Characterizing Inequalities for Biwarped Product Submanifolds of Sasakian Space Forms

Khan, Meraj Ali; Al-dayel, Ibrahim

doi:https://doi.org/10.1155/2021/9966248

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Volume 2021 | Article ID 9966248 | https://doi.org/10.1155/2021/9966248

Characterizing Inequalities for Biwarped Product Submanifolds of Sasakian Space Forms

Meraj Ali Khan¹and Ibrahim Al-dayel²

Academic Editor: Francesco Lolli

Received31 Mar 2021

Revised04 Jul 2021

Accepted09 Aug 2021

Published20 Aug 2021

Abstract

The biwarped product submanifolds generalize the class of product submanifolds and are particular case of multiply warped product submanifolds. The present paper studies the biwarped product submanifolds of the type in Sasakian space forms , where , , and are the invariant, anti-invariant, and pointwise slant submanifolds of . Some characterizing inequalities for the existence of such type of submanifolds are proved; besides these inequalities, we also estimated the norm of the second fundamental form.

1. Introduction

Because of its expected applications in material science and relativistic theory, the investigation of warped product manifolds has obtained a conspicuous subject in the field of differential geometry; for example, warped products give numerous major solutions for Einstein field equations [1]. The theory of warped product manifolds is being used to demonstrate space-time close to the black holes [2]. The warped product represents Schwartzschild space-time, with base , , and fibre that is sphere with radius one. However, the Schwartzschild space-time will transform into a black hole under some instances [3].

In the paper [4], some of the inherent properties of warped product manifolds were investigated. Chen (see [5]) undertook the very first extrinsic study of warped product manifolds in the almost complex setting while acquiring certain existence results for CR-submanifolds to be CR-warped product submanifolds in Kaehler manifolds. Hasegawa and Mihai [6], on the other hand, analyzed contact CR-warped product submanifolds in almost contact environments. Many other people have investigated warped product manifolds in contact geometry, yielding an assortment of existence outcomes for instance (see [7–10]).

Another general class of warped product semislant submanifolds and contact CR-warped product submanifolds is the warped product pointwise semislant submanifold. The analysis was then continued by I. Mihai and S. Uddin in the framework of Sasakian manifolds, and few ideal inequalities relating to the second fundamental form and warping function were obtained. In the papers (see [11–13]), warped product pointwise semislant submanifolds for almost contact and almost complex manifolds were investigated.

One more generalized class of product manifolds is biwarped product manifolds, which are a subclass of multiply warped product manifolds. Chen and Dillen [14] looked at multiply warped product submanifolds immersed in Kaehler manifolds and found the remarkable optimum inequalities for them. Biwarped product submanifolds have recently been investigated (cf., see [15, 16]). Also, there is a recent paper [17], which initiates the study of inequalities for biwarped product submanifolds of nearly trans-Sasakian manifolds.

In this manuscript, authors established some inequalities for the squared norm of the second fundamental forms. These inequalities generalize several results available in the literature. Since Sasakian manifolds are particular cases of the nearly trans-Sasakian manifolds, therefore, a natural question arises that the inequalities obtained in the present paper may be particular case of inequalities obtained in [17]. Although in the paper [17] authors studied whole norm of second fundamental form, in the present study, the inequalities for second fundamental form are obtained by taking the restriction on the distribution , which is a part of the normal distribution. Therefore, our main results are different from the results obtained in [17] except some initial results.

Basically, in this manuscript, we look at biwarped product submanifolds of Sasakian space forms and determine some interesting inequalities. In terms of warping functions and slant functions, we estimate the norm of the second fundamental form. As a result, the equity case is taken into account.

The article is structured as follows. Second section is contributed to fundamental concepts, formulae, and results that are essential for the paper’s next analysis. We prove our key findings in Section 3 by looking into the nature of biwarped product submanifolds in Sasakian space forms.

Throughout the text, we used some abbreviations like Biwarped Product BW-P, Sasakian space form S-S-F, totally Geodesic T-G, and totally umbilical T-U.

2. Preliminaries

A dimensional manifold is said to have an almost contact structure if on there exist a tensor field of type , a vector field , and a 1-form satisfying the following properties [18]:

The manifold with the structure is called almost contact metric manifold. There exists a Riemannian metric on an almost contact metric manifold , satisfying the following:for all where is the tangent bundle of .

