Abstract

We study the geometry of half lightlike submanifolds 𝑀 of an indefinite cosymplectic manifold 𝑀. First, we construct two types of half lightlike submanifolds according to the form of the structure vector field of 𝑀, 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 [14]. Half lightlike submanifold is a special case of 𝑟-lightlike submanifold [5, 6] such that 𝑟=1 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 𝑟-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 [79].

The objective of this paper is to study the geometry of half lightlike submanifolds 𝑀 of an indefinite cosymplectic manifold 𝑀. There are many different types of half lightlike submanifolds of an indefinite cosymplectic manifold 𝑀 according to the form of the structure vector field of 𝑀. 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 𝑀 induced by the contact metric structure of 𝑀.

2. Half Lightlike Submanifolds

An odd dimensional smooth manifold (𝑀,𝑔) is called a contact metric manifold if there exists a contact metric structure (𝐽,𝜃,𝜁,𝑔), where 𝐽 is a (1,1)-type tensor field, 𝜁 a vector field which is called the structure vector field of 𝑀 and 𝜃 a 1-form satisfying 𝐽2𝑋=𝑋+𝜃(𝑋)𝜁,𝐽𝜁=0,𝜃𝐽=0,𝜃(𝜁)=1,𝑔(𝜁,𝜁)=1,𝑔(𝐽𝑋,𝐽𝑌)=𝑔(𝑋,𝑌)𝜃(𝑋)𝜃(𝑌),𝜃(𝑋)=𝑔(𝜁,𝑋),𝑑𝜃(𝑋,𝑌)=𝑔(𝐽𝑋,𝑌),(2.1) for any vector fields 𝑋, 𝑌 on 𝑀. We say that 𝑀 has a normal contact structure if 𝑁𝐽+𝑑𝜃𝜁=0, where 𝑁𝐽 is the Nijenhuis tensor field of 𝐽. A normal contact metric manifold is called a cosymplectic [10, 11] for which we have 𝑋𝜃=0,𝑋𝐽=0,(2.2) for any vector field 𝑋 on 𝑀, where is the Levi-Civita connection of 𝑀. A cosymplectic manifold 𝑀=(𝑀,𝐽,𝜁,𝜃,𝑔) is called an indefinite cosymplectic manifold [79] if (𝑀,𝑔) is a semi-Riemannian manifold of index 𝜇(>0).

For any indefinite cosymplectic manifold 𝑀, applying 𝑋 to 𝐽𝜁=0 and using (2.2), we have 𝐽(𝑋𝜁)=0. Applying 𝐽 to this and using the fact 𝜃(𝑋𝜁)=0, we get 𝑋𝜁=0.(2.3)

A submanifold 𝑀 of a semi-Riemannian manifold 𝑀 of codimension 2 is called a half lightlike submanifold if the rank of the radical distribution Rad(𝑇𝑀)=𝑇𝑀𝑇𝑀 is 1, where 𝑇𝑀 and 𝑇𝑀 are the tangent and normal bundles of 𝑀, respectively. Then there exist complementary nondegenerate distributions 𝑆(𝑇𝑀) and 𝑆(𝑇𝑀) of Rad(𝑇𝑀) in 𝑇𝑀 and 𝑇𝑀, respectively, which are called the screen and coscreen distribution on 𝑀: 𝑇𝑀=Rad(𝑇𝑀)orth𝑆(𝑇𝑀),𝑇𝑀=Rad(𝑇𝑀)orth𝑆𝑇𝑀,(2.4) where the symbol orth denotes the orthogonal direct sum. We denote such a half lightlike submanifold by 𝑀=(𝑀,𝑔,𝑆(𝑇𝑀)). Denote by 𝐹(𝑀) the algebra of smooth functions on 𝑀 and by Γ(𝐸) the 𝐹(𝑀) module of smooth sections of a vector bundle 𝐸 over 𝑀. Choose 𝐿 as a unit vector field of 𝑆(𝑇𝑀) such that 𝑔(𝐿,𝐿)=±1. In this paper we may assume that 𝑔(𝐿,𝐿)=1 without loss of generality. Consider the orthogonal complementary distribution 𝑆(𝑇𝑀) to 𝑆(𝑇𝑀) in 𝑇𝑀. Certainly Rad(𝑇𝑀) and 𝑆(𝑇𝑀) are vector subbundles of 𝑆(𝑇𝑀). Thus we have the following orthogonal decomposition: 𝑆(𝑇𝑀)=𝑆𝑇𝑀orth𝑆𝑇𝑀,(2.5) where 𝑆(𝑇𝑀) is the orthogonal complementary to 𝑆(𝑇𝑀) in 𝑆(𝑇𝑀). It is well-known [1, 2] that, for any null section 𝜉 of Rad(𝑇𝑀) on a coordinate neighborhood 𝒰𝑀, there exists a uniquely defined null vector field 𝑁Γ(ltr(𝑇𝑀)) satisfying 𝑔(𝜉,𝑁)=1,𝑔(𝑁,𝑁)=𝑔(𝑁,𝑋)=𝑔(𝑁,𝐿)=0,𝑋Γ(𝑆(𝑇𝑀)).(2.6) Let tr(𝑇𝑀)=𝑆(𝑇𝑀)orthltr(𝑇𝑀). We say that 𝑁, ltr(𝑇𝑀) and tr(𝑇𝑀) are the lightlike transversal vector field, lightlike transversal vector bundle and transversal vector bundle of 𝑀 with respect to 𝑆(𝑇𝑀), respectively. Therefore 𝑇𝑀 is decomposed as 𝑇𝑀=𝑇𝑀tr(𝑇𝑀)={Rad(𝑇𝑀)tr(𝑇𝑀)}orth𝑆(𝑇𝑀)={Rad(𝑇𝑀)ltr(𝑇𝑀)}orth𝑆(𝑇𝑀)orth𝑆𝑇𝑀.(2.7)

