The Scientific World Journal

Volume 2014 (2014), Article ID 696231, 12 pages

http://dx.doi.org/10.1155/2014/696231

Research Article

On Lightlike Geometry of Para-Sasakian Manifolds

Bilal Eftal Acet,¹ Selcen Yüksel Perktaş,¹ and Erol Kılıç²

¹Faculty of Arts and Sciences, Department of Mathematics, Adıyaman University, 02040 Adıyaman, Turkey

²Faculty of Arts and Sciences, Department of Mathematics, İnönü University, 44280 Malatya, Turkey

Received 4 February 2014; Accepted 3 April 2014; Published 28 April 2014

Academic Editor: Michele Zappalorto

Copyright © 2014 Bilal Eftal Acet 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 lightlike hypersurfaces of para-Sasakian manifolds tangent to the characteristic vector field. In particular, we define invariant lightlike hypersurfaces and screen semi-invariant lightlike hypersurfaces, respectively, and give examples. Integrability conditions for the distributions on a screen semi-invariant lightlike hypersurface of para-Sasakian manifolds are investigated. We obtain a para-Sasakian structure on the leaves of an integrable distribution of a screen semi-invariant lightlike hypersurface.

1. Introduction

It is well known that the main difference between the geometry of submanifolds in Riemannian manifolds and in semi-Riemannian manifolds is that in the latter case the induced metric tensor field by the semi-Riemannian metric on the ambient space is not necessarily nondegenerate. If the induced metric tensor field is degenerate the classical theory of Riemannian submanifolds fails since the normal bundle and the tangent bundle of the submanifold have a nonzero intersection. In particular, from the point of physics lightlike hypersurfaces are important as they are models of various types of horizons, such as Killing, dynamical and conformal horizons, studied in general relativity.

Lightlike submanifolds of semi-Riemannian manifolds were introduced by Duggal and Bejancu in [1]. Since then many authors studied lightlike hypersurfaces of semi-Riemannian manifolds and especially of indefinite Sasakian manifolds (for differential geometry of lightlike submanifolds we refer to the book [2]).

The study of paracontact geometry was initiated by Kaneyuki and Konzai in [3]. The authors defined almost paracontact structure on a pseudo-Riemannian manifold of dimension and constructed the almost paracomplex structure on . Recently, Zamkovoy [4] studied paracontact metric manifolds and some remarkable subclasses like para-Sasakian manifolds. In particular, in the recent years, many authors [5–9] have pointed out the importance of paracontact geometry and, in particular, of para-Sasakian geometry, by several papers giving the relationships with the theory of para-Kähler manifolds and its role in pseudo-Riemannian geometry and mathematical physics.

These circumstances motivated us to initiate the study of lightlike geometry of submanifolds in almost paracontact metric manifolds. As a first step, in the present paper, we study the lightlike hypersurfaces of almost paracontact metric manifolds. We introduce the invariant lightlike hypersurfaces and screen semi-invariant lightlike hypersurfaces of almost paracontact metric manifolds, respectively, and give examples. Moreover, integrability conditions for the distributions involved in the definition of a screen semi-invariant lightlike hypersurface are investigated in case of the ambient manifold being para-Sasakian manifold.

2. Preliminaries

2.1. Almost Paracontact Metric Manifolds

A differentiable manifold of dimension is called almost paracontact manifold with the almost paracontact structure if it admits a tensor field of type , a vector field , and a -form satisfying the following conditions [3]:

where denotes the identity transformation. Moreover, the tensor field induces an almost paracomplex structure on the paracontact distribution ; that is, the eigen distributions corresponding to the eigenvalues of are both -dimensional.

If a -dimensional almost paracontact manifold with an almost paracontact structure admits a pseudo-Riemannian metric such that [4]

then we say that is an almost paracontact metric manifold with an almost paracontact metric structure and such metric is called compatible metric. Any compatible metric is necessarily of signature .

From (5) it can be easily seen that [4]

for any . The fundamental -form of is defined by

An almost paracontact metric structure becomes a paracontact metric structure if , for all vector fields , where .

For a -dimensional manifold with an almost paracontact metric structure one can also construct a local orthonormal basis which is called -basis [4].