An almost contact metric manifold is said to be Sasakian manifold if it satisfies the following relation [18]:for any , where denotes the Riemannian connection of the metric . More details of almost contact metric manifold can be seen in [18]. For a Sasakian manifold, we have

A Sasakian manifold is said to be a Sasakian space form if it has constant -holomorphic sectional curvature and is denoted by . The curvature tensor of S-S-F is given byfor all vector fields on .

Let be a submanifold of an A-C-M manifold with induced metric . The Riemannian connection of induces canonically the connections and on the tangent bundle and the normal bundle of , respectively, and then the Gauss and Weingarten formulae are governed byfor each and , where and are the second fundamental form and the shape operator, respectively, for the immersion of into . They are related aswhere is the Riemannian metric on as well as the induced metric on .

For a submanifold , the equation of Codazzi is provided bywhere is the normal component of the curvature tensor .

If and represent the tangential and normal part of , respectively, for any , one can write

Similarly, for any , we writewhere and are the tangential and normal parts of , respectively. Thus, (resp. ) is 1-1 tensor field on (respectively, ), and (respectively, ) is a tangential (respectively, normal) valued 1-form on (respectively, ). The covariant derivatives of the tensor fields , , and are defined as

From equations (3), (6), (7), (10), and (11), we have

The mean curvature vector of is defined aswhere is the dimension of and is a local orthonormal basis of . The squared norm of the second fundamental form is defined as

A submanifold of is said to be a T-G submanifold if and T-U submanifold if , for each .

The notion of slant submanifolds in contact geometry was first defined by Lotta [19]. Later, these submanifolds were studied by Cabrerizo et al. [20]. Now, we have following definition of slant submanifolds.

Definition 1. A submanifold of an almost contact metric manifold is said to be slant submanifold if for any and , the angle between and is constant. The constant angle is then called slant angle of in . If , the submanifold is invariant submanifold, and if , then it is anti-invariant submanifold. If , it is proper slant submanifold.
Etayo [21] presented the idea of pointwise slant submanifolds as a generalization of slant submanifolds in the context of almost Hermitian manifolds. Further, Chen and Garay [22] looked into pointwise slant submanifolds for almost Hermitian manifolds and came up with some important results. However, Park [23] expanded the definition of pointwise slant submanifolds in almost contact metric manifolds, which was an important development in this direction. Nevertheless, for almost contact metric manifolds, Uddin and Alkhalidi [24] revised the concept of pointwise slant submanifolds. More specifically, a submanifolds of an almost contact metric manifold are claimed to be pointwise slant submanifold if for any in the sense that is tangential to , and the angle between and is independent of the choice of nonzero vector field . In this case, is viewed as the slant function of the pointwise slant submanifold, which is a function on . We now have the following descriptive theorem.

Theorem 1 (see [24]). Let be a submanifold of an A-C-M manifold such that . Then, S is pointwise slant iffwhere is the real valued function on .

As a result, the above formula has the following implications:.

One can conceive the warped product of manifolds as a generalization of the product manifolds, which are explained as follows.

Consider two Riemannian manifolds and with corresponding Riemannian metrics and and be a positive differentiable function. If and are projection maps such that and , which are defined as and , then is called warped product manifold if the Riemannian structure on satisfiesfor all . The function represents the warping function of . We can generalize this definition to multiply W-P manifolds as follows.

Let be Riemannian manifolds with respective Riemannian metrics , and let be positive valued functions on . Then, the product manifold equipped with Riemannian metric given byis said to multiply W-P manifold denoted by where are the projection maps of onto , respectively, and are their respective tangent maps for . The functions are known as the warping functions [14]. If the warping functions are constants, the warped product is simply a Riemannian product, known as a trivial multiply warped product.

The analysis of multiply warped product manifolds has recently gained attention in both complex and almost contact settings [14, 25]. We may define biwarped product manifolds as a special case of multiply warped product manifolds by using in the above description. For the BW-P manifold with the Levi-Civita connection, and denote the Levi-Civita connection of for . Some formulae relating to covariant derivatives for a BW-P manifold are given in the following lemma.

Lemma 1 (see [26]). Let be a BW-P manifold. Then, we havefor and , .

is the gradient of and is defined as . Let be an dimensional Riemannian manifold with the Riemannian metric , and let be an orthogonal basis of TS. As a consequence of (24), we have

The Laplacian of is defined by

Hopf’s lemma is now described.

