Journal of Function Spaces and Applications

Volume 2012 (2012), Article ID 636242, 16 pages

doi:10.1155/2012/636242

Research Article

Special Half Lightlike Submanifolds of an Indefinite Cosymplectic Manifold

Dae Ho Jin

Department of Mathematics, Dongguk University, Gyeongju 780-714, Republic of Korea

Received 1 May 2012; Revised 17 August 2012; Accepted 25 August 2012

Academic Editor: Dashan Fan

Copyright © 2012 Dae Ho Jin. 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 the geometry of half lightlike submanifolds of an indefinite cosymplectic manifold . First, we construct two types of half lightlike submanifolds according to the form of the structure vector field of , named by tangential and ascreen half lightlike submanifolds. Next, we characterize the lightlike geometries of such two types of half lightlike submanifolds.

1. Introduction

The class of codimension 2 lightlike submanifolds of a semi-Riemannian manifold is composed entirely of two classes by virtue of the rank of its radical distribution, called half lightlike and coisotropic submanifolds [1–4]. Half lightlike submanifold is a special case of -lightlike submanifold [5, 6] such that and its geometry is more general than that of coisotropic submanifold. Moreover much of the works on half lightlike submanifolds will be immediately generalized in a formal way to arbitrary -lightlike submanifolds. Recently several authors studied the geometry of lightlike submanifolds of indefinite cosymplectic manifolds. Much of them have studied so-called CR-types (CR, SCR, GCR, QCR, etc) lightlike submanifolds of indefinite cosymplectic manifolds. Unfortunately, an intrinsic study of lightlike submanifolds of indefinite cosymplectic manifolds is slight as yet. Only there are some limited papers on particular subcases recently studied [7–9].

The objective of this paper is to study the geometry of half lightlike submanifolds of an indefinite cosymplectic manifold . There are many different types of half lightlike submanifolds of an indefinite cosymplectic manifold according to the form of the structure vector field of . We study two types of them here: tangential and ascreen half lightlike submanifolds. We provide several new results on each types by using the structure of induced by the contact metric structure of .

2. Half Lightlike Submanifolds

An odd dimensional smooth manifold is called a contact metric manifold if there exists a contact metric structure , where is a -type tensor field, a vector field which is called the structure vector field of and a 1-form satisfying

for any vector fields , on . We say that has a normal contact structure if , where is the Nijenhuis tensor field of . A normal contact metric manifold is called a cosymplectic [10, 11] for which we have

for any vector field on , where is the Levi-Civita connection of . A cosymplectic manifold is called an indefinite cosymplectic manifold [7–9] if is a semi-Riemannian manifold of index .

For any indefinite cosymplectic manifold , applying to and using (2.2), we have . Applying to this and using the fact , we get

A submanifold of a semi-Riemannian manifold of codimension is called a half lightlike submanifold if the rank of the radical distribution is , where and are the tangent and normal bundles of , respectively. Then there exist complementary nondegenerate distributions and of in and , respectively, which are called the screen and coscreen distribution on :

where the symbol denotes the orthogonal direct sum. We denote such a half lightlike submanifold by . Denote by the algebra of smooth functions on and by the module of smooth sections of a vector bundle over . Choose as a unit vector field of such that . In this paper we may assume that without loss of generality. Consider the orthogonal complementary distribution to in . Certainly and are vector subbundles of . Thus we have the following orthogonal decomposition:

where is the orthogonal complementary to in . It is well-known [1, 2] that, for any null section of on a coordinate neighborhood , there exists a uniquely defined null vector field satisfying

Let . We say that , and are the lightlike transversal vector field, lightlike transversal vector bundle and transversal vector bundle of with respect to , respectively. Therefore is decomposed as

Let be the projection morphism of on with respect to the decomposition (2.4). The local Gauss and Weingarten formulas for and are given by

for all , , where and are induced linear connections on and , respectively, and are called the local second fundamental forms of , is called the local second fundamental form on . , , and are linear operators on and ,  , and are 1-forms on . Since is torsion-free, is also torsion-free, and and are symmetric. From the facts and , we know that and are independent of the choice of the screen distribution and

We say that is the second fundamental tensor of . The induced connection of is not metric and satisfies

for all , , , where is a 1-form on such that

But the connection on is metric. The above three local second fundamental forms of and are related to their shape operators by

By (2.16) and (2.17), we show that and are -valued shape operators related to and , respectively, and is self-adjoint on and

Replacing by to (2.8) and using (2.12) and (2.13), we have

3. Tangential Half Lightlike Submanifolds

Let be a half lightlike submanifold of an indefinite cosymplectic manifold . In general the structure vector field of , defined by (2.1), belongs to . Thus, from the decomposition (2.7) of , the structure vector field is decomposed as follows:

where is a smooth vector field on , and , , and are smooth functions on . First of all, we introduce the following result.

