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Journal of Function Spaces and Applications
VolumeΒ 2012Β (2012), Article IDΒ 636242, 16 pages
http://dx.doi.org/10.1155/2012/636242
Research Article

Special Half Lightlike Submanifolds of an Indefinite Cosymplectic Manifold

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: DashanΒ Fan

Copyright Β© 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 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 [1–4]. Half lightlike submanifold is a special case of π‘Ÿ-lightlike submanifold [5, 6] such that π‘Ÿ=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 π‘Ÿ-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 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,πœƒ,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 πœƒ a 1-form satisfying 𝐽2𝑋=βˆ’π‘‹+πœƒ(𝑋)𝜁,𝐽𝜁=0,πœƒβˆ˜π½=0,πœƒ(𝜁)=1,𝑔(𝜁,𝜁)=1,𝑔(𝐽𝑋,π½π‘Œ)=𝑔(𝑋,π‘Œ)βˆ’πœƒ(𝑋)πœƒ(π‘Œ),πœƒ(𝑋)=𝑔(𝜁,𝑋),π‘‘πœƒ(𝑋,π‘Œ)=𝑔(𝐽𝑋,π‘Œ),(2.1) for any vector fields 𝑋, π‘Œ on 𝑀. We say that 𝑀 has a normal contact structure if 𝑁𝐽+π‘‘πœƒβŠ—πœ=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 βˆ‡π‘‹πœƒ=0,βˆ‡π‘‹π½=0,(2.2) for any vector field 𝑋 on 𝑀, where βˆ‡ is the Levi-Civita connection of 𝑀. A cosymplectic manifold 𝑀=(𝑀,𝐽,𝜁,πœƒ,𝑔) is called an indefinite cosymplectic manifold [7–9] if (𝑀,𝑔) is a semi-Riemannian manifold of index πœ‡(>0).

For any indefinite cosymplectic manifold 𝑀, applying βˆ‡π‘‹ to 𝐽𝜁=0 and using (2.2), we have 𝐽(βˆ‡π‘‹πœ)=0. Applying 𝐽 to this and using the fact πœƒ(βˆ‡π‘‹πœ)=0, we get βˆ‡π‘‹πœ=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)=π‘‡π‘€βˆ©π‘‡π‘€βŸ‚ is 1, where 𝑇𝑀 and π‘‡π‘€βŸ‚ are the tangent and normal bundles of 𝑀, respectively. Then there exist complementary nondegenerate distributions 𝑆(𝑇𝑀) and 𝑆(π‘‡π‘€βŸ‚) of Rad(𝑇𝑀) in 𝑇𝑀 and π‘‡π‘€βŸ‚, respectively, which are called the screen and coscreen distribution on 𝑀: 𝑇𝑀=Rad(𝑇𝑀)βŠ•orth𝑆(𝑇𝑀),π‘‡π‘€βŸ‚=Rad(𝑇𝑀)βŠ•orthπ‘†ξ€·π‘‡π‘€βŸ‚ξ€Έ,(2.4) where the symbol βŠ•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 Ξ“(𝐸) the 𝐹(𝑀) module of smooth sections of a vector bundle 𝐸 over 𝑀. Choose 𝐿 as a unit vector field of 𝑆(π‘‡π‘€βŸ‚) such that 𝑔(𝐿,𝐿)=Β±1. In this paper we may assume that 𝑔(𝐿,𝐿)=1 without loss of generality. Consider the orthogonal complementary distribution 𝑆(𝑇𝑀)βŸ‚ to 𝑆(𝑇𝑀) in 𝑇𝑀. Certainly Rad(𝑇𝑀) and 𝑆(π‘‡π‘€βŸ‚) are vector subbundles of 𝑆(𝑇𝑀)βŸ‚. Thus we have the following orthogonal decomposition: 𝑆(𝑇𝑀)βŸ‚ξ€·=π‘†π‘‡π‘€βŸ‚ξ€ΈβŠ•orthπ‘†ξ€·π‘‡π‘€βŸ‚ξ€ΈβŸ‚,(2.5) where 𝑆(π‘‡π‘€βŸ‚)βŸ‚ is the orthogonal complementary to 𝑆(π‘‡π‘€βŸ‚) in 𝑆(𝑇𝑀)βŸ‚. It is well-known [1, 2] that, for any null section πœ‰ of Rad(𝑇𝑀) on a coordinate neighborhood π’°βŠ‚π‘€, there exists a uniquely defined null vector field π‘βˆˆΞ“(ltr(𝑇𝑀)) satisfying 𝑔(πœ‰,𝑁)=1,𝑔(𝑁,𝑁)=𝑔(𝑁,𝑋)=𝑔(𝑁,𝐿)=0,βˆ€π‘‹βˆˆΞ“(𝑆(𝑇𝑀)).(2.6) Let tr(𝑇𝑀)=𝑆(π‘‡π‘€βŸ‚)βŠ•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=π‘‡π‘€βŠ•tr(𝑇𝑀)={Rad(𝑇𝑀)βŠ•tr(𝑇𝑀)}βŠ•orth𝑆(𝑇𝑀)={Rad(𝑇𝑀)βŠ•ltr(𝑇𝑀)}βŠ•orth𝑆(𝑇𝑀)βŠ•orthπ‘†ξ€·π‘‡π‘€βŸ‚ξ€Έ.(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 βˆ‡π‘‹π‘Œ=βˆ‡π‘‹π‘Œ+𝐡(𝑋,π‘Œ)𝑁+𝐷(𝑋,π‘Œ)𝐿,(2.8)βˆ‡π‘‹π‘=βˆ’π΄π‘π‘‹+𝜏(𝑋)𝑁+𝜌(𝑋)𝐿,(2.9)βˆ‡π‘‹πΏ=βˆ’π΄πΏβˆ‡π‘‹+πœ™(𝑋)𝑁;(2.10)π‘‹π‘ƒπ‘Œ=βˆ‡βˆ—π‘‹βˆ‡π‘ƒπ‘Œ+𝐢(𝑋,π‘ƒπ‘Œ)πœ‰,(2.11)π‘‹πœ‰=βˆ’π΄βˆ—πœ‰π‘‹βˆ’πœ(𝑋)πœ‰,(2.12) for all 𝑋, π‘ŒβˆˆΞ“(𝑇𝑀), where βˆ‡ and βˆ‡βˆ— are induced linear connections on 𝑇𝑀 and 𝑆(𝑇𝑀), respectively, 𝐡 and 𝐷 are called the local second fundamental forms of 𝑀, 𝐢 is called the local second fundamental form on 𝑆(𝑇𝑀). 𝐴𝑁, π΄βˆ—πœ‰, and 𝐴𝐿 are linear operators on 𝑇𝑀 and 𝜏,β€‰β€‰πœŒ, and πœ™ are 1-forms on 𝑇𝑀. Since βˆ‡ is torsion-free, βˆ‡ is also torsion-free, and 𝐡 and 𝐷 are symmetric. From the facts 𝐡(𝑋,π‘Œ)=𝑔(βˆ‡π‘‹π‘Œ,πœ‰) and 𝐷(𝑋,π‘Œ)=𝑔(βˆ‡π‘‹π‘Œ,𝐿), we know that 𝐡 and 𝐷 are independent of the choice of the screen distribution 𝑆(𝑇𝑀) and 𝐡(𝑋,πœ‰)=0,𝐷(𝑋,πœ‰)=βˆ’πœ™(𝑋),βˆ€π‘‹βˆˆΞ“(𝑇𝑀).(2.13) We say that β„Ž(𝑋,π‘Œ)=𝐡(𝑋,π‘Œ)𝑁+𝐷(𝑋,π‘Œ)𝐿 is the second fundamental tensor of 𝑀. The induced connection βˆ‡ of 𝑀 is not metric and satisfies ξ€·βˆ‡π‘‹π‘”ξ€Έ(π‘Œ,𝑍)=𝐡(𝑋,π‘Œ)πœ‚(𝑍)+𝐡(𝑋,𝑍)πœ‚(π‘Œ),(2.14) for all 𝑋, π‘Œ, π‘βˆˆΞ“(𝑇𝑀), where πœ‚ is a 1-form on 𝑇𝑀 such that πœ‚(𝑋)=𝑔(𝑋,𝑁),βˆ€π‘‹βˆˆΞ“(𝑇𝑀).(2.15) But the connection βˆ‡βˆ— on 𝑆(𝑇𝑀) is metric. The above three local second fundamental forms of 𝑀 and 𝑆(𝑇𝑀) are related to their shape operators by 𝐴𝐡(𝑋,π‘Œ)=π‘”βˆ—πœ‰ξ‚,𝑋,π‘Œπ‘”ξ‚€π΄βˆ—πœ‰ξ‚ξ€·π΄π‘‹,𝑁=0,(2.16)𝐢(𝑋,π‘ƒπ‘Œ)=𝑔𝑁,𝑋,π‘ƒπ‘Œπ‘”ξ€·π΄π‘ξ€Έξ€·π΄π‘‹,𝑁=0,(2.17)𝐷(𝑋,π‘ƒπ‘Œ)=𝑔𝐿,𝑋,π‘ƒπ‘Œπ‘”ξ€·π΄πΏξ€Έξ€·π΄π‘‹,𝑁=𝜌(𝑋),(2.18)𝐷(𝑋,π‘Œ)=𝑔𝐿𝑋,π‘Œβˆ’πœ™(𝑋)πœ‚(π‘Œ),βˆ€π‘‹,π‘ŒβˆˆΞ“(𝑇𝑀).(2.19) By (2.16) and (2.17), we show that π΄βˆ—πœ‰ and 𝐴𝑁 are Ξ“(𝑆(𝑇𝑀))-valued shape operators related to 𝐡 and 𝐢, respectively, and π΄βˆ—πœ‰ is self-adjoint on 𝑇𝑀 and π΄βˆ—πœ‰πœ‰=0.(2.20) Replacing π‘Œ by πœ‰ to (2.8) and using (2.12) and (2.13), we have βˆ‡π‘‹πœ‰=βˆ’π΄βˆ—πœ‰π‘‹βˆ’πœ(𝑋)πœ‰βˆ’πœ™(𝑋)𝐿,βˆ€π‘‹βˆˆΞ“(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=πœ”+π‘Žπœ‰+𝑏𝑁+𝑒𝐿,(3.1) where πœ” is a smooth vector field on 𝑆(𝑇𝑀), and π‘Ž=πœƒ(𝑁), 𝑏=πœƒ(πœ‰), and 𝑒=πœƒ(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 𝐽𝑆(𝑇𝑀)βŸ‚ξ€ΈβŠ‚π‘†(𝑇𝑀).(3.2)

