Research Article | Open Access

Siraj Uddin, Bernardine R. Wong, A. A. Mustafa, "Warped Product Pseudo-Slant Submanifolds of a Nearly Cosymplectic Manifold", *Abstract and Applied Analysis*, vol. 2012, Article ID 420890, 13 pages, 2012. https://doi.org/10.1155/2012/420890

# Warped Product Pseudo-Slant Submanifolds of a Nearly Cosymplectic Manifold

**Academic Editor:**Liming Ge

#### Abstract

We study warped product pseudo-slant submanifolds of a nearly cosymplectic manifold. We obtain some characterization results on the existence or nonexistence of warped product pseudo-slant submanifolds of a nearly cosymplectic manifold in terms of the canonical structures and .

#### 1. Introduction

To study the manifolds with negative curvature, Bishop and O'Neill [1] introduced the notion of warped product manifolds by homothetically warping the product metric of a product manifold onto the fibers for each . Later on, the geometrical aspect of these manifolds has been studied by many researchers (cf. [2–4]). Pseudo-slant submanifolds were introduced by Carriazo [5] as a special case of bislant submanifolds.

Almost contact manifolds with Killing structure tensors were defined in [6] as nearly cosymplectic manifolds, and it was shown that the normal nearly cosymplectic manifolds are cosymplectic (see also [7]). Later on, Blair and Showers [8] studied nearly cosymplectic structure on a Riemannian manifold with closed from the topological viewpoint.

Recently, Sahin [9] studied the warped product hemislant (pseudo-slant) submanifolds of Kaehler manifolds. He proved that the warped product submanifolds of the type of a Kaehler manifold do not exist and obtained some characterization results on the existence of warped product submanifold , where and are totally real and proper slant submanifolds of a Kaehler manifold , respectively. After that, we have extended this study to the more general setting of nearly Kaehler manifolds [4]. The warped product semi-invariant submanifolds of a nearly cosymplectic manifold had been studied in [10].

In this paper, we study warped product pseudo-slant submanifolds of a nearly cosymplectic manifold. We obtain some characterization results of warped product submanifolds of the types and in terms of the canonical structures and , where and are anti-invariant and proper slant submanifolds of a nearly cosymplectic manifold , respectively.

#### 2. Preliminaries

A dimensional manifold is said to have an *almost contact structure* if there exist on a tensor field of type , a vector field , and a form satisfying [8]
There always exists a Riemannian metric on an almost contact manifold satisfying the following compatibility condition:
where and are vector fields on [8].

An almost contact structure is said to be * normal* if the almost complex structure on the product manifold given by
where is a -function on has no torsion, that is, is integrable, the condition for normality in terms of and is on , where is the Nijenhuis tensor of . Finally the *fundamental 2-form * is defined by .

An almost contact metric structure is said to be *cosymplectic*, if it is normal and both and are closed [8]. The structure is said to be *nearly cosymplectic* if is Killing, that is, if
for any , where is the tangent bundle of and denotes the Riemannian connection of the metric . Equation (2.4) is equivalent to , for each . The structure is said to be *closely cosymplectic* if is Killing and is closed. It is well known that an almost contact metric manifold is *cosymplectic* if and only if vanishes identically, that is, and .

Proposition 2.1 (see [8]). *On a nearly cosymplectic manifold the vector field is Killing.*

From the above proposition, one has , for any vector field tangent to , where is a nearly cosymplectic manifold.

Let be submanifold of an almost contact metric manifold with induced metric and if and are the induced connections on the tangent bundle and the normal bundle of , respectively, then Gauss and Weingarten formulae are given by for each and , where and are the second fundamental form and the shape operator (corresponding to the normal vector field ), respectively, for the immersion of into . They are related as where denotes the Riemannian metric on as well as induced on .

For any , one writes where is the tangential component and is the normal component of .

Similarly for any , one writes where is the tangential component and is the normal component of .

