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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 752650, 8 pages
http://dx.doi.org/10.1155/2013/752650
Research Article

Neutral Slant Submanifolds of a Para-Kähler Manifold

Department of Mathematics, Dicle University, 21280 Diyarbakır, Turkey

Received 27 May 2013; Accepted 9 August 2013

Academic Editor: Jaume Giné

Copyright © 2013 Yılmaz Gündüzalp. 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 define and study both neutral slant and semineutral slant submanifolds of an almost para-Hermitian manifold and, in particular, of a para-Kähler manifold. We give characterization theorems for neutral slant and semi-neutral slant submanifolds. We also investigate the integrability conditions for the distributions involved in the definition of a semi-neutral slant submanifold when the ambient manifold is a para-Kähler manifold.

1. Introduction

The geometry of slant submanifolds was initiated by Chen, as a generalization of both holomorphic and totally real submanifolds in complex geometry [1, 2]. Since then, many mathematicians have studied these submanifolds. Slant submanifolds have been studied by many geometers in various manifolds [35]. In particular, Papaghiuc [6] introduced semislant submanifolds. Lotta [7, 8] defined and studied slant submanifolds in contact geometry. Cabrerizo et al. studied slant, semislant, and bislant submanifolds in contact geometry [9, 10]. Recently, Arslan et al. [11] studied these submanifolds in the setting of neutral Kähler manifolds.

In this paper we define and study both neutral slant and semineutral slant submanifolds of an almost para-Hermitian manifold and, in particular, of a para-Kähler manifold. The paper is organized as follows. In Section 2, we review some formulas and definitions for an almost para-Hermitian manifold and their submanifolds. In Section 3, we define neutral slant submanifolds for an almost para-Hermitian manifold and give theorem for a neutral slant submanifold. In the last section, we define and study semineutral slant submanifolds of an almost para-Hermitian manifold. We give theorems for a semineutral slant submanifold. In the last part of Section 4, we obtain that the distributions are integrable and their leaves are totally geodesic in semineutral slant submanifold under the condition . Finally, the paper contains some examples.

2. Preliminaries

An almost para-Hermitian manifold is a smooth manifold endowed with an almost paracomplex structure and a pseudo-Riemannian metric compatible in the sense that where is the module of differentiable vector fields on . It follows that the metric is neutral; that is, it has signature , and the eigenbundles are totally isotropic with respect to .

An almost para-Hermitian manifold is called a para-Kähler manifold if where is the Levi-Civita connection on [12, 13].

Let be an isometrically immersed submanifold of an almost para-Hermitian manifold . We denote the Levi-Civita connections on and by and , respectively. Then, the Gauss and Weingarten formulas are given by for any and , where is the connection in the normal bundle is the second fundamental form of , and is the shape operator. The second fundamental form and the shape operator are related by where the induced pseudo-Riemannian metric on is denoted by the same symbol .

Let us consider that is an immersed submanifold of an almost para-Hermitian manifold . For any and , we put where (resp., ) is tangential (resp., normal) part of and (resp., ) is tangential (resp., normal) part of . From (1) and (5), we have 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 submanifold if is identically zero; that is, , for any .

For any , by a direct calculation, we have

Similarly, for any , we have

Now, let be an immersed submanifold of an almost para-Kähler manifold . For any , from , taking into account (3), (5), and (6), then we have

Comparing the tangential and normal components of (11), respectively, we get where the covariant derivations of and are, respectively, defined by for any .

Let be a submanifold of a para-Hermitian manifold . A tangent vector is said to be spacelike (resp., timelike) if (resp., . If is a spacelike vector (resp., timelike), we have (resp., ) [11].

3. Neutral Slant Submanifolds of Almost Para-Hermitian Manifolds

In this section, we study neutral slant immersions of an almost para-Hermitian manifold . First, we present definition of a neutral slant submanifold of an almost para-Hermitian manifold following Chen’s [1] definition for a Hermitian manifold. Let be a semi-Riemannian manifold isometrically immersed in an almost para-Hermitian manifold . For each nonzero spacelike vector tangent to at , the angle , between and is called the Wirtinger angle of . Then, is said to be neutral slant if the angle is a constant, which is independent of the choice of and . The angle of a neutral slant immersion is called the slant angle of the immersion. Thus, the invariant and anti-invariant immersions are neutral slant immersions with slant angle and , respectively. A neutral slant immersion which is neither invariant nor anti-invariant is called a proper neutral slant immersion.