Note 1. Although, in general, 𝑆(𝑇𝑀) is not unique, it is canonically isomorphic to the factor vector bundle 𝑆(𝑇𝑀)βˆ—=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)βŸ‚)βŠ‚π‘†(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=π‘Žπœ‰ and π‘Žβ‰ 0 (or 𝜁=𝑏𝑁 and 𝑏≠0). From this, we have 1=𝑔(𝜁,𝜁)=π‘Ž2𝑔(πœ‰,πœ‰)=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). CΔƒlin [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, π‘Ž=𝑏=𝑒=0 by Note 2. Then there exists a nondegenerate almost complex distribution π»π‘œ on 𝑀 with respect to 𝐽, that is, 𝐽(π»π‘œ)=π»π‘œ, such that 𝑆(𝑇𝑀)={𝐽(Rad(𝑇𝑀))βŠ•π½(ltr(𝑇𝑀))}βŠ•orthπ½ξ€·π‘†ξ€·π‘‡π‘€βŸ‚βŠ•ξ€Έξ€Έorthπ»π‘œ.(3.4) Thus the general decompositions (2.4) and (2.7) reduce, respectively, to 𝑇𝑀=π»βŠ•π»ξ…ž,𝑇𝑀=π»βŠ•π»β€²βŠ•tr(𝑇𝑀),(3.5) where 𝐻 and 𝐻′ are 2- and 1-lightlike distributions on 𝑀 such that 𝐻=Rad(𝑇𝑀)βŠ•orth𝐽(Rad(𝑇𝑀))βŠ•orthπ»π‘œ,𝐻′=𝐽(ltr(𝑇𝑀))βŠ•orthπ½ξ€·π‘†ξ€·π‘‡π‘€βŸ‚.ξ€Έξ€Έ(3.6)𝐻 is an almost complex distribution of 𝑀 with respect to 𝐽. Consider a pair of local null vector fields {π‘ˆ,𝑉} and a local nonnull vector field π‘Š on 𝑆(𝑇𝑀) defined by π‘ˆ=βˆ’π½π‘,𝑉=βˆ’π½πœ‰,π‘Š=βˆ’π½πΏ.(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+𝑒(𝑋)π‘ˆ+𝑀(𝑋)π‘Š,𝐽𝑋=𝐹𝑋+𝑒(𝑋)𝑁+𝑀(𝑋)𝐿,(3.8) where 𝑒, 𝑣, and 𝑀 are 1-forms locally defined on 𝑀 by 𝑒(𝑋)=𝑔(𝑋,𝑉),𝑣(𝑋)=𝑔(𝑋,π‘ˆ),𝑀(𝑋)=𝑔(𝑋,π‘Š)(3.9) and 𝐹 is a tensor field of type (1,1) globally defined on 𝑀 by 𝐹=π½βˆ˜π‘†. Applying the operator βˆ‡π‘‹ 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,π‘ŒβˆˆΞ“(𝑇𝑀), we have βˆ‡π΅(𝑋,π‘ˆ)=𝐢(𝑋,𝑉),𝐡(𝑋,π‘Š)=𝐷(𝑋,𝑉),𝐢(𝑋,π‘Š)=𝐷(𝑋,π‘ˆ),(3.10)π‘‹ξ€·π΄π‘ˆ=πΉπ‘π‘‹ξ€Έβˆ‡+𝜏(𝑋)π‘ˆ+𝜌(𝑋)π‘Š,(3.11)𝑋𝐴𝑉=πΉβˆ—πœ‰π‘‹ξ‚βˆ‡βˆ’πœ(𝑋)π‘‰βˆ’πœ™(𝑋)π‘Š,(3.12)π‘‹ξ€·π΄π‘Š=πΉπΏπ‘‹ξ€Έξ€·βˆ‡+πœ™(𝑋)π‘ˆ,(3.13)𝑋𝐹(π‘Œ)=𝑒(π‘Œ)𝐴𝑁𝑋+𝑀(π‘Œ)π΄πΏπ‘‹βˆ’π΅(𝑋,π‘Œ)π‘ˆβˆ’π·(𝑋,π‘Œ)π‘Š.(3.14)

