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 .