Abstract and Applied Analysis

Volume 2012 (2012), Article ID 420890, 13 pages

doi:10.1155/2012/420890

Research Article

Warped Product Pseudo-Slant Submanifolds of a Nearly Cosymplectic Manifold

Siraj Uddin, Bernardine R. Wong, and A. A. Mustafa

Institute of Mathematical Sciences, Faculty of Science, University of Malaya, 50603 Kuala Lumpur, Malaysia

Received 17 March 2012; Accepted 23 June 2012

Academic Editor: Liming Ge

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.

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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