Inequalities for the Class of Warped Product Submanifold of Para-Cosymplectic Manifolds

Mofarreh, Fatemah; Srivastava, Sachin Kumar; Dhiman, Mayrika; Othman, Wan Ainun Mior; Ali, Akram

doi:https://doi.org/10.1155/2022/7270446

Advances in Mathematical Physics

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Geometry of Warped Product Manifolds: Theory and Applications 2022

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Research Article | Open Access

Volume 2022 | Article ID 7270446 | https://doi.org/10.1155/2022/7270446

Inequalities for the Class of Warped Product Submanifold of Para-Cosymplectic Manifolds

Fatemah Mofarreh,¹Sachin Kumar Srivastava,²Mayrika Dhiman,²Wan Ainun Mior Othman,³and Akram Ali⁴

Academic Editor: Mohammad Alomari

Received29 May 2022

Accepted11 Aug 2022

Published26 Sept 2022

Abstract

The aim of this paper is to study the warped product pointwise semislant submanifolds in the para-cosymplectic manifold with the semi-Riemannian metric. For which, firstly we provide the more generalized definition of pointwise slant submanifolds and related characterization results followed by the definition of pointwise slant distributions and pointwise semislant submanifolds. We also derive some results for different foliations on distribution, and lastly, we defined pointwise semislant warped product submanifold, given existence and nonexistence results, basic lemmas, theorems, and optimal inequalities for the ambient manifold.

1. Introduction

To generalize the Riemannian product manifolds, Bishop and O’Neil [1] introduced the concept of warped product for the manifolds with negative curvature and showed the surface of revolution as the simplest example of warped product manifold. The authors of [2–5] studied the warped product submanifolds for different manifolds. Warped product plays the beneficial role in encoding the universe, and the inequalities related to the second fundamental form with the warping function cover the wide as well as important section of it. These were firstly formulated by Chen in [6, 7]. Warped product for lightlike manifolds for the first time was studied in [8] and for semi-Riemannian manifold under the name PR-warped product on para-Kähler manifold in [9], where he derived the aforesaid inequalities for the case of semi-Riemannian metric. From there, the study on warped product escalates among geometers with also in view that the same has so many applications in the physics mainly in general relativity and black hole theory [10].

Beside this, the name slant submanifolds were introduced as the generalized version of holomorphic and totally real cases of submanifolds by Chen in [11]. Further, the theory extended to various manifolds with Riemannian as well as semi-Riemannian metric by many geometers. Later, in 2017, the authors in [12] defined the slant submanifolds irrespective of the writinger angle for the semi-Riemannian manifold and formulated three cases which are separately explained and achieved some effective results with bunch of examples. They defined it in terms of quotient which is constant for the case of slant submanifolds for every vector field (spacelike or timelike) on the submanifold of manifold . As slant and semislant submanifolds generalized to pointwise slant submanifolds (former called quasi slant) by Etayo in [13], Chen and Garay studied the same for the almost Hermitian case [14]. Sahin [15] defined pointwise semislant notion of submanifolds with an example. Recently, there are many interesting papers related with submanifold theory, singularity theory, classical differential geometry, etc. The readers can find more details about those techniques and theories in a series of papers [16–29]. Moreover, interdisciplinary research is one of the hottest trends in science; in the future work, we intend to apply and combine the techniques and results presented in [16–25] alongside with the methods in this paper to obtain more new results.

The paper is structured as follows: Section 2 contains the preliminary knowledge about ambient manifold, submanifold, and warped product with some important lemmas. Section 3 defines the pointwise slant submanifold, characterization lemma, and an example. Section 4 and Section 5 deal with the study of the pointwise slant distributions and pointwise semislant submanifolds, respectively. Section 6 includes the definition of warped product, some nonexistence results, lemmas, and theorems provided with an example. Finally, inequalities for the same submanifold are given in Section 7.

2. Preliminaries

Definition 1. A -dimensional smooth manifold admits structure with as a -tensor field, as a characteristic vector field, as a globally differential 1-form, and as a semi-Riemannian metric named as an almost paracontact semi-Riemannian manifold which satisfies where represents an identity transformation of tangent space of and represents a tensor product. A structure compatible semi-Riemannian metric “” relates to as [30].

