A Study of Doubly Warped Product Immersions in a Nearly Trans-Sasakian Manifold with Slant Factor

Alkhaldi, Ali H.; Siddiqui, Aliya Naaz; Ahmad, Kamran; Ali, Akram

doi:https://doi.org/10.1155/2021/5065333

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Volume 2021 | Article ID 5065333 | https://doi.org/10.1155/2021/5065333

A Study of Doubly Warped Product Immersions in a Nearly Trans-Sasakian Manifold with Slant Factor

Ali H. Alkhaldi,¹Aliya Naaz Siddiqui,²Kamran Ahmad,²and Akram Ali¹

Academic Editor: Zine El Abiddine Fellah

Received10 Jul 2021

Accepted14 Sept 2021

Published04 Oct 2021

Abstract

In this article, we discuss the de Rham cohomology class for bislant submanifolds in nearly trans-Sasakian manifolds. Moreover, we give a classification of warped product bislant submanifolds in nearly trans-Sasakian manifolds with some nontrivial examples in the support. Next, it is of great interest to prove that there does not exist any doubly warped product bislant submanifolds other than warped product bislant submanifolds in nearly trans-Sasakian manifolds. Some immediate consequences are also obtained.

1. Introduction and Motivations

The most inventive topic in the field of differential geometry currently is the theory of warped product manifolds. These manifolds are the most fruitful and natural generalization of Riemannian product manifolds. Due to the important roles of the warped product in mathematical physics and geometry, it has become the most active and interesting topic for researchers, and many nice results are available in the literature (see [1–3]).

Chen [4, 5] initiates the concept of warped product submanifolds by proving the nonexistence result of warped product CR-submanifolds of type in Kähler manifolds, where and are anti-invariant and invariant submanifolds, respectively. Moreover, he considers warped product CR-submanifolds of type and gives an inequality involving a warping function and the squared norm of the second fundamental form .

On the other hand, the concept of ordinary warped products can be extended to doubly warped products. By using this generalization, Sahin [6] shows that there exist no doubly warped product CR-submanifolds in Kähler manifolds other than warped product CR-submanifolds. He also investigates the existence of doubly twisted product CR-submanifolds in the same ambient. Many geometers have obtained several results on warped products and doubly warped products [7–12].

The concept of bislant submanifolds is defined by Cabrerizo et al. [13] as the natural generalization of contact CR-, slant, and semislant submanifolds. Such submanifolds generalize invariant, anti-invariant, and pseudoslant submanifolds as well. Recently, the warped product bislant submanifolds in nearly trans-Sasakian manifolds is studied by Siddiqui et al. in [1]. They obtain several inequalities for the squared norm of the second fundamental form in terms of a warping function .

In this paper, firstly, we discuss the de Rham cohomology class for closed bislant submanifolds in a nearly trans-Sasakian manifold. Secondly, in view of embedding theorem of Nash [14], we study an isometric immersion of a warped product bislant submanifold into an arbitrary nearly trans-Sasakian manifold. Then, we investigate the existence of doubly warped products in the same ambient.

2. Nearly Trans-Sasakian Manifolds and their Submanifolds

Definition 1 (see [15]). A -dimensional differentiable manifold is said to have an almost contact structure if there exists on , where (i)a tensor field of type (ii)a vector field (iii)a 1-form (iv)a Riemannian metric such that for any .

The covariant derivative of the tensor field is given by

for any .

In 2000, Gherghe introduced a notion of nearly trans-Sasakian structure of type , which generalizes the trans-Sasakian structure. A nearly trans-Sasakian structure of type is called nearly -Sasakian (resp. nearly -Kenmotsu) if (resp. ).

Definition 2 (see [16]). An almost contact metric structure on is called a nearly trans-Sasakian structure if for any .

Remark 3. (i)A nearly trans-Sasakian structure of type is(a)nearly Sasakian if [17](b)nearly Kenmotsu if [18](c)nearly cosymplectic if [19](ii)Remark that every Kenmotsu manifold is a nearly Kenmotsu manifold but the converse is not true. Also, a nearly Kenmotsu manifold is not a Sasakian manifold. On another hand, every nearly Sasakian manifold of dimension greater than five is a Sasakian manifold.

