Mathematical Problems in EngineeringVolume 2011 (2011), Article ID 516238, 9 pagesdoi:10.1155/2011/516238

Research Article

Totally Umbilical Proper Slant and Hemislant Submanifolds of an LP-Cosymplectic Manifold

Siraj Uddin,¹ Meraj Ali Khan,² and Khushwant Singh³

¹Institute of Mathematical Sciences, Faculty of Science, University of Malaya, Kuala Lumpur 50603, Malaysia²Department of Mathematics, University of Tabouk, Tabouk, Saudi Arabia³School of Mathematics and Computer Applications, Thapar University, Patiala 147 004, India

Received 13 December 2010; Accepted 15 April 2011

Academic Editor: Alex Elias-Zuniga

Copyright © 2011 Siraj Uddin et al. Siraj Uddin et al. This is an open access paper 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.

In the present note, we study slant and hemislant submanifolds of an LP-cosymplectic manifold which are totally umbilical. We prove that every totally umbilical proper slant submanifold of an LP-cosymplectic manifold is either totally geodesic or if is not totally geodesic in then we derive a formula for slant angle of . Also, we obtain the integrability conditions of the distributions of a hemi-slant submanifold, and then we give a result on its classification.

1. Introduction

A manifold with Lorentzian paracontact metric structure satisfying is called an LP-cosymplectic manifold, where is the Levi-Civita connection corresponding to the Lorentzian metric on . The study of slant submanifolds was initiated by Chen [1]. Since then, many research papers have appeared in this field. Slant submanifolds are the natural generalization of both holomorphic and totally real submanifolds. Lotta [2] defined and studied these submanifolds in contact geometry. Later on, Cabrerizo et al. studied slant, semi-slant, and bislant submanifolds in contact geometry [3, 4]. In particular, totally umbilical proper slant submanifold of a Kaehler manifold has also been studied in [5]. Recently, Khan et al. [6] studied these submanifolds in the setting of Lorentzian paracontact manifolds.

The idea of hemi-slant submanifolds was introduced by Carriazo as a particular class of bislant submanifolds, and he called them antislant submanifolds [7]. Recently, these submanifolds are studied by Sahin for their warped products [8]. In this paper, we study slant and hemi-slant submanifolds of an LP-cosymplectic manifold. We prove that a totally umbilical proper slant submanifold is either totally geodesic in or if it is not totally geodesic, then the slant angle . Also, we define hemi-slant submanifolds of an LP-contact manifold. After we find integrability conditions of the distributions, we investigate a classification of totally umbilical hemi-slant submanifolds of an LP-cosymplectic manifold.

2. Preliminaries

Let be a -dimensional paracontact manifold with the Lorentzian paracontact metric structure , that is, is a tensor field, is a contravariant vector field, is a 1-form, and is a Lorentzian metric with signature on , satisfying [9], for all .

A Lorentzian paracontact metric structure on is called a Lorentzian para-cosymplectic structure if , where denotes the Levi-Civita connection with respect to . The manifold in this case is called a Lorentzian para-cosymplectic (in brief, an LP-cosymplectic) manifold [10]. From formula , it follows that .

Let be a submanifold of a Lorentzian almost paracontact manifold with Lorentzian almost paracontact structure . Let the induced metric on also be denoted by , then Gauss and Weingarten formulae are given by for any in and in , where is the Lie algebra of vector field in and is the set of all vector fields normal to . is the connection in the normal bundle, is the second fundamental form, and is the Weingarten endomorphism associated with . It is easy to see that

For any , we write where is the tangential component and is the normal component of . Similarly for , we write where is the tangential component and is the normal component of .

The covariant derivatives of the tensor fields , , and are defined as Moreover, for an LP-cosymplectic manifold, one has

A submanifold is said to be totally umbilical if where is the mean curvature vector. Furthermore, if for all , then is said to be totally geodesic, and if , then is minimal in .

A submanifold of a paracontact manifold is said to be a slant submanifold if for any and , the angle between and is constant. The constant angle is then called slant angle of . The tangent bundle of is decomposed as where the orthogonal complementary distribution of is known as the slant distribution on . If is invariant subspace of the normal bundle , then

Khan et al. [6] proved the following theorem for a slant submanifold of a Lorentzian paracontact manifold with slant angle .