Note 3. From now on, 𝑀=(π‘…π‘ž2π‘š+1,𝐽,𝜁,πœƒ,𝑔) will denote the semi-Euclidean manifold π‘…π‘ž2π‘š+1 equipped with its usual cosymplectic structure given by πœƒ=𝑑𝑧,𝜁=πœ•π‘§,𝑔=πœƒβŠ—πœƒβˆ’π‘ž/2𝑖=1𝑑π‘₯π‘–βŠ—π‘‘π‘₯𝑖+π‘‘π‘¦π‘–βŠ—π‘‘π‘¦π‘–ξ€Έ+π‘šξ“π‘–=π‘ž+1𝑑π‘₯π‘–βŠ—π‘‘π‘₯𝑖+π‘‘π‘¦π‘–βŠ—π‘‘π‘¦π‘–ξ€Έ,π½ξƒ©π‘šξ“π‘–=1ξ€·π‘‹π‘–πœ•π‘₯𝑖+π‘Œπ‘–πœ•π‘¦π‘–ξ€Έξƒͺ=+π‘πœ•π‘§π‘šξ“π‘–=1ξ€·π‘Œπ‘–πœ•π‘₯π‘–βˆ’π‘‹π‘–πœ•π‘¦π‘–ξ€Έ,(3.15) where (π‘₯𝑖,𝑦𝑖,𝑧) are the Cartesian coordinates and 𝑔 is a semi-Euclidean metric of signature (βˆ’,+,…,+;βˆ’,+,…,+;+) with respect to the canonical basis ξ€½πœ•π‘₯1,πœ•π‘₯2,…,πœ•π‘₯π‘š;πœ•π‘¦1,πœ•π‘¦2,…,πœ•π‘¦π‘šξ€Ύ;πœ•π‘§.(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,πœƒ,𝑔) given by the equations π‘₯1=𝑦4,π‘₯2=1βˆ’π‘¦22,𝑦2β‰ Β±1.(3.17) Then a local frame fields of 𝑇𝑀 are given by πœ‰=πœ•π‘₯1+πœ•π‘¦4,π‘ˆ1=πœ•π‘₯4βˆ’πœ•π‘¦1,π‘ˆ2=πœ•π‘₯3,π‘ˆ3=πœ•π‘¦3,π‘ˆ4𝑦=βˆ’2π‘₯2πœ•π‘₯2+πœ•π‘¦2,π‘ˆ5=πœ•π‘₯4+πœ•π‘¦1,π‘ˆ6=𝜁=πœ•π‘§.(3.18) This implies Rad(𝑇𝑀)=Span{πœ‰}, π½πœ‰=π‘ˆ1, and Rad(𝑇𝑀)∩𝐽(Rad(𝑇𝑀))={0}. Next, π½π‘ˆ2=βˆ’π‘ˆ3 implies that π»π‘œ={π‘ˆ2,π‘ˆ3} invariant with respect to the almost contact structure tensor 𝐽. By direct calculations, we have π‘†ξ€·π‘‡π‘€βŸ‚ξ€Έξ‚»=Span𝐿=πœ•π‘₯2+𝑦2π‘₯2πœ•π‘¦21,ltr(𝑇𝑀)=Span𝑁=2ξ€·βˆ’πœ•π‘₯1+πœ•π‘¦4.(3.19) We show that 𝐽𝐿=βˆ’π‘ˆ4, 𝐽𝑁=(1/2)π‘ˆ5, 𝐽𝜁=0 and βˆ‡π‘‹πœ=0 for all π‘‹βˆˆΞ“(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 βˆ‡ and βˆ‡βˆ—. Furthermore, 𝜁 is conjugate to any vector field of 𝑀 with respect to β„Ž and 𝐢.