Proposition 3.1 (see [3]). Let be a half lightlike submanifold of an indefinite almost contact metric manifold . Then there exists a screen distribution such that

Note 1. Although, in general, is not unique, it is canonically isomorphic to the factor vector bundle considered by Kupeli [12]. Thus all screen distributions are mutually isomorphic. For this reason, we consider only half lightlike submanifold equipped with a screen distribution such that , such a screen distribution is called a generic screen distribution [8] of .

Proposition 3.2. Let be a half lightlike submanifold of an indefinite cosymplectic manifold . Then the structure vector field does not belong to and .

Proof. Assume that belongs to (or ). Then (3.1) deduces to and (or and ). From this, we have

It is a contradiction. Thus does not belong to and .

Note 2. If the structure vector field is tangent to , that is, , then does not belong to by Proposition 3.2. This enables one to choose a screen distribution which contains . This implies that if is tangent to , then it belongs to . Călin [13] also proved this result which we assume in this section.

Definition 3.3. A half lightlike submanifold of an indefinite cosymplectic manifold is said to be a tangential half lightlike submanifold [4] of if is tangent to .

For any tangential half lightlike submanifold , we show that belongs to , that is, by Note 2. Then there exists a nondegenerate almost complex distribution on with respect to , that is, , such that

Thus the general decompositions (2.4) and (2.7) reduce, respectively, to

where and are 2- and 1-lightlike distributions on such that

is an almost complex distribution of with respect to . Consider a pair of local null vector fields and a local nonnull vector field on defined by

Denote by the projection morphism of on with respect to the decomposition (3.5). Then any vector field on and its action by are expressed as follows:

where , , and are 1-forms locally defined on by

and is a tensor field of type globally defined on by . Applying the operator to (3.7) and the second equation of (3.8) (denote (3.8)₂) and using (2.2), (2.8), (2.9), (2.10), (2.21), (3.7), (3.8) and (3.9), for all , we have

Note 3. From now on, will denote the semi-Euclidean manifold equipped with its usual cosymplectic structure given by

where are the Cartesian coordinates and is a semi-Euclidean metric of signature with respect to the canonical basis

This construction will help in understanding how the indefinite cosymplectic structure is recovered in examples of this paper.

Example 3.4. Consider a submanifold of given by the equations

Then a local frame fields of are given by

This implies , , and . Next, implies that invariant with respect to the almost contact structure tensor . By direct calculations, we have

We show that , , and for all . Therefore is a tangential half-lightlike submanifold of an indefinite cosymplectic manifold .

Theorem 3.5. Let be a tangential half lightlike submanifold of an indefinite cosymplectic manifold . Then the structure vector field is parallel with respect to the connections and . Furthermore, is conjugate to any vector field of with respect to and .

Proof. Replacing by to (2.8) and using (2.3) and the fact , we get

Taking the scalar product with and to this equation by turns, we have

From (3.21)₁, we see that is parallel with respect to the induced connection . (3.21)2,3 implies that is conjugate to any vector field on with respect to the second fundamental form . Replacing by to (2.11) and using (3.21)₁ and the fact , we get

Taking the scalar product with to this equation, we have

Thus is also parallel with respect to the lieasr connection and conjugate to any vector field on with respect to . Thus we have our assertions.

Definition 3.6. A half lightlike submanifold of is totally umbilical [5] if there is a smooth vector field on on any coordinate neighborhood such that

In case , that is, on , we say that is totally geodesic.

It is easy to see that is totally umbilical if and only if there exist smooth functions and on each coordinate neighborhood such that

Theorem 3.7. Any totally umbilical tangential half lightlike submanifold of an indefinite cosymplectic manifold is totally geodesic.

Proof. Assume that is totally umbilical. From (3.21) and (3.25), we have

Replacing by in this equations and using the fact , we have , that is, . Thus we have and is totally geodesic.

Definition 3.8. Ascreen distribution is called totally umbilical [5] (in ) if there is a smooth function on any coordinate neighborhood in such that

In case on , we say that is totally geodesic (in ).

Theorem 3.9. Let be a tangential half lightlike submanifold of an indefinite cosymplectic manifold such that is totally umbilical. Then is totally geodesic.

