Ricci Curvature for Warped Product Submanifolds of Sasakian Space Forms and Its Applications to Differential Equations

Mofarreh, Fatemah; Ali, Akram; Alluhaibi, Nadia; Belova, Olga

doi:https://doi.org/10.1155/2021/1207646

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Volume 2021 | Article ID 1207646 | https://doi.org/10.1155/2021/1207646

Ricci Curvature for Warped Product Submanifolds of Sasakian Space Forms and Its Applications to Differential Equations

Fatemah Mofarreh,¹Akram Ali,²Nadia Alluhaibi,³and Olga Belova⁴

Academic Editor: Hang Xu

Received24 Aug 2020

Revised06 Nov 2020

Accepted22 Dec 2020

Published15 Jan 2021

Abstract

In the present paper, we establish a Chen–Ricci inequality for a C-totally real warped product submanifold of Sasakian space forms . As Chen–Ricci inequality applications, we found the characterization of the base of the warped product via the first eigenvalue of Laplace–Beltrami operator defined on the warping function and a second-order ordinary differential equation. We find the necessary conditions for a base of a C-totally real-warped product submanifold to be an isometric to the Euclidean sphere .

1. Introduction and Motivations

For geometric analysis, the work of Obata [1] becomes an essential tool of investigation. Obata [1] provided a characterization theorem for a standard sphere in terms of a differential equation, known as the Obata equation. If is a complete manifold with , then the function is nonconstant and fulfills the ordinary differential equation:if and only if there is an isometry between and the sphere , where denotes the sectional curvature. If , then and the unit sphere are congruent. A large number of investigations on this subject are studied. Therefore, the characterization of these spaces the Euclidean space , the Euclidean sphere , and the complex projective space are recognized fields in the study of differential geometry and are studied in research works such as [2–18]. In particular, the Euclidean space is designated through the differential equation , where is a positive constant, which is proven by Tashiro [19]. In [20], Lichnerowicz has established that if the first nonzero eigenvalue of the Laplace operator of the compact manifold with is , then is isometric to the sphere . Thus, Obata’s theorem can be utilized to address Lichnerowicz’s eigenvalue equality condition derived in [20]. Deshmukh and Al-Solamy [21] proved that an -dimensional connected Riemannian manifold which is compact and has a Ricci curvature satisfying for a constant , where is the first eigenvalue of the Laplacian, is isometric to if is allowed to be a nonzero conformal gradient vector field. The authors also proved that if is an Einstein manifold, meaning that the Einstein constant is , then is isometric to with if a conformal gradient vector field is allowed. Taking account of ODE (1), Barros et al. [6] showed that the gradient of an almost Ricci soliton that is compact is isometric to the Euclidean sphere when the Ricci tensor is Codazzi, and a constant sectional curvature is present. For more information regarding the Obata equation, see [1]. Motivated by the previous studies, we establish a number of results in the present paper which realize as characterizations of spheres. More precisely, we have the following.

Theorem 1. Assume that is a C-totally real isometric embedding from a warped product submanifold into a Sasakian space form with a nonnegative Ricci curvature. Then, the compact and minimal base is isometric to the Euclidean sphere if the following equality holds:where is the eigenvalue connected to an eigenfunction for the Laplacian operator and is a Hessian tensor for the function . Moreover, here, the constant curvature is equal to . In a particular case, if satisfies the conditionthen, the base is isometric to the standard sphere .

From the Bochner formula, we are able to prove the following result:

Theorem 2. Let be a C-totally real isometric embedding from a warped product submanifold into a Sasakian space form having the nonnegative Ricci curvature. Then, compact and minimal base is isometric to the sphere with a constant sectional curvature equal to if the following equality holds:where is a positive eigenvalue associated with the eigenfunction . Moreover, , , and .

The paper is organized as follows. In Section 2, we study some preliminaries formulas, notations, and definitions related to our study. In the same section, we prove a lemma, a key result for our main theorem. In Section 3, we demonstrate our main conclusions and provide several consequences from our main findings. We also give an example for existence -totally real warped product submanifold in Sasakian manifolds. In Section 4, we give concluding remarks.