We put dim and dim . The Riemannian metric for and is denoted by the same symbol . Let and denote the Lie algebra of the vector field and set of all normal vector fields on , respectively. The operator of covariant differentiation with respect to the Levi-Civita connection in and is denoted by and , respectively. The Gauss and Weingarten formulae are respectively given as [15]

for any and . Here, is the second fundamental form, is the shape operator, and is the operator of covariant differentiation with respect to the linear connection induced in the normal bundle .

The second fundamental form and the shape operator are related as [15]

for any and . Here, denote the induced metric on as well as the Riemannian metric on .

Let and be a local orthonormal frame of and be a local orthonormal frame of . The mean curvature vector of a submanifold at is given by [15]

A submanifold of is said to be [15] (i)totally umbilical if , for any (ii)totally geodesic if , for any (iii)minimal if , that is, trace

For any , we put [15]

where and . Then is an endomorphism of , and is the normal bundle valued -form on . In the same way, for any , we put [15]

where and . It is easy to see that and are skew-symmetric and

for any and .

Definition 4. A submanifold of an almost contact metric manifold is said to be invariant if , that is, , and anti-invariant if , that is, , for any .

In contact geometry, Lotta introduced slant immersions as follows [20].

Definition 5. Let be a submanifold of an almost contact metric manifold . For each nonzero vector and , the angle between and is called slant angle of . If slant angle is constant for each , then the submanifold is called the slant submanifold.

For slant submanifolds, the following facts are known:

for any . Here, is slant angle of .

Remark 6. If we assume (i), then is an invariant submanifold(ii), then is an anti-invariant submanifold(iii), then is a proper slant submanifold

There are some other important classes of submanifolds which are determined by the behavior of tangent bundle of the submanifold under the action of an almost contact metric structure of [1]: (i)A submanifold of is called a contact CR-submanifold of if there exists a differentiable distribution on whose orthogonal complementary distribution is anti-invariant(ii)A submanifold of is called a semislant submanifold of if there exists a pair of orthogonal distributions and such that is invariant and is proper slant(iii)A submanifold of is called pseudoslant submanifold of if there exists a pair of orthogonal distributions and such that is anti-invariant and is proper slant

Definition 7 (see [13]). A submanifold of an almost contact metric manifold is said to be a bislant submanifold if there exists a pair of orthogonal distributions and of such that (i) and (ii)Each distribution is slant with slant angle for

Remark 8. If we assume (i) and , then is a CR-submanifold(ii) and , then is a semislant submanifold(iii) and , then is a pseudoslant submanifold(iv), then is a proper bislant submanifoldFor a bislant submanifold of an almost contact metric manifold, the normal bundle of is decomposed as where is a -invariant normal subbundle of .

3. Cohomology Class for Bislant Submanifolds of Nearly Trans-Sasakian Manifolds

Chen [21] introduces a canonical de Rham cohomology class for a closed CR-submanifold in a Kähler manifold. So, in this section, we discuss the de Rham cohomology class for a closed bislant submanifold of a nearly trans-Sasakian manifold with minimal horizontal distribution . We put dim and dim. Let us assume the following: (i) is a local orthonormal frame of (ii) is a local orthonormal frame of (iii) is a local orthonormal frame of

We choose as the dual frame of -forms to the above local orthonormal frame. Then, we define a -form on by . It is globally defined on . In the same way, we again define a -form on by , which is globally defined on .

We prepare some preliminary lemmas.

Lemma 9. Let be a submanifold of an arbitrary nearly trans-Sasakian manifold , then for any .

Proof. For any vector fields , making use of the structure equation and (2), we obtain which gives Comparing the tangential and normal components of the above equation, we get the desired relations (15) and (16).
The next lemma gives the integrability condition of slant distribution .☐

Lemma 10. Let be a bislant submanifold of an arbitrary nearly trans-Sasakian manifold . Then, slant distribution is integrable if and only if for any .

Proof. Making use of Lemma 9, we obtain for any . Thus, the assertion follows from the fact that and are mutually perpendicular. In this way, we proved the integrability condition of slant distribution .☐

We prove the following.

Theorem 11. For any closed bislant submanifold of an arbitrary nearly trans-Sasakian manifold with minimal and for any , the -form is closed and defines a canonical de Rham cohomology class , where dim.
Moreover, the cohomology group is nontrivial if is minimal and is integrable.