Proof. Assume that is totally umbilical in . Replacing by to (3.27) and using (3.23), we have for all . Replacing by to this equation and using the fact , we obtain . Thus is totally geodesic in .

Theorem 3.10. Let be a tangential half lightlike submanifold of an indefinite cosymplectic manifold . Then is an integrable distribution on if and only if

Moreover, if is totally umbilical, then is a parallel distribution on .

Proof. Taking , we show that . Applying to and using (2.3), (2.8), (3.7), (3.8)₂, and (3.9), we have

By direct calculations from two equations of (3.29), we have

If is integrable, then for any , . This implies . Thus we get for all , . Conversely if for all , , then we have . This imply for all ,. Thus is an integrable distribution of .

If is totally umbilical, from Theorem 3.7 and (3.29), we have

This imply for all , , that is, is a parallel distribution on .

Theorem 3.11. Let be a tangential half lightlike submanifold of an indefinite cosymplectic manifold . Then is parallel on with respect to the connection if and only if is a parallel distribution on .

Proof. Assume that is parallel on with respect to . For any , , we have . Taking the scalar product with and to (3.30) with , we have and for all , , respectively. From (3.29), we have and . This imply for all ,. Thus is a parallel distribution on .

Conversely if is a parallel distribution on , from (3.29) we have

For any , we show that . Replacing by to (3.33) and using (3.21), we have and for any . Thus is parallel on with respect to by (3.30).

Theorem 3.12. Let be a tangential half lightlike submanifold of an indefinite cosymplectic manifold . If is parallel with respect to the induced connection , then is a parallel distribution on and is locally a product manifold , where and are null curves tangent to and , respectively, and is a leaf of .

Proof. Assume that is parallel on with respect to . Then is parallel on with respect to . By Theorem 3.11, is a parallel distribution on . Applying the operator to (3.14) with , we have

due to . Replacing by and to this equation by turns and using (3.9), we have and . Taking the scalar product with and to (3.14) with by turns, we have

Replacing by to (3.35), we get due to (2.13)₂. Also replacing by to (3.36), we have due to (2.18)₂. From thess results, (3.11) and (3.13), we get and for all . Thus and are also parallel distributions on . By the decomposition theorem of de Rham [14], we show that , where and are null curves tangent to and , respectively, and is a leaf of .

Definition 3.13. A half lightlike submanifold of a semi-Riemannian manifold is said to be irrotational [12] if for any .

Note 4. From (2.21) we see that a necessary and sufficient condition for to be irrotational is for all .

Theorem 3.14. Let be a tangential half lightlike submanifold of an indefinite cosymplectic manifold . Then one has the following assertions.

(i)If is parallel with respect to , then is irrotational, and

(ii)If is parallel with respect to , then one has and

(iii)If is parallel with respect to , then is irrotational and

Moreover, if all of , , and are parallel on with respect to , then is totally geodesic in and on . In this case, the null transversal vector field of is a constant on .

Proof. If is parallel with respect to , then, taking the scalar product with and to (3.12) by turns, we have and ( is irrotational), respectively. Thus we have for all . From this result and (3.8), we obtain . Applying to this equation and using , we obtain (i). In a similar way, by using (3.11), (3.13), (3.21), and (3.23), we have (ii) and (iii).

Assume that all of , , and are parallel on with respect to . Substituting the equation of (i) into (3.10)-1, we have

Also, substituting the equation of (iii) into (3.10)-3, we have

From the last two equations and the equation of (ii), we see that . Thus is totally geodesic in and the 1-forms , , and , defined by (2.9) and (2.10), satisfy on . Using this results, we see that is a constant on .

Theorem 3.15. Let be a totally umbilical tangential half lightlike submanifold of an indefinite cosymplectic manifold such that is totally umbilical. Then is locally a product manifold , where and are leaves of and , respectively.

Proof. By Theorem 3.10, is a parallel distribution . Thus, for all , , we have . From (2.11) and (3.30), we have

due to . If is totally umbilical in , then we have due to Theorem 3.7. By (2.11) and (3.42), we get

These results and (3.29) imply for all , . Thus is a parallel distribution on and , where . By Theorems 3.5 and 3.7, we have and . Thus (2.12) and (3.11)(3.13) deduce, respectively, to

Thus is also a parallel distribution on . Thus we have , where is a leaf of and is a leaf of .