Let 𝑃 be the projection morphism of 𝑇𝑀 on 𝑆(𝑇𝑀) with respect to the decomposition (2.4). The local Gauss and Weingarten formulas for 𝑀 and 𝑆(𝑇𝑀) are given by 𝑋𝑌=𝑋𝑌+𝐵(𝑋,𝑌)𝑁+𝐷(𝑋,𝑌)𝐿,(2.8)𝑋𝑁=𝐴𝑁𝑋+𝜏(𝑋)𝑁+𝜌(𝑋)𝐿,(2.9)𝑋𝐿=𝐴𝐿𝑋+𝜙(𝑋)𝑁;(2.10)𝑋𝑃𝑌=𝑋𝑃𝑌+𝐶(𝑋,𝑃𝑌)𝜉,(2.11)𝑋𝜉=𝐴𝜉𝑋𝜏(𝑋)𝜉,(2.12) for all 𝑋, 𝑌Γ(𝑇𝑀), where and are induced linear connections on 𝑇𝑀 and 𝑆(𝑇𝑀), respectively, 𝐵 and 𝐷 are called the local second fundamental forms of 𝑀, 𝐶 is called the local second fundamental form on 𝑆(𝑇𝑀). 𝐴𝑁, 𝐴𝜉, and 𝐴𝐿 are linear operators on 𝑇𝑀 and 𝜏,  𝜌, and 𝜙 are 1-forms on 𝑇𝑀. Since is torsion-free, is also torsion-free, and 𝐵 and 𝐷 are symmetric. From the facts 𝐵(𝑋,𝑌)=𝑔(𝑋𝑌,𝜉) and 𝐷(𝑋,𝑌)=𝑔(𝑋𝑌,𝐿), we know that 𝐵 and 𝐷 are independent of the choice of the screen distribution 𝑆(𝑇𝑀) and 𝐵(𝑋,𝜉)=0,𝐷(𝑋,𝜉)=𝜙(𝑋),𝑋Γ(𝑇𝑀).(2.13) We say that (𝑋,𝑌)=𝐵(𝑋,𝑌)𝑁+𝐷(𝑋,𝑌)𝐿 is the second fundamental tensor of 𝑀. The induced connection of 𝑀 is not metric and satisfies 𝑋𝑔(𝑌,𝑍)=𝐵(𝑋,𝑌)𝜂(𝑍)+𝐵(𝑋,𝑍)𝜂(𝑌),(2.14) for all 𝑋, 𝑌, 𝑍Γ(𝑇𝑀), where 𝜂 is a 1-form on 𝑇𝑀 such that 𝜂(𝑋)=𝑔(𝑋,𝑁),𝑋Γ(𝑇𝑀).(2.15) But the connection on 𝑆(𝑇𝑀) is metric. The above three local second fundamental forms of 𝑀 and 𝑆(𝑇𝑀) are related to their shape operators by 𝐴𝐵(𝑋,𝑌)=𝑔𝜉,𝑋,𝑌𝑔𝐴𝜉𝐴𝑋,𝑁=0,(2.16)𝐶(𝑋,𝑃𝑌)=𝑔𝑁,𝑋,𝑃𝑌𝑔𝐴𝑁𝐴𝑋,𝑁=0,(2.17)𝐷(𝑋,𝑃𝑌)=𝑔𝐿,𝑋,𝑃𝑌𝑔𝐴𝐿𝐴𝑋,𝑁=𝜌(𝑋),(2.18)𝐷(𝑋,𝑌)=𝑔𝐿𝑋,𝑌𝜙(𝑋)𝜂(𝑌),𝑋,𝑌Γ(𝑇𝑀).(2.19) By (2.16) and (2.17), we show that 𝐴𝜉 and 𝐴𝑁 are Γ(𝑆(𝑇𝑀))-valued shape operators related to 𝐵 and 𝐶, respectively, and 𝐴𝜉 is self-adjoint on 𝑇𝑀 and 𝐴𝜉𝜉=0.(2.20) Replacing 𝑌 by 𝜉 to (2.8) and using (2.12) and (2.13), we have 𝑋𝜉=𝐴𝜉𝑋𝜏(𝑋)𝜉𝜙(𝑋)𝐿,𝑋Γ(𝑇𝑀).(2.21)

3. Tangential Half Lightlike Submanifolds

Let 𝑀 be a half lightlike submanifold of an indefinite cosymplectic manifold 𝑀. In general the structure vector field 𝜁 of 𝑀, defined by (2.1), belongs to 𝑇𝑀. Thus, from the decomposition (2.7) of 𝑇𝑀, the structure vector field 𝜁 is decomposed as follows: 𝜁=𝜔+𝑎𝜉+𝑏𝑁+𝑒𝐿,(3.1) where 𝜔 is a smooth vector field on 𝑆(𝑇𝑀), and 𝑎=𝜃(𝑁), 𝑏=𝜃(𝜉), and 𝑒=𝜃(𝐿) are smooth functions on 𝑀. First of all, we introduce the following result.

Proposition 3.1 (see [3]). Let 𝑀 be a half lightlike submanifold of an indefinite almost contact metric manifold 𝑀. Then there exists a screen distribution 𝑆(𝑇𝑀) such that 𝐽𝑆(𝑇𝑀)𝑆(𝑇𝑀).(3.2)

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

Proposition 3.2. Let 𝑀 be a half lightlike submanifold of an indefinite cosymplectic manifold 𝑀. Then the structure vector field 𝜁 does not belong to Rad(𝑇𝑀) and ltr(𝑇𝑀).

Proof. Assume that 𝜁 belongs to Rad(𝑇𝑀) (or ltr(𝑇𝑀)). Then (3.1) deduces to 𝜁=𝑎𝜉 and 𝑎0 (or 𝜁=𝑏𝑁 and 𝑏0). From this, we have 1=𝑔(𝜁,𝜁)=𝑎2𝑔(𝜉,𝜉)=0or1=𝑔(𝜁,𝜁)=𝑏2𝑔.(𝑁,𝑁)=0(3.3) It is a contradiction. Thus 𝜁 does not belong to Rad(𝑇𝑀) and ltr(𝑇𝑀).