Proof. Replacing π‘Œ by 𝜁 to (2.8) and using (2.3) and the fact πœβˆˆΞ“(𝑇𝑀), we get βˆ‡π‘‹πœ+𝐡(𝑋,𝜁)𝑁+𝐷(𝑋,𝜁)𝐿=0,βˆ€π‘‹βˆˆΞ“(𝑇𝑀).(3.20) Taking the scalar product with πœ‰ and 𝐿 to this equation by turns, we have βˆ‡π‘‹πœ=0,𝐡(𝑋,𝜁)=0,𝐷(𝑋,𝜁)=0,βˆ€π‘‹βˆˆΞ“(𝑇𝑀).(3.21) From (3.21)1, we see that 𝜁 is parallel with respect to the induced connection βˆ‡. (3.21)2,3 implies that 𝜁 is conjugate to any vector field on 𝑀 with respect to the second fundamental form β„Ž. Replacing π‘ƒπ‘Œ by 𝜁 to (2.11) and using (3.21)1 and the fact πœβˆˆΞ“(𝑆(𝑇𝑀)), we get βˆ‡βˆ—π‘‹πœ+𝐢(𝑋,𝜁)πœ‰=0,βˆ€π‘‹βˆˆΞ“(𝑇𝑀).(3.22) Taking the scalar product with 𝑁 to this equation, we have βˆ‡βˆ—π‘‹πœ=0,𝐢(𝑋,𝜁)=0,βˆ€π‘‹βˆˆΞ“(𝑇𝑀).(3.23) Thus 𝜁 is also parallel with respect to the lieasr connection βˆ‡βˆ— and conjugate to any vector field on 𝑀 with respect to 𝐢. 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 β„‹ on tr(𝑇𝑀) on any coordinate neighborhood 𝒰 such that β„Ž(𝑋,π‘Œ)=ℋ𝑔(𝑋,π‘Œ),βˆ€π‘‹,π‘ŒβˆˆΞ“(𝑇𝑀).(3.24) In case β„‹=0, that is, β„Ž=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 𝐡(𝑋,π‘Œ)=𝛽𝑔(𝑋,π‘Œ),𝐷(𝑋,π‘Œ)=𝛿𝑔(𝑋,π‘Œ),βˆ€π‘‹,π‘ŒβˆˆΞ“(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,βˆ€π‘‹βˆˆΞ“(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, 𝐡=𝐷=0. Thus we have β„Ž=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 𝐢(𝑋,π‘ƒπ‘Œ)=𝛾𝑔(𝑋,π‘Œ),βˆ€π‘‹,π‘ŒβˆˆΞ“(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 π‘Œ by 𝜁 to (3.27) and using (3.23), we have 𝛾𝑔(𝑋,𝜁)=0 for all π‘‹βˆˆΞ“(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 β„Ž(𝑋,πΉπ‘Œ)=β„Ž(𝐹𝑋,π‘Œ),βˆ€π‘‹,π‘ŒβˆˆΞ“(𝐻).(3.28) Moreover, if 𝑀 is totally umbilical, then 𝐻 is a parallel distribution on 𝑀.

Proof. Taking π‘ŒβˆˆΞ“(𝐻), we show that πΉπ‘Œ=π½π‘ŒβˆˆΞ“(𝐻). Applying βˆ‡π‘‹ to πΉπ‘Œ=π½π‘Œ and using (2.3), (2.8), (3.7), (3.8)2, and (3.9), we have π΅ξ€·βˆ‡(𝑋,πΉπ‘Œ)=π‘”π‘‹ξ€Έξ€·βˆ‡π‘Œ,𝑉,𝐷(𝑋,πΉπ‘Œ)=π‘”π‘‹ξ€Έξ€·βˆ‡π‘Œ,π‘Š,(3.29)𝑋𝐹(π‘Œ)=βˆ’π΅(𝑋,π‘Œ)π‘ˆβˆ’π·(𝑋,π‘Œ)π‘Š,βˆ€π‘‹βˆˆΞ“(𝑇𝑀).(3.30) By direct calculations from two equations of (3.29), we have [][]β„Ž(𝑋,πΉπ‘Œ)βˆ’β„Ž(𝐹𝑋,π‘Œ)=𝑔(𝑋,π‘Œ,𝑉)𝑁+𝑔(𝑋,π‘Œ,π‘Š)𝐿.(3.31) If 𝐻 is integrable, then [𝑋,π‘Œ]βˆˆΞ“(𝐻) for any 𝑋, π‘ŒβˆˆΞ“(𝐻). This implies 𝑔([𝑋,π‘Œ],𝑉)=𝑔([𝑋,π‘Œ],π‘Š)=0. Thus we get β„Ž(𝑋,πΉπ‘Œ)=β„Ž(𝐹𝑋,π‘Œ) for all 𝑋, π‘ŒβˆˆΞ“(𝐻). Conversely if β„Ž(𝑋,πΉπ‘Œ)=β„Ž(𝐹𝑋,π‘Œ) for all 𝑋, π‘ŒβˆˆΞ“(𝐻), then we have 𝑔([𝑋,π‘Œ],𝑉)=𝑔([𝑋,π‘Œ],π‘Š)=0. This imply [𝑋,π‘Œ]βˆˆΞ“(𝐻) for all 𝑋,π‘ŒβˆˆΞ“(𝐻). Thus 𝐻 is an integrable distribution of 𝑀.
If 𝑀 is totally umbilical, from Theorem 3.7 and (3.29), we have π‘”ξ€·βˆ‡π‘‹ξ€Έξ€·βˆ‡π‘Œ,𝑉=π‘”π‘‹ξ€Έπ‘Œ,π‘Š=0,βˆ€π‘‹βˆˆΞ“(𝑇𝑀),βˆ€π‘ŒβˆˆΞ“(𝐻).(3.32) This imply βˆ‡π‘‹π‘ŒβˆˆΞ“(𝐻) for all 𝑋, π‘ŒβˆˆΞ“(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 βˆ‡ if and only if 𝐻 is a parallel distribution on 𝑀.