Lemma 2 (see [27]). Let be an n-dimensional connected compact Riemannian manifold. If is a differentiable function on such that everywhere on (or everywhere on ), then is a constant function.

3. Main Results

In the present section, first we trace the existence of BW-P submanifolds for any Riemannian submanifolds , , and in Sasakian manifolds with warping functions and and then we demonstrate our key findings. Hasegawa and Mihai [6] set up the following result.

Theorem 2. Let be a dimensional Sasakian manifold. Then, there do not exist W-P submanifolds such that is an anti-invariant submanifold tangent to and an invariant submanifold of .

We draw the conclusion based on the above result; that is, if , , and are invariant, anti-invariant, and pointwise proper slant submanifolds, then BW-P submanifolds of the forms and in a Sasakian manifold do not exist. From [23], we have the following observation.

Theorem 3. Let be a dimensional Sasakian manifold. Then, there do not exist W-P submanifolds tangential to such that is pointwise proper slant submanifold and is invariant submanifold of , respectively.

It can be deduced by Theorem 2 that BW-P submanifolds of the types and in a Sasakian manifold are trivial.

Park identified the existence of the warped product pointwise semislant submanifolds of Sasakian manifolds of the form in his paper [23], with warping function , where and are the holomorphic and pointwise slant submanifolds of , as well as proving the next lemma.

Lemma 3. Let be a W-P pointwise semislant submanifold of a Sasakian manifold such that , where and are invariant and pointwise slant submanifolds of , respectively. Then,for any and .

Consider the biwarped product submanifolds of the type of a Sasakian manifold with warping functions and such that , , and are the invariant, anti-invariant, and pointwise slant submanifolds of correspondingly. To address the question of which factor of the biwarped product submanifold is parallel to , we have the following.for all and .where and are the basis of the orthonormal vector fields on and , respectively.Proof. Choosing unit vector fields , , and and using (5) and (19), we have

Theorem 4. Let be a Sasakian manifold. If is a biwarped product submanifold of such that , , and are the invariant, anti-invariant, and pointwise slant submanifolds of , respectively, then we have the following:(i)If , is constant(ii)If , is constant

Proof. The proof of this theorem can be deduced directly from Proposition 1 in [17] for .

Remark 1. Proposition 1 in [17] was proved for slant distributions, and the same proof is also valid for pointwise slant distributions.
As a consequence of the above, we can point out that there are no any nontrivial BW-P submanifolds of the type of a Sasakian manifold if the vector field is tangential to or .
Now, let be a BW-P submanifold of a Sasakian manifold and consider the vector field tangent to be . If is invariant distribution, is anti-invariant and is pointwise slant distribution with the slant function . The following decomposition refers to the tangent bundle :The normal bundle is decompounded aswhere is the invariant orthogonal complementary distribution of in .
The second fundamental form can be written as a consequence of the above direct decomposition.for , where , , and are the components of in the normal sub-bundles , and , respectively. Moreover if be a local orthonormal frame of vector fields of , thenwhereWe create an example of a BW-P submanifold of the form in Sasakian manifold with .

Example 1. It is well known that denotes a Sasakian manifold with its standard Sasakian structure given byConsider the submanifold as follows:And consider a frame of orthogonal vector fields tangent to asIt is then simple to note that span , span , and span defined as the invariant, anti-invariant, and pointwise slant distributions with the slant function on the Sasakian manifold . If we denote the integral manifold of , , and by , , and correspondingly, then the metric on is given byThen, is a BW-P submanifold with the warping functions , .
First, we demonstrate some preliminary findings.

Lemma 4. Let be a BW-P submanifold of a Sasakian manifold . Then,(i) and (ii)(iii)(iv)(v)for all , , and , where is the component of the second fundamental form .

Proof. The part (i) can be deduced from Proposition 2 of [17]. Moreover, the parts (ii) and (iii) can be concluded from equations (30) and (20) in [17], respectively.
Using (6) and (3), we can prove part (iv) asBy applying (23), the above equation can now be written asComparing the normal parts,On taking Riemannian product with , we findCalculating the last term of (40) by using (3) and (6) and (23),Utilizing part (ii), we getUsing the above equation in (40), we get the required result. Part (v) of the lemma can also be verified in a similar way.