An almost paracontact metric structure is a para-Sasakian manifold if and only if [4]

where and is a Levi-Civita connection on .

From (9), it can be seen that

Example 1. Let be the -dimensional real number space with standard coordinate system . Defining

where , the set is an almost paracontact metric structure on .

2.2. Lightlike Hypersurfaces

In this section, we recall some basic results about lightlike hypersurfaces of a semi-Riemannian manifold [1].

Let be a -dimensional semi-Riemannian manifold with index    and a hypersurface of . Assume that the induced metric on the hypersurface is degenerate on . Then there exists a vector field on such that

The radical space [10] of is defined by

whose dimension is called the nullity degree of and is called a lightlike hypersurface of . Since is degenerate and any null vector is perpendicular to itself, is also degenerate and

For a lightlike hypersurface , implies that

We call the radical distribution and it is spanned by the null vector field .

Consider complementary vector bundle of in . This means that

where denotes the orthogonal direct sum. is called the screen distribution on . Since the screen distribution is nondegenerate, there exists a complementary orthogonal vector subbundle to in which is called screen transversal subbundle; that is,

The rank of is 2.

Theorem 2 (see [1]). Let be a lightlike hypersurface of an almost paracontact manifold . Then there exists a unique rank one vector subbundle of  , with base space , such that, for any nonzero section of on a coordinate neighborhood , there exists a unique section of on satisfying

for all . is called the lightlike transversal vector bundle of with respect to .

One can consider the following decompositions:

Let be the Levi-Civita connection on . Using (20) we deduce

for any and . Then and are called the induced connection on and , respectively, and, as in the classical theory of Riemannian hypersurfaces, and are called the second fundamental form and the shape operator, respectively. The above equations are cited as the Gauss and Weingarten equation, respectively [1].

Locally, let , , and be as in Theorem 2. Then, for any , putting

we can write

is called the local second fundamental form of , because it determines on . Moreover, is degenerate and

for any .

The decomposition (16) allows to define a canonical projection . For each , we may write

where is a 1-form given by

From (24), for all , we get

which implies that the induced connection is a nonmetric connection on .

Then for any and we can write

where and are linear connections on the bundles and , respectively. Further, and are called the second fundamental form and the shape operator of the screen distribution, respectively. Locally, let be a coordinate neighborhood of and , sections on , as in Theorem 2. Then, putting , for any , one has

and, locally on , (30) and (31) become

The local second fundamental forms and , respectively, of and on are related to their shape operators by

Furthermore, one has , . (For more details we refer to [1, 2].)

3. Lightlike Hypersurfaces of Para-Sasakian Manifolds

Let be a -dimensional para-Sasakian manifold and a lightlike hypersurface of such that the structure vector field is tangent to . For local sections and of and , respectively, in view of (7) we have

From (6) it is easy to see that and are lightlike vector fields and

Now, for any , we write

where and

Proposition 3. Let be a -dimensional para-Sasakian manifold and a lightlike hypersurface of such that the structure vector field is tangent to . Then one has

where is any local section of and is any local section of .

Proof. From (10) and (18) we have

which gives (40) by virtue of (22).

Remark 4. From (6) we get , which implies that there is no component of in and so . Moreover, (40) implies that there may be a component of in . Thus, in view of (27), we observe that

Proposition 5. Let be a -dimensional para-Sasakian manifold and a lightlike hypersurface of such that the structure vector field is tangent to . Then one has

for any .

Proof. By using (38) and (39) we obtain

Hence in view of (6) we get (43). From (38) we have

Thus, by using the last equation above and (5), we complete the proof.

Corollary 6. Let be a lightlike hypersurface of a para-Sasakian manifold such that the structure vector field is tangent to . Then, for all , one has .

Proposition 7. Let be a lightlike hypersurface of a -dimensional para-Sasakian manifold such that the structure vector field is tangent to . Then, for any , one has

Proof. From (1) and (38), we get (47). Next, by using (10), (24), and (38), we have

Then by equating the tangential and transversal parts in the previous equation we get (48) and (49), respectively.

