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

Li, Yanlin; Alkhaldi, Ali H.; Ali, Akram

doi:https://doi.org/10.1155/2021/5900801

Advances in Mathematical Physics

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Geometry of Warped Product Manifolds: Theory and Applications

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Research Article | Open Access

Volume 2021 | Article ID 5900801 | https://doi.org/10.1155/2021/5900801

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

Yanlin Li,¹Ali H. Alkhaldi,²and Akram Ali³

Academic Editor: John D. Clayton

Received22 Aug 2021

Accepted27 Sept 2021

Published05 Nov 2021

Abstract

In this study, we develop a general inequality for warped product semi-slant submanifolds of type 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.

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. Let and be two Riemannian manifolds, where and ; the orthogonal projection maps are defined as and for any . Then, the warped product is a product manifold that is associated with the Riemannian structure; in other words, for any , where the tangent map is denoted by and represents a warping function of .

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 , where is a proper slant submanifold and is a complex submanifold.

The researchers next looked into warped products of the type 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 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, . 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 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.

2. Preliminaries

An almost Hermitian manifold of a -dimensional space, such that is an almost complex structure and is a Riemannian metric, satisfies for any on , where the identity map is denoted by . Let denote the set of all vector fields tangent to ; the Levi-Civita connection defined on is denoted by . Then, a Kaehler manifold with an almost complex structure satisfies for any and . Moreover, if the almost complex structure is such that for any vector field tangent to , then the manifold represents a nearly Kaehler manifold [25–27]. The above equation is similar to the following:

Assume that is a complex space form of constant holomorphic sectional curvature and it is denoted by . The curvature tensor of can be expressed as for all . Based on the cases , is the complex hyperbolic space , complex Euclidean space , or the complex projective space . 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 -manifold of constant type with constant holomorphic sectional curvature . Moreover, it is denoted by . Hence, the curvature tensor for generalized complex space is given by for any . Thus, for the more classifications of generalized complex space forms, we refer to [10, 11, 28–32]. The curvature tensor for a nearly Kaehler -sphere is given by for any .

A submanifold is denoted by the of an almost Hermitian manifold with an induced Riemannian metric . However, and represent the induced Riemannian connections on the normal bundle and tangent bundle of , respectively. Thus, the Gauss and Weingarten formulas are defined as for every and , where and denote the shape operator and second fundamental form for an immersion of into , respectively. Now, for any and , we have where and are the normal and tangential components of , respectively. From (2), it can be clearly seen that, for each , we have for each tangent to . Assuming to be a Riemannian manifold and a submanifold of , the Gauss equation can be defined as for any , where and represent the curvature tensors on and , respectively. Furthermore, totally umbilical and totally geodesic submanifolds satisfy and , respectively, for any , where is the mean curvature vector of . If , then is called a minimal submanifold. The mean curvature vector is expressed in terms of , which is the so-called orthonormal frame of the tangent space ; it is defined as where . Moreover, we have for which and are orthonormal frames tangent to and normal to , respectively. The scalar curvature for a submanifold of an almost complex manifold is given by

In the above equation, and represent the span of the plane section, and its sectional curvature is denoted by . Let be an -plane section on and be any orthonormal basis of . Then, the scalar curvature of is defined as

Let be a differential function defined on . Thus, the gradient is given as

Thus, from the above equation, the Hamiltonian in a local orthonormal frame is defined as

Moreover, the Laplacian of is also given by

Similarly, the Hessian tensor of function is given by where denotes the Hessian tensor. The compact manifold is considered as being without a boundary; that is, . Thus, we have the following lemma.

Lemma 2 (see [18]; Hopf’s lemma). Let be a connected and compact Riemannian manifold and a smooth function on such that . Then, is a constant function on .

Moreover, the integration of the Laplacian of the smooth function, defined on a compact-orientated Riemannian manifold without boundary, vanishes with respect to the volume element of such a manifold, and we obtain the following formula: where denotes the volume of (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 has a boundary.

