Generic Lightlike Submanifolds of an Indefinite Cosymplectic Manifold

Abstract

Lightlike geometry has its applications in general relativity, particularly in black hole theory. Indeed, it is known that lightlike hypersurfaces are examples of physical models of Killing horizons in general relativity (Galloway, 2007). In this paper, we introduce the definition of generic lightlike submanifolds of an indefinite cosymplectic manifold. We investigate new results on a class of generic lightlike submanifolds 𝑀 of an indefinite cosymplectic manifold 𝑀.

1. Introduction

In the generalization from Riemannian to semi-Riemannian manifolds, the induced metric may be degenerate (lightlike) therefore there is a natural existence of lightlike submanifolds and for which the local and global geometry is completely different than nondengerate case. In lightlike case, the standard textbook definitions do not make sense- and one fails to use the theory of non-degenerate geometry in the usual way. The primary difference between the lightlike submanifolds and non-degenerate submanifolds is that in the first case, the normal vector bundle intersects with the tangent bundle. Thus, the study of lightlike submanifolds becomes more difficult and different from the study of non-degenerate submanifolds. Moreover, the geometry of lightlike submanifolds is used in mathematical physics, in particular, in general relativity since lightlike submanifolds produce models of different types of horizons (event horizons, Cauchy’s horizons, and Kruskal’s horizons). The universe can be represented as a four-dimensional submanifold embedded in a (4+𝑛)-dimensional spacetime manifold. Lightlike hypersurfaces are also studied in the thoery of electromagnetism [1]. Thus, large number of applications but limited information available, motivated us to do research on this subject matter. Kupeli [2] and Bejancu and Duggal [1] developed the general theory of degenerate (lightlike) submanifolds. They constructed a transversal vector bundle of lightlike submanifold and investigated various properties of these manifolds. The geometry of both lightlike hypersurfaces and half lightlike submanifolds of indefinite cosymplectic manifolds was studied by Jin ([3, 4]). However, a general notion of generic lightlike submanifolds of an indefinite cosymplectic manifold has not been introduced as yet.

The objective of this paper is to study generic 𝑟-lightlike submanifolds 𝑀 of an indefinite cosymplectic manifold 𝑀 subject to the conditions: (1) 𝑀 is totally umbilical, or (2) 𝑆(𝑇𝑀) is totally umbilical in 𝑀. In Section 1, we first of all recall some of fundamental formulas in the theory of 𝑟-lightlike submanifolds. In Section 2, we newly define generic lightlike submanifolds. After that, we prove some basic theorems which will be used in the rest of this paper. In Section 3, we study generic 𝑟-lightlike submanifolds of 𝑀.

2. Lightlike Submanifolds

Let (𝑀,𝑔) be an 𝑚-dimensional lightlike submanifold of an (𝑚+𝑛)-dimensional semi-Riemannian manifold (𝑀,𝑔). Then the radical distribution Rad(𝑇𝑀)=𝑇𝑀∩𝑇𝑀^⟂ is a vector subbundle of the tangent bundle 𝑇𝑀 and the normal bundle 𝑇𝑀^⟂, of rank 𝑟(1≤𝑟≤min{𝑚,𝑛}). In general, there exist two complementary non-degenerate distributions 𝑆(𝑇𝑀) and 𝑆(𝑇𝑀^⟂) of Rad(𝑇𝑀) in 𝑇𝑀 and 𝑇𝑀^⟂, respectively, called the screen and coscreen distributions on 𝑀, such that𝑇𝑀=Rad(𝑇𝑀)⊕_orth𝑆(𝑇𝑀),𝑇𝑀^⟂=Rad(𝑇𝑀)⊕_orth𝑆𝑇𝑀^⟂,(2.1) where the symbol ⊕_orth denotes the orthogonal direct sum. We denote such a lightlike submanifold by (𝑀,𝑔,𝑆(𝑇𝑀),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. We use the same notation for any other vector bundle. We use the following range of indices:𝑖,𝑗,𝑘,…∈{1,…,𝑟},𝛼,𝛽,𝛾,…∈{𝑟+1,…,𝑛}.(2.2)