Equations (1) and (2) easily ensure the following:

Let be the fundamental 2-form on ; then,

Basis. An almost paracontact semi-Riemannian manifold always exists with a , a certain type of local pseudoorthonormal basis which includes , as space-like, and as timelike vector fields.

Definition 2 (see [31]). An almost paracontact semi-Riemannian manifold is termed as para-cosymplectic if the forms and are parallel with respect to the Levi-Civita connection by

Lemma 3. Let be a para-cosymplectic manifold with structure vector field ; then,

Proof. Directly follow with the help of Equations (4) and (7) and covariant differentiation.

2.1. Submanifold

Let be an isometrically immersed submanifold of a para-cosymplectic manifold with an induced nondegenerate metric (denoted metric by same symbol as on ), denoting as Levi-Civita connection and as the second fundamental form on . Thus, the Gauss-Weingarten formulas are for (tangent bundle), and (normal bundle); denotes normal connection, and denotes shape operator associated with the normal section on . The metric relation of and is given as

Every is split as

Similarly, every is split as where and ( and ) are the tangential parts (normal parts) of and , respectively. Based on Equation (12), the submanifold classifies as anti-invariant if or invariant if on . After using Equation (12) in Equation (5), we get

Now, from Lemma 3 and Equation (11), we have our next result.

Lemma 4. If is a submanifold immersed in a para-cosymplectic manifold with structure vector field , then for every and .

Next, let us take two semi-Riemannian manifolds and and a positive smooth function on . Taking as the product manifold along with canonical projections, such that and for any point . Then, the product manifold is called warped product if metric called the warped metric on can be formulated as For every , “” represents the derivation map, and we call as a warping function. Abstractly, the metric can be written as where the warped product is split into a product of the base space and the fiber space , except that the fiber is warped [1, 32].

Proposition 5 (see [32]). The warped product submanifold satisfies (i)(ii)(iii)for and , where is the Levi-Civita connection on , is the connection on , and is the gradient of defined as .

Further, let be a local orthonormal basis on among which are tangent to and are normal to . If we set then we get where are the coefficients of . Accordingly, squared norm of the second fundamental form is defined as

3. Pointwise Slant Submanifolds

The semi-Riemannian manifold has difficulty of defining the Writinger angle as the vector fields may be timelike. Thus, the next definition is in the view of [12], generalizing the slant submanifold in our ambient semi-Riemannian manifold.

Definition 6. An isometrically immersed submanifold of an almost paracontact manifold is termed as pointwise slant if at every point , the quotient for is independent of the choice of any nonzero spacelike or timelike vector , where . For slant angle , we say a slant coefficient.

Remark 7. The value of can be (i) for ; is timelike or spacelike of each spacelike or timelike vector field adding (ii) for ; is timelike or spacelike of each spacelike or timelike vector field adding (iii) for is timelike or spacelike for any timelike or spacelike vector field adding

Remark 8. The special cases are as follows: (i)The constant value of throughout implies is slant submanifold [11, 12](ii)The point is called a complex point if , which means that the slant coefficient is equal to . The submanifold with every point as complex point is complex or holomorphic submanifold(iii)The point is called a totally real point if , which means that the slant coefficient is equal to . The submanifold with every point as totally real point is totally real submanifold

Furthermore, let us take the union of all ’s and denoting the same by

Lemma 9. The submanifold isometrically immersed in para-cosymplectic manifold is a pointwise slant submanifold if and only if on every point ; there exists for some such that for each spacelike (or timelike) vector field .

Proof. For each point of a pointwise slant submanifold , the definition (22) follow as for . With the use of Equations (5) and (14) and the condition that in Equation (23), we get the desired result.

Proceeding further with some results which are not hard to prove, any pointwise slant submanifold satisfies for .

Proposition 10. The submanifold of a para-cosymplectic manifold is pointwise slant submanifold if and only if (i) and for any spacelike (or timelike) vector field (ii) for nonlightlike normal vector field , where is the slant coefficient of

Proof. Assume as a pointwise slant submanifold. (i)Then for every , . On other way, Equating tangential and normal parts and using Lemma 9, we can attain the result(ii)Since , thus there exists as is pointwise slant submanifold such that . Now, . The converse can be easily derived using same equations

Theorem 11 (see [33]). A totally geodesic and connected pointwise slant submanifold of a para-cosymplectic manifold is a slant submanifold.