Note 2. If the structure vector field 𝜁 is tangent to 𝑀, that is, 𝑏=𝑒=0, then 𝜁 does not belong to Rad(𝑇𝑀) by Proposition 3.2. This enables one to choose a screen distribution 𝑆(𝑇𝑀) which contains 𝜁. This implies that if 𝜁 is tangent to 𝑀, then it belongs to 𝑆(𝑇𝑀). Călin [13] also proved this result which we assume in this section.

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

For any tangential half lightlike submanifold 𝑀, we show that 𝜁 belongs to 𝑆(𝑇𝑀), that is, 𝑎=𝑏=𝑒=0 by Note 2. Then there exists a nondegenerate almost complex distribution 𝐻𝑜 on 𝑀 with respect to 𝐽, that is, 𝐽(𝐻𝑜)=𝐻𝑜, such that 𝑆(𝑇𝑀)={𝐽(Rad(𝑇𝑀))𝐽(ltr(𝑇𝑀))}orth𝐽𝑆𝑇𝑀orth𝐻𝑜.(3.4) Thus the general decompositions (2.4) and (2.7) reduce, respectively, to 𝑇𝑀=𝐻𝐻,𝑇𝑀=𝐻𝐻tr(𝑇𝑀),(3.5) where 𝐻 and 𝐻 are 2- and 1-lightlike distributions on 𝑀 such that 𝐻=Rad(𝑇𝑀)orth𝐽(Rad(𝑇𝑀))orth𝐻𝑜,𝐻=𝐽(ltr(𝑇𝑀))orth𝐽𝑆𝑇𝑀.(3.6)𝐻 is an almost complex distribution of 𝑀 with respect to 𝐽. Consider a pair of local null vector fields {𝑈,𝑉} and a local nonnull vector field 𝑊 on 𝑆(𝑇𝑀) defined by 𝑈=𝐽𝑁,𝑉=𝐽𝜉,𝑊=𝐽𝐿.(3.7) Denote by 𝑆 the projection morphism of 𝑇𝑀 on 𝐻 with respect to the decomposition (3.5). Then any vector field 𝑋 on 𝑀 and its action 𝐽𝑋 by 𝐽 are expressed as follows: 𝑋=𝑆𝑋+𝑢(𝑋)𝑈+𝑤(𝑋)𝑊,𝐽𝑋=𝐹𝑋+𝑢(𝑋)𝑁+𝑤(𝑋)𝐿,(3.8) where 𝑢, 𝑣, and 𝑤 are 1-forms locally defined on 𝑀 by 𝑢(𝑋)=𝑔(𝑋,𝑉),𝑣(𝑋)=𝑔(𝑋,𝑈),𝑤(𝑋)=𝑔(𝑋,𝑊)(3.9) and 𝐹 is a tensor field of type (1,1) globally defined on 𝑀 by 𝐹=𝐽𝑆. Applying the operator 𝑋 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 𝑋,𝑌Γ(𝑇𝑀), we have 𝐵(𝑋,𝑈)=𝐶(𝑋,𝑉),𝐵(𝑋,𝑊)=𝐷(𝑋,𝑉),𝐶(𝑋,𝑊)=𝐷(𝑋,𝑈),(3.10)𝑋𝐴𝑈=𝐹𝑁𝑋+𝜏(𝑋)𝑈+𝜌(𝑋)𝑊,(3.11)𝑋𝐴𝑉=𝐹𝜉𝑋𝜏(𝑋)𝑉𝜙(𝑋)𝑊,(3.12)𝑋𝐴𝑊=𝐹𝐿𝑋+𝜙(𝑋)𝑈,(3.13)𝑋𝐹(𝑌)=𝑢(𝑌)𝐴𝑁𝑋+𝑤(𝑌)𝐴𝐿𝑋𝐵(𝑋,𝑌)𝑈𝐷(𝑋,𝑌)𝑊.(3.14)

Note 3. From now on, 𝑀=(𝑅𝑞2𝑚+1,𝐽,𝜁,𝜃,𝑔) will denote the semi-Euclidean manifold 𝑅𝑞2𝑚+1 equipped with its usual cosymplectic structure given by 𝜃=𝑑𝑧,𝜁=𝜕𝑧,𝑔=𝜃𝜃𝑞/2𝑖=1𝑑𝑥𝑖𝑑𝑥𝑖+𝑑𝑦𝑖𝑑𝑦𝑖+𝑚𝑖=𝑞+1𝑑𝑥𝑖𝑑𝑥𝑖+𝑑𝑦𝑖𝑑𝑦𝑖,𝐽𝑚𝑖=1𝑋𝑖𝜕𝑥𝑖+𝑌𝑖𝜕𝑦𝑖=+𝑍𝜕𝑧𝑚𝑖=1𝑌𝑖𝜕𝑥𝑖𝑋𝑖𝜕𝑦𝑖,(3.15) where (𝑥𝑖,𝑦𝑖,𝑧) are the Cartesian coordinates and 𝑔 is a semi-Euclidean metric of signature (,+,,+;,+,,+;+) with respect to the canonical basis 𝜕𝑥1,𝜕𝑥2,,𝜕𝑥𝑚;𝜕𝑦1,𝜕𝑦2,,𝜕𝑦𝑚;𝜕𝑧.(3.16) This construction will help in understanding how the indefinite cosymplectic structure is recovered in examples of this paper.

Example 3.4. Consider a submanifold 𝑀 of 𝑀=(𝑅92,𝐽,𝜁,𝜃,𝑔) given by the equations 𝑥1=𝑦4,𝑥2=1𝑦22,𝑦2±1.(3.17) Then a local frame fields of 𝑇𝑀 are given by 𝜉=𝜕𝑥1+𝜕𝑦4,𝑈1=𝜕𝑥4𝜕𝑦1,𝑈2=𝜕𝑥3,𝑈3=𝜕𝑦3,𝑈4𝑦=2𝑥2𝜕𝑥2+𝜕𝑦2,𝑈5=𝜕𝑥4+𝜕𝑦1,𝑈6=𝜁=𝜕𝑧.(3.18) This implies Rad(𝑇𝑀)=Span{𝜉}, 𝐽𝜉=𝑈1, and Rad(𝑇𝑀)𝐽(Rad(𝑇𝑀))={0}. Next, 𝐽𝑈2=𝑈3 implies that 𝐻𝑜={𝑈2,𝑈3} invariant with respect to the almost contact structure tensor 𝐽. By direct calculations, we have 𝑆𝑇𝑀=Span𝐿=𝜕𝑥2+𝑦2𝑥2𝜕𝑦21,ltr(𝑇𝑀)=Span𝑁=2𝜕𝑥1+𝜕𝑦4.(3.19) We show that 𝐽𝐿=𝑈4, 𝐽𝑁=(1/2)𝑈5, 𝐽𝜁=0 and 𝑋𝜁=0 for all 𝑋Γ(𝑇𝑀). Therefore 𝑀 is a tangential half-lightlike submanifold of an indefinite cosymplectic manifold 𝑀.