We note that our definition is quite different from Chen’s definition for slant submanifold [1], and the slant submanifold is given by Arslan et al. [11].

Next we give a useful characterization of neutral slant submanifolds in an almost para-Hermitian manifold.

Theorem 1. Let be a submanifold of a para-Hermitian manifold . Then,(i)is neutral slant if and only if there exists a constant such that . Furthermore, in this case, if is the slant angle of , it satisfies ;(ii) is a neutral slant submanifold if and only if there exists a constant such that . Furthermore, in this case, if is the slant angle of , it satisfies .

Proof. (i) Suppose that is a neutral slant submanifold. For any , we can write where is the slant angle. By using (7), (15), and (1), we get for all . Since is a neutral metric, from (16), we have
Let . Then it is obvious that .
Conversely, let us assume that there exists a constant such that is satisfied. From (7), (17), and (1), we get for all . Thus we have Since , then by using the last equation we obtain , which implies that is a constant and so is a neutral slant.
(ii) From (8) and (i), we have (ii).

Corollary 2. Let be a neutral slant submanifold of an almost para-Hermitian manifold with slant angle . Then, for any , we have

Proof. From Theorem 1(i) and (7), we get for any . On the other hand, from (1), (5), and (20), we obtain This completes the proof.

Now, we give some examples of the neutral slant submanifolds in almost para-Hermitian manifolds inspirited by Chen [1].

Note that given a semi-Euclidean space with coordinates on , we can naturally choose an almost paracomplex structure on as follows: where . Let be a semi-Euclidean space of signature with respect to the canonical basis .

Example 3. Consider a submanifold in given by
It is easy to see that is a neutral slant submanifold with the slant angle .

Example 4. Consider a submanifold in given by where and are constant. Then is a neutral slant submanifold with the slant angle .

Remark 5. Consider a neutral submanifold of an almost para-Hermitian manifold , in fact a neutral manifold , with is called a neutral slant submanifold if the Wirtinger angle between and is constant, for all a spacelike vector field and all . It is well defined, because that angle can be measured as usual, it the same angle between and , and they both are timelike vector fields.
In fact, if that conditions hold, it would be the same angle between and for a timelike vector, both and would be spacelike vector fields. This condition is equivalent to or , in fact it is equivalent to Theorem 1 condition .

4. Semineutral Slant Submanifolds of Almost Para-Hermitian Manifolds

Definition 6. Let be an almost para-Hermitian manifold with an almost paracomplex structure . A differentiable distribution on is called a neutral slant distribution if for each and each nonzero spacelike vector , the angle between and is a constant, that is, independent of the choice of and . In this case, we call the constant angle the slant angle of the distribution .
Let be an immersed submanifold of an almost para-Hermitian manifold and a differentiable distribution on . We denote the orthogonal distribution to on by . Then, for all , we write where and are orthogonal projections on and , respectively.
Next, we will give a sufficient and necessary condition for a distribution to be slant.

Theorem 7. Let be a submanifold of an almost para-Hermitian manifold and a differentiable distribution on . Then is a neutral slant distribution if and only if there exists a constant such that
Furthermore, in such case, if is the slant angle of then .

Proof. We suppose that is a neutral slant distribution on . Then, from (29), we have which implies that for any . By using (29), (32), and (1), we have
Since is a neutral metric, we obtain
If we put , then we have (30).
Conversely, let be a constant such that (30) is satisfied. Then, from (1) we have for any . Thus we get
On the other hand, since , then we obtain , which implies that is a constant and is a neutral slant distribution. This completes the proof.

Definition 8. is called a bineutral slant submanifold of an almost para-Hermitian manifold if there exist two orthogonal distributions and on such that (i) admits the orthogonal direct decomposition ;(ii) is a neutral slant distribution with slant angle for . Given a bineutral slant submanifold , we can write, for any , where denotes the component of in for any . In particular, if , then we obtain . If we define , then we have for any .
We note that semi-invariant submanifolds are particular cases of bineutral slant submanifolds with slant angles and .

Theorem 9. Let be a bineutral slant submanifold with angles . If , for any and , then is slant with angle .

Proof. Since , for any and , we have ; that is, . Similarly, for , we find. Then for any can be written as follows: such that and and , . Since , we get which gives assertion of the theorem.
Now, as a generalization of semi-invariant submanifolds, we can define semineutral slant submanifolds of an almost para-Hermitian manifold.