Proof. From the definition of , we have , which implies that if and only if for any and . Thus, by simple computation, we find that (22) is satisfied if and only if is integrable. On the other hand, (23) is satisfied if and only if is minimal. However, the integrability condition of holds due to Lemma 10, and by the hypothesis of the theorem, we have is minimal. Hence, the form is closed. It defines a canonical de Rham cohomology class .
Next, we prove that the cohomology class is nontrivial. Since is minimal and is integrable, then in this case, we need to show that is harmonic. By definition of and the similar argument for , we see that , that is, is closed, if is integrable and is minimal. This further proves that , that is, is coclosed. From , and is a closed submanifold, we deduce that is harmonic -form. Hence, the cohomology group is nontrivial if is minimal and is integrable.☐

4. Warped Product Bislant Submanifolds

Definition 12 (see [22]). Let and be two Riemannian manifolds and be a differentiable function on . Consider two projections on , and . The projection maps given by and for . Then, the warped product is the product manifold equipped with the Riemannian structure such that for any , where is the symbol for the tangent maps. The function is called the warping function of .

Example 13. A surface of revolution is a warped product manifold.

Example 14. The standard space-time models of the universe are warped products as the simplest models of neighbourhoods of stars and black holes.

Remark 15. In particular, a warped product manifold is said to be trivial if its warping function is constant. In such a case, we call the warped product manifold a Riemannian product manifold. If is a warped product manifold, then is totally geodesic and is totally umbilical submanifold of [22].
Let be a warped product manifold with a warping function . Then, for any and , where is the gradient of and and denote the Levi-Civita connections on and , respectively.

The definition of warped product bislant submanifolds in a nearly trans-Sasakian manifold is as follows.

Definition 16. A warped product of two slant submanifolds and of a nearly trans-Sasakian manifold is called a warped product bislant submanifold.

Remark 17. A warped product bislant submanifold is called proper if and are proper slant in . Otherwise, the warped product bislant submanifold is called nonproper.

For a warped product bislant submanifold in a nearly trans-Sasakian manifold such that , we have the following result.

Theorem 18. Let be a warped product bislant submanifold with bislant angles in a nearly trans-Sasakian manifold such that . If, for any and , holds, then one of the following cases must occur: (i) is a warped product pseudoslant submanifold such that is a totally real submanifold of (ii)If is nearly Sasakian manifold, that is, , then is a Riemannian product(iii)If , then

Proof. For any vector fields and , we have On the other hand, we have By adding (27) and (28), we get Interchanging by in (29), we find By subtracting (30) from (29) and by applying our assumption, we obtain For , we get From the last expression, any one of the following holds: if , then is constant, or if , then or . Thus, our assertions follow.
Now, we have the following theorem for a warped product bislant submanifold in a nearly trans-Sasakian manifold such that .☐

Theorem 19. Let be a warped product bislant submanifold with bislant angles in a nearly trans-Sasakian manifold such that . If, for any and , holds, then one of the following cases must occur: (i) is a warped product pseudoslant submanifold such that is a totally real submanifold of (ii) is a Riemannian product

Proof. For any vector fields and , we have On the other hand, we have By adding (34) and (35), we get Interchanging by in (36), we find By subtracting (37) from (36) and by applying our assumption, we obtain For , we get Therefore, either is constant or holds. Consequently, either is a Riemannian product or . In the latter case, is a warped product pseudoslant submanifold.☐

We give some nontrivial examples of warped product bislant submanifold of the form whose bislant angles and . Such warped product bislant submanifolds are called pseudoslant submanifolds.

Example 20. Let be the complex Euclidean space with its usual Kähler structure and the real global coordinates and be a warped product manifold between the product real line of and the complex space . Let be the Euclidean metric tensor of . An almost contact structure of is defined by such that On the other hand, we define a submanifold by immersion as follows: Therefore, it is easy to choose tangent bundle of which is spanned by the following: Thus, is a slant distribution with slant angle . Also, it is easy to verify that is a totally real distribution. Hence, the submanifold defined by is a bislant submanifold, which is tangent to the structure vector and whose bislant angles satisfy and . It is easy to check that the distributions and are integrable. Then, it can be verified that is a warped product bislant submanifold of with warping function , .

Example 21. We consider any submanifold in a nearly trans-Sasakian manifold The tangent bundle of is spanned by Furthermore, we have It is easy to check that is orthogonal to . Then, the proper slant and anti-invariant distributions of are respectively defined by with slant angle and . Also, is tangent to . Hence, defines a proper -dimensional pseudoslant submanifold (bislant submanifold with bislant angles ) in . It is easy to check that the distributions and are integrable.