2. Preliminaries and Notations

Let be the odd-dimensional -manifold equipped with an almost contact structure such thatfor any . Of course, the notations are well known: is a structure vector, the -type tensor field is denoted by , and is the dual one-form. Moreover, the tensorial equation for a Sasakian manifold [22–24] with the structure is given by

If we choose two vector fields at , such that is the Riemannian connection regarding , and assume that is a Sasakian space form with a constant -sectional curvature , then its curvature tensor isfor all . The odd-dimensional Euclidean space and odd-dimensional sphere with sectional curvatures of and are remarkable examples of Sasakian space forms in [25]. Moreover, if the structure vector field belongs to the normal space of , then is said to be a C-totally real submanifold; for more details, see the work presented in [22–24, 26, 27]. It should be noted that the curvature tensor for in Sasakian space form is defined as

Suppose is a Riemannian submanifold of a Riemannian manifold considering the induced metric and and are connections along and of , where is a tangent bundle and is a normal bundle of . Therefore, for any and , the Gauss and Weingarten formulas are written as , respectively. Note that and denote the second fundamental form as well as the shape operator, respectively. They are governed by the relation . The Gauss equation isfor any , where the curvature tensors of and are represented by and . Furthermore, , which is the mean curvature of , is calculated as . is totally umbilical if and totally geodesic if , for any . Furthermore, is minimal if . Here,gives the second fundamental form kernel of over . If the plane section is spanned by and over in , then such a curvature is called a sectional curvature and is denoted by . The relation between the scalar curvature of and at some in is represented by

The first equality in (11) is reciprocal to the following:

The previous relation will be utilized in the subsequent proofs. Similarly, the scalar curvature of an -plan is expressed as

Let be an orthonormal frame of the tangent space and be an orthonormal frame of the normal space . Thus, we have

Let and be the sectional curvature of a submanifold and ; then, we have following relation due to the Gauss equation (9):

Furthermore, the Ricci tensor is defined as

Fixing the distinct indices for vector fields from on by , which is governed by , the Ricci curvature is given as

Therefore, equation (12) can be written as

Thus,which will be frequently used in future studies. The gradient-squared norm of the positive smooth function for an orthonormal basis is given by

Assume that and are Riemannian manifolds with Riemannian metrics and , respectively. Suppose is a differentiable function on . Then, the manifold endorsed by the Riemannian metric is referred to a warped product manifold and classified as notation [28]. Assume that is a warped product, then we have and . It was proved in Section 3.3 in [28] that the following relation holds:

Remark 1. is Riemannian product manifold if is a constant.

Remark 2. Sometimes we will use the following abbreviation throughout the paper: “WPS” for Warped product submanifold, “WF” for warping function, “RM” for Riemannian manifold, and “SSF” for Sasakian space form.

3. Ricci Curvature for C-Totally Real Warped Products

Inspired by the work [2, 3, 9], we prove the following proposition which we will use in further result.

Proposition 1. Let be a C-totally real warped product submanifold into a Sasakian space form having the minimal base . Then, for all unit vectors , the following Ricci inequality holds:where and .(i)In case , for , there exists a unit vector satisfying the equality in (23) if and only if is mixed totally geodesic and lies in at .(ii)If is -minimal, thus(a)The equality in (23) remains for any unit tangent vectors at and any is totally geodesic and -totally geodesic WPS in .(b)The equality in (23) remains for any unit tangent vectors at and any is totally geodesic, either a -totally geodesic WPS or a -totally umbilical WPS in such that .(iii)The equality in (23) is satisfied for any unit tangent vectors at and any is either totally geodesic, or totally umbilical, mixed totally geodesic, and -totally geodesic WPS such that .

