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

In this study, we develop a general inequality for warped product semi-slant submanifolds of typeMn =N1 T × f N n2 θ 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.


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. Let ðN 1 , g 1 Þ and ðN 2 , g 2 Þ be two Riemannian manifolds, where f : N 1 ⟶ ð0,∞Þ and γ 1 : N 1 × N 2 ⟶ N 1 , γ 2 : N 1 × N 2 ⟶ N 2 ; the orthogonal projection maps are defined as γ 1 ðt, sÞ = t and γ 2 ðt, sÞ = s for any ðt, sÞ ∈ N 1 × N 2 . Then, the warped product N 1 × f N 2 is a product manifold that is N 1 × N 2 associated with the Riemannian structure; in other words, for any X, Y ∈ TM n , where the tangent map is denoted by * and f represents a warping function of M n .
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 M n = N T is a complex submanifold. The researchers next looked into warped products of the type M n = N n 1 T × f N n 2 ϑ 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 M n = N n 2 ϑ × f N n 1 T 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 CRwarped product of complex space forms in [7]. Furthermore, in [8][9][10][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][13][14][15][16][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][21][22][23][24].
The following is a breakdown of the paper's structure: We review some basic formulas and definitions in Section 2 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.

Preliminaries
An almost Hermitian manifold ðM, J, gÞ of a 2m-dimensional space, such that J is an almost complex structure and g is a Riemannian metric, satisfies for any X, Y onM 2m , where the identity map is denoted by I.
for any vector field X, Y tangent toM 2m , then the manifold M represents a nearly Kaehler manifold [25][26][27]. The above equation is similar to the following: Assume thatM n is a complex space form of constant holomorphic sectional curvature 4κ and it is denoted bỹ M n ð4κÞ. The curvature tensorR ofM n ð4κÞ can be expressed as for all W 1 , W 2 , W 3 ∈ ΓðTÑÞ. Based on the cases κ < 0, κ = 0, and κ > 0, e ℕ n ð4κÞ is the complex hyperbolic space ℂ H n , complex Euclidean space ℂ n , or the complex projective space ℂP n . 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 byM 2m ðκ, αÞ.
Hence, the curvature tensorR for generalized complex space is given bỹ for any X, Y, Z, W ∈ ΓðTM 2m Þ. Thus, for the more classifications of generalized complex space forms, we refer to [10,11,[28][29][30][31][32]. The curvature tensorR for a nearly Kaehler 6-sphere is given bỹ for any X, Y, Z, WðTMðS 6 Þ. A submanifold is denoted by the M n of an almost Hermitian manifoldM 2m with an induced Riemannian metric g. However, ∇ ⊥ and ∇ represent the induced Riemannian connections on the normal bundle T ⊥ M n and tangent bundle TM n of M n , respectively. Thus, the Gauss and Weingarten formulas are defined as for every N ∈ ΓðT ⊥ M n Þ and X, Y ∈ ΓðTM n Þ, where A N and h denote the shape operator and second fundamental form for an immersion of M intoM 2m , respectively. Now, for any N ∈ ΓðT ⊥ M n Þ and X ∈ ΓðTM n Þ, we have where FXð f NÞ and PXðtNÞ are the normal and tangential components of JNðJXÞ, respectively. From (2), it can be clearly seen that, for each X, Y ∈ ΓðTM n Þ, we have for each e i , i = 1, ⋯, n tangent to M n . AssumingM 2m to be a Riemannian manifold and M n a submanifold ofM 2m , the Gauss equation can be defined as for any X, Y, Z, W ∈ ΓðTM n Þ, whereR and R represent the curvature tensors onM 2m and M n , respectively. Furthermore, totally umbilical and totally geodesic submanifolds satisfy hðX, YÞ = gðX, YÞH and hðX, YÞ = 0, respectively, for any X, Y ∈ ΓðTM n Þ, where H is the mean curvature vector of M n . If H = 0, then M n is called a minimal submanifold. The mean curvature vector H is expressed in terms of fe 1 , e 2 ⋯,e n g, which is the so-called orthonormal frame of the tangent space TM n ; it is defined as where n = dim M. Moreover, we have for which fe i g i=1,⋯n and fe r g r=n+1,⋯2m are orthonormal frames tangent to M n and normal to M n , respectively. The scalar curvature τ for a submanifold M n of an almost complex manifoldM 2m is given by