Theorem 4 (see [18]). Let be a connected and compact manifold and a positive differentiable function on such that . Thus, , where is the boundary of .

Moreover, let be a compact Riemannian manifold and be a positive differentiable function on . Then, the Dirichlet energy function is defined as described in [18]; that is,

If is compact, then . We provide the following definition of a slant submanifold.

Definition 5 (see [4]). Assume to be a set containing all nonzero tangent vector fields of immersion in an almost Hermitian manifold at a point . Then, for each vector at point , the angle between and the tangent space is considered to be the Wirtinger angle of at ; this is denoted as . In this case, a submanifold of is called a slant submanifold such that 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. Let be an almost Hermitian manifold and be a submanifold of . Then, is slant if and only if there exists a constant such that where for a slant angle defined on the tangent bundle of .

Hence, we have the following consequences of Theorem 6: for any .

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 submanifold of an almost Hermitian manifold is defined as a semi-slant submanifold if there exist two complementary distributions and such that (i)(ii) is a holomorphic distribution; that is, (iii) is called a slant distribution if slant angle

Remark 8. For a semi-slant submanifold, let us consider the dimensions of and in and ; then, is holomorphic if and slant if . Furthermore, if and , then is represented as an antiholomorphic (totally real) submanifold. Moreover, is referred to as a proper semi-slant submanifold if differs from and . We can also define as proper if and .

Remark 9. If is an invariant normal subspace under an almost complex structure of the normal bundle , then this normal bundle is decomposed as

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 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 and , the following is satisfied: where denotes the Levi-Civita connection on . We recall the following lemma obtained in [3].

Lemma 10. A warped product manifold . Thus, (i)(ii)for any and , where is the Levi-Civita connection on .

Remark 11. If the warping function is constant, then is a trivial warped product or a simple Riemannian product.

Remark 12. In a nontrivial warped product manifold , the manifold is totally geodesic, and is a totally umbilical submanifold in .

Let be an isometric immersion of a warped product manifold in an arbitrary Riemannian manifold . Furthermore, let , and represent the real dimensions of , and , respectively. Then, for any unit tangent vectors and on and , respectively, we have

If we consider the local orthonormal frame such that the vectors are tangential to and are tangential to , then in view of the Gauss equation (12), we can deduce that for each .

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 form in a nearly Kaehler manifold such that is a proper slant submanifold and is a holomorphic submanifold of .

Lemma 14. For a nontrivial warped product semi-slant submanifold in a nearly Kaehler manifold , we have the following equalities: (i)(ii)for any and .

Proof. For the first part of the proof, using (9) (i) and the orthogonality of vector fields, we establish that From the covariant derivative of the almost complex structure and from (27), we derive that 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 wit in (i) and using (2) (i), we obtain the required result (ii). The lemma is proven completely.

Lemma 15. Assume that is a nontrivial warped product semi-slant submanifold in a nearly Kaehler manifold; thus, we have (i)(ii)for any and .

Proof. Replacing and by and , 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 substitute in Lemma 10, then Lemma 10 coincides with Lemma 3.1 in [5].

Lemma 17. Let be a warped product semi-slant submanifold of a nearly Kaehler manifold . Then, (i)(ii)for any and .

Proof. Replacing by 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 that is a nontrivial warped product semi-slant submanifold in a nearly Kaehler manifold. For any and , we have (i)(ii)

Proof. Similarly, by replacing with 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. Let be a warped product semi-slant submanifold in a nearly Kaehler manifold . Thus, (i)(ii)for any , , and .

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. Let be an isometric immersion of a warped product semi-slant submanifold in a nearly Kaehler manifold . Then, is a minimal submanifold of , and the squared norm of the mean curvature of is given by where denotes the mean curvature vector. Moreover, , and are dimensions of , and , 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. Let be an isometric immersion of a warped product semi-slant submanifold in a nearly Kaehler manifold . Thus, (i)The squared norm of the second fundamental form of satisfieswhere is the dimension of the slant submanifold , and is the Laplacian operator of . (ii)The equality holds in (35) if and only if is totally geodesic and is totally umbilical in and if is a minimal submanifold of

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.