4. Invariant Lightlike Hypersurfaces of Almost Paracontact Metric Manifolds

We begin with the following.

Definition 8. Let be a -dimensional para-Sasakian manifold and a lightlike hypersurface of such that the structure vector field is tangent to . If , then will be called an invariant lightlike hypersurface of .

Example 9. Let be the -dimensional real number space with a coordinate system . Define a frame of vector fields on given by Weƚyczko [11]:

Defining

the set is an almost paracontact metric structure on with index . Consider a hypersurface of given by

It is easy to check that is a lightlike hypersurface and

Then the lightlike transversal vector bundle is spanned by

It follows that corresponding screen distribution is spanned by

We easily check that

which gives . Thus is an invariant lightlike hypersurface of .

Now we give a characterization of an invariant lightlike hypersurface.

Theorem 10. Let be a -dimensional para-Sasakian manifold and a lightlike hypersurface of such that the structure vector field is tangent to . Then is an invariant lightlike hypersurface of if and only if

Proof. Let be an invariant lightlike hypersurface of . From (27) and (42), for any , we get ; that is, there is no component of in . Moreover, it is obvious from (6) that has no component in and so .

On the other hand, for any local section of , we can write

By using the previous equation we have , for any , which implies that has no component in . Since , then it is also seen that there is no component of in . Hence .

Conversely, let and . For any we have

which implies that has no component in . Similarly, we get

thus there is no component of in . The proof is completed.

Corollary 11. Let be a -dimensional para-Sasakian manifold and a lightlike hypersurface of such that the structure vector field is tangent to . Then is an invariant lightlike hypersurface of if and only if

Theorem 12. Let be a -dimensional almost paracontact metric manifold and an invariant lightlike hypersurface of . Then is an almost paracontact metric manifold.

Proof. Let be an invariant lightlike hypersurface of . For any , from (38), we get

Using (1) and (63), we have

Also from (63), it follows that

Next, in view of (64) and (65) one can easily see that

Moreover, from (44), we have

From (64)-(67) we complete the proof.

Proposition 13. Let be an invariant lightlike hypersurface of a para-Sasakian manifold . Then we have

for any .

Proof. Since , using (10), we get

From (25), we have the assertion of the proposition.

Theorem 14. An invariant lightlike hypersurface of a para-Sasakian manifold is always para-Sasakian. Moreover,

for any .

Proof. We have

which in view of (9) gives

Equating tangential parts in (73) provides

In view of (74) and Theorem 12, we see that is a para-Sasakian manifold. Equating transversal parts in (73) yields (70). Next, using (9) and (25), we have

In the last equation, if we equate the tangential parts, we get (71).

This completes the proof.

Remark 15. It is well known that, if there exists a lightlike hypersurface in an indefinite Sasakian manifold, then the dimension of the indefinite Sasakian manifold must be equal or greater than . But in a paracontact metric manifold there is not such a restriction in the dimension of the ambient manifold for the existence of lightlike hypersurfaces.

Let be a -dimensional almost paracontact metric manifold and a lightlike surface of such that the structure vector field is tangent to . Since is a nonnull vector field, it belongs to the screen distribution . Thus, is a quasi-orthonormal frame of . Also with respect to the quasi-orthonormal frame of we can write

where , , , , , and . Thus we have

Hence, a lightlike surface of , tangent to the structure vector field , is always an invariant lightlike surface.

Example 16. Let be a -dimensional almost paracontact metric manifold with the structure given in Example 1. Consider a surface of given by . It is easy to check that is a lightlike surface and and are given by

respectively. It follows that corresponding screen distribution is spanned by

Then and , which imply that is an invariant lightlike surface.

Example 17. Let be the standard flat para-Kähler structure on ,

for , where are the Cartesian coordinates on . Consider the hypersurface in given by the equation

Let be the normal vector field of . Then .

Define a vector field , a tensor field of type , a 1-form , and a pseudo-Riemannian metric on by assuming

Then we get an almost paracontact metric structure on . Moreover, this structure is para-Sasakian [11]. Now cut by the hyperplane and obtain a lightlike surface of with

where , . It follows that the lightlike transversal bundle is spanned by

Furthermore,

Thus is an invariant surface of .