4. Ascreen Half Lightlike Submanifolds

Definition 4.1. A half lightlike submanifold of an indefinite cosymplectic manifold is said to be an ascreen half lightlike submanifold [4] of if the structure vector field of belongs to the distribution .

For any ascreen half lightlike submanifold , the vector field is decomposed as

In this case, we show that and by Proposition 3.2.

Definition 4.2. A half lightlike submanifold is called screen conformal [2, 3] if there exists a nonvanishing smooth function such that , or equivalently,

Theorem 4.3. Let be an ascreen half lightlike submanifold of an indefinite cosymplectic manifold . Then is screen conformal.

Proof. Applying to (4.1) and using (2.3), (2.9), and (2.21), we have

Taking the product with , , and by turns and using (2.16)₂ and (2.17)₂, we get

for all , where we set . Thus is screen conformal.

Substituting (4.1) into , we have . Consider the local unit timelike vector field on and its 1-form defined by

Let . Then is a unit timelike vector field on such that . Applying to (4.1) and using (2.1) and , we have

From this we show that . Using this and Proposition 3.1, the tangent bundle of is decomposed as follows:

where is a nondegenerate and almost complex distribution on with respect to the indefinite cosymplectic structure tensor , otherwise is degenerate. Consider the local unit spacelike vector field on and its 1-form defined by

Denote by the projection morphism of on . Using (4.7), for any vector field on , the vector field is expressed as follows:

because , where is a tensor field of type (1, 1) defined by

Applying to (2.10) and (2.21) and using (2.2), (2.8) and (4.4)(4.9), we get

Example 4.4. Consider a submanifold of given by the equation

By direct calculations we easily check that

We obtain the lightlike transversal and transversal vector bundles

From this results, we show that , , , , and , and . Thus is an ascreen half lightlike submanifold of an indefinite cosymplectic manifold .

Theorem 4.5. Let be an ascreen half lightlike submanifold of an indefinite cosymplectic manifold . If is totally umbilical, then and are totally geodesic.

Proof. Assume that is totally umbilical. From (3.25) and (4.13), we have

Replacing by and to this equation by turns, we have and , respectively. Thus we have and is totally geodesic. By (4.2), we also have . Thus is also totally geodesic in .

Taking . Then we have due to (4.9). Applying to and using (2.2), (2.8), (4.2), (4.5), and (4.9), we have

for all . By the procedure same as the proofs of Theorem 3.10 and Theorem 3.11 and by using (4.18) and (4.19), instead of (3.29) and (3.30), and that is integrable due to (4.2), the following two theorems hold.

Theorem 4.6. Let be an ascreen half lightlike submanifold of an indefinite cosymplectic manifold . is an integrable distribution on if and only if one has

Moreover, if is totally umbilical, then is a parallel distribution on .

Theorem 4.7. Let be an ascreen half lightlike submanifold of an indefinite cosymplectic manifold . Then is parallel on with respect to the induced connection if and only if is a parallel distribution on .

Theorem 4.8. Let be an ascreen half lightlike submanifold of an indefinite cosymplectic manifold . If is totally umbilical, then is locally a product manifold , where , , and are null, timelike, and spacelike curves tangent to , , and , respectively, and is a leaf of .

Proof. If is totally umbilical, then is a parallel distribution on by Theorem 4.6 and we have by Theorem 4.5. From (4.4)₁, we also have . Using (2.12), (4.11), and (4.12), we have and due to . This implies that all of the distributions , , and are parallel on . Thus we have , where , , and are null, timelike, and spacelike curves tangent to , and , respectively, and is a leaf of .

By straightforward calculations from (4.11) and (4.12) and the same method as the proof of Theorem 3.14, the following theorem holds.

Theorem 4.9. Let be an ascreen half lightlike submanifold of an indefinite cosymplectic manifold . Then one has the following assertions.

(i)If is parallel with respect to on , then is irrotational and

(ii)If is parallel with respect to on , then is irrotational and

Moreover, if and are parallel with respect to , then one sees that and the screen distribution is totally geodesic in .

Theorem 4.10. Let be an ascreen half lightlike submanifold of an indefinite cosymplectic manifold . If and are parallel with respect to , then is locally a product manifold , where , , and are null, timelike, and spacelike curves tangent to , , and , respectively, and is a leaf of .

Proof. If is parallel with respect to , for any , we have

Thus we get because . Also if is parallel with respect to , then, for any , we have

From these results and (4.19), we show that is parallel on with respect to . Thus, by Theorem 4.7, we see that is a parallel distribution on . As and are parallel with respect to and due to , we have our theorem.

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