3.1. 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.

Theorem 22. Assume that be an isometric immersion of a warped product semi-slant into generalized complex space form admitting a nearly Kaehler structure with constant holomorphic sectional curvature of constant type . Then, (i)The squared norm of the second fundamental form of satisfieswhere and . (ii)The equality holds in (36), if and only if and are totally umbilical and totally geodesic submanifolds in , respectively. Moreover, is a minimal submanifold of

Proof. Letting us substitute and in (8), we get Taking the summation over the basis vector fields of such that , one shows that is a warped product of holomorphic and proper slant submanifolds in a generalized complex space form . Thus, we set the following frame of orthonormal vector fields as From using the above orthonormal frame, we obtain Thus, it is easily seen that From (38) and (40), it follows that Similarly, for , we derive Now using fact that , for slant bundle , one derives 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 function for a warped product submanifold.

Corollary 23. Let be an isometric immersion of a warped product into generalized complex space form admitting a nearly Kaehler structure. Then, where is the Hessian tensor of the warping function .

4. Compact-Orientated Warped Product semi-slant Submanifolds

In this section, we consider compact Riemannian manifolds without boundaries; that is, . Applying these to warped product semi-slant submanifolds, and using integration theory on the manifold, we obtain several characterizations.

Theorem 24. Let be a compact-orientated warped product semi-slant submanifold generalized complex space form admitting a nearly Kaehler structure. Then, is a trivial warped product if and only if

Proof. From Theorem 21, we have This implies that Applying integration theory on the compact-orientated Riemannian manifold 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, ; that is, is a constant function on . Thus, by Remark 11 on warped product manifolds, is trivial. The converse proof is straightforward.

To prove the equality case, we must prove the following theorem for later use.

Theorem 25. Let be a -minimal isometric immersion of a warped product semi-slant submanifold in a nearly Kaehler manifold . If is totally umbilical in , then is -totally geodesic.

Proof. Let us assume that the second fundamental forms of and are denoted by and , respectively; we define for any vector fields and that are tangential to . Thus, the above hypothesis and Remark 12 show that is totally umbilical in , owing to its being totally umbilical in . Then, following Lemma 10 (ii), equation (47) can be written as where the vector field is normal to and is such that . Assuming that is an orthonormal frame of the slant submanifold , then by taking a summation over the vector fields of in equation (49), we obtain The left-hand side of the above equation identically vanishes due to the -minimality of , such that . Then, equation (50) takes the following form: This implies that is nonempty, such that Thus, from (50) and (53), it follows that for every . This means that is -totally geodesic. This completes the proof of the theorem.

Theorem 26. Let be a compact-orientated warped product semi-slant submanifold in a generalized complex space form admitting a nearly Kaehler structure. Then, is a trivial warped product if and only if where and . Moreover, is a component of in .

Proof. We assume that the equality sign holds in (36); then, we have However, the equality case of inequality (36) implies that is totally geodesic in a nearly Kaehler manifold; this means that for any . Moreover, is totally umbilical and can be written as for any . Furthermore, is a minimal submanifold of a nearly Kaehler manifold; thus, its mean curvature vector should be zero; that is, ; hence, through Theorem 25We assume that is a compact submanifold; thus, is closed and bounded; hence, by integrating the above equation over the volume element of and using (21), we find that Now, let and for and , 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 -component and the second term is the -component. Let us suppose that is a warped product semi-slant submanifold of an -dimension in a nearly Kaehler manifold of dimensions, such that and . We assume that the tangent spaces of and are and , respectively. We further assume that is a local orthonormal frame of and that is a local orthonormal frame of . Thus, the orthonormal frames of the normal subbundles and are and , respectively. Taking a summation over the vector fields on and and using adapted frame fields, we obtain Then, using Lemmas 14 to 18 in the above equations, we derive From (17), the last equation can be expressed as which implies that Then, from (58) and (63), it follows that If (54) holds identically, then from (66), we find that either is constant on or . However, is a proper semi-slant submanifold; thus, is a simple Riemannian product. The converse proof follows immediately from (66). Hence, the theorem is proven completely.