Proof. Assume that is a -minimal C-totally real warped product. An analogous technique will be used for similar cases. Utilizing the Gauss equation (9), we deriveAssume is the local orthonormal frame field of in which the basis are tangent to and are tangent to . Thus, for the unit tangent vector , we can expand (24):which is equivalent to the following by using (8):As we assumed that the base of the warped product submanifold is minimal, we deriveIn view of (15), we obtainFrom the fact that the base is minimal and putting (28) in (27), we deduceOn the contrary, using (11), we defineFrom (22) and (11), we obtainFrom (29)–(31) and using (14), we deduceNow, we note that is either tangent to the base or to the fiber . After that, the proof of the former case is introduced.

Case 1. Let be tangent to . We fix the unit tangent vector from to be and consider . Then, from (17) and (32), we obtainSubstituting and for in (7) and summarizing, we obtainTherefore, using (34) in equation (33), we obtainThe calculation of the last two terms of (35) impliesIn similar way, we obtainUsing (37) in (33) leads toAs for the warped product submanifold such that the base is minimal in , we compute the following simplification:Similarly, we haveAt the same time, utilizing the minimality of the base manifold , we deduce thatUtilizing (39)–(41), equation (38) will take the formThe above inequality is equivalent to the following:Using (22), this gives inequality (23). For the second case, we have the following.

Case 2. Assume that is tangent . We fix a unit tangent vector field from in which . Utilizing (17) to (33) and following a similar technique from (33)–(42), it impliesUsing (34), we obtainAs the base of is minimal, thenBy using a similar technique to the first case, using (46) in (45), we obtainAfter some calculations, we obtainPerforming other calculations for the last two terms givesThus, (48) can be reduced, using the above relation, as follows:Therefore, using (50) in inequality (47), we deduce thatFrom the minimality of the base of warped product submanifold , we obtainThis gives the proof of inequality (23). We will use the technique adopted for case (i) to determine inequality (23) when is -minimal. Now, equality (23) can be verified in a similar manner as in [2, 3, 29].

For a completely minimal submanifold, Proposition 1 presents the following result.

Lemma 1. Assume is a C-totally real minimal isometric embedding from a warped product into a Sasakian space form . Therefore, for any unit vector , the following Ricci inequality is satisfied:where and .

4. Application to Differential Equations

4.1. Proof of Theorem 1

Consider the following equation with :

However, we know that as well as . Then, the proceeding equation takes the form

If is an eigenvalue of the eigenfunction, then . Thus, we obtain

On the contrary, we obtain

Again, using , we have

It follows from (56) and (58) that

In particular, taking on (59) and integrating, we obtain

Again, integrating (23) and including the Green lemma, we have

From (60) and (61), we derive

Under the assumption that the Ricci curvature is greater than or equal to zero, i.e., , the above equation implieswhich is equivalent to the following:

If the following equality holds by assumption (2),then equations (64) and (65) imply that

However, it is clear that

Combining equations (66) and (67), we obtain

Since the WF of a nontrivial WPS is a nonconstant, then equation (68), reduces to Obata’s differential equation where by . Thus, is isometric to . This is a complete proof of the first part. On the contrary, if we have , then, from (68), we havefor any . The proof of this theorem is completed.

4.2. Proof of Theorem 2

If is the positive differential function on the Riemannian manifold , then Bochner formula is defined as

Taking the integration along the volume element and using the Stokes Theorem, we obtain

Assume that is an eigenvalue of the eigenfunction; then, , and we have

Inserting the above equation into (60), we find that

Utilizing (61) in the above equation, we arrive at

This can be simplified as

Following our assumption that the Ricci curvature is greater than or equal to zero, i.e., , we derive that

If the hypothesis of the theorem regarding the extrinsic condition (4) holds, from the above equation, we obtainfor any . This is again Obata’s ODE [1], which implies that the base is isometric to the Euclidean sphere . The proof is completed.

Using the fact that the warped product submanifold is minimal, we give the following corollary derived from Theorem 1.

Corollary 1. Let be a Sasakian space form, and let be a C-totally real minimal isometric embedding of the warped product submanifold into with a nonnegative Ricci curvature. Then, there is an isometry between the compact base and the sphere if it satisfies the following:

Proof. Assuming is minimal and , then, from (64), we obtainIf assumption (78) holds, we get the following from (79):for a nonconstant function . Thus, [1] completes the proof of the corollary.