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In the above equation, e i and e j represent the span of the plane section, and its sectional curvature is denoted by K ðe i ∧ e j Þ. Let G r be an r-plane section on TM n and fe 1 , e 2 ⋯ e r g be any orthonormal basis of G r . Then, the scalar curvature τðG r Þ of G r is defined as Let ϕ be a differential function defined on M n . Thus, the gradient ∇ ! ϕ is given as Thus, from the above equation, the Hamiltonian in a local orthonormal frame is defined as Moreover, the Laplacian Δf of f is also given by Similarly, the Hessian tensor of function f is given by where H ϕ denotes the Hessian tensor. The compact manifold M n is considered as being without a boundary; that is, ∂M n = ∅. Thus, we have the following lemma.
Lemma 2 (see [18]; Hopf's lemma). Let M n be a connected and compact Riemannian manifold and ϕ a smooth function on M n such that Δϕ ≥ 0ðΔϕ ≤ 0Þ. Then, ϕ is a constant function on M n .
Moreover, the integration of the Laplacian of the smooth function, defined on a compact-orientated Riemannian manifold M n without boundary, vanishes with respect to the volume element of such a manifold, and we obtain the following formula: ð where dV denotes the volume of M n (see [33]).
Theorem 3 (see [18]). The Euler-Lagrange equation for the Lagrangian is Hopf's lemma becomes the uniqueness theorem for the Dirichlet problem if manifold M n has a boundary.
Theorem 4 (see [18]). Let M n be a connected and compact manifold and f a positive differentiable function on M n such that Δϕ = 0, OnM n . Thus, ϕ = 0, where ∂M n is the boundary of M.
Moreover, let M n be a compact Riemannian manifold and f be a positive differentiable function on M n . Then, the Dirichlet energy function is defined as described in [18]; that is, If M n is compact, then 0 ≤ EðϕÞ < ∞. We provide the following definition of a slant submanifold.
Definition 5 (see [4]). Assume T x M n − f0g to be a set containing all nonzero tangent vector fields of immersion M n in an almost Hermitian manifoldM 2m at a point x ∈ M n .
Then, for each vector X ∈ ðT x M n Þ at point x ∈ M n , the angle between JX and the tangent space T x M is considered to be the Wirtinger angle of X at x ∈ M n ; this is denoted as ϑðXÞ.
In this case, a submanifold M n ofM 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. if there exists a constant λ ∈ ½0, 1 such that where λ = cos 2 ϑ for a slant angle ϑ defined on the tangent bundle TM n of M n .
Hence, we have the following consequences of Theorem 6: for any X, Y ∈ ΓðTM n Þ.
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.

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][36][37][38][39][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 be trivial 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 ∈ ΓðTN 1 Þ and Z ∈ ΓðTN 2 Þ, the following is satisfied: where ∇ denotes the Levi-Civita connection on M n . We recall the following lemma obtained in [3]. Thus, for any Z, W ∈ ΓðTN 2 Þ and X, Y ∈ ΓðTN 1 Þ, where ∇ ′ is the Levi-Civita connection on N 2 . Remark 11. If the warping function f is constant, then M = N 1 × f N 2 is a trivial warped product or a simple Riemannian product.
Remark 12. In a nontrivial warped product manifold M n = N 1 × f N 2 , the manifold N 1 is totally geodesic, and N 2 is a totally umbilical submanifold in M n .
2m be an isometric immersion of a warped product manifold N 1 × f N 2 in an arbitrary Riemannian manifoldM 2m . Furthermore, let n 1 , n 2 , and n represent the real dimensions of N 1 , N 2 ,, and M n , respectively. Then, for any unit tangent vectors Y and W on N 1 and N 2 , respectively, we have If we consider the local orthonormal frame fe 1 , e 2 ,⋯,e n g such that the vectors e 1 , e 2 ⋯ , e n 1 are tangential to N 1 and e n 1 +1 ⋯ , e n are tangential to N 2 , then in view of the Gauss equation (12), we can deduce that for each j = n 1 + 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] Using the structure equation (4) in the above equation and in (10) (i), we find that Thus, from (27) and (10) (ii), we finally obtain which is (i). Replacing X wit JX in (i) and using (2) (i), we obtain the required result (ii). The lemma is proven completely. Proof. Similarly, by replacing X 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. 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].
where H denotes the mean curvature vector. Moreover, n 1 , n 2 , n, and 2m are dimensions of N 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. 6 Advances in Mathematical Physics where δ =τðTM n Þ −τðTN Proof. The proof proceeds in a similar manner to the proof of Theorem 29 [31] if we consider the nearly Kaehler manifold, instead of nearly trans-Sasakian manifold.