Corollary 27. Assume that is a warped product semi-slant submanifold in a generalized complex space form admitting a nearly Kaehler structure. Suppose that is a compact invariant submanifold and is a nonzero eigenvalue of the Laplacian on . Then, where is the volume element on . The equality sign holds if and only if the following are satisfied: (i)(ii)A warped product semi-slant submanifold is both - and -totally geodesic

Proof. Thus, using the minimum principle property, we have The equality holds if and only if one has . 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 - and -totally geodesic. This completes the proof of the corollary.

Corollary 28. Let be a warped product semi-slant submanifold in a generalized complex space form admitting a nearly Kaehler structure such that is compact, and let be a nonzero eigenvalue of the Laplacian on . Then, The equality sign holds if and only if the following are satisfied: (a)(b)A warped product semi-slant submanifold is both - and -totally geodesic

Proof. 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, . 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 that is an isometric immersion of a warped product semi-slant in a generalized complex space form admitting a nearly Kaehler structure. A connected and compact warped product is trivial if and only if the Dirichlet energy function satisfies where represents the Dirichlet energy of the warping function and is the volume element on .

Proof. Combining equations (57) and (63), we obtain Taking an integration on over the volume element 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 which implies that The theorem hypothesizes as a connected, compact warped product semi-slant submanifold; thus, Theorem 4 implies that which means that is constant on . Thus, the theorem is proven completely.

In a similar manner, we derive several characterizations in terms of the Hamiltonian.

Theorem 30. Let be a connected and compact warped product semi-slant submanifold in a generalized complex space form admitting a nearly Kaehler structure. Then, is a trivial warped product submanifold of and if and only if the Hamiltonian of the warping function satisfies the following equality:

Proof. Using (15) in (69), we derive Equation (72) is obtained if and only if on ; thus, by Theorem 4, the warped product submanifold is trivial. This completes the proof of the theorem.

The analogy of Theorem 3 for this case is classified below.

Theorem 31. Assume that is an isometric immersion of a compact warped product semi-slant submanifold in a generalized complex space form admitting a nearly Kaehler structure. Let the warping function be the solution of the Euler–Lagrange equation; then, is necessarily a trivial warped product if

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 Suppose that inequality (74) holds; then, (76) implies that the warping function must be constant on . This completes the proof of the theorem.

Theorem 32. Assume that is an isometric immersion of a compact warped product semi-slant submanifold in a generalized complex space form 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 being trivial are as follows:

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.

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 any -tensor on with a -tensor via for all Thus, we obtain for all . In particular, we have . Moreover, the following general facts are well established in the literature:

We consider to be a compact Riemannian manifold with boundaries and obtain the following classification results.

Theorem 33. Assume that is an isometric immersion of a compact warped product semi-slant submanifold in a generalized complex space form admitting a nearly Kaehler structure. If is satisfied for the warped product submanifold , then at least one of the following statements is true for : (i)The warped product semi-slant submanifold becomes a CR-warped product that is isometrically immersed in a nearly Kaehler manifold(ii)The nontrivial warped product semi-slant submanifold in a nearly Kaehler manifold is a simple Riemannian product of and

Proof. Using the first identity of (79) and setting , we derive from the hypothesis of the theorem; assuming that is a compact warped product submanifold with a boundary and integrating along the volume element , we obtain We use the Green theorem on the compact manifold ; given a smooth function , we have . We can apply the results of Yano and Kon (see [33]) immediately as . From the Green lemma, for any arbitrary vector field on . Thus, we obtain ; is the Hessian tensor of the warped function (or the Laplacian of ); hence, (82) implies that Meanwhile, if we assume that the equality holds in the inequality (36), then from (57) and (63), we have From (83) and (84), we find the following equation: Further simplifications give If the equality (80) is satisfied, then from (86), we obtain the following condition: Therefore, from the above equation, we derive two cases such that Case I: We consider , which implies that From Remark 8, we conclude that becomes a totally real submanifold; hence, 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 which means that and . This implies that is a constant function on . Hence, from Remark 11, we conclude that is a trivial warped product semi-slant submanifold of a nearly Kaehler manifold. This is the second part (ii) of Theorem 33.