Now, we assume that and are the integral manifolds of and , respectively. Then, it follows from Definition 12 and (44) that the induced metric tensor of is given by

where and are respectively the metric tensors of and . As a consequence, is a warped product pseudoslant submanifold of with a warping function, that is, such that is tangent to .

5. Doubly Warped Product Bislant Submanifolds

In general, doubly warped products can be considered as a generalization of warped products.

Definition 22 (see [23, 24]). Let and be Riemannian manifolds. A doubly warped product is a product manifold which is of the form with the metric where and are smooth maps. More precisely, if and are natural projections, the metric is defined by for any , where is the symbol for the tangent maps. The functions and are called the warping functions of .

Remark 23. If we assume (i)either or , but not both, then we obtain a warped product(ii)both and , then we have a product manifold(iii)neither nor is constant, then we have a nontrivial doubly warped product

For doubly warped product manifold with warping functions and , we have the following:

for any and .

Now, we define the notion of doubly warped product bislant submanifolds in nearly trans-Sasakian manifolds as follows.

Definition 24. The doubly warped product of two slant submanifolds, , is called the doubly warped product bislant submanifold of slant submanifolds and with slant angles and , respectively, of a nearly trans-Sasakian manifold with warping functions and if only depend on the points of and , respectively.

First we have the following theorem for doubly warped product submanifolds in nearly trans-Sasakian manifolds such that .

Theorem 25. Let be a doubly warped product submanifold in a nearly trans-Sasakian manifold , where and are Riemannian submanifolds of and . Then, is a warped product bislant submanifold of type if and only if for any and .

Proof. From Lemma 9, we get for any and . Applying (49), we derive Taking the inner product with , we obtain Using relation (10) in the above equation, we get Thus, from (54), we conclude that if and only if for any and . shows that is constant, that is, depends only on the points of . Thus, it follows that is a warped product bislant submanifold of type . This proves the theorem completely.☐

Secondly, we prove the following theorem for doubly warped product bislant submanifolds in nearly trans-Sasakian manifolds such that .

Theorem 26. Let be a doubly warped product bislant submanifold in a nearly trans-Sasakian manifold , where and are proper slant submanifolds with respect to and , respectively, and . Then, is a warped product bislant submanifold of type if and only if for any and .

Proof. For any vector fields and , we have Replacing by in the last relation, we obtain Thus, from (54), we conclude that if and only if for any and .
implies that is constant, that is, depends only on the points of . Hence, is a warped product bislant submanifold of type . This proves the theorem completely.☐

6. Conclusion

From Theorems 25 and 26, we conclude that there exist no doubly warped product bislant submanifolds in nearly trans-Sasakian manifolds, other than warped product bislant submanifolds, under some additional conditions.

7. Some Applications of Theorem 25 for Different Kinds of Ambient Manifolds

Let be a doubly warped product submanifold, where and are Riemannian submanifolds of and . The following corollaries are the immediate consequences of Theorem 25.

Corollary 27. There does not exist any doubly warped product submanifold in a nearly Sasakian manifold , other than the warped product bislant submanifold, if and only if (50) holds.

Corollary 28. There does not exist a doubly warped product submanifold in a nearly Kenmotsu manifold , other than the warped product bislant submanifold, if and only if (50) holds.

Corollary 29. There does not exist a doubly warped product submanifold in a nearly cosymplectic manifold , other than the warped product bislant submanifold, if and only if (50) holds.

8. Some Applications of Theorem 26 for Different Kinds of Ambient Manifolds

Let be a doubly warped product bislant submanifold, where and are proper slant submanifolds with respect to and , respectively, and . The following corollaries are the immediate consequences of Theorem 26.

Corollary 30. There is no doubly warped product bislant submanifold in a nearly Sasakian manifold , other than the warped product bislant submanifold, if and only if (56) holds.

Corollary 31. There is no doubly warped product bislant submanifold in a nearly Kenmotsu manifold , other than the warped product bislant submanifold, if and only if (56) holds.

Corollary 32. There is no doubly warped product bislant submanifold in a nearly cosymplectic manifold , other than the warped product bislant submanifold, if and only if (56) holds.

Data Availability

There is no data used for this manuscript.

Conflicts of Interest

The authors declare no competing of interest.

Authors’ Contributions

All authors have equal contribution and finalized.

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.

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Copyright © 2021 Ali H. Alkhaldi 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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