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

Proof. Replacing 𝑌 by 𝜁 to (2.8) and using (2.3) and the fact 𝜁Γ(𝑇𝑀), we get 𝑋𝜁+𝐵(𝑋,𝜁)𝑁+𝐷(𝑋,𝜁)𝐿=0,𝑋Γ(𝑇𝑀).(3.20) Taking the scalar product with 𝜉 and 𝐿 to this equation by turns, we have 𝑋𝜁=0,𝐵(𝑋,𝜁)=0,𝐷(𝑋,𝜁)=0,𝑋Γ(𝑇𝑀).(3.21) From (3.21)1, we see that 𝜁 is parallel with respect to the induced connection . (3.21)2,3 implies that 𝜁 is conjugate to any vector field on 𝑀 with respect to the second fundamental form . Replacing 𝑃𝑌 by 𝜁 to (2.11) and using (3.21)1 and the fact 𝜁Γ(𝑆(𝑇𝑀)), we get 𝑋𝜁+𝐶(𝑋,𝜁)𝜉=0,𝑋Γ(𝑇𝑀).(3.22) Taking the scalar product with 𝑁 to this equation, we have 𝑋𝜁=0,𝐶(𝑋,𝜁)=0,𝑋Γ(𝑇𝑀).(3.23) Thus 𝜁 is also parallel with respect to the lieasr connection and conjugate to any vector field on 𝑀 with respect to 𝐶. Thus we have our assertions.

Definition 3.6. A half lightlike submanifold 𝑀 of 𝑀 is totally umbilical [5] if there is a smooth vector field on tr(𝑇𝑀) on any coordinate neighborhood 𝒰 such that (𝑋,𝑌)=𝑔(𝑋,𝑌),𝑋,𝑌Γ(𝑇𝑀).(3.24) In case =0, that is, =0 on 𝒰, we say that 𝑀 is totally geodesic.

It is easy to see that 𝑀 is totally umbilical if and only if there exist smooth functions 𝛽 and 𝛿 on each coordinate neighborhood 𝒰 such that 𝐵(𝑋,𝑌)=𝛽𝑔(𝑋,𝑌),𝐷(𝑋,𝑌)=𝛿𝑔(𝑋,𝑌),𝑋,𝑌Γ(𝑇𝑀).(3.25)

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

Proof. Assume that 𝑀 is totally umbilical. From (3.21) and (3.25), we have 𝛽𝑔(𝑋,𝜁)=0,𝛿𝑔(𝑋,𝜁)=0,𝑋Γ(𝑇𝑀).(3.26) Replacing 𝑋 by 𝜁 in this equations and using the fact 𝑔(𝜁,𝜁)=1, we have 𝛽=𝛿=0, that is, 𝐵=𝐷=0. Thus we have =0 and 𝑀 is totally geodesic.

Definition 3.8. Ascreen distribution 𝑆(𝑇𝑀) is called totally umbilical [5] (in 𝑀) if there is a smooth function 𝛾 on any coordinate neighborhood 𝒰 in 𝑀 such that 𝐶(𝑋,𝑃𝑌)=𝛾𝑔(𝑋,𝑌),𝑋,𝑌Γ(𝑇𝑀).(3.27) In case 𝛾=0 on 𝒰, we say that 𝑆(𝑇𝑀) is totally geodesic (in 𝑀).

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

Proof. Assume that 𝑆(𝑇𝑀) is totally umbilical in 𝑀. Replacing 𝑌 by 𝜁 to (3.27) and using (3.23), we have 𝛾𝑔(𝑋,𝜁)=0 for all 𝑋Γ(𝑇𝑀). Replacing 𝑋 by 𝜁 to this equation and using the fact 𝑔(𝜁,𝜁)=1, we obtain 𝛾=0. Thus 𝑆(𝑇𝑀) is totally geodesic in 𝑀.

Theorem 3.10. Let 𝑀 be a tangential half lightlike submanifold of an indefinite cosymplectic manifold 𝑀. Then 𝐻 is an integrable distribution on 𝑀 if and only if (𝑋,𝐹𝑌)=(𝐹𝑋,𝑌),𝑋,𝑌Γ(𝐻).(3.28) Moreover, if 𝑀 is totally umbilical, then 𝐻 is a parallel distribution on 𝑀.