Theorem 34. Let be an isometric immersion of a compact warped product semi-slant submanifold in a generalized complex space form admitting a nearly Kaehler structure, such that the warping function is the first eigenfunction of the Laplacian of ; this is associated with the first eigenvalue , which satisfies the following:

Proof. is the first eigenfunction of the Laplacian of and is associated with the first eigenvalue ; that is, . Thus, we recall the Bochner formula (see, e.g., [42]), which states that for a differentiable function defined on a Riemannian manifold, the following relation holds: Integrating the above equation with the help of the Stokes theorem, we obtain Now, by using and rearranging the above equation, we derive Integrating equation (84), we obtain It follows from (92) and (93) that The above equation and (75) imply that , which implies that Again, from Remark 8, we conclude that 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. Let be an isometric immersion of a compact warped product semi-slant submanifold in a generalized complex space form admitting a nearly Kaehler structure, such that the warping function is the first eigenfunction of the Laplacian of and is associated with the first eigenvalue ; then, is Ricci-flat if and only if

Proof. Thus, from (94) and (95), we obtain . This means that is Ricci-flat. The converse proof is straightforward. This completes the proof of the theorem.

Data Availability

There is no data used for this manuscript.

Conflicts of Interest

The authors declare no competing of interest.

Authors’ Contributions

All authors contributed equally to this work. All authors finalized the manuscript.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through a research group program under grant number R.G.P.2/74/42. This work was also funded by the National Natural Science Foundation of China (Grant No. 12101168).

