in a generalized complex space form M~2mκ,α admitting a nearly Kaehler structure and that the warping function is a solution of the Euler–Lagrange equation. Then, the necessary and sufficient conditions for the warped product NTn1×fNϑn2 being trivial are as follows:
(79)∑i=1n1∑j=1n2hνei,ej2=κ+3α4n2n1.Proof.
The proof of the above theorem is the same as that of Theorem31; it uses (57), (63), and Theorem 3. This completes the proof of the theorem.


## 6. Classification of Ricci Curvature and Divergence of the Hessian Tensor

In this section, we study several applications of the derived inequality by considering equality cases. Let us identify any0,2-tensor T on M with a 1,1-tensor via
(80)gTZ,Y=TZ,Y,for all Y,Z∈ΓTM. Thus, we obtain
(81)divϕT=ϕdivT+T∇ϕ,·,∇ϕT=ϕ∇T+dϕ⊗T,for all ϕ∈C∞M. In particular, we have divϕg=dϕ. Moreover, the following general facts are well established in the literature:
(82)1div∇2ϕ=Ric∇ϕ,·+dΔϕ,ii12d∇2=∇2ϕ∇,·.We considerMn to be a compact Riemannian manifold with boundaries and obtain the following classification results.Theorem 33.
Assume thatφ:Mn=NTn1×fNϑn2 is an isometric immersion of a compact warped product semi-slant submanifold NTn1×fNϑn2 in a generalized complex space form M~2mκ,α admitting a nearly Kaehler structure. If
(83)∑i=1n1∑j=1n2hνei,ej∗2=κ+3α4n2n1+n2∫MnRic∇lnf,·dV,is satisfied for the warped product submanifold Mn, then at least one of the following statements is true for Mn:
(i)
The warped product semi-slant submanifoldNTn1×fNϑn2 becomes a CR-warped product that is isometrically immersed in a nearly Kaehler manifold(ii)
The nontrivial warped product semi-slant submanifoldNTn1×fNϑn2 in a nearly Kaehler manifold is a simple Riemannian product of NTn1 and Nϑn2Proof.
Using the first identity of (79) and setting ϕ=lnf, we derive
(84)div∇2lnf=Ric∇lnf,·+dΔlnf,from the hypothesis of the theorem; assuming that Mn is a compact warped product submanifold with a boundary and integrating along the volume element dV, we obtain
(85)Δlnf=∫Mdiv∇2lnfdV−∫MRic∇lnf,·dV.
We use the Green theorem on the compact manifoldMn; given a smooth function f:M⟶ℝ, we have ∫MΔfdV=0. We can apply the results of Yano and Kon (see [33]) immediately as Δf=−div∇f. From the Green lemma, ∫MdivXdV=0 for any arbitrary vector field X on Mn. Thus, we obtain ∫Mdiv∇2lnfdV=0; ∇2lnf is the Hessian tensor of the warped function (or the Laplacian of lnf); hence, (82) implies that
(86)Δlnf=−∫MRic∇lnf,·dV.
Meanwhile, if we assume that the equality holds in the inequality (36), then from (57) and (63), we have
(87)n2Δlnf+109n2cot2ϑ∇lnf2=κ+3α4n2n1−∑i=12d1∑j=12d2hνei,ej∗2.
From (83) and (84), we find the following equation:
(88)109cot2ϑ∇lnf2=n1−1n2∑i=12d1∑j=12d2hνei,ej∗2+∫MRic∇lnf,·dV.
Further simplifications give(89)∫MnRic∇lnf,·dV+κ+3α4n1=1n2∑i=12d1∑j=12d2hνei,ej∗2+109cot2ϑ∇lnf2.
If the equality (80) is satisfied, then from (86), we obtain the following condition:
(90)109cot2ϑ∇lnf2=0.
Therefore, from the above equation, we derive two cases such that(91)cot2ϑ=0,or∇lnf2=0.