Remark 3. We know that every simply connected Sasakian space form is isometric to the odd-dimensional sphere and odd-dimensional Euclidean space with -constant sectional curvature and , respectively. For more details and examples, see the work presented in [24, 26, 27].

Another consequence of Theorem 1 is as follows.

Corollary 2. Let be a Sasakian space form and be the warped product submanifold having the nonnegative Ricci curvature. If is a C-totally real minimal isometric embedding of to . Then, there is an isometric between the compact base and the sphere if it satisfies the following:

Proof. Now, substituting in (79), we derive thatApplying equation (81) into the above equation, we obtainThe result follows from [1]. This completes the proof of the corollary.

One result deduced from Theorem 2 is as follows.

Corollary 3. Assume that that is a Sasakian space form and is a warped product submanifold having the nonnegative Ricci curvature and the compact base is minimal. If is a C-totally real isometric embedding from into , then is an isometric to the sphere such that if the following equality holds:

Proof. Inserting into (76), we obtainFollowing condition (84) and combining it with (85), we derive thatThen, the Obata Theorem [1] leads to the desired result.

If we choose that is the Sasakian space form is nothing but the Euclidean space by using Theorem 1, we obtain

Corollary 4. Suppose is a C-totally real isometric embedding of the warped product submanifold into of a nonnegative Ricci curvature. Then, a compact minimal base is isometric to the sphere if the following condition holds:

Proof. Putting into (64), we obtainEquations (87) and (88) imply thatAgain, from Obata’s Theorem [1], we reached the desired result.

Similarly, from Theorem 2, we derive

Corollary 5. Let be a warped product submanifold and be a C-totally real minimal isometric embedding from to the Sasakian space form . Then, the compact base is isometric to the sphere .

Proof. According to the hypothesis of the theorem, we know that is minimal and . Therefore, from (76), we determined the ordinary differential equation (89). Thus, we have completed the proof.

Remark 4. For examples of C-totally real isometric immersions from warped product manifolds, see the work presented in [27, 30].

For existence such a warped product, we provided an example is the following

Example 1. (see [30]). At point , there exists no less than two types C-totally real submanifolds going within of the non-Sasakian-manifold. Assume the foliations are denoted by eigendistributions of ; thus, their leaves would be totally geodesic C-totally real submanifolds for the given non-Sasakian -manifold. Suppose a C-totally real submanifold in a contact metric manifold. Let us consider , so , such that is tangential part of to any . Assume that is the minimal C-totally real surface of . Construct the warped product manifold . Then, the embedding such that , where is the unit vector perpendicular with the linear subspace including , which is a C-totally real isometric embedding. For more classification, see [31–33].

Remark 5. We provided example 1 which shows the existence of C-totally real isometric immersion from warped product into an almost contact metric manifolds. However, we are not claiming that this example will satisfy our assumption.

5. Concluding Remarks

The paper dealed with ordinary differential equation on C-totally real warped product submanifolds from the optimization on the warping function of a C-totally real warped product submanifold of Sasakian space forms. First, we obtained a Ricci curvature inequality in the setting a C-totally real warped product submanifold which is a generalization of Theorem 2.1 in [24]. Then, we studied some characterizations theorems for a C-totally real warped product submanifold of a Sasakian space forms. Therefore, the paper has excellent combinations of ordinary differential equation with Riemnnian geometry. We hope that the paper will get influence in mathematical science because we first applied differential equation in product manifolds.

Data Availability

No data were used to support the findings of the study.

Conflicts of Interest

The authors declare no conflicts of interest.

Authors’ Contributions

All authors equally contributed to the work and approved the final version.

Acknowledgments

This research was funded by the Deanship of Scientific Research at Princess Nourah Bint Abdulrahman University through the Fast-track Research Funding Program.

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Copyright © 2021 Fatemah Mofarreh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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