Proof. Taking 𝑌Γ(𝐻), we show that 𝐹𝑌=𝐽𝑌Γ(𝐻). Applying 𝑋 to 𝐹𝑌=𝐽𝑌 and using (2.3), (2.8), (3.7), (3.8)2, and (3.9), we have 𝐵(𝑋,𝐹𝑌)=𝑔𝑋𝑌,𝑉,𝐷(𝑋,𝐹𝑌)=𝑔𝑋𝑌,𝑊,(3.29)𝑋𝐹(𝑌)=𝐵(𝑋,𝑌)𝑈𝐷(𝑋,𝑌)𝑊,𝑋Γ(𝑇𝑀).(3.30) By direct calculations from two equations of (3.29), we have [][](𝑋,𝐹𝑌)(𝐹𝑋,𝑌)=𝑔(𝑋,𝑌,𝑉)𝑁+𝑔(𝑋,𝑌,𝑊)𝐿.(3.31) If 𝐻 is integrable, then [𝑋,𝑌]Γ(𝐻) for any 𝑋, 𝑌Γ(𝐻). This implies 𝑔([𝑋,𝑌],𝑉)=𝑔([𝑋,𝑌],𝑊)=0. Thus we get (𝑋,𝐹𝑌)=(𝐹𝑋,𝑌) for all 𝑋, 𝑌Γ(𝐻). Conversely if (𝑋,𝐹𝑌)=(𝐹𝑋,𝑌) for all 𝑋, 𝑌Γ(𝐻), then we have 𝑔([𝑋,𝑌],𝑉)=𝑔([𝑋,𝑌],𝑊)=0. This imply [𝑋,𝑌]Γ(𝐻) for all 𝑋,𝑌Γ(𝐻). Thus 𝐻 is an integrable distribution of 𝑀.
If 𝑀 is totally umbilical, from Theorem 3.7 and (3.29), we have 𝑔𝑋𝑌,𝑉=𝑔𝑋𝑌,𝑊=0,𝑋Γ(𝑇𝑀),𝑌Γ(𝐻).(3.32) This imply 𝑋𝑌Γ(𝐻) for all 𝑋, 𝑌Γ(𝐻), that is, 𝐻 is a parallel distribution on 𝑀.

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

Proof. Assume that 𝐹 is parallel on 𝐻 with respect to . For any 𝑋, 𝑌Γ(𝐻), we have (𝑋𝐹)𝑌=0. Taking the scalar product with 𝑉 and 𝑊 to (3.30) with (𝑋𝐹)𝑌=0, we have 𝐵(𝑋,𝑌)=0 and 𝐷(𝑋,𝑌)=0 for all 𝑋, 𝑌Γ(𝐻), respectively. From (3.29), we have 𝑔(𝑋𝑌,𝑉)=0 and 𝑔(𝑋𝑌,𝑊)=0. This imply 𝑋𝑌Γ(𝐻) for all 𝑋,𝑌Γ(𝐻). Thus 𝐻 is a parallel distribution on 𝑀.
Conversely if 𝐻 is a parallel distribution on 𝑀, from (3.29) we have 𝐵(𝑋,𝐹𝑌)=0,𝐷(𝑋,𝐹𝑌)=0,𝑋,𝑌Γ(𝐻).(3.33) For any 𝑌Γ(𝐻), we show that 𝐹2𝑌=𝐽2𝑌=𝑌+𝜃(𝑌)𝜁. Replacing 𝑌 by 𝐹𝑌 to (3.33) and using (3.21), we have 𝐵(𝑋,𝑌)=0 and 𝐷(𝑋,𝑌)=0 for any 𝑋,𝑌Γ(𝐻). Thus 𝐹 is parallel on 𝐻 with respect to by (3.30).

Theorem 3.12. Let 𝑀 be a tangential half lightlike submanifold of an indefinite cosymplectic manifold 𝑀. If 𝐹 is parallel with respect to the induced connection , then 𝐻 is a parallel distribution on 𝑀 and 𝑀 is locally a product manifold 𝐿𝑈×𝐿𝑊×𝑀𝑇, where 𝐿𝑈 and 𝐿𝑊 are null curves tangent to 𝐽(ltr(𝑇𝑀)) and 𝐽(𝑆(𝑇𝑀)), respectively, and 𝑀𝑇 is a leaf of 𝐻.

Proof. Assume that 𝐹 is parallel on 𝑇𝑀 with respect to . Then 𝐹 is parallel on 𝐻 with respect to . By Theorem 3.11, 𝐻 is a parallel distribution on 𝑀. Applying the operator 𝐹 to (3.14) with (𝑋𝐹)𝑌=0, we have 𝑢𝐴(𝑌)𝐹𝑁𝑋𝐴+𝑤(𝑌)𝐹𝐿𝑋=0,𝑋,𝑌Γ(𝑇𝑀),(3.34) due to 𝐹𝑈=𝐹𝑊=0. Replacing 𝑌 by 𝑈 and 𝑊 to this equation by turns and using (3.9), we have 𝐹(𝐴𝑁𝑋)=0 and 𝐹(𝐴𝐿𝑋)=0. Taking the scalar product with 𝑊 and 𝑁 to (3.14) with (𝑋𝐹)𝑌=0 by turns, we have 𝐷𝐴(𝑋,𝑌)=𝑢(𝑌)𝑤𝑁𝑋𝐴+𝑤(𝑌)𝑤𝐿𝑋𝐴,(3.35)𝑤(𝑌)𝑔𝐿𝑋,𝑁=0,𝑋,𝑌Γ(𝑇𝑀).(3.36) Replacing 𝑌 by 𝜉 to (3.35), we get 𝜙=0 due to (2.13)2. Also replacing 𝑌 by 𝑊 to (3.36), we have 𝜌=0 due to (2.18)2. From thess results, (3.11) and (3.13), we get 𝑋𝑈=𝜏(𝑋)𝑈 and 𝑋𝑊=0 for all 𝑋Γ(𝐻). Thus 𝐽(ltr(𝑇𝑀)) and 𝐽(𝑆(𝑇𝑀)) are also parallel distributions on 𝑀. By the decomposition theorem of de Rham [14], we show that 𝑀=𝐿𝑈×𝐿𝑊×𝑀𝑇, where 𝐿𝑈 and 𝐿𝑊 are null curves tangent to 𝐽(ltr(𝑇𝑀)) and 𝐽(𝑆(𝑇𝑀)), respectively, and 𝑀𝑇 is a leaf of 𝐻.

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

Note 4. From (2.21) we see that a necessary and sufficient condition for 𝑀 to be irrotational is 𝐷(𝑋,𝜉)=0=𝜙(𝑋) for all 𝑋Γ(𝑇𝑀).