Case I: We considercot2ϑ=0cos2ϑ/sin2ϑ=0, which implies that cosϑ=0ϑ=π/2. From Remark 8, we conclude that Nϑn2 becomes a totally real submanifold; hence, Mn becomes a CR-warped product submanifold of a nearly Kaehler manifold. This completes the proof of (i) from Theorem 33.
Case II: We assume that∇lnf2=0, which means that ∇lnf=0 and gradlnf=0. This implies that f is a constant function on Mn. Hence, from Remark 11, we conclude that Mn is a trivial warped product semi-slant submanifold of a nearly Kaehler manifold. This is the second part (ii) of Theorem 33.Theorem 34.
Letφ:Mn=NTn1×fNϑn2 be an isometric immersion of a compact warped product semi-slant submanifold NTn1×fNϑn2 in a generalized complex space form M~2mκ,α admitting a nearly Kaehler structure, such that the warping function lnf is the first eigenfunction of the Laplacian of NTn1; this is associated with the first eigenvalue λ1, which satisfies the following:
(92)∫Hesslnf2dV+∫Ric∇lnf,∇lnfdV=9λ1tan2ϑ10n2∫κ+3α4n2n1−∑i=1n1∑j=1n2hνei,ej∗2dV.Proof.
lnf is the first eigenfunction of the Laplacian of Mn and is associated with the first eigenvalue λ1; that is, Δlnf=λ1lnf. Thus, we recall the Bochner formula (see, e.g., [42]), which states that for a differentiable function f defined on a Riemannian manifold, the following relation holds:
(93)12Δ∇lnf2=Hesslnf2+Ric∇lnf,∇lnf+g∇lnf,∇Δlnf.
Integrating the above equation with the help of the Stokes theorem, we obtain(94)∫Hesslnf2dV+∫Ric∇lnf,∇lnfdV+∫g∇lnf,∇ΔlnfdV=0.
Now, by usingΔlnf=λ1lnf and rearranging the above equation, we derive
(95)∫∇lnf2dV=−1λ1∫Hesslnf2dV+∫Ric∇lnf,∇lnfdV.
Integrating equation (84), we obtain
(96)10n29cot2ϑ∫∇lnf2dV=∫κ+3α4n2n1−∑i=12d1∑j=12d2hνei,ej∗2dV.
It follows from (92) and (93) that
(97)10n2cot2ϑ9λ1∫Hesslnf2dV+∫Ric∇lnf,∇lnfdV=∫κ+3α4n2n1−∑i=12d1∑j=12d2hνei,ej∗2dV.
The above equation and (75) imply that cot2ϑ=0cos2ϑ/sin2ϑ=0, which implies that cosϑ=0ϑ=π/2. Again, from Remark 8, we conclude that Nϑn2 becomes a totally real submanifold; by using the statement of Theorem 34, we obtain our desired result.Riemannian manifolds with no Ricci curvature are known as Ricci-flat manifolds. Ricci-flat manifolds are Einstein manifolds that do not require the cosmological constant to vanish. In a Ricci-flat manifold (particularly in Euclidean space), a circle, for example, can be deformed into an ellipse of equal area. We get the following result after taking into account the fact that warped product submanifolds are Ricci-flat.Theorem 35.
Letφ:Mn=NTn1×fNϑn2 be an isometric immersion of a compact warped product semi-slant submanifold NTn1×fNϑn2 in a generalized complex space form M~2mκ,α admitting a nearly Kaehler structure, such that the warping function lnf is the first eigenfunction of the Laplacian of NTn1 and is associated with the first eigenvalue λ1; then, NTn1 is Ricci-flat if and only if
(98)∫Hesslnf2dV=9λ1tan2ϑ10n2∫κ+3α4n2n1−∑i=12d1∑j=12d2hνei,ej∗2dV.Proof.
Thus, from (94) and (95), we obtain ∫Ric∇lnf,∇lnfdV=0. This means that NTn1 is Ricci-flat. The converse proof is straightforward. This completes the proof of the theorem.



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