Proof. Assume that 𝐹 is parallel on 𝐻 with respect to βˆ‡. For any 𝑋, π‘ŒβˆˆΞ“(𝐻), we have (βˆ‡π‘‹πΉ)π‘Œ=0. Taking the scalar product with 𝑉 and π‘Š to (3.30) with (βˆ‡π‘‹πΉ)π‘Œ=0, we have 𝐡(𝑋,π‘Œ)=0 and 𝐷(𝑋,π‘Œ)=0 for all 𝑋, π‘ŒβˆˆΞ“(𝐻), respectively. From (3.29), we have 𝑔(βˆ‡π‘‹π‘Œ,𝑉)=0 and 𝑔(βˆ‡π‘‹π‘Œ,π‘Š)=0. This imply βˆ‡π‘‹π‘ŒβˆˆΞ“(𝐻) for all 𝑋,π‘ŒβˆˆΞ“(𝐻). Thus 𝐻 is a parallel distribution on 𝑀.
Conversely if 𝐻 is a parallel distribution on 𝑀, from (3.29) we have 𝐡(𝑋,πΉπ‘Œ)=0,𝐷(𝑋,πΉπ‘Œ)=0,βˆ€π‘‹,π‘ŒβˆˆΞ“(𝐻).(3.33) For any π‘ŒβˆˆΞ“(𝐻), we show that 𝐹2π‘Œ=𝐽2π‘Œ=βˆ’π‘Œ+πœƒ(π‘Œ)𝜁. Replacing π‘Œ by πΉπ‘Œ to (3.33) and using (3.21), we have 𝐡(𝑋,π‘Œ)=0 and 𝐷(𝑋,π‘Œ)=0 for any 𝑋,π‘ŒβˆˆΞ“(𝐻). Thus 𝐹 is parallel on 𝐻 with respect to βˆ‡ 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 βˆ‡, then 𝐻 is a parallel distribution on 𝑀 and 𝑀 is locally a product manifold πΏπ‘ˆΓ—πΏπ‘ŠΓ—π‘€π‘‡, where πΏπ‘ˆ and πΏπ‘Š are null curves tangent to 𝐽(ltr(𝑇𝑀)) and 𝐽(𝑆(π‘‡π‘€βŸ‚)), respectively, and 𝑀𝑇 is a leaf of 𝐻.

Proof. Assume that 𝐹 is parallel on 𝑇𝑀 with respect to βˆ‡. Then 𝐹 is parallel on 𝐻 with respect to βˆ‡. By Theorem 3.11, 𝐻 is a parallel distribution on 𝑀. Applying the operator 𝐹 to (3.14) with (βˆ‡π‘‹πΉ)π‘Œ=0, we have 𝑒𝐴(π‘Œ)𝐹𝑁𝑋𝐴+𝑀(π‘Œ)𝐹𝐿𝑋=0,βˆ€π‘‹,π‘ŒβˆˆΞ“(𝑇𝑀),(3.34) due to πΉπ‘ˆ=πΉπ‘Š=0. Replacing π‘Œ by π‘ˆ and π‘Š to this equation by turns and using (3.9), we have 𝐹(𝐴𝑁𝑋)=0 and 𝐹(𝐴𝐿𝑋)=0. Taking the scalar product with π‘Š and 𝑁 to (3.14) with (βˆ‡π‘‹πΉ)π‘Œ=0 by turns, we have 𝐷𝐴(𝑋,π‘Œ)=𝑒(π‘Œ)𝑀𝑁𝑋𝐴+𝑀(π‘Œ)𝑀𝐿𝑋𝐴,(3.35)𝑀(π‘Œ)𝑔𝐿𝑋,𝑁=0,βˆ€π‘‹,π‘ŒβˆˆΞ“(𝑇𝑀).(3.36) Replacing π‘Œ by πœ‰ to (3.35), we get πœ™=0 due to (2.13)2. Also replacing π‘Œ by π‘Š to (3.36), we have 𝜌=0 due to (2.18)2. From thess results, (3.11) and (3.13), we get βˆ‡π‘‹π‘ˆ=𝜏(𝑋)π‘ˆ and βˆ‡π‘‹π‘Š=0 for all π‘‹βˆˆΞ“(𝐻′). Thus 𝐽(ltr(𝑇𝑀)) and 𝐽(𝑆(π‘‡π‘€βŸ‚)) are also parallel distributions on 𝑀. By the decomposition theorem of de Rham [14], we show that 𝑀=πΏπ‘ˆΓ—πΏπ‘ŠΓ—π‘€π‘‡, where πΏπ‘ˆ and πΏπ‘Š are null curves tangent to 𝐽(ltr(𝑇𝑀)) and 𝐽(𝑆(π‘‡π‘€βŸ‚)), 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 βˆ‡π‘‹πœ‰βˆˆΞ“(𝑇𝑀) for any π‘‹βˆˆΞ“(𝑇𝑀).

Note 4. From (2.21) we see that a necessary and sufficient condition for 𝑀 to be irrotational is 𝐷(𝑋,πœ‰)=0=πœ™(𝑋) for all π‘‹βˆˆΞ“(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 βˆ‡, then 𝑀 is irrotational, 𝜏=0 and π΄βˆ—πœ‰ξ‚€π΄π‘‹=π‘’βˆ—πœ‰π‘‹ξ‚ξ‚€π΄π‘ˆ+π‘€βˆ—πœ‰π‘‹ξ‚π‘Š,βˆ€π‘‹βˆˆΞ“(𝑇𝑀).(3.37)(ii)If π‘ˆ is parallel with respect to βˆ‡, then one has 𝜏=𝜌=0 and 𝐴𝑁𝐴𝑋=π‘’π‘π‘‹ξ€Έξ€·π΄π‘ˆ+π‘€π‘π‘‹ξ€Έπ‘Š,βˆ€π‘‹βˆˆΞ“(𝑇𝑀).(3.38)(iii)If π‘Š is parallel with respect to βˆ‡, then 𝑀 is irrotational and 𝐴𝐿𝐴𝑋=π‘’πΏπ‘‹ξ€Έξ€·π΄π‘ˆ+π‘€πΏπ‘‹ξ€Έπ‘Š,βˆ€π‘‹βˆˆΞ“(𝑇𝑀).(3.39)Moreover, if all of 𝑉, π‘ˆ, and π‘Š are parallel on 𝑇𝑀 with respect to βˆ‡, then 𝑆(𝑇𝑀) is totally geodesic in 𝑀 and 𝜏=πœ™=𝜌=0 on Ξ“(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 βˆ‡, then, taking the scalar product with π‘ˆ and π‘Š to (3.12) by turns, we have 𝜏=0 and πœ™=0 (𝑀 is irrotational), respectively. Thus we have 𝐹(π΄βˆ—πœ‰π‘‹)=0 for all π‘‹βˆˆΞ“(𝑇𝑀). From this result and (3.8), we obtain 𝐽(π΄βˆ—πœ‰π‘‹)=𝑒(π΄βˆ—πœ‰π‘‹)𝑁+𝑀(π΄βˆ—πœ‰π‘‹)𝐿. Applying 𝐽 to this equation and using πœƒ(π΄βˆ—πœ‰π‘‹)=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, π‘ˆ, and π‘Š are parallel on 𝑇𝑀 with respect to βˆ‡. Substituting the equation of (i) into (3.10)-1, we have 𝑒𝐴𝑁𝑋𝐴=π‘£βˆ—πœ‰π‘‹ξ‚ξ‚€π΄=π‘”βˆ—πœ‰ξ‚π‘‹,π‘ˆ=0,βˆ€π‘‹βˆˆΞ“(𝑇𝑀).(3.40) Also, substituting the equation of (iii) into (3.10)-3, we have 𝑀𝐴𝑁𝑋𝐴=𝑣𝐿𝑋𝐴=𝑔𝐿𝑋,π‘ˆ=0,βˆ€π‘‹βˆˆΞ“(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, πœ™, and 𝜌, defined by (2.9) and (2.10), satisfy 𝜏=πœ™=𝜌=0 on Ξ“(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=𝑀4Γ—π‘€π‘‡π‘œ, where 𝑀4 and π‘€π‘‡π‘œ are leaves of π»βŸ‚π‘œ and π»π‘œ, respectively.