Applications of Theorem 21 to Generalized Complex
Space Forms. In this section, we prove our main theorem using Theorem 21 for a generalized complex space form. Then, we give the following result. where Taking the summation over the basis vector fields of T M n such that 1 ≤ i ≠ j ≤ n, one shows that = sec ϑPe 2d 1 +1 , ⋯e 2d 1 +2d 2 −1 e 2d 1 +2d 2 = sec ϑPe d 1 −1 : ð39Þ From using the above orthonormal frame, we obtain Thus, it is easily seen that 〠 n i,j=1 g 2 Pe i , e j À Á = n 1 + n 2 cos 2 ϑ: From (38) and (40), it follows that Similarly, for TN Now using fact that kPk 2 = n 2 cos 2 ϑ, for slant bundle TN n 2 ϑ , one derives Therefore, substituting (41), (42), and (40) in Theorem 7 Advances in Mathematical Physics 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 function ln f for a warped product submanifold.
where H ln f is the Hessian tensor of the warping function ln f .

Compact-Orientated Warped Product semislant Submanifolds
In this section, we consider compact Riemannian manifolds without boundaries; that is, ∂M n = ∅. Applying these to warped product semi-slant submanifolds, and using integration theory on the manifold, we obtain several characterizations.
Proof. From Theorem 21, we have This implies that Applying integration theory on the compact-orientated Riemannian manifold M n without boundary, and then using (21), we obtain Now, if the inequality (44) holds, then from (47), we find that which is impossible for a positive integrable function; hence, ∇ln f = 0; that is, f is a constant function on M n . Thus, by Remark 11 on warped product manifolds, M n is trivial. The converse proof is straightforward.
To prove the equality case, we must prove the following theorem for later use. Proof. Let us assume that the second fundamental forms of M n andM 2m are denoted by h * andh, respectively; we define for any vector fields Z and W that are tangential to N n 2 ϑ . Thus, the above hypothesis and Remark 12 show that N n 2 ϑ is totally umbilical inM 2m , owing to its being totally umbilical in M n . Then, following Lemma 10 (ii), equation (47) can be written as where the vector field ξ is normal to ΓðTN ϑ Þ and is such that ξ ∈ ΓðTM n Þ. Assuming that fe * 1 ,⋯e * n 2 g is an orthonormal frame of the slant submanifold N n 2 ϑ , then by taking a summation over the vector fields of N n 2 ϑ in equation (49), we obtain The left-hand side of the above equation identically vanishes due to the D ϑ -minimality of φ, such that ∑ Then, equation (50) takes the following form: This implies that N n 2 ϑ is nonempty, such that Thus, from (50) and (53), it follows that hðZ, WÞ = 0, for every Z, W ∈ ΓðTN where n 1 = dim N T and n 2 = dim R N ϑ . Moreover, h ν is a component of h in ΓðνÞ.
Proof. We assume that the equality sign holds in (36); then, we have However, the equality case of inequality (36) implies that N n 1 T is totally geodesic in a nearly Kaehler manifold; this means that hðe i , e j Þ = 0, for any 1 ≤ i, j ≤ 2d 1 . Moreover, N n 2 ϑ is totally umbilical and can be written as for any 1 ≤ t, s ≤ 2d 2 . Furthermore, M n is a minimal submanifold of a nearly Kaehler manifold; thus, its mean curvature vector H should be zero; that is, H = 0; hence, hðe * t , e * s Þ = 0, for every 1 ≤ t, s ≤ 2d 2 through Theorem 25 We assume that M n is a compact submanifold; thus, M n is closed and bounded; hence, by integrating the above equation over the volume element dV of M n and using (21), we find that Now, let X = e i and Z = e j for 1 ≤ i ≤ n 1 and 1 ≤ j ≤ n 2 , respectively; then, using (14) and expressing in terms of the orthonormal frame, we have In the above equation, the first term on the right-hand side is the FD ϑ -component and the second term is the ν -component. Let us suppose that M n = N n 1 T × f N n 2 ϑ is a warped product semi-slant submanifold of an n-dimension in a nearly Kaehler manifoldM 2m of 2m dimensions, such that dim N T = n 1 = 2d 1 and dim R N ϑ = n 2 = 2d 2 . We assume that the tangent spaces of N n 1 T and N n 2 ϑ are D and D ϑ , respectively. We further assume that fe 1 , e 2 ,⋯, e d 1 , e d 1 +1 = Je 1 ,⋯,e 2d 1 = Je d 1 g is a local orthonormal frame of TN n 1 T and that fe 2d 1 +1 = e * 1 ,⋯,e 2d 1 +d 2 = e * d 2 , e 2d 1 +d 2 +1 = e * d 2 +1 = sec ϑPe * 1 ,⋯,e n 1 +n 2 = e * n 2 = sec ϑPe * d 2 g is a local orthonormal frame of TN n 2 ϑ . Thus, the orthonormal frames of the normal subbundles FD ϑ and ν are fe n+1 =ẽ 1 = csc ϑF e * 1 ,⋯,e n+d 2 =ẽ d 2 = csc ϑFe * 1 , e n+d 2 +1 =ẽ d 2 +1 = csc ϑ sec ϑFPe * 1 , ⋯,e n+2d 2 =ẽ 2d 2 = csc ϑ sec ϑFPe * d 2 g and fe n+2d 2 +1 ,⋯,e 2m g, respectively. Taking a summation over the vector fields on N n 1 T and N n 2 ϑ and using adapted frame fields, we obtain Then, using Lemmas 14 to 18 in the above equations, we derive h D, D ϑ 2 = 2 csc 2 ϑ + 1 9 e i ln f ð Þ 2 g e * j , e * j 2 + 2 csc 2 ϑ + 1 9 Je i ln f ð Þ 2 g e * j , e * j 2 g h e i , e j À Á , e r À Á 2 : which implies that h D, D ϑ 2 = n 2 1 + 10 9 Then, from (58) and (63), it follows that h μ e i , e j À Á 2 + 10 9 If (54) holds identically, then from (66), we find that either f is constant on M n or cot ϑ = 0. However, M n is a proper semi-slant submanifold; thus, M n is a simple Riemannian product. The converse proof follows immediately from (66). Hence, the theorem is proven completely. Proof. Thus, using the minimum principle property, we have ð The equality holds if and only if one has ∇ln f = λ T ln f . Thus, from (44) The equality sign holds if and only if the following are satisfied:

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.
where Eðln f Þ represents the Dirichlet energy of the warping function ln f and dV is the volume element on M n .

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Proof. Combining equations (57) and (63), we obtain κ + 3α 4 n 2 n 1 = n 2 Δ ln f ð Þ+ 〠 h ν e i , e j À Á 2 + n 2 10 9 cot 2 ϑ ∇ln f k k 2 : Taking an integration on M n over the volume element dV with a nonempty boundary in the above equation, we find that Then, from (23) and (70), it follows that Equality (68) is satisfied if and only if we obtain from (71) the condition that Ð M n Δðln f ÞdV = 0, which implies that Δðln f Þ = 0: The theorem hypothesizes M n as a connected, compact warped product semi-slant submanifold; thus, Theorem 4 implies that ln f = 0 ⟹ f = 1, which means that f is constant on M n . Thus, the theorem is proven completely.
In a similar manner, we derive several characterizations in terms of the Hamiltonian.
Proof. Using (15) in (69) Equation (72) is obtained if and only if Δðln f Þ = 0 on M n ; thus, by Theorem 4, the warped product submanifold M n is trivial. This completes the proof of the theorem.
The analogy of Theorem 3 for this case is classified below.
Proof. If the warping function satisfies the conditions of the Euler-Lagrange equation, then from Theorem 3, we obtain Thus, from (44) and (75), we derive h k k 2 ≥ 2 κ + 3α 4 n 2 n 1 + n 2 ∇ln f k k 2 : Suppose that inequality (74) holds; then, (76) implies that the warping function must be constant on M n . This completes the proof of the theorem. h ν e i , e j À Á 2 = κ + 3α 4 n 2 n 1 : Proof. The proof of the above theorem is the same as that of Theorem 31; it uses (57), (63), and Theorem 3. This completes the proof of the theorem.