Definition 10. is called a semineutral slant submanifold of an almost para-Hermitian manifold if there exist two orthogonal distributions and on such that (i) admits the orthogonal direct sum ,(ii)the distribution is invariant; that is, ,(iii)the distribution is neutral slant with slant angle . In this case, we call the slant angle of submanifold .
It is obvious that the invariant and anti-invariant distributions of a semineutral slant submanifold are neutral slant distributions with the slant angles and , respectively.
Now, let be a semineutral slant submanifold of an almost para-Hermitian manifold . Let be a semislant submanifold with and . Then we have the following particular cases.(i)If , then is an invariant submanifold. (ii)If and , then is an anti-invariant submanifold. (iii)If and , then is a proper neutral slant submanifold with slant angle .(iv)If and , then is a proper semineutral slant submanifold.
We now give an example of bineutral slant submanifolds.

Example 11. Let , , , where and are constant. Then, is a 4-dimensional submanifold of .
By defining we have that and are neutral slant with slant angles and , respectively. Thus is a bineutral slant submanifold of .
Now, let be a semineutral slant submanifold of an almost para-Hermitian manifold and , , denoting the orthogonal projections on , . Then, for any , applying to (37), we have where
From (41) and (42), we have
By putting in (20) and in (21), we get respectively.
We give a characterization for the semineutral slant submanifolds of an almost para-Hermitian manifold.

Theorem 12. Let be an immersed submanifold of an almost para-Hermitian manifold . Then is a semineutral slant submanifold if and only if there exists a constant such that is a distribution. Furthermore, in this case, , where denotes slant angle of .

Proof. Let be a semineutral slant submanifold and , where is invariant and is neutral slant. We put , where denotes slant angle of . For any , if , then we have which implies that . But this is a contradiction that . Therefore we obtain . On the other hand, since is a neutral slant distribution, it follows from Theorem 7 that , which means that . Thus is a distribution.
Conversely, we can consider the orthogonal direct decomposition . It is obvious that , from which we have for any and . Hence is an invariant distribution. Finally, Theorem 7 imply that is a neutral slant distribution, with slant angle satisfying .

We can easily present some examples of the above situation.

Example 13. , , , defines a four-dimensional proper semineutral slant submanifold , with slant angle , in .
Moreover, it is easy to see that from a local orthogonal frame of . Then, we can define and .

Example 14. , , defines a four-dimensional proper semineutral slant submanifold , with slant angle , in , where and are constant.
Moreover it is easy to see that from a local orthogonal frame of . Then we can define and .
Then, it is easy to show that all conditions of Theorem 12 are satisfied.

Next, we will give useful characterizations for integrable conditions of distributions.

Theorem 15. Let be a semineutral slant submanifold of a para-Kähler manifold . Then we have the following: (a)the distribution is integrable if and only if for any ,(b)the distribution is integrable if and only if for any .

Proof. From (2), we get for all .(a) By using Gauss-Weingarten formulas, (5), and (6) in (50), we have
for any . From (41) and (51), we obtain
By equating the normal part of the last equation, we have
If we change the role of and in (53), we write
Since is symmetric, from (53) and (54), we get
Assume that the distribution is integrable. Then, for any , we have which implies that . Thus from (55) we obtain (48).
Conversely, if (48) is satisfied, then from (55), we have , for any , which implies that . Then we conclude that .(b) From (41) and Gauss-Weingarten formulae, we have for all . On the other hand, by using (5) and (6), we write
By using (56) and (57) in (50), we get for any . Since is symmetric we obtain which gives
Let the distribution be integrable. Then , for all , and hence from (60), the equation (49) is obvious.
Conversely, if (49) is satisfied then ; that is, for any . This completes the proof.

Definition 16. Let be a semi-invariant submanifold of an almost para-Hermitian manifold . Then we say that(i) is -geodesic if (ii) is -geodesic if (iii) is mixed geodesic if

Lemma 17. Let be a mixed-geodesic semineutral slant submanifold of a para-Kähler manifold . Then the distribution is integrable if and only if for any and .

Proof. Since is a mixed-geodesic submanifold, from (4) we find that has no component on . By using (4) and (1), we obtain
Thus, we can write for all . Taking into account Theorem 15(a) and the last equation, the proof is completed.