Proof. By Theorem 3.10, 𝐻 is a parallel distribution 𝑀. Thus, for all 𝑋, π‘ŒβˆˆΞ“(π»π‘œ), we have βˆ‡π‘‹π‘ŒβˆˆΞ“(𝐻). From (2.11) and (3.30), we have πΆξ€·βˆ‡(𝑋,πΉπ‘Œ)=π‘”π‘‹ξ€Έβˆ‡πΉπ‘Œ,𝑁=π‘”ξ€·ξ€·π‘‹πΉξ€Έξ€·βˆ‡π‘Œ+πΉπ‘‹π‘Œξ€Έξ€Έξ€·πΉξ€·βˆ‡,𝑁=π‘”π‘‹π‘Œξ€Έξ€Έξ€·βˆ‡,𝑁=βˆ’π‘”π‘‹ξ€Έξ€·βˆ‡π‘Œ,𝐽𝑁=𝑔𝑋,π‘Œ,π‘ˆ(3.42) due to πΉπ‘ŒβˆˆΞ“(π»π‘œ). If 𝑆(𝑇𝑀) is totally umbilical in 𝑀, then we have 𝐢=0 due to Theorem 3.7. By (2.11) and (3.42), we get π‘”ξ€·βˆ‡π‘‹ξ€Έξ€·βˆ‡π‘Œ,𝑁=0,π‘”π‘‹ξ€Έξ€·π»π‘Œ,π‘ˆ=0,βˆ€π‘‹βˆˆΞ“(𝑇𝑀),βˆ€π‘ŒβˆˆΞ“π‘œξ€Έ.(3.43) These results and (3.29) imply βˆ‡π‘‹π‘ŒβˆˆΞ“(π»π‘œ) for all 𝑋, π‘ŒβˆˆΞ“(π»π‘œ). Thus π»π‘œ is a parallel distribution on 𝑀 and 𝑇𝑀=π»π‘œβŠ•orthπ»βŸ‚π‘œ, where π»βŸ‚π‘œ=Span{πœ‰,𝑉,π‘ˆ,π‘Š}. By Theorems 3.5 and 3.7, we have 𝐡=𝐷=𝐴𝑁=πœ™=0 and 𝐴𝐿𝑋=𝜌(𝑋)πœ‰. Thus (2.12) and (3.11)∼(3.13) deduce, respectively, to βˆ‡π‘‹πœ‰=βˆ’πœ(𝑋)πœ‰,βˆ‡π‘‹βˆ‡π‘ˆ=𝜏(𝑋)π‘ˆ+𝜌(𝑋)π‘Š,𝑋𝑉=βˆ’πœ(𝑋)𝑉,βˆ‡π‘‹ξ€·π»π‘Š=βˆ’πœŒ(𝑋)𝑉,βˆ€π‘‹βˆˆΞ“βŸ‚π‘œξ€Έ.(3.44) Thus π»βŸ‚π‘œ is also a parallel distribution on 𝑀. Thus we have 𝑀=𝑀4Γ—π‘€π‘‡π‘œ, where 𝑀4 is a leaf of π»βŸ‚π‘œ and π‘€π‘‡π‘œ is a leaf of π»π‘œ.

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)βŠ•ltr(𝑇𝑀).

For any ascreen half lightlike submanifold 𝑀, the vector field 𝜁 is decomposed as 𝜁=π‘Žπœ‰+𝑏𝑁.(4.1) In this case, we show that π‘Žβ‰ 0 and 𝑏≠0 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 πœ‘ such that 𝐴𝑁=πœ‘π΄βˆ—πœ‰, or equivalently, 𝐢(𝑋,π‘ƒπ‘Œ)=πœ‘π΅(𝑋,π‘Œ),βˆ€π‘‹,π‘ŒβˆˆΞ“(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 βˆ‡π‘‹ to (4.1) and using (2.3), (2.9), and (2.21), we have π‘Žπ΄βˆ—πœ‰π‘‹+𝑏𝐴𝑁𝑋={π‘‹π‘Žβˆ’π‘Žπœ(𝑋)}πœ‰+{𝑋𝑏+π‘πœ(𝑋)}𝑁+{π‘πœŒ(𝑋)βˆ’π‘Žπœ™(𝑋)}𝐿.(4.3) Taking the product with πœ‰, 𝑁, and 𝐿 by turns and using (2.16)2 and (2.17)2, we get 𝐴𝑁𝑋=πœ‘π΄βˆ—πœ‰π‘‹,π‘‹π‘Ž=π‘Žπœ(𝑋),𝑋𝑏=βˆ’π‘πœ(𝑋),π‘Žπœ™(𝑋)=π‘πœŒ(𝑋),(4.4) for all π‘‹βˆˆΞ“(𝑇𝑀), where we set πœ‘=βˆ’π‘Ž/𝑏. Thus 𝑀 is screen conformal.