Lemma 5. Let be a BW-P submanifold of a Sasakian manifold . Then,

Proof. The proof of the present lemma can be concluded from equation (33) in [17] for .

Lemma 6. On a BW-P submanifold of a Sasakian manifold , we have

Proof. First, we will modify the left-hand term as follows:On using part (ii) of Lemmas 4 and 5 and combining (25), we findReplacing by in the above equation, we getWe get the required result while subtracting the above two findings.
The following characterization is now established.

Theorem 5. Let be a BW-P submanifold of a Sasakian space form such that is a compact submanifold. The following characterization is now established. If the following inequalities hold, is a Riemannian product submanifold.where represents the projection of in and , , and are the dimensions of , , and subsequently while is a local orthonormal basis of .

Then, again by the Codazzi equation,

The estimations of the terms engaged with (51) are presently processed. Foremost, we have

Applying the part (ii) of Lemma 4 in the last equation, we find

Calculating the last term of (53) and using (10), we have

By the use of (6) and (12), the previous equation changes to

By the application of (3), (6), and (23) and part (ii) and (iii) of Lemma 4, we get

Making use of (56) in (53), we find

In similar fashion, we are able to write

We have the following from part (ii) of Lemma 4:

Changing out by (applying the totally geodesicness of , ) in the previous equation, we obtain

By (6), the equation above has the following form:

It is simple to verify that , for all in by using the fact that the first factor is totally geodesic in . Substituting this and (12) in the last equation, we obtain

Adopting similar steps, we can put

By part (ii) of Lemma 4 and (23), we get

Substituting values of (49) and (57)–(65) in (51), we obtain

On using (30), (32), (6), and (3), the previous equation becomes

Let be the orthonormal frame on and be an orthonormal frame on . Taking sum of the above equation with the indices and and making use of (25) and (26) and part (iii) of Lemma 4, we get

In the similar way, for and again using the Codazzi equation, we can prove the following:

Let be an orthonormal frame of . Taking sum by using and and simultaneously using (24) and (25), the above equation yields the following:

From (69),

This means , so by application of Hopf’s Lemma, is constant that indicates is constant. Moreover, in (70), ifthen , so by Hopf’s lemma, is constant that implies that the W–F is constant. We get the necessary result when these two statements are combined.

The squared norm of the second fundamental form is obtained using the warping functions and the slant function in the following theorem.

Theorem 6. Let be a dimensional S-C-F and be an dimensional BW-P submanifold such that is a dimensional invariant submanifold, is a dimensional anti-invariant submanifold, and be a dimensional proper pointwise slant submanifold of . Ifthen(i)The squared norm of the second fundamental form satisfies(ii)The equality sign of (74) satisfies identically if and only if(i) is T-G invariant submanifold of . Hence, is a S–C-F.(ii) and are T-U submanifolds of .(iii).

Proof. From (69), we haveFor the orthonormal frames and , in view of formulae (31) and (32) and part (ii) of Lemma 4, we getFurther, using Lemma 5 and (25), the above equation is reduced toNow, for any and , from part (iii) of Lemma 1, we haveBy the above equations for the frame and , it is simple to conclude thatMoreover, from (3) and (23), we getfor all , , and . We can deduce the following from these two findings:From (75), (77), (79), (81), and (82), we get the required inequality.
To prove the part (ii), let and be the second fundamental forms for the immersion of and in , respectively. Then, for any and , using the Gauss formula, we haveBy (24), we obtainorAccordingly, for any and , we haveIf the equality sign of (74) holds identically, then we haveThe first condition of (87) suggests that is T-G submanifold in S. Then, again it is not difficult to see that and, for all It follows that is T-G in and hence is a S-C-F. The second condition of (87) with (86) implies that is T-U. Besides, the third condition of (87) along with (85) suggests that is a T-U submanifold. This demonstrates the proof.

4. Conclusion

In this paper, by utilizing Hopf’s Lemma, we acquired the describing inequalities for the existence of biwarped product submanifolds of Sasakian space forms. Besides, we additionally worked out an assessment for the squared norm of the second fundamental form in terms of the warping function and slant function. To fortify our study, we gave a nontrivial example of a biwarped product submanifold in a Sasakian manifold.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that there are no conflicts of interest.

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Copyright © 2021 Meraj Ali Khan and Ibrahim Al-dayel. 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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