References

B. Chen, “Geometry of warped product CR-submanifolds in Kaehler manifolds,” Monatshefte für Mathematik, vol. 133, no. 3, pp. 177–195, 2001.
View at: Publisher Site | Google Scholar
P. Ehrlich, Y. T. Jung, S. B. Kim, and C. G. Shin, “Partial differential equations and scalar curvature of warped product manifolds,” Nonlinear Analysis, vol. 44, no. 4, pp. 545–553, 2001.
View at: Publisher Site | Google Scholar
R. Bishop and B. O’Neill, “Manifolds of negative curvature,” Transactions of the American Mathematical Society, vol. 145, pp. 1–9, 1969.
View at: Publisher Site | Google Scholar
B. Chen, “Slant immersions,” Bulletin of the Australian Mathematical Society, vol. 41, no. 1, pp. 135–147, 1990.
View at: Publisher Site | Google Scholar
F. R. al-Solamy, V. Khan, and S. Uddin, “Geometry of warped product semi-slant submanifolds of nearly Kaehler manifolds,” Results in Mathematics, vol. 71, no. 3-4, pp. 783–799, 2017.
View at: Publisher Site | Google Scholar
B. Chen, “Geometry of warped product submanifolds: a survey,” Journal of Advanced Mathematical Studies, vol. 6, no. 2, pp. 1–43, 2013.
View at: Google Scholar
B. Chen, “Another general inequality for warped product CR-warped product submanifold in complex space forms,” Hokkaido Mathematical Journal, vol. 32, pp. 415–444, 2003.
View at: Google Scholar
M. Atceken, “Contact CR-warped product submanifolds in cosymplectic space forms,” Collectanea Mathematica, vol. 62, no. 1, pp. 17–26, 2011.
View at: Publisher Site | Google Scholar
I. Hasegawa and I. Mihai, “Contact CR-warped product submanifolds in Sasakian manifolds,” Geometriae Dedicata, vol. 102, no. 1, pp. 143–150, 2003.
View at: Publisher Site | Google Scholar
A. Mihai, “Shape operator for slant submanifolds in generalized complex space forms,” Turkish Journal of Mathematics, vol. 27, no. 4, pp. 509–524, 2004.
View at: Google Scholar
I. Munteanu, “Warped product contact CR-submanifolds of Sasakian space forms,” Universitatis Debreceniensis, vol. 66, pp. 75–120, 2005.
View at: Google Scholar
A. Ali, P. Laurian-Ioan, and A. Alkhaldi, “Ricci curvature on warped product submanifolds in spheres with geometric applications,” Journal of Geometry and Physics, vol. 146, article 103510, 2019.
View at: Publisher Site | Google Scholar
A. Ali, J. W. Lee, and A. H. Alkhaldi, “Geometric classification of warped product submanifolds of nearly Kaehler manifolds with a slant fiber,” International Journal of Geometric Methods in Modern Physics., vol. 16, no. 2, article 1950031, 2019.
View at: Publisher Site | Google Scholar
R. Ali, F. Mofarreh, N. Alluhaibi, A. Ali, and I. Ahmad, “On differential equations characterizing Legendrian submanifolds of Sasakian space forms,” Mathematics., vol. 8, no. 2, p. 150, 2020.
View at: Publisher Site | Google Scholar
A. Ali and P. Laurian-Ioan, “Geometry of warped product immersions of Kenmotsu space forms and its applications to slant immersions,” Journal of Geometry and Physics, vol. 114, pp. 276–290, 2017.
View at: Publisher Site | Google Scholar
A. Ali and C. Ozel, “Geometry of warped product pointwise semi-slant submanifolds of cosymplectic manifolds and its applications,” International Journal of Geometric Methods in Modern Physics, vol. 14, no. 3, article 1750042, 2017.
View at: Publisher Site | Google Scholar
I. al-Dayel and M. Ali Khan, “Ricci curvature of contact CR-warped product submanifolds in generalized Sasakian space forms admitting nearly Sasakian structure,” AIMS Mathematics, vol. 6, no. 3, pp. 2132–2151, 2021.
View at: Publisher Site | Google Scholar
O. Calin and D. Chang, Geometric mechanics on Riemannian manifolds: applications to partial differential equations, Springer Science & Business Media, 2006.
D. Kim and Y. Kim, “Compact Einstein warped product spaces with nonpositive scalar curvature,” Proceedings of the American Mathematical Society, vol. 131, no. 8, pp. 2573–2576, 2003.
View at: Publisher Site | Google Scholar
Y. Li, S. Liu, and Z. Wang, “Tangent developables and Darboux developables of framed curves,” Topology and its Applications, vol. 301, article 107526, 2021.
View at: Publisher Site | Google Scholar