Substituting (4.1) into 𝑔(𝜁,𝜁)=1, we have 2π‘Žπ‘=1. Consider the local unit timelike vector field π‘‰βˆ— on 𝑀 and its 1-form π‘£βˆ— defined by π‘‰βˆ—=βˆ’π‘βˆ’1π½πœ‰,π‘£βˆ—ξ€·(𝑋)=βˆ’π‘”π‘‹,π‘‰βˆ—ξ€Έ,βˆ€π‘‹βˆˆΞ“(𝑇𝑀).(4.5) Let π‘ˆβˆ—=βˆ’π‘Žβˆ’1𝐽𝑁. Then π‘ˆβˆ— is a unit timelike vector field on 𝑆(𝑇𝑀) such that 𝑔(π‘‰βˆ—,π‘ˆβˆ—)=1. Applying 𝐽 to (4.1) and using (2.1) and 2π‘Žπ‘=1, we have 𝑉0=π‘Žπ½πœ‰+𝑏𝐽𝑁=βˆ’βˆ—+π‘ˆβˆ—2i.e.,π‘ˆβˆ—=βˆ’π‘‰βˆ—.(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)βŠ•orth𝐽(Rad(𝑇𝑀))βŠ•orthπ½ξ€·π‘†ξ€·π‘‡π‘€βŸ‚βŠ•ξ€Έξ€Έorthπ»βˆ—ξ€Ύ,(4.7) where π»βˆ— 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 π‘Šβˆ— on 𝑆(𝑇𝑀) and its 1-form π‘€βˆ— defined by π‘Šβˆ—=βˆ’π½πΏ,π‘€βˆ—ξ€·(𝑋)=𝑔𝑋,π‘Šβˆ—ξ€Έ,βˆ€π‘‹βˆˆΞ“(𝑇𝑀).(4.8) Denote by π‘†βˆ— the projection morphism of 𝑇𝑀 on π»βˆ—. Using (4.7), for any vector field 𝑋 on 𝑀, the vector field 𝐽𝑋 is expressed as follows: 𝐽𝑋=𝑓𝑋+π‘Žπ‘£βˆ—(𝑋)πœ‰βˆ’π‘πœ‚(𝑋)π‘‰βˆ—βˆ’π‘π‘£βˆ—(𝑋)𝑁+π‘€βˆ—(𝑋)𝐿,(4.9) because π½π‘‰βˆ—=π‘Žπœ‰βˆ’π‘π‘, where 𝑓 is a tensor field of type (1, 1) defined by 𝑓𝑋=π½π‘†βˆ—π‘‹,βˆ€π‘‹βˆˆΞ“(𝑇𝑀).(4.10) Applying 𝐽 to (2.10) and (2.21) and using (2.2), (2.8) and (4.4)∼(4.9), we get π‘βˆ‡π‘‹π‘‰βˆ—ξ‚€π΄=π‘“βˆ—πœ‰π‘‹ξ‚ξ€·βˆ’π‘Žπ΅π‘‹,π‘‰βˆ—ξ€Έπœ‰βˆ’πœ™(𝑋)π‘Šβˆ—βˆ‡,(4.11)π‘‹π‘Šβˆ—ξ€·π΄=π‘“πΏπ‘‹ξ€Έξ€·βˆ’π‘Žπ·π‘‹,π‘‰βˆ—ξ€Έπœ‰βˆ’2π‘Žπœ™(𝑋)π‘‰βˆ—,ξ€·(4.12)𝑏𝐷𝑋,π‘‰βˆ—ξ€Έξ€·=𝐡𝑋,π‘Šβˆ—ξ€Έ,βˆ€π‘‹βˆˆΞ“(𝑇𝑀).(4.13)

Example 4.4. Consider a submanifold 𝑀 of 𝑀=(𝑅52,𝐽,𝜁,πœƒ,𝑔) given by the equation 𝑋𝑒1,𝑒2,𝑒3ξ€Έ=𝑒1,𝑒2,𝑒3,𝑒2,1√2𝑒1+𝑒3ξ€Έξƒͺ.(4.14) By direct calculations we easily check that 𝑇𝑀=Spanπœ‰=πœ•π‘₯1+πœ•π‘¦1+√2πœ•π‘§,π‘ˆ=πœ•π‘₯1βˆ’πœ•π‘¦1,𝑉=πœ•π‘₯2+πœ•π‘¦2,π‘‡π‘€βŸ‚ξ€½=Spanπœ‰,𝐿=πœ•π‘₯2βˆ’πœ•π‘¦2ξ€Ύ,Rad(𝑇𝑀)=Span{πœ‰}.(4.15) We obtain the lightlike transversal and transversal vector bundles 1ltr(𝑇𝑀)=Span𝑁=4ξ‚€βˆ’πœ•π‘₯1βˆ’πœ•π‘¦1+√2πœ•π‘§ξ‚ξ‚‡,tr(𝑇𝑀)=Span{𝑁,𝐿}.(4.16) From this results, we show that π½πœ‰=π‘ˆ, Rad(𝑇𝑀)∩𝐽(Rad(𝑇𝑀))={0}, 𝐽𝑁=βˆ’(1/4)π‘ˆ, 𝐽𝐿=βˆ’π‘‰, 𝐽𝑁=βˆ’(1/4)π½πœ‰ and 𝐽(Rad(𝑇𝑀)=𝐽(ltr(𝑇𝑀), √𝜁=(1/2√2)πœ‰+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,π‘‰βˆ—ξ€Έξ€·=𝛽𝑔𝑋,π‘Šβˆ—ξ€Έ,βˆ€π‘‹βˆˆΞ“(𝑇𝑀).(4.17) Replacing 𝑋 by π‘Šβˆ— and π‘‰βˆ— to this equation by turns, we have 𝛽=0 and 𝛿=0, respectively. Thus we have β„Ž=0 and 𝑀 is totally geodesic. By (4.2), we also have 𝐢=0. Thus 𝑆(𝑇𝑀) is also totally geodesic in 𝑀.