Theorem 3.14. Let 𝑀 be a tangential half lightlike submanifold of an indefinite cosymplectic manifold 𝑀. Then one has the following assertions.(i)If 𝑉 is parallel with respect to , then 𝑀 is irrotational, 𝜏=0 and 𝐴𝜉𝐴𝑋=𝑢𝜉𝑋𝐴𝑈+𝑤𝜉𝑋𝑊,𝑋Γ(𝑇𝑀).(3.37)(ii)If 𝑈 is parallel with respect to , then one has 𝜏=𝜌=0 and 𝐴𝑁𝐴𝑋=𝑢𝑁𝑋𝐴𝑈+𝑤𝑁𝑋𝑊,𝑋Γ(𝑇𝑀).(3.38)(iii)If 𝑊 is parallel with respect to , then 𝑀 is irrotational and 𝐴𝐿𝐴𝑋=𝑢𝐿𝑋𝐴𝑈+𝑤𝐿𝑋𝑊,𝑋Γ(𝑇𝑀).(3.39)Moreover, if all of 𝑉, 𝑈, and 𝑊 are parallel on 𝑇𝑀 with respect to , then 𝑆(𝑇𝑀) is totally geodesic in 𝑀 and 𝜏=𝜙=𝜌=0 on Γ(𝑇𝑀). In this case, the null transversal vector field 𝑁 of 𝑀 is a constant on 𝑀.

Proof. If 𝑉 is parallel with respect to , then, taking the scalar product with 𝑈 and 𝑊 to (3.12) by turns, we have 𝜏=0 and 𝜙=0 (𝑀 is irrotational), respectively. Thus we have 𝐹(𝐴𝜉𝑋)=0 for all 𝑋Γ(𝑇𝑀). From this result and (3.8), we obtain 𝐽(𝐴𝜉𝑋)=𝑢(𝐴𝜉𝑋)𝑁+𝑤(𝐴𝜉𝑋)𝐿. Applying 𝐽 to this equation and using 𝜃(𝐴𝜉𝑋)=0, 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 𝑉, 𝑈, and 𝑊 are parallel on 𝑇𝑀 with respect to . Substituting the equation of (i) into (3.10)-1, we have 𝑢𝐴𝑁𝑋𝐴=𝑣𝜉𝑋𝐴=𝑔𝜉𝑋,𝑈=0,𝑋Γ(𝑇𝑀).(3.40) Also, substituting the equation of (iii) into (3.10)-3, we have 𝑤𝐴𝑁𝑋𝐴=𝑣𝐿𝑋𝐴=𝑔𝐿𝑋,𝑈=0,𝑋Γ(𝑇𝑀).(3.41) From the last two equations and the equation of (ii), we see that 𝐴𝑁=0. Thus 𝑆(𝑇𝑀) is totally geodesic in 𝑀 and the 1-forms 𝜏, 𝜙, and 𝜌, defined by (2.9) and (2.10), satisfy 𝜏=𝜙=𝜌=0 on Γ(𝑇𝑀). Using this results, we see that 𝑁 is a constant on 𝑀.

Theorem 3.15. Let 𝑀 be a totally umbilical tangential half lightlike submanifold of an indefinite cosymplectic manifold 𝑀 such that 𝑆(𝑇𝑀) is totally umbilical. Then 𝑀 is locally a product manifold 𝑀=𝑀4×𝑀𝑇𝑜, where 𝑀4 and 𝑀𝑇𝑜 are leaves of 𝐻𝑜 and 𝐻𝑜, respectively.

Proof. By Theorem 3.10, 𝐻 is a parallel distribution 𝑀. Thus, for all 𝑋, 𝑌Γ(𝐻𝑜), we have 𝑋𝑌Γ(𝐻). From (2.11) and (3.30), we have 𝐶(𝑋,𝐹𝑌)=𝑔𝑋𝐹𝑌,𝑁=𝑔𝑋𝐹𝑌+𝐹𝑋𝑌𝐹,𝑁=𝑔𝑋𝑌,𝑁=𝑔𝑋𝑌,𝐽𝑁=𝑔𝑋,𝑌,𝑈(3.42) due to 𝐹𝑌Γ(𝐻𝑜). If 𝑆(𝑇𝑀) is totally umbilical in 𝑀, then we have 𝐶=0 due to Theorem 3.7. By (2.11) and (3.42), we get 𝑔𝑋𝑌,𝑁=0,𝑔𝑋𝐻𝑌,𝑈=0,𝑋Γ(𝑇𝑀),𝑌Γ𝑜.(3.43) These results and (3.29) imply 𝑋𝑌Γ(𝐻𝑜) for all 𝑋, 𝑌Γ(𝐻𝑜). Thus 𝐻𝑜 is a parallel distribution on 𝑀 and 𝑇𝑀=𝐻𝑜orth𝐻𝑜, where 𝐻𝑜=Span{𝜉,𝑉,𝑈,𝑊}. By Theorems 3.5 and 3.7, we have 𝐵=𝐷=𝐴𝑁=𝜙=0 and 𝐴𝐿𝑋=𝜌(𝑋)𝜉. Thus (2.12) and (3.11)(3.13) deduce, respectively, to 𝑋𝜉=𝜏(𝑋)𝜉,𝑋𝑈=𝜏(𝑋)𝑈+𝜌(𝑋)𝑊,𝑋𝑉=𝜏(𝑋)𝑉,𝑋𝐻𝑊=𝜌(𝑋)𝑉,𝑋Γ𝑜.(3.44) Thus 𝐻𝑜 is also a parallel distribution on 𝑀. Thus we have 𝑀=𝑀4×𝑀𝑇𝑜, where 𝑀4 is a leaf of 𝐻𝑜 and 𝑀𝑇𝑜 is a leaf of 𝐻𝑜.

4. Ascreen Half Lightlike Submanifolds

Definition 4.1. A half lightlike submanifold 𝑀 of an indefinite cosymplectic manifold 𝑀 is said to be an ascreen half lightlike submanifold [4] of 𝑀 if the structure vector field 𝜁 of 𝑀 belongs to the distribution Rad(𝑇𝑀)ltr(𝑇𝑀).

For any ascreen half lightlike submanifold 𝑀, the vector field 𝜁 is decomposed as 𝜁=𝑎𝜉+𝑏𝑁.(4.1) In this case, we show that 𝑎0 and 𝑏0 by Proposition 3.2.