Now, denote by and the tangential and normal parts of , that is, for all . Making use of (2.8), (2.10), and the Gauss and Weingarten formulae, the following equations may easily be obtained: Similarly, for any , denoting tangential and normal parts of by and , respectively, one obtains where the covariant derivatives of , and are defined by for all and .

It is straightforward to verify the following properties of and , which one enlists here for later use

(*p*_{1})
(i) , (ii) ,(*p*_{2})
(i) , (ii) ,(*p*_{3})
(i) , (ii) ,(*p*_{4}) ,

for all and .

On a submanifold of a nearly cosymplectic manifold, by (2.4) and (2.10), one has

for any .

The submanifold is said to be *invariant* if is identically zero, that is, for any . On the other hand, is said to be *anti-invariant* if is identically zero, that is, , for any .

One will always consider to be tangent to the submanifold . There is another class of submanifolds that is called the * slant submanifold*. For each nonzero vector tangent to at any , such that is not proportional to , one denotes by , the angle between and is called the slant angle. If the slant angle is constant for all and , then M is said to be a slant submanifold [11]. Obviously, if , then is an invariant submanifold and if = , then is an anti-invariant submanifold. A slant submanifold is said to be *proper slant* if it is neither invariant nor anti-invariant.

One recalls the following result for a slant submanifold.

Theorem 2.2 (see [11]). *Let M be a submanifold of an almost contact metric manifold , such that . Then M is slant if and only if there exists a constant such that
**
Furthermore, if is slant angle, then .*

The following relations are straightforward consequence of (2.18): for all .

A submanifold of an almost contact manifold is said to be a *pseudo-slant submanifold* if there exist two orthogonal complementary distributions and satisfying: (i), (ii) is a slant distribution with slant angle ,(iii) is anti-invariant that is, .

A pseudo-slant submanifold of an almost contact manifold is * mixed geodesic* if
for any and .

If is the invariant subspace of the normal bundle , then in the case of pseudo-slant submanifold, the normal bundle can be decomposed as follows:

#### 3. Warped Product Pseudo-Slant Submanifolds

Bishop and O'Neill [1] introduced the notion of warped product manifolds. These manifolds are the natural generalizations of Riemannian product manifolds. They defined these manifolds as follows Let and be two Riemannian manifolds and , a positive differentiable function on . The warped product of and is the Riemannian manifold , where
A warped product manifold is said to be *trivial* if the warping function is constant. We recall the following general formula on a warped product manifold [1]:
where is tangential to and is tangential to .

Let be a warped product manifold. This means that is totally geodesic and is a totally umbilical submanifold of , respectively [1].

Throughout this section, we consider the warped product pseudo-slant submanifolds which are either in the form or in a nearly cosymplectic manifold , where and are proper slant and anti-invariant submanifolds of a nearly cosymplectic manifold , respectively. On a warped product submanifold of a nearly cosymplectic manifold , we have the following result.

Theorem 3.1 (see [10]). * A warped product submanifold of a nearly cosymplectic manifold is an usual Riemannian product if the structure vector field is tangential to , where and are the Riemannian submanifolds of .*

Now, one considers the warped product pseudo-slant submanifolds in the form of a nearly cosymplectic manifold . If one considers the structure vector field then by Theorem 3.1, the warping function is constant and hence one will considers .

Proposition 3.2. *Let be a warped product pseudo-slant submanifold of a nearly cosymplectic manifold . Then,
**
for any and , where and are proper slant and anti-invariant submanifolds of , respectively.*

*Proof. * For any and , by (2.8), we have
Using (2.5), (2.6), and the covariant derivative property of , we obtain
Then from (2.2), (2.4), and the fact that is a Killing vector field on , thus we obtain
Using the property of , we get
Then by (2.5) and (3.2), we derive
Interchanging by in (3.8) and using (2.18), (2.19), and the fact that , we obtain
Thus, the result follows from (3.8) and (3.9).