Let tr(𝑇𝑀) and ltr(𝑇𝑀) be complementary (but not orthogonal) vector bundles to 𝑇𝑀 in 𝑇𝑀_∣𝑀 and 𝑇𝑀^⟂ in 𝑆(𝑇𝑀)^⟂, respectively, and let {𝑁₁,…,𝑁_𝑟} be a lightlike basis of Γ(ltr(𝑇𝑀)_∣𝒰) consisting of smooth sections of 𝑆(𝑇𝑀)^⟂∣_𝒰, where 𝒰 is a coordinate neighborhood of 𝑀, such that𝑔𝑁_𝑖,𝜉_𝑗=𝛿_𝑖𝑗,𝑔𝑁_𝑖,𝑁_𝑗=0,(2.3) where {𝜉₁,…,𝜉_𝑟} is a lightlike basis of Γ(Rad(𝑇𝑀)). Then we have𝑇𝑀=={𝑇R𝑀ad⊕(𝑇t𝑀r()𝑇⊕𝑀l)tr=(𝑇{𝑀Ra)d}(⊕𝑇𝑀_orth)⊕tr(𝑇𝑀)}⊕_orth𝑆(𝑇𝑀)𝑆(𝑇𝑀)⊕_orth𝑆𝑇𝑀^⟂.(2.4)

We say that a lightlike submanifold (𝑀,𝑔,𝑆(𝑇𝑀),𝑆(𝑇𝑀^⟂)) of 𝑀 is(1)𝑟-lightlike if 1≤𝑟<𝑚𝑖𝑛{𝑚,𝑛};(2)coisotropic if 1≤𝑟=𝑛<𝑚;(3)isotropic if 1≤𝑟=𝑚<𝑛;(4)totally lightlike if 1≤𝑟=𝑚=𝑛.

The above three classes (2)–(4) are particular cases of the class (1) as follows: 𝑆(𝑇𝑀^⟂)={0}, 𝑆(𝑇𝑀)={0}, and 𝑆(𝑇𝑀)=𝑆(𝑇𝑀^⟂)={0}, respectively. The geometry of 𝑟-lightlike submanifolds is more general form than that of the other three type submanifolds. For this reason, in this paper we consider only 𝑟-lightlike submanifolds 𝑀≡(𝑀,𝑔,𝑆(𝑇𝑀),𝑆(𝑇𝑀^⟂)), with the following local quasiorthonormal field of frames of 𝑀: 𝜉₁,…,𝜉_𝑟,𝑁₁,…,𝑁_𝑟,𝐹_𝑟+1,…,𝐹_𝑚,𝐸_𝑟+1,…,𝐸_𝑛,(2.5) where the sets {𝐹_𝑟+1,…,𝐹_𝑚} and {𝐸_𝑟+1,…,𝐸_𝑛} are orthonormal basis of Γ(𝑆(𝑇𝑀)) and Γ(𝑆(𝑇𝑀^⟂)), respectively.