Y. Li and Z. Wang, “Lightlike tangent developables in de Sitter 3-space,” Journal of Geometry and Physics, vol. 164, article 104188, 2021.
View at: Publisher Site | Google Scholar
Y. Li, Z. Wang, and T. Zhao, “Geometric Algebra of Singular Ruled Surfaces,” Advances in Applied Clifford Algebras, vol. 31, no. 1, pp. 1–19, 2021.
View at: Publisher Site | Google Scholar
Y. Li, Y. Zhu, and Q. Sun, “Singularities and dualities of pedal curves in pseudo-hyperbolic and de sitter space,” International Journal of Geometric Methods in Modern Physics, vol. 18, no. 1, article 2150008, 2021.
View at: Publisher Site | Google Scholar
Y. Li, Z. Wang, and T. Zhao, “Slant helix of order n and sequence of Darboux developables of principal-directional curves,” Mathematicsl Methods in the Applied Sciences, vol. 43, no. 17, pp. 9888–9903, 2020.
View at: Publisher Site | Google Scholar
B. Sahin, “Non-existence of warped product semi-slant submanifolds of Kaehler manifolds,” Geometriae Dedicata, vol. 117, no. 1, pp. 195–202, 2006.
View at: Publisher Site | Google Scholar
B. Sahin and R. Gunes, “CR-warped product submanifolds of nearly Kaehler manifolds,” Beiträge zur Algebra und Geometrie, vol. 49, pp. 383–397, 2008.
View at: Google Scholar
F. Urbano, CR-submanifolds of nearly Kaehler manifolds, [PhD Thesis], Granada, 1980.
I. Mihai, “Ideal Kaehlerian slant submanifolds in complex space forms,” The Rocky Mountain Journal of Mathematics, vol. 35, pp. 941–952, 2005.
View at: Google Scholar
A. Mihai and B. Y. Chen, “Inequalities for slant submanifolds in generalized complex space forms,” Radovi Mathematicki, vol. 2, pp. 215–231, 2004.
View at: Google Scholar
A. Mihai, “Warped product submanifolds in generalized complex space forms,” Acta Mathematica Academiae Paedagogicae Ny´ıregyh´aziensis, vol. 21, pp. 79–87, 2005.
View at: Google Scholar
A. Mustafa, S. Uddin, and R. Wong, “Generalized inequalities on warped product submanifolds in nearly trans-Sasakian manifolds,” Journal of Inequalities and Applications, vol. 2014, no. 1, Article ID 346, p. 16, 2014.
View at: Publisher Site | Google Scholar
I. Mihai, “Contact CR-warped product submanifolds in Sasakian space forms,” Geometriae Dedicata, vol. 109, no. 1, pp. 165–173, 2004.
View at: Publisher Site | Google Scholar
K. Yano and M. Kon, Structures on Manifolds, World Scientific, 1985.
N. Papaghiuc, “Semi-slant submanifold of Kaehlerian manifold,” Annals of the "Alexandru Ioan Cuza" University of Iaşi, vol. 40, pp. 55–61, 1994.
View at: Google Scholar
Y. Li, M. Lone, and U. Wani, “Biharmonic submanifolds of Kaehler product manifolds,” AIMS Mathematics, vol. 6, pp. 309–9321, 2021.
View at: Google Scholar
Y. Li, A. Ali, and R. Ali, “A general inequality for CR-warped products in generalized Sasakian space form and its applications,” Advances in Mathematical Physics, vol. 2021, Article ID 5777554, 6 pages, 2021.
View at: Publisher Site | Google Scholar
Y. Li, P. Laurian-Ioan, A. Ali, and A. Alkhaldi, “Null homology groups and stable currents in warped product submanifolds of Euclidean spaces,” Symmetry, vol. 13, no. 9, p. 1587, 2021.
View at: Publisher Site | Google Scholar
Y. Li, A. Ali, F. Mofarreh, and N. Alluhaibi, “Homology groups in warped product submanifolds in hyperbolic spaces,” Journal of Mathematics, vol. 2021, Article ID 8554738, 10 pages, 2021.
View at: Publisher Site | Google Scholar
A. Ali, S. Uddin, and W. Othman, “Geometry of warped product pointwise semi-slant submanifolds of Kaehler manifolds,” Univerzitet u Nišu, vol. 31, no. 12, pp. 3771–3788, 2017.
View at: Google Scholar
N. S. al-Luhaibi, F. R. al-Solamy, and V. Khan, “CR-warped product submanifolds of nearly Kaehler manifolds,” Journal of the Korean Mathematical Society, vol. 46, no. 5, pp. 979–995, 2009.
View at: Publisher Site | Google Scholar
V. Khan and K. Khan, “Generic warped product submanifolds in nearly Kaehler manifolds,” Contributions to Algebra and Geometry, vol. 50-2, pp. 337–352, 2009.
View at: Google Scholar
M. Berger, “Les variétés riemanniennes -pincées,” Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, vol. 14, no. 4, pp. 161–170, 1960.
View at: Google Scholar

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