5. Screen Semi-Invariant Lightlike Hypersurfaces of Almost Paracontact Metric Manifolds

If is local section of , one has ; therefore and we get a 1-dimensional distribution on .

Definition 18. Let be a -dimensional almost paracontact metric manifold and a lightlike hypersurface of . If

then will be called a screen semi-invariant lightlike hypersurface of .

Example 19. Let be a -dimensional almost paracontact metric manifold with the structure given in Example 1. Consider a hypersurface of given by

Then the tangent bundle of is spanned by

The radical distribution and the lightlike transversal bundle are given by

It follows that the screen distribution is spanned by , where

Furthermore,

Thus, is a screen semi-invariant lightlike hypersurface of .

From now on, we will write to denote a screen semi-invariant lightlike hypersurface, together with the choices of a fixed nonzero section of , a fixed screen distribution , , and as in Theorem 2.

Since is a screen semi-invariant lightlike hypersurface, then we have and

which imply that is orthogonal to by virtue of (19). Also, from (5), we obtain

Therefore, is a nondegenerate vector subbundle of of rank 2.

In the following, being and nondegenerate, we can define the unique nondegenerate distribution such that [2]

Then and is invariant under ; that is, . Moreover, from (16), (20), and (94) we write

For an almost paracontact metric manifold , we construct a useful local orthonormal basis. Let be a coordinate neighborhood on and any unit vector field on orthogonal to . Then is a vector field orthogonal to both and , and . Choose a unit vector field orthogonal to , , and . Then is also orthogonal to , , and and . Proceeding in this way we obtain a set of local orthonormal vector fields . Now construct the unit vector field orthogonal to , , and   . Then is also orthogonal to , , and    and . By a similar way set a unit vector field orthogonal to , , and . It is easy to see that is also orthogonal to , , and . Hence, from a quasi-orthonormal basis of , we obtain a local orthonormal basis

where , called -basis. Thus we have the following.

Proposition 20. Let be a -dimensional almost paracontact metric manifold and a screen semi-invariant lightlike hypersurface of . Then there always exists a -basis on generated from a quasi-orthonormal basis , of .

Now, we consider the distributions , on . Then is a -invariant distribution and we have

Thus, every can be expressed by

where and are the projections of into and , respectively. Hence, we may write , for any . Let us consider the local lightlike vector fields and . From (1), (38), and (39), we obtain

By comparing the tangential and transversal parts in (100) we get

respectively. Next, from (3) one can easily see that

Since , by using (38), we also have

Furthermore, from (4), we have

Finally, we get

which gives

Thus we have the following.

Proposition 21. Let be a screen semi-invariant lightlike hypersurface of an almost paracontact metric manifold . Then possesses a para -structure; that is,

Theorem 22. Let be a screen semi-invariant lightlike hypersurface of a para-Sasakian manifold . Then one has

for all .

Proposition 23. Let be a para-Sasakian manifold and a screen semi-invariant lightlike hypersurface of . Then M is totally geodesic if and only if, for any and for ,

Proof. Let assume that is totally geodesic; that is, for any , . Then, for using in (109) we have

Similarly, using (109) we have

which gives

by virtue of and .

Conversely, suppose that the conditions (111) and (112) hold and we will prove that vanishes. If , using the decomposition (98), there exists such that

and for any we obtain

For , using (109) and (111), we find

which implies that .

Also, for any from (109), using (112), we get

which implies . This completes the proof.

Proposition 24. Let be a para-Sasakian manifold and a screen semi-invariant lightlike hypersurface of . Then one has, for any ,

(i)if the vector field is parallel, then

(ii)if the vector field is parallel, then

where and .

6. Integrability of Distributions on a Screen Semi-Invariant Lightlike Hypersurface of a Para-Sasakian Manifold

6.1. The Distribution

Firstly, we consider the distribution , defined in (94). Using (95) and putting , for any , , and we have

where is a linear connection on the bundle , is an bilinear, is an linear operator on , respectively, and is a linear connection on .