Definition 4.2. A half lightlike submanifold 𝑀 is called screen conformal [2, 3] if there exists a nonvanishing smooth function 𝜑 such that 𝐴𝑁=𝜑𝐴𝜉, or equivalently, 𝐶(𝑋,𝑃𝑌)=𝜑𝐵(𝑋,𝑌),𝑋,𝑌Γ(𝑇𝑀).(4.2)

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

Proof. Applying 𝑋 to (4.1) and using (2.3), (2.9), and (2.21), we have 𝑎𝐴𝜉𝑋+𝑏𝐴𝑁𝑋={𝑋𝑎𝑎𝜏(𝑋)}𝜉+{𝑋𝑏+𝑏𝜏(𝑋)}𝑁+{𝑏𝜌(𝑋)𝑎𝜙(𝑋)}𝐿.(4.3) Taking the product with 𝜉, 𝑁, and 𝐿 by turns and using (2.16)2 and (2.17)2, we get 𝐴𝑁𝑋=𝜑𝐴𝜉𝑋,𝑋𝑎=𝑎𝜏(𝑋),𝑋𝑏=𝑏𝜏(𝑋),𝑎𝜙(𝑋)=𝑏𝜌(𝑋),(4.4) for all 𝑋Γ(𝑇𝑀), where we set 𝜑=𝑎/𝑏. Thus 𝑀 is screen conformal.

Substituting (4.1) into 𝑔(𝜁,𝜁)=1, we have 2𝑎𝑏=1. Consider the local unit timelike vector field 𝑉 on 𝑀 and its 1-form 𝑣 defined by 𝑉=𝑏1𝐽𝜉,𝑣(𝑋)=𝑔𝑋,𝑉,𝑋Γ(𝑇𝑀).(4.5) Let 𝑈=𝑎1𝐽𝑁. Then 𝑈 is a unit timelike vector field on 𝑆(𝑇𝑀) such that 𝑔(𝑉,𝑈)=1. Applying 𝐽 to (4.1) and using (2.1) and 2𝑎𝑏=1, we have 𝑉0=𝑎𝐽𝜉+𝑏𝐽𝑁=+𝑈2i.e.,𝑈=𝑉.(4.6) From this we show that 𝐽(Rad(𝑇𝑀))=𝐽(ltr(𝑇𝑀)). Using this and Proposition 3.1, the tangent bundle 𝑇𝑀 of 𝑀 is decomposed as follows: 𝑇𝑀=Rad(𝑇𝑀)orth𝐽(Rad(𝑇𝑀))orth𝐽𝑆𝑇𝑀orth𝐻,(4.7) where 𝐻 is a nondegenerate and almost complex distribution on 𝑀 with respect to the indefinite cosymplectic structure tensor 𝐽, otherwise 𝑆(𝑇𝑀) is degenerate. Consider the local unit spacelike vector field 𝑊 on 𝑆(𝑇𝑀) and its 1-form 𝑤 defined by 𝑊=𝐽𝐿,𝑤(𝑋)=𝑔𝑋,𝑊,𝑋Γ(𝑇𝑀).(4.8) Denote by 𝑆 the projection morphism of 𝑇𝑀 on 𝐻. Using (4.7), for any vector field 𝑋 on 𝑀, the vector field 𝐽𝑋 is expressed as follows: 𝐽𝑋=𝑓𝑋+𝑎𝑣(𝑋)𝜉𝑏𝜂(𝑋)𝑉𝑏𝑣(𝑋)𝑁+𝑤(𝑋)𝐿,(4.9) because 𝐽𝑉=𝑎𝜉𝑏𝑁, where 𝑓 is a tensor field of type (1, 1) defined by 𝑓𝑋=𝐽𝑆𝑋,𝑋Γ(𝑇𝑀).(4.10) Applying 𝐽 to (2.10) and (2.21) and using (2.2), (2.8) and (4.4)(4.9), we get 𝑏𝑋𝑉𝐴=𝑓𝜉𝑋𝑎𝐵𝑋,𝑉𝜉𝜙(𝑋)𝑊,(4.11)𝑋𝑊𝐴=𝑓𝐿𝑋𝑎𝐷𝑋,𝑉𝜉2𝑎𝜙(𝑋)𝑉,(4.12)𝑏𝐷𝑋,𝑉=𝐵𝑋,𝑊,𝑋Γ(𝑇𝑀).(4.13)

Example 4.4. Consider a submanifold 𝑀 of 𝑀=(𝑅52,𝐽,𝜁,𝜃,𝑔) given by the equation 𝑋𝑢1,𝑢2,𝑢3=𝑢1,𝑢2,𝑢3,𝑢2,12𝑢1+𝑢3.(4.14) By direct calculations we easily check that 𝑇𝑀=Span𝜉=𝜕𝑥1+𝜕𝑦1+2𝜕𝑧,𝑈=𝜕𝑥1𝜕𝑦1,𝑉=𝜕𝑥2+𝜕𝑦2,𝑇𝑀=Span𝜉,𝐿=𝜕𝑥2𝜕𝑦2,Rad(𝑇𝑀)=Span{𝜉}.(4.15) We obtain the lightlike transversal and transversal vector bundles 1ltr(𝑇𝑀)=Span𝑁=4𝜕𝑥1𝜕𝑦1+2𝜕𝑧,tr(𝑇𝑀)=Span{𝑁,𝐿}.(4.16) From this results, we show that 𝐽𝜉=𝑈, Rad(𝑇𝑀)𝐽(Rad(𝑇𝑀))={0}, 𝐽𝑁=(1/4)𝑈, 𝐽𝐿=𝑉, 𝐽𝑁=(1/4)𝐽𝜉 and 𝐽(Rad(𝑇𝑀)=𝐽(ltr(𝑇𝑀), 𝜁=(1/22)𝜉+2𝑁 and 𝐽𝜁=0. Thus 𝑀 is an ascreen half lightlike submanifold of an indefinite cosymplectic manifold 𝑀.

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

Proof. Assume that 𝑀 is totally umbilical. From (3.25) and (4.13), we have 𝑏𝛿𝑔𝑋,𝑉=𝛽𝑔𝑋,𝑊,𝑋Γ(𝑇𝑀).(4.17) Replacing 𝑋 by 𝑊 and 𝑉 to this equation by turns, we have 𝛽=0 and 𝛿=0, respectively. Thus we have =0 and 𝑀 is totally geodesic. By (4.2), we also have 𝐶=0. Thus 𝑆(𝑇𝑀) is also totally geodesic in 𝑀.