4. Pointwise Slant Distributions

Analogous to [34], we generalize slant distributions by defining pointwise slant distributions in . Furthermore, we study some basic characterizations for the distributions on our ambient manifold.

Definition 12. A pointwise slant distribution on is a differentiable distribution for which the quotient is independent of the choice of any spacelike or timelike vector field . Here, (i) is the distribution at point (ii) is the projection of on the distribution (iii) is the slant coefficient corresponding to the distribution on for , and the value of may be , , or

Remark 13. (1)A pointwise slant distribution is invariant if with or anti-invariant for with . Other than these two cases, we call the distribution to be proper pointwise slant distribution [12](2)The distribution on is as follows [9, 31]: (i)totally geodesic: if (ii)involutive: if for every .

Corollary 14. The distribution on the submanifold is pointwise slant distribution if and only if there exists for such that = for any nonlightlike vector field .

Proof. The result follows similar to Lemma 9.

5. Pointwise Semislant Submanifold

Definition 15. A submanifold of a para-cosymplectic is named as pointwise semislant submanifold if the set of complementary orthogonal distributions exists on and fulfills the listed conditions: (i)(ii) is -invariant distribution, i.e., (iii) is a pointwise slant distribution having as a slant coefficient for

Remark 16. Further, submanifold is (i)proper pointwise semislant when with nonconstant (ii)proper slant submanifold when and with globally constant for [35](iii)proper semi-invariant when and such that for any [12](iv)invariant submanifold when [35](v)anti-invariant submanifold when and for every [35]

Remark 17. The decomposition of the tangent space can be expressed in two ways: (i)If , the (ii)If , the . Here, . Thus, we have either or [4]

Denote and as the projections, respectively, on the distributions and . Then, any is split as

Operating , using Equation (12) and the case distribution which is -invariant on the previous equation, we concluded that

As is pointwise slant distribution, by the consequences of Corollary 14, we obtain that for with as the slant coefficient. Clearly, for any point , if , then where is the projection on the distribution . But this does not affect our result as disappears when operates on .

However, the normal bundle denoted as may be written as where represents the subspace of normal bundle that is invariant under .

Lemma 18 (see [31]). The shape operator of a proper pointwise semislant submanifold of para-cosymplectic manifold ensures the listed conditions: for , , , and .

Both when is normal or tangent to , the integrability and geodesic conditions brought out to be same after calculations for both the distributions, thus denoting them as common .

Lemma 19. If is a proper pointwise semislant submanifold of para-cosymplectic manifold , for or , the invariant distribution on is (i)integrable if and only if (ii)totally geodesic if and only if for and .

Proof. Equation (2) expands as for every nonzero vector fields and . Using Equation (12) for the in Equation (35) and followed by using Equations (5), (7), and (9) and Lemma 9, we arrive at Result (i) is clear using remark (28) as is nonconstant in Equation (36). Again, from Gauss formula and Equation (2), Employing Equations (7), (9), (11), (12), and (28) and Remark 16 in Equation (37), result (ii) follows.

Lemma 20. If is a proper pointwise semislant submanifold of para-cosymplectic manifold , for or , the pointwise slant distribution on is (i)involutive if and only if (ii)totally geodesic if and only if for and .

Proof. Equation (2) implies for every nonzero vector fields and . Solving separately the term using Equations (5), (7), (10), (12), and (28), we receive where is the first derivative of . Surely, are orthogonal to after using this fact in Equation (41), and substituting in Equation (40), we get For , one can replace by in Equation (42), and consequently, we get Using Lemma 4 in Equation (43) and for reason that a nonconstant, we get Therefore, in Equation (42) using Equation (44) along with the facts that is proper, we arrived at the desired result (i).
Further, using Gauss formula and employing Equations (2), (7), (9), (11), (12), and (28) give Since , we can replace by in Equation (45), and consequently, we get Using Lemma 4 in above expression, we get Hence, Equation (45) implies that Thus, from (48) and as proper, as nonnull vector fields, the proof of the (ii) directly follows.