Proposition 3.3. *Let be a warped product pseudo-slant submanifold of a nearly cosymplectic manifold . Then,
**
for any and , where and are proper slant and anti-invariant submanifolds of , respectively.*

* Proof. *For any and by (2.14), we have
Using (2.20), (2.5), and the fact that is killing vector field, we obtain
Then from (3.2), we derive
Now, from (3.8) and (3.13), we obtain
Interchanging by in (3.14) and then using (2.18), (2.19), and the fact that , we get
From (3.14) and (3.15), we arrive at
Hence, the result is proved.

Lemma 3.4. *Let be a warped product pseudo-slant submanifold of a nearly cosymplectic manifold . Then,
**
for any and , where and are proper slant and anti-invariant submanifolds of , respectively.*

*Proof. *For any and by (2.5), we have
Then from (2.8), we derive
From the covariant derivative property of and (2.5), we obtain
By (2.2), (2.10), and (3.2), we derive
Using (2.5), (2.8), (2.17), (2.19) and the fact that , we get
Thus, by property , (2.18), and (3.2) and the fact that , we obtain
Hence, the above equation takes the form
which proves our assertion.

Theorem 3.5. * Let be a warped product submanifold of a nearly cosymplectic manifold . Then is Riemannian product of and if and only if , for any , where and are proper slant and anti-invariant submanifolds of , respectively.*

*Proof. *If the structure vector field , then, by Theorem 3.1, is Riemannian product of and . Now, we consider , then for any and from (2.5), we have
Then by (2.2), we get
Using the covariant derivative formula of , we derive
Then from (2.10) and the property of , we obtain
Thus by (2.5), (2.8), and (2.17)(a), we arrive at
Using (3.2) and then (2.19) and the fact that , we get
By property , we derive
Interchanging by in (3.30) and then using (2.18), (2.19), and the fact that , we obtain
Using the property and then (2.17), we arrive at

Then from (3.30) and (3.33), we obtain

Thus, by Lemma 3.4, we conclude that
Since is proper slant, thus we get , if and only if lies in for all and . This proves the theorem completely.

Now, we discuss the other case, that is, the warped product submanifold of a nearly cosymplectic manifold . In this case also, if the structure vector filed then the warping function is constant (by Theorem 3.1), thus we consider .

Proposition 3.6. *Let be a warped product pseudo-slant submanifold of a nearly cosymplectic manifold . Then,
**
for any and , where and are proper slant and anti-invariant submanifolds of , respectively.*

*Proof. * For any and , by (2.2) we have
Using the property of the connection and the fact that is a Killing vector field, then, from (2.5), we obtain
Thus by (3.2) and the covariant derivative formula of , we derive
Then form (2.6), (2.8), (2.10), and by the orthogonality of two distributions, we get
Thus, on using (2.7) and (2.17)(b), the above equation takes the form
Now, for any and from (2.14), we have
Using (3.2), we obtain
By orthogonality of two normal distributions, we get
Then, from (3.41) and (3.44), we obtain
Interchanging by in (3.45) and using (2.18) and the fact that , for any on a nearly cosymplectic manifold , hence we get
Using property and (2.17), we derive
Again, by property , we obtain
Thus, the result follows from (3.45) and (3.48).

Theorem 3.7. *Let be a warped product submanifold of a nearly cosymplectic manifold . Then is Riemannian product of and if and only if
**
for any and , where and are proper slant and anti-invariant submanifolds of , respectively.*

*Proof. * If , then by Theorem 3.1, is constant on . Now, we consider . In this case, for any and by (2.5), we have
Using (2.2), we get
Thus, on using the covariant derivative property of , we obtain
Then from (2.8) and (2.10), we get
Using property and the property of the connection , we derive
As we have from (2.4) and (2.10), then by (2.18) the above equation reduced to
Since is a Killing vector field on , then by (2.5), (3.2), and the property of the connection , the above equation takes the form
Interchanging by in (3.56) and using (2.18), we obtain
Since , for nearly cosymplectic, then the above equation reduces to
Thus, from (3.58), we obtain if and only if . This proves the theorem completely.

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#### Copyright

Copyright © 2012 Siraj Uddin 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.