Let ∇ be the Levi-Civita connection of 𝑀 and 𝑃 the projection morphism of Γ(𝑇𝑀) on Γ(𝑆(𝑇𝑀)) with respect to (2.1). For an 𝑟-lightlike submanifold, the local Gauss-Weingarten formulas are given by∇_𝑋𝑌=∇_𝑋𝑌+^𝑟_𝑖=1ℎ^ℓ𝑖(𝑋,𝑌)𝑁_𝑖+^𝑛_𝛼_=𝑟+1ℎ^𝑠𝛼(𝑋,𝑌)𝐸_𝛼,(2.6)∇_𝑋𝑁_𝑖=−𝐴_𝑁𝑖𝑋+^𝑟_𝑗=1𝜏_𝑖𝑗(𝑋)𝑁_𝑗+^𝑛_𝛼_=𝑟+1𝜌_𝑖𝛼(𝑋)𝐸_𝛼,(2.7)∇_𝑋𝐸_𝛼=−𝐴_𝐸𝛼𝑋+^𝑟_𝑖=1𝜙_𝛼𝑖(𝑋)𝑁_𝑖+^𝑛_𝛽_=𝑟+1∇𝜎_𝛼𝛽(𝑋)𝐸_𝛽,(2.8)_𝑋𝑃𝑌=∇^∗𝑋𝑃𝑌+^𝑟_𝑖=1∇ℎ^∗𝑖(𝑋,𝑃𝑌)𝜉_𝑖,(2.9)_𝑋𝜉_𝑖=−𝐴^∗𝜉_𝑖𝑋−^𝑟_𝑗=1𝜏_𝑗𝑖(𝑋)𝜉_𝑗,(2.10) for any 𝑋,𝑌∈Γ(𝑇𝑀), where ∇ and ∇^∗ are induced linear connections on 𝑇𝑀 and 𝑆(𝑇𝑀), respectively, the bilinear forms ℎ^ℓ𝑖 and ℎ^𝑠𝛼 on 𝑀 are called the local lightlike and screen second fundamental forms on 𝑇𝑀, respectively, ℎ^∗𝑖 are called the local radical second fundamental forms on 𝑆(𝑇𝑀). 𝐴_𝑁𝑖, 𝐴^∗𝜉_𝑖, and 𝐴_𝐸𝛼 are linear operators on Γ(𝑇𝑀) and 𝜏_𝑖𝑗, 𝜌_𝑖𝛼, 𝜙_𝛼𝑖 and 𝜎_𝛼𝛽 are 1-forms on 𝑇𝑀. Since ∇ is torsion-free, ∇ is also torsion-free and both ℎ^ℓ𝑖 and ℎ^𝑠𝛼 are symmetric. From the fact ℎ^ℓ𝑖(𝑋,𝑌)=𝑔(∇_𝑋𝑌,𝜉_𝑖), we know that ℎ^ℓ𝑖 are independent of the choice of a screen distribution. We say that ℎ(𝑋,𝑌)=^𝑟_𝑖=1ℎ^ℓ𝑖(𝑋,𝑌)𝑁_𝑖+^𝑛_𝛼_=𝑟+1ℎ^𝑠𝛼(𝑋,𝑌)𝐸_𝛼(2.11) is the second fundamental tensor of 𝑀.