Lemma 25. Let be a screen semi-invariant lightlike hypersurface of a para-Sasakian manifold and a coordinate neighborhood as fixed in Theorem 2. Then, for any , one has

Proof. Calculation is straightforward by using (9).

Let be a coordinate neighborhood as fixed in Theorem 2. Then according to decomposition given by (95) we set

for any . So (122) can be written locally as

We will express , , and in terms of and . Firstly, we compute

Then, by using (125), (24) and (25), we get

in view of being -invariant and being a metric connection.

Next we have

And, using (124), (24), and (31) in the previous equation, we obtain

By a similar way, we compute

Since , then we get

Therefore using the expressions of , and in (127) we write

which implies that

Theorem 26. Let be a screen semi-invariant lightlike hypersurface of a para-Sasakian manifold . Then the distribution is integrable if and only if

where .

Proof. Since is torsion free connection, by using (134), we have

for any . Now suppose that is integrable. Then the components of with respect to , , and vanish. So, we get (136).

Conversely, if (136) is satisfied, then we get (137)

for any , which implies that . This completes the proof.

Corollary 27. Let be a para-Sasakian manifold and a screen semi-invariant lightlike hypersurface of . Then is symmetric on if and only if is integrable.

Theorem 28. Let be a screen semi-invariant lightlike hypersurface of a para-Sasakian manifold and the distribution on is integrable. Then is minimal with respect to the symmetric connection on ; that is, , if and only if

for any .

Proof. Using decomposition (94), we find . Now, from Proposition 20, consider an orthonormal -basis of , . Then we have

Also from (25) and (35) we have

Hence, using integrability condition of we get

which completes the proof.

Now, using the decomposition (96) and putting , for any , , and , we write

where is a linear connection on the bundle , is bilinear and is an linear operator on , and is a linear connection on .

Let be a coordinate neighborhood as fixed in Theorem 2. Then, by using (96), we set

for any . Hence, (143) can be written locally:

From the expression of , , in terms of and , we obtain

Theorem 29. Let be a screen semi-invariant lightlike hypersurface of a para-Sasakian manifold with the integrable distribution . Then is minimal; that is, , if and only if

for any .

Proof. Let , , be an orthonormal -basis. Then, we have

It is easy to see that . Also, using (147), we get

by virtue of integrability condition of . This completes the proof.

Theorem 30. Let be a para-Sasakian manifold and a screen semi-invariant lightlike hypersurface of . If is integrable, then the leaves of have a para-Sasakian structure.

Proof. Let be a screen semi-invariant lightlike hypersurface and a leaf of . Then, for any , we have . Since and , we get

for any .

By putting and , one can see that defines an -tensor field on because is -invariant. From (101), we get

for any . Hence is an almost paracontact manifold.

Next, by using (38), for any , we have

which implies that is an almost paracontact metric manifold.

Moreover, for any , , and  . Also and coincide on .

Finally, we have

for any , which implies that is a Levi-Civita connection and using (5) we get

This completes the proof.

6.2. Integrability of

In this section, we consider the distribution , which is defined by . Firstly we have the following.

Lemma 31. Let be a para-Sasakian manifold anda screen semi-invariant lightlike hypersurface of . Then, for any and , one has

that is, the component of along vanishes.

Proposition 32. Let be a para-Sasakian manifold anda screen semi-invariant lightlike hypersurface of . Then the distribution is integrable if and only if satisfies the following conditions:

(i), for any ,(ii), for any ,(iii),

where .

Proof. For any , we obtain the component of along as

From the definition of the distribution , we set

Since is -invariant, from the previous expressions of and from (26), we have

for any .

Now assume that is integrable. Since , , , and are sections of , then we get

If , we find

and if we get

Consequently using (160) with (i), (ii), and (iii), it is easy to check that in . This completes the proof.

Proposition 33. Let be a para-Sasakian manifold andbe a screen semi-invariant lightlike hypersurface of . If is totally geodesic, then the following statements hold.

(i)The distribution is integrable.(ii)The distribution is parallel with respect to the induced connection .

Proof. (i) Assume that is totally geodesic. Then we can state that the distribution is integrable from Proposition 32.

(ii) For any and , using (157), we get

This completes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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