Theorem 2.1. Let be a submanifold of an -contact manifold such that , then is slant submanifold if and only if there exists a constant such that Furthermore, if is slant angle of , then . Thus, one has the following consequences of formula (2.16): for any .

3. Totally Umbilical Proper Slant Submanifold

In this section, we consider as a totally umbilical proper slant submanifold of an LP-cosymplectic manifold . Such submanifolds we always consider tangent to the structure vector field .

Theorem 3.1. A nontrivial totally umbilical proper slant submanifold of an LP-cosymplectic manifold is either totally geodesic or if it is not totally geodesic in , then the slant angle , for any .

Proof. For any , (2.11) gives Taking the product with and using (2.9), we obtain Using (2.5) and the fact that is totally umbilical, the above equation takes the form Then, from the characteristic equation of LP-cosymplectic manifold, we obtain Thus, from (3.4), it follows that either or is trivial.Now, for an LP-cosymplectic manifold, one has, from (2.8), for any . From (2.3) and (2.6), we obtain Again using (2.3), (2.4), and (2.6), we get As is totally umbilical, then Taking the inner product with and using the fact that , we obtain Then from (2.2) and (2.13), we get Again, using (2.2) and the fact that , then is also lies in ; thus, we obtain Then, from (2.4), we derive Now, for any , one has Using the fact that as is an LP-cosymplectic manifold, we obtain Using (2.4), (2.6), and (2.7), we obtain Taking the product in (3.15) with for any and using the fact , the above equation gives Using (2.18), we obtain then, from (2.5) and (2.13), we get Thus, from (3.12) and (3.18), we derive Hence, (3.19) gives either or if , then the slant angle of is . This proves the theorem completely.

4. Hemislant Submanifolds

In the following section, we assume that is a hemi-slant submanifold of an LP-cosymplectic manifold such that the structure vector field tangent to . First, we define a hemi-slant submanifold, and then we obtain the integrability conditions of the involved distributions and in the definition of a hemi-slant submanifold of an LP-cosymplectic manifold .

Definition 4.1. A submanifold of an LP-contact manifold is said to be a hemi-slant submanifold if there exist two orthogonal complementary distributions and satisfying(i),(ii) is a slant distribution with slant angle ,(iii) is totally real that is, .

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

In the following, we obtain the integrability conditions of involved distributions in the definition of hemi-slant submanifold.

Proposition 4.2. Let be a hemi-slant submanifold of an LP-cosymplectic manifold , then the anti-invariant distribution is integrable if and only if for any .

Proof. For any , one has Using (2.8), we obtain Then, from (2.4), we derive As is an anti-invariant distribution, then the tangential part of (4.5) should be identically zero; hence, we obtain the required result.

Proposition 4.3. Let be a hemi-slant submanifold of an LP-cosymplectic manifold , then the invariant distribution is integrable if and only if for any and .

Proof. For any , one has Then, from (2.8) and the fact that is LP-cosymplectic, we obtain Using (2.6), we get Thus, from (2.3) and (2.4), we derive Taking the product in (4.10) with , for any , we obtain Thus, the assertion follows from (4.11) after using (2.2) and the fact that is tangential to .

Now, we consider as a totally umbilical hemi-slant submanifold of an LP-cosymplectic manifold . For any , one has Using this fact, if we take for any , then from (2.3) and (2.4), the above equation takes the form Thus, on using (2.6) and (2.7), we obtain Equating the tangential components, we get Taking the product with , we obtain Using (2.2), (2.5), and the fact that , for any , thus, the above equation takes the form As is totally umbilical, we derive Thus, (4.18) has a solution if either , that is, or or . Hence, we state the following theorem.

Theorem 4.4. Let be a totally umbilical hemi-slant submanifold of an LP-cosymplectic manifold , then at least one of the following statements is true: (i)the dimension of anti-invariant distribution is one, that is, ,(ii)the mean curvature vector ,(iii) is proper slant submanifold of .

Acknowledgments

The authors are thankful to the referee for his valuable suggestion and comments. The first author is supported by the research Grant no. RG117/10AFR (University Malaya).

References

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