Taking π‘ŒβˆˆΞ“(π»βˆ—). Then we have π‘“π‘Œ=π½π‘ŒβˆˆΞ“(π»βˆ—) due to (4.9). Applying βˆ‡π‘‹ to π½π‘Œ=π‘“π‘Œ and using (2.2), (2.8), (4.2), (4.5), and (4.9), we have π΅ξ€·βˆ‡(𝑋,π‘“π‘Œ)=π‘π‘”π‘‹π‘Œ,π‘‰βˆ—ξ€Έξ€·βˆ‡,𝐷(𝑋,π‘“π‘Œ)=π‘”π‘‹π‘Œ,π‘Šβˆ—ξ€Έξ€·βˆ‡,(4.18)π‘‹π‘“ξ€Έξ€·βˆ‡π‘Œ=βˆ’π‘Žπ‘”π‘‹π‘Œ,π‘‰βˆ—ξ€Έπœ‰+2π‘Žπ΅(𝑋,π‘Œ)π‘‰βˆ—βˆ’π·(𝑋,π‘Œ)π‘Šβˆ—,(4.19) for all π‘‹βˆˆΞ“(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. π»βˆ— is an integrable distribution on 𝑀 if and only if one has β„Žξ€·π»(𝑋,π‘“π‘Œ)=β„Ž(𝑓𝑋,π‘Œ),βˆ€π‘‹,π‘ŒβˆˆΞ“βˆ—ξ€Έ.(4.20) Moreover, if 𝑀 is totally umbilical, then π»βˆ— 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 π»βˆ— with respect to the induced connection βˆ‡ if and only if π»βˆ— 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 πΏπœ‰Γ—πΏπ‘‰βˆ—Γ—πΏπ‘Šβˆ—Γ—π‘€βˆ—, where πΏπœ‰, πΏπ‘‰βˆ—, and πΏπ‘Šβˆ— are null, timelike, and spacelike curves tangent to Rad(𝑇𝑀), 𝐽(Rad(𝑇𝑀)), and 𝐽(𝑆(π‘‡π‘€βŸ‚)), respectively, and π‘€βˆ— is a leaf of π»βˆ—.

Proof. If 𝑀 is totally umbilical, then π»βˆ— is a parallel distribution on 𝑀 by Theorem 4.6 and we have 𝐡=𝐷=π΄βˆ—πœ‰=πœ™=0;𝐴𝐿𝑋=𝜌(𝑋)πœ‰ by Theorem 4.5. From (4.4)1, we also have 𝐴𝑁=0. Using (2.12), (4.11), and (4.12), we have βˆ‡π‘‹πœ‰=βˆ’πœ(𝑋)πœ‰ and βˆ‡π‘‹π‘‰βˆ—=βˆ‡π‘‹π‘Šβˆ—=0 due to π‘“πœ‰=0. This implies that all of the distributions Rad(𝑇𝑀), 𝐽(Rad(𝑇𝑀)), and 𝐽(𝑆(π‘‡π‘€βŸ‚)) are parallel on 𝑀. Thus we have 𝑀=πΏπœ‰Γ—πΏπ‘‰βˆ—Γ—πΏπ‘Šβˆ—Γ—π‘€βˆ—, where πΏπœ‰, πΏπ‘‰βˆ—, and πΏπ‘Šβˆ— are null, timelike, and spacelike curves tangent to Rad(𝑇𝑀), 𝐽(Rad(𝑇𝑀)) and 𝐽(𝑆(π‘‡π‘€βŸ‚)), respectively, and π‘€βˆ— is a leaf of π»βˆ—.

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 π‘‰βˆ— is parallel with respect to βˆ‡ on 𝑀, then 𝑀 is irrotational and π΄βˆ—πœ‰ξ€·π‘‹=𝐡𝑋,π‘Šβˆ—ξ€Έπ‘Šβˆ—ξ€·,𝐡𝑋,π‘‰βˆ—ξ€Έ=0,𝜌(𝑋)=0,βˆ€π‘‹βˆˆΞ“(𝑇𝑀).(4.21)(ii)If π‘Šβˆ— is parallel with respect to βˆ‡ on 𝑀, then 𝑀 is irrotational and 𝐴𝐿𝑋=𝐷𝑋,π‘Šβˆ—ξ€Έπ‘Šβˆ—ξ€·,𝐷𝑋,π‘‰βˆ—ξ€Έ=0,𝜌(𝑋)=0,βˆ€π‘‹βˆˆΞ“(𝑇𝑀).(4.22)Moreover, if π‘‰βˆ— and π‘Šβˆ— are parallel with respect to βˆ‡, then one sees that π΄βˆ—πœ‰=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 π‘‰βˆ— and π‘Šβˆ— are parallel with respect to βˆ‡, then 𝑀 is locally a product manifold πΏπœ‰Γ—πΏπ‘‰βˆ—Γ—πΏπ‘Šβˆ—Γ—π‘€βˆ—, where πΏπœ‰, πΏπ‘‰βˆ—, and πΏπ‘Šβˆ— are null, timelike, and spacelike curves tangent to Rad(𝑇𝑀), 𝐽(Rad(𝑇𝑀)), and 𝐽(𝑆(π‘‡π‘€βŸ‚)), respectively, and π‘€βˆ— is a leaf of π»βˆ—.

Proof. If π‘‰βˆ— is parallel with respect to βˆ‡, for any π‘ŒβˆˆΞ“(π»βˆ—), we have 𝐴𝐡(𝑋,π‘Œ)=π‘”βˆ—πœ‰ξ‚ξ€·π‘‹,π‘Œ=𝐡𝑋,π‘Šβˆ—ξ€Έπ‘”ξ€·π‘Œ,π‘Šβˆ—ξ€Έ=0,βˆ€π‘‹βˆˆΞ“(𝑇𝑀).(4.23) Thus we get 𝑔(βˆ‡π‘‹π‘Œ,π‘‰βˆ—)=π‘βˆ’1𝐡(𝑋,π‘“π‘Œ)=0 because π‘“π‘ŒβˆˆΞ“(π»βˆ—). Also if π‘Šβˆ— is parallel with respect to βˆ‡, then, for any π‘ŒβˆˆΞ“(π»βˆ—), we have 𝐷𝐴(𝑋,π‘Œ)=𝑔𝐿𝑋,π‘Œ=𝐷𝑋,π‘Šβˆ—ξ€Έπ‘”ξ€·π‘Œ,π‘Šβˆ—ξ€Έ=0,βˆ€π‘‹βˆˆΞ“(𝑇𝑀).(4.24) From these results and (4.19), we show that 𝑓 is parallel on π»βˆ— with respect to βˆ‡. Thus, by Theorem 4.7, we see that π»βˆ— is a parallel distribution on 𝑀. As π‘‰βˆ— and π‘Šβˆ— are parallel with respect to βˆ‡ and βˆ‡π‘‹πœ‰=βˆ’πœ(𝑋)πœ‰ due to π΄βˆ—πœ‰=0, we have our theorem.

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