6. Pointwise Semislant Warped Product Submanifold

Definition 21. A pointwise semislant warped product submanifold of a para-cosymplectic manifold is a warped product of an invariant submanifold and a proper pointwise slant submanifold either in the form or , where is a positive smooth function taken on first submanifold in the product and slant coefficient of is . A trivial product is the case of such submanifold for which warping function is constant.

Proposition 22 (see [33]). A nontrivial pointwise semislant warped product submanifold of the form with does not exist on a para-cosymplectic manifold .

Proposition 23 (see [33]). A nontrivial pointwise semislant warped product submanifold of the form with does not exist on a para-cosymplectic manifold .

Proposition 24. A nontrivial pointwise semislant warped product submanifold of the form with does not exist on a para-cosymplectic manifold .

Proof. Directly follow from Lemma 4 and Proposition 5.

Lemma 25. For a nontrivial pointwise semislant warped product submanifold of a para-cosymplectic manifold , and .

Proof. (1)Since for and , on right side, using Equations (5), (7), and (12), Proposition 5, and Lemma 9, we have Equations (10), (11), and (14) and Lemma 9 further help to achieve (49)(2)As , next substituting applying Equations (5) and (7), Proposition 5, and Lemma 9 and the facts that is invariant, we get (50)

Proposition 26. If is a nontrivial pointwise semislant warped product submanifold of a para-cosymplectic manifold , then and .

Proof. As , . Using Equations (7), (9), and (12) and Lemma 9 in above expression gives (52). In similar way, we can prove (53).

Lemma 27. A nontrivial proper pointwise semislant submanifold of a para-cosymplectic manifold satisfies and .

Proof. Result (54) is not hard to prove using Equations (7), (12), and (14) and Proposition 5. Substituting in Equation (49) gives ; one can replace , and using Equations (5) and (7) and Proposition 5 gives (55). Putting and , respectively, in Equation (55) gives results (56) and (57).

Lemma 28. If is a nontrivial pointwise semislant warped product submanifold of a para-cosymplectic manifold , then (i)for , (ii)for , and .

Proof. Replacing in Equation (55) and having the fact that submanifold is invariant, both results directly follow.

Proposition 29. Let be nontrivial pointwise semislant warped product submanifold of a para-cosymplectic manifold , then and , and is the first derivative of slant coefficient.

Proof. Using metric and para-cosymplectic condition this expression under the effect of Equations (5), (9), (12), and (14) turns as Further, using Propositions 10 and (5) ended with the desired result.

Proposition 30. A nontrivial pointwise semislant warped product submanifold of a para-cosymplectic manifold satisfies the following: (i)For , (ii)For , and .

Proof. Lemma 28 and Equation (57) of Proposition 26 directly give the results.

Definition 31. The submanifold is named as mixed totally geodesic if for every and ,

Theorem 32. If is a mixed totally geodesic pointwise semislant warped product submanifold of a para-cosymplectic manifold , following cases arise: (i)If , then is either a trivial product or a warped product of a holomorphic (complex) submanifold and a totally real submanifold(ii)If , then is either a trivial product or a warped product of two complex submanifolds

Proof. Using definition (59), satisfies as well as (as is -invariant) for and . Using this condition in proposition (58) when , we get Indicate either implies the trivial case or , after taking inner product with , and in the view of Remark 8, the condition for the totally real holds for the submanifold . Following similar way for the second case, we ended up with which is the condition for complex submanifold.

Theorem 33. A mixed totally geodesic pointwise semislant warped product submanifold of a para-cosymplectic manifold satisfies spacelike (or timelike) vector fields and .

Proof. As , under the effects of Equation (14), Proposition 5, and Lemma 9, it turns Other way, ; with this expression under the use of Equations (5), (7), and (12), Proposition 5, and Lemma 9, and as mixed totally geodesic, we have This expression with the use of Equation (68) and for the reason vector field can be spacelike or timelike yields the result.