Taking 𝑌Γ(𝐻). Then we have 𝑓𝑌=𝐽𝑌Γ(𝐻) due to (4.9). Applying 𝑋 to 𝐽𝑌=𝑓𝑌 and using (2.2), (2.8), (4.2), (4.5), and (4.9), we have 𝐵(𝑋,𝑓𝑌)=𝑏𝑔𝑋𝑌,𝑉,𝐷(𝑋,𝑓𝑌)=𝑔𝑋𝑌,𝑊,(4.18)𝑋𝑓𝑌=𝑎𝑔𝑋𝑌,𝑉𝜉+2𝑎𝐵(𝑋,𝑌)𝑉𝐷(𝑋,𝑌)𝑊,(4.19) for all 𝑋Γ(𝑇𝑀). 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 𝑆(𝑇𝑀) is integrable due to (4.2), the following two theorems hold.

Theorem 4.6. Let 𝑀 be an ascreen half lightlike submanifold of an indefinite cosymplectic manifold 𝑀. 𝐻 is an integrable distribution on 𝑀 if and only if one has 𝐻(𝑋,𝑓𝑌)=(𝑓𝑋,𝑌),𝑋,𝑌Γ.(4.20) Moreover, if 𝑀 is totally umbilical, then 𝐻 is a parallel distribution on 𝑀.

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

Theorem 4.8. Let 𝑀 be an ascreen half lightlike submanifold of an indefinite cosymplectic manifold 𝑀. If 𝑀 is totally umbilical, then 𝑀 is locally a product manifold 𝐿𝜉×𝐿𝑉×𝐿𝑊×𝑀, where 𝐿𝜉, 𝐿𝑉, and 𝐿𝑊 are null, timelike, and spacelike curves tangent to Rad(𝑇𝑀), 𝐽(Rad(𝑇𝑀)), and 𝐽(𝑆(𝑇𝑀)), respectively, and 𝑀 is a leaf of 𝐻.

Proof. If 𝑀 is totally umbilical, then 𝐻 is a parallel distribution on 𝑀 by Theorem 4.6 and we have 𝐵=𝐷=𝐴𝜉=𝜙=0;𝐴𝐿𝑋=𝜌(𝑋)𝜉 by Theorem 4.5. From (4.4)1, we also have 𝐴𝑁=0. Using (2.12), (4.11), and (4.12), we have 𝑋𝜉=𝜏(𝑋)𝜉 and 𝑋𝑉=𝑋𝑊=0 due to 𝑓𝜉=0. This implies that all of the distributions Rad(𝑇𝑀), 𝐽(Rad(𝑇𝑀)), and 𝐽(𝑆(𝑇𝑀)) are parallel on 𝑀. Thus we have 𝑀=𝐿𝜉×𝐿𝑉×𝐿𝑊×𝑀, where 𝐿𝜉, 𝐿𝑉, and 𝐿𝑊 are null, timelike, and spacelike curves tangent to Rad(𝑇𝑀), 𝐽(Rad(𝑇𝑀)) and 𝐽(𝑆(𝑇𝑀)), respectively, and 𝑀 is a leaf of 𝐻.

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 𝑀 be an ascreen half lightlike submanifold of an indefinite cosymplectic manifold 𝑀. Then one has the following assertions.(i)If 𝑉 is parallel with respect to on 𝑀, then 𝑀 is irrotational and 𝐴𝜉𝑋=𝐵𝑋,𝑊𝑊,𝐵𝑋,𝑉=0,𝜌(𝑋)=0,𝑋Γ(𝑇𝑀).(4.21)(ii)If 𝑊 is parallel with respect to on 𝑀, then 𝑀 is irrotational and 𝐴𝐿𝑋=𝐷𝑋,𝑊𝑊,𝐷𝑋,𝑉=0,𝜌(𝑋)=0,𝑋Γ(𝑇𝑀).(4.22)Moreover, if 𝑉 and 𝑊 are parallel with respect to , then one sees that 𝐴𝜉=0 and the screen distribution 𝑆(𝑇𝑀) is totally geodesic in 𝑀.

Theorem 4.10. Let 𝑀 be an ascreen half lightlike submanifold of an indefinite cosymplectic manifold 𝑀. If 𝑉 and 𝑊 are parallel with respect to , then 𝑀 is locally a product manifold 𝐿𝜉×𝐿𝑉×𝐿𝑊×𝑀, where 𝐿𝜉, 𝐿𝑉, and 𝐿𝑊 are null, timelike, and spacelike curves tangent to Rad(𝑇𝑀), 𝐽(Rad(𝑇𝑀)), and 𝐽(𝑆(𝑇𝑀)), respectively, and 𝑀 is a leaf of 𝐻.

Proof. If 𝑉 is parallel with respect to , for any 𝑌Γ(𝐻), we have 𝐴𝐵(𝑋,𝑌)=𝑔𝜉𝑋,𝑌=𝐵𝑋,𝑊𝑔𝑌,𝑊=0,𝑋Γ(𝑇𝑀).(4.23) Thus we get 𝑔(𝑋𝑌,𝑉)=𝑏1𝐵(𝑋,𝑓𝑌)=0 because 𝑓𝑌Γ(𝐻). Also if 𝑊 is parallel with respect to , then, for any 𝑌Γ(𝐻), we have 𝐷𝐴(𝑋,𝑌)=𝑔𝐿𝑋,𝑌=𝐷𝑋,𝑊𝑔𝑌,𝑊=0,𝑋Γ(𝑇𝑀).(4.24) From these results and (4.19), we show that 𝑓 is parallel on 𝐻 with respect to . Thus, by Theorem 4.7, we see that 𝐻 is a parallel distribution on 𝑀. As 𝑉 and 𝑊 are parallel with respect to and 𝑋𝜉=𝜏(𝑋)𝜉 due to 𝐴𝜉=0, we have our theorem.