Theorem 18. Let be a semineutral slant submanifold of a para-Kähler manifold . If , then is a mixed-geodesic submanifold. Furthermore,(a)if , then either is a -geodesic submanifold or is an eigenvector of with the eigenvalue 1,(b)if , then either is a -geodesic submanifold or is an eigenvector of with the eigenvalue .

Proof. If , then from (13) we get , for all . Since is an invariant and is a neutral slant distribution with the slant angle , we obtain for any . By using (67) we get which implies that , for any , , that is, is mixed-geodesic. Similarly, we obtain for all , and for all . This completes the proof.

Proposition 19. Let be a semineutral slant submanifold of a para-Kähler manifold . Then if and only if for all .

Proof. From (13) and (1),we get for any . Taking into account (4), we get which completes the proof.

Proposition 20. Let be a semineutral slant submanifold of a para-Kähler manifold . Then if and only if for all .

Proof. From (12) and (1) we have for any . Since the is symmetric then we obtain from the last equation This completes the proof.

Proposition 21. Let be a semineutral slant submanifold of a para-Kähler manifold . If   then the distributions are integrable and their leaves are totally geodesic in .

Proof. Since , then from (12) we obtain for any and . By using (1) and (5), we have where and . Thus one can easily see that
Since is a para-Kähler manifold, taking into account (78), we get which gives ; that is, . Now, let and . Since is orthogonal to , the induced metric on is the neutral metric, and it is easy to see that . Hence the proof is complete.

Acknowledgment

The author is grateful to the referees for their valuable comments and suggestions.

References

  1. B.-Y. Chen, “Slant immersions,” Bulletin of the Australian Mathematical Society, vol. 41, no. 1, pp. 135–147, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. B.-Y. Chen, Geometry of Slant Submanifolds, Katholieke Universiteit Leuven, Louvain, Belgium, 1990. View at Zentralblatt MATH · View at MathSciNet
  3. M. Atçeken, “Semi-slant submanifolds of an almost paracontact metric manifold,” Canadian Mathematical Bulletin, vol. 53, no. 2, pp. 206–217, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  4. H. Li and X. Liu, “Semineutral Slant Submanifolds of Almost Para-Hermitian Manifolds,” Georgian Mathematical Journal, vol. 12, no. 2, pp. 273–282, 2005. View at Zentralblatt MATH · View at MathSciNet
  5. B. Şahin and S. Keleş, “Slant submanifolds of Kaehler product manifolds,” Turkish Journal of Mathematics, vol. 31, no. 1, pp. 65–77, 2007. View at Zentralblatt MATH · View at MathSciNet
  6. N. Papaghiuc, “Semi-slant submanifolds of a Kaehlerian manifold,” Analele Ştiinţifice ale Universităţii Al. I. Cuza din Iaşi. Serie Nouă. Matematică, vol. 40, no. 1, pp. 55–61, 1994. View at Zentralblatt MATH · View at MathSciNet
  7. A. Lotta, “Slant submanifolds in contact geometry,” Bulletin Mathématique de la Société des Sciences Mathématiques de Roumanie, vol. 39, pp. 183–198, 1996.
  8. A. Lotta, “Three-dimensional slant submanifolds of K-contact manifolds,” Balkan Journal of Geometry and its Applications, vol. 3, no. 1, pp. 37–51, 1998. View at MathSciNet
  9. J. L. Cabrerizo, A. Carriazo, L. M. Fernández, and M. Fernández, “Slant submanifolds in Sasakian manifolds,” Glasgow Mathematical Journal, vol. 42, no. 1, pp. 125–138, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. J. L. Cabrerizo, A. Carriazo, L. M. Fernández, and M. Fernández, “Semi-slant submanifolds of a Sasakian manifold,” Geometriae Dedicata, vol. 78, no. 2, pp. 183–199, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. K. Arslan, A. Carriazo, B.-Y. Chen, and C. Murathan, “On slant submanifolds of neutral Kaehler manifolds,” Taiwanese Journal of Mathematics, vol. 14, no. 2, pp. 561–584, 2010. View at Zentralblatt MATH · View at MathSciNet
  12. V. Cruceanu, P. Fortuny, and P. M. Gadea, “A survey on paracomplex geometry,” The Rocky Mountain Journal of Mathematics, vol. 26, no. 1, pp. 83–115, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. S. Ivanov and S. Zamkovoy, “Parahermitian and paraquaternionic manifolds,” Differential Geometry and Its Applications, vol. 23, no. 2, pp. 205–234, 2005. View at Publisher · View at Google Scholar · View at MathSciNet