Example 34. Consider a -dimensional smooth manifold having standard Cartesian coordinates as and defining a structure as for as a local orthonormal frame on . Obviously, over fufills the condition of para-cosymplectic manifold. Let be a submanifold of with tangent to it and defined for , and some (constant) as

The vector fields that generates the tangent bundle are

After calculations, it is found that the invariant distributions is the span of subspace and pointwise slant distribution is the span of subspace with such that ; then, the slant coefficient is (i)cosh for (ii)cos for

As the distributions and are integrable, let and be their respective integral manifolds such that turns a nontrivial 5-dimensional pointwise semislant warped product submanifold of with induced metric (semi-Riemannian) as with warping function .

Let be an another submanifold of with normal to it and defined for as

Then, the vector fields that generates are

The invariant distribution is the span of subspace , and pointwise slant distribution is the span of subspace with such that ; then, the slant coefficient is (i)cosh for (ii)cos for

As the distributions and are integrable, let and be their respective integral manifolds such that turns a nontrivial 4-dimensional pointwise semislant warped product submanifold of with induced metric (semi-Riemannian) as with warping function .

7. Inequalities

This section includes the geometric sharp inequalities for the aforesaid submanifold for the case tangent and normal to .

Lemma 35 (see [31]). Let be a pointwise semislant warped product submanifold of a para-cosymplectic manifold . Then, ensures , , and .

Theorem 36. Let be a pointwise semislant warped product submanifold of a para-cosymplectic manifold with . If is an invariant submanifold of -dimension and is a proper pointwise slant submanifold -dimension satisfying , the succeeding inequalities holds for where is the slant coefficient corresponding to , is the gradient of , with its component and invariant distribution , , and .

Proof. For , choose the local orthonormal frame on the following: (a) by for and for such that implies and (b) by for and such that implies (c) by for , having and on by such that implies Compute which is given as The first term can be expanded as As is totally geodesic and Equation (54) of Lemma 27 directs that using which, we can write Simplifying these expressions in Equation (82) and using Equation (19) and Lemma 4 and in view of orthonormal frame, we get The integrable condition of the and Equation (77) of the Lemma 35 implies that After substitution of Equation (84) in (86), we get For the second part, we have where After simplifying Equation (88) using expressions in Equation (91), we get Using Equations (55), (56), (58), (77), and (78), we have Substituting above values in Equation (92), we have Since and using the condition that in formula (78), we concluded that , above equation leads to Lastly, where the included expressions are as below: Employing these expressions in Equation (96) in view of the chosen frame and simplifying, we get Using the condition that in formula (79), we concluded that , which implies the Equation (98) with Result directly follows by letting and .

Remark 37. Equality holds if .

Theorem 38. Let be a pointwise semislant warped product submanifold of a para-cosymplectic manifold with normal to such that . If is an invariant submanifold of -dimension and is a proper pointwise slant submanifold of -dimension satisfying , the succeeding inequalities holds for where is the slant coefficient corresponding to , is the gradient of , with its component and invariant distribution , , and .

Proof. For , choose the local orthonormal frame on the following: (a) by for such that implies (b) by for and such that implies (c) by for , having and on by for and for such that implies and Further, result can be acquired carrying the same steps as above proof and using Equations (33) and (34) of Lemma 18.

Data Availability

There is no data used for this manuscript.

Conflicts of Interest

The authors declare no competing interest.

Authors’ Contributions

M.D. and S.K.S conceptualized the study. A.A. was responsible for the methodology. F.M. was responsible for the software. W.A.M.O, F.M., and M.D. were responsible for the validation. A.A. was responsible for the formal analysis. S.K.S. was responsible for the investigation. S.K.S was responsible for the resources. A.A. was responsible for the data curation. S.K.S and M.D wrote the original draft. A.A. and F.M. wrote, reviewed, and edited the manuscript. F.M. was responsible for the visualization. S.K.S. supervised the study. F.M. was responsible for the project administration. A.A was responsible for the funding acquisition. All authors have read and agreed to the published version of the manuscript.

Acknowledgments

The last author extends his appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through the research group program under grant number R.G.P.2/130/43. The authors also express their gratitude to the Princess Nourah Bint Abdulrahman University Researchers Supporting Project number (PNURSP2022R27), Princess Nourah Bint Abdulrahman University, Riyadh, Saudi Arabia.

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Copyright © 2022 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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