The induced connection ∇ on 𝑇𝑀 is not metric and satisfies∇_𝑋𝑔(𝑌,𝑍)=^𝑟_𝑖=1ℎ^ℓ𝑖(𝑋,𝑌)𝜂_𝑖(𝑍)+ℎ^ℓ𝑖(𝑋,𝑍)𝜂_𝑖(𝑌),(2.12) for all 𝑋,𝑌∈Γ(𝑇𝑀), where 𝜂_𝑖𝑠 are the 1-forms such that𝜂_𝑖(𝑋)=𝑔𝑋,𝑁_𝑖,∀𝑋∈Γ(𝑇𝑀).(2.13) But the connection ∇^∗ on 𝑆(𝑇𝑀) is metric. The above three local second fundamental forms are related to their shape operators byℎ^ℓ𝑖(𝑋,𝑌)=𝑔𝐴^∗𝜉_𝑖𝑋,𝑌−^𝑟_𝑘=1ℎℎ^ℓ𝑘𝑋,𝜉_𝑖𝜂_𝑘(𝑌),(2.14)_ℓ𝑖𝜖(𝑋,𝑃𝑌)=𝑔𝐴^∗𝜉_𝑖𝑋,𝑃𝑌,𝑔𝐴^∗𝜉_𝑖𝑋,𝑁_𝑗=0,(2.15)_𝛼ℎ^𝑠𝛼(𝑋,𝑌)=𝑔𝐴_𝐸𝛼𝑋,𝑌−^𝑟_𝑖=1𝜖𝜙_𝛼𝑖(𝑋)𝜂_𝑖(𝑌),(2.16)_𝛼ℎℎ^𝑠𝛼(𝑋,𝑃𝑌)=𝑔𝐴_𝐸𝛼𝑋,𝑃𝑌,𝑔𝐴_𝐸𝛼𝑋,𝑁_𝑖=𝜖_𝛼𝜌_𝑖𝛼(𝑋),(2.17)_∗𝑖(𝑋𝜖,𝑃𝑌)=𝑔𝐴_𝑁𝑖𝑋,𝑃𝑌,𝜂_𝑗𝐴_𝑁𝑖𝑋+𝜂_𝑖𝐴^𝑁_𝑗𝑋=0,(2.18)_𝛽𝜎_𝛼𝛽=−𝜖_𝛼𝜎_𝛽𝛼,∀𝑋,𝑌∈Γ(𝑇𝑀),(2.19) where 𝜖_𝛼=𝑔(𝐸_𝛼,𝐸_𝛼)(=±1) is the sign of the vector field 𝐸_𝛼. From (2.18), we know that each 𝐴_𝑁𝑖 is shape operator related to the local second fundamental form ℎ^∗𝑖 on 𝑆(𝑇𝑀). Replace 𝑌 by 𝜉_𝑗 in (2.14), we haveℎ^ℓ𝑖𝑋,𝜉_𝑗+ℎ^ℓ𝑗𝑋,𝜉_𝑖=0,(2.20) for all 𝑋∈Γ(𝑇𝑀). It followsℎ^ℓ𝑖𝑋,𝜉_𝑖=0,ℎ^ℓ𝑖𝜉_𝑗,𝜉_𝑘=0.(2.21) Also, replace 𝑋 by 𝜉_𝑗 in (2.14) and use (2.21), we haveℎ^ℓ𝑖𝑋,𝜉_𝑗=𝑔𝑋,𝐴^∗𝜉_𝑖𝜉_𝑗,𝐴^∗𝜉_𝑖𝜉_𝑗+𝐴^∗𝜉_𝑗𝜉_𝑖=0,𝐴^∗𝜉_𝑖𝜉_𝑖=0.(2.22) For an 𝑟-lightlike submanifold, replace 𝑌 by 𝜉_𝑖 in (2.16), we haveℎ^𝑠𝛼𝑋,𝜉_𝑖=−𝜖_𝛼𝜙_𝛼𝑖(𝑋),∀𝑋∈Γ(𝑇𝑀).(2.23) From (2.6), (2.10), and (2.23), for all 𝑋∈Γ(𝑇𝑀), we have∇_𝑋𝜉_𝑖=−𝐴^∗𝜉_𝑖𝑋−^𝑟_𝑗=1𝜏_𝑗𝑖(𝑋)𝜉_𝑗−^𝑛_𝛼_=𝑟+1𝜖_𝛼𝜙_𝛼𝑖(𝑋)𝐸_𝛼+^𝑟_𝑗=1ℎ^ℓ𝑗𝑋,𝜉_𝑖𝑁_𝑗.(2.24)

Definition 2.1. A lightlike submanifold 𝑀 of a semi-Riemannian manifold (𝑀,𝑔) is said to be irrotational if ∇_𝑋𝜉_𝑖∈Γ(𝑇𝑀) for any 𝑋∈Γ(𝑇𝑀) and 𝜉_𝑖∈Γ(Rad(𝑇𝑀)) for all 𝑖.

Note 1. For an 𝑟-lightlike 𝑀, the above definition is equivalent to ℎ^ℓ𝑗𝑋,𝜉_𝑖=0,ℎ^𝑠𝛼𝑋,𝜉_𝑖=𝜙_𝛼𝑖(𝑋)=0,∀𝑋∈Γ(𝑇𝑀).(2.25)

Denote by 𝑅 and 𝑅 the curvature tensors of ∇ and ∇, respectively. Using the local Gauss-Weingarten formulas for 𝑀, we obtain𝑅(𝑋,𝑌)𝑍=+𝑅(𝑋,𝑌)𝑍_𝑟_𝑖=1ℎ^ℓ𝑖+_𝑛(𝑋,𝑍)𝐴_𝑁𝑖𝑌−ℎ^ℓ𝑖(𝑌,𝑍)𝐴_𝑁𝑖𝑋_𝛼_=𝑟1ℎ^𝑠𝛼+_𝑟(𝑋,𝑍)𝐴_𝐸𝛼𝑌−ℎ^𝑠𝛼(𝑌,𝑍)𝐴_𝐸𝛼𝑋_𝑖=1∇^𝑋+ℎ^ℓ𝑖(𝑌,𝑍)−∇_𝑌ℎ^ℓ𝑖(𝑋,𝑍)_𝑟_𝑗=1𝜏_𝑗𝑖+(𝑋)ℎ^ℓ𝑗(𝑌,𝑍)−𝜏_𝑗𝑖(𝑌)ℎ^ℓ𝑗(𝑋,𝑍)_𝑛_𝛼_=𝑟+1𝜙_𝛼𝑖(𝑋)ℎ^𝑠𝛼(𝑌,𝑍)−𝜙_𝛼𝑖(𝑌)ℎ^𝑠𝛼(X,𝑍)𝑁_𝑖+^𝑛_𝛼_=𝑟+1∇^𝑋+ℎ^𝑠𝛼(𝑌,𝑍)−∇_𝑌ℎ^𝑠𝛼(𝑋,𝑍)_𝑟_𝑖=1𝜌_𝑖𝛼+(𝑋)ℎ^ℓ𝑖(𝑌,𝑍)−𝜌_𝑖𝛼(𝑌)ℎ^𝑠𝛼(𝑋,𝑍)_𝑛_𝛽_=𝑟+1𝜎^𝛽𝛼(𝑋)ℎ^𝑠𝛽(𝑌,𝑍)−𝜎_𝛽𝛼(𝑌)ℎ^𝑠𝛽(𝑋,𝑍)𝐸_𝛼,(2.26) for all 𝑋,𝑌,𝑍∈Γ(𝑇𝑀). Assume that 𝑀 is irrotational. Replace 𝑍 by 𝜉_𝑘 in (2.26) and use (2.10), (2.15), (2.17), and (2.25), then we have𝑅(𝑋,𝑌)𝜉_𝑘=𝑅(𝑋,𝑌)𝜉_𝑘+^𝑟_𝑖=1𝑔𝐴^∗𝜉_𝑖𝑌,𝐴^∗𝜉_𝑘𝑋−𝑔𝐴^∗𝜉_𝑖𝑋,𝐴^∗𝜉_𝑘𝑌𝑁_𝑖+^𝑛_𝛼_=𝑟+1𝜖_𝛼𝑔𝐴^𝐸_𝛼𝑌,𝐴^∗𝜉_𝑘𝑋−𝑔𝐴_𝐸𝛼𝑋,𝐴^∗𝜉_𝑘𝑌𝐸_𝛼.(2.27) Using (2.27) and the fact 𝑅(𝑋,𝑌)𝑍∈Γ(𝑇𝑀) for 𝑋,𝑌,𝑍∈Γ(𝑇𝑀), we get𝑔𝑅(𝑋,𝑌)𝑍,𝜉_𝑘==−−𝑔𝑔𝑅𝑅((𝑋𝑋,,𝑌𝑌))𝜉𝜉_𝑘𝑘,𝑍,𝑍+^𝑟_𝑖=1𝑔𝐴^∗𝜉_𝑖=𝑔𝑅(𝑋,𝑌)𝑍,𝜉_𝑘𝑋,𝐴^∗𝜉_𝑘𝑌−𝑔𝐴^∗𝜉_𝑖𝑌,𝐴^∗𝜉_𝑘𝑋𝜂_𝑖(𝑍)=+^𝑟_𝑖=1𝑔𝐴^∗𝜉_𝑖𝑋,𝐴^∗𝜉_𝑘𝑌−𝑔𝐴^∗𝜉_𝑖𝑌,𝐴^∗𝜉_𝑘𝑋𝜂_𝑖(𝑍)_𝑟_𝑖=1𝑔𝐴^∗𝜉_𝑖𝑋,𝐴^∗𝜉_𝑘𝑌−𝑔𝐴^∗𝜉_𝑖𝑌,𝐴^∗𝜉_𝑘𝑋𝜂_𝑖(𝑍),∀𝑋,𝑌,𝑍∈Γ(𝑇𝑀).(2.28)

3. Indefinite Cosymplectic Manifolds

An odd dimensional smooth manifold (𝑀,𝑔) is called a contact metric manifold [5, 6] if there exists a (1,1)-type tensor field 𝐽, a vector field 𝜁, called the characteristic vector field, and its 1-form 𝜃 satisfying𝐽²𝑋=−𝑋+𝜃(𝑋)𝜁,𝐽𝜁=0,𝜃∘𝐽=0,𝜃(𝜁)=1,𝑔𝜃((𝑋𝜁,)𝜁=)=𝜖𝜖,𝑔(𝐽𝑋,𝐽𝑌)=𝑔(𝑋,𝑌)−𝜖𝜃(𝑋)𝜃(𝑌),𝑔(𝜁,𝑋),𝑑𝜃(𝑋,𝑌)=𝑔(𝐽𝑋,𝑌),𝜖=±1,(3.1) for any vector fields 𝑋,𝑌 on 𝑀. Then the set (𝐽,𝜃,𝜁,𝑔) is called a contact metric structure on 𝑀. Note that we may assume that 𝜖=1 without loss of generality [7]. We say that 𝑀 has a normal contact structure [5, 8] if 𝑁_𝐽+𝑑𝜃⊗𝜁=0, where 𝑁_𝐽 is the Nijenhuis tensor field of 𝐽. A normal contact metric manifold is called a cosymplectic [9, 10] for which we have∇_𝑋𝜃=0,∇_𝑋𝐽=0,(3.2) for any vector field 𝑋 on 𝑀. A cosymplectic manifold 𝑀=(𝑀,𝐽,𝜁,𝜃,𝑔) is called an indefinite cosymplectic manifold [3, 4] if (𝑀,𝑔) is a semi-Riemannian manifold of index 𝜇(>0). For any indefinite cosymplectic manifold, apply ∇_𝑋 to 𝐽𝜁=0 for any vector field 𝑋 on 𝑀 and use (3.2), then we have 𝐽(∇_𝑋𝜁)=0. Apply 𝐽 to this and use (3.1) and 𝜃(∇_𝑋𝜁)=0, we get∇_𝑋𝜁=0.(3.3)

An indefinite cosymplectic manifold 𝑀 is called an indefinite cosymplectic space form, denoted by 𝑀(𝑐), if it has the constant 𝐽-sectional curvature 𝑐 [3, 9, 10]. The curvature tensor 𝑅 of this space form 𝑀(𝑐) is given by𝑅(𝑋,𝑌)𝑍=𝑐4−𝑔(𝜃𝑌(,𝑌𝑍))𝜃𝑋(𝑍−)𝑔𝑋(𝑋+,𝑍)𝑌+𝜃(𝑋)𝜃(𝑍)𝑌+𝑔(𝑋,𝑍)𝜃(𝑌)𝜁−𝑔(𝑌,𝑍)𝜃(𝑋)𝜁𝑔(𝐽𝑌,𝑍)𝐽𝑋+𝑔(𝐽𝑍,𝑋)𝐽𝑌−2𝑔(𝐽𝑋,𝑌)𝐽𝑍,(3.4) for any vector fields 𝑋,𝑌, and 𝑍 in 𝑀.

Let 𝑀 be an 𝑚-dimensional 𝑟-lightlike submanifold of an (𝑚+𝑛)-dimensional indefinite cosymplectic manifold 𝑀 and 𝑃 the projection morphism of Γ(𝑇𝑀) on Γ(𝑆(𝑇𝑀)) with respect to (2.4). The characteristic vector field 𝜁 of 𝑀 from (2.4) is decomposed by𝜁=𝑃𝜁+^𝑟_𝑖=1𝑎_𝑖𝜉_𝑖+^𝑟_𝑖=1𝑏_𝑖𝑁_𝑖+^𝑛_𝛼_=𝑟+1𝑒_𝛼𝐸_𝛼,(3.5) where 𝑎_𝑖=𝜃(𝑁_𝑖), 𝑏_𝑖=𝜃(𝜉_𝑖) and 𝑒_𝛼=𝜖_𝛼𝜃(𝐸_𝛼) are smooth functions on 𝑀.

Note 2. Although 𝑆(𝑇𝑀) is not unique, it is canonically isomorphic to the factor vector bundle 𝑆(𝑇𝑀)^∗=𝑇𝑀/Rad(𝑇𝑀) considered by Kupeli [2]. Thus all screen distributions 𝑆(𝑇𝑀) are mutually isomorphic. For this reason, the following definition is well defined.

Definition 3.1 (see [6]). One says that 𝑀 is generic lightlike submanifold of 𝑀 if there exists a screen distribution 𝑆(𝑇𝑀) of 𝑀 such that 𝐽𝑆(𝑇𝑀)^⟂⊂𝑆(𝑇𝑀).(3.6)

Proposition 3.2 (see [3]). Let 𝑀 be a lightlike hypersurface of an indefinite cosymplectic manifold 𝑀. Then 𝑀 is a generic lightlike submanifold of 𝑀.

Proposition 3.3 (see [4]). Let 𝑀 be a 1-lightlike submanifold of codimension 2 of an indefinite cosymplectic manifold 𝑀 such that the coscreen distribution 𝑆(𝑇𝑀^⟂) is spacelike. Then 𝑀 is a generic lightlike submanifold of 𝑀.

Theorem 3.4. Let 𝑀 be an irrotational generic 𝑟-lightlike submanifold of an indefinite cosymplectic space form 𝑀(𝑐). Then one has 𝑐=0.

Proof. Assume that 𝑏_𝑘≠0 in (3.5). Note that (3.1) implies 𝑔(𝐽𝑋,𝜉_𝑘)=−𝑔(𝑋,𝐽𝜉_𝑘) for all 𝑋∈Γ(𝑇𝑀). Then, taking the scalar product with 𝜉_𝑘 to (3.4), and using (2.28), we get 4^𝑟_𝑖=1𝑔𝐴^∗𝜉_𝑖=𝑐𝑏_𝑘𝑋,𝐴^∗𝜉_𝑘𝑌−𝑔𝐴^∗𝜉_𝑖𝑌,𝐴^∗𝜉_𝑘𝑋𝜂_𝑖(𝑍)−𝑔(𝑋,𝑍)𝜃(𝑌)−𝑏_𝑘𝑔(𝑌,𝑍)𝜃(𝑋)−𝑔(𝐽𝑌,𝑍)𝑔𝑋,𝐽𝜉_𝑘𝑔(𝐽𝑍,𝑋)𝑔𝑌,𝐽𝜉_𝑘+2𝑔(𝐽𝑋,𝑌)𝑔𝑍,𝐽𝜉_𝑘,∀𝑋,𝑌,𝑍∈Γ(𝑇𝑀).(3.7) Replace 𝑍 by 𝐽𝜉_𝑘 and 𝑌 by 𝜉_𝑘 in the equation and use (3.1), then we have 𝑏^2𝑘𝑐𝑔𝑋,𝐽𝜉_𝑘=0,∀𝑋∈Γ(𝑇𝑀),(3.8) because 𝜂_𝑖(𝐽𝜉_𝑘)=0 by (3.6). Replacing 𝑋 by 𝐽𝜉_𝑘 in this equation, we obtain 𝑏^4𝑘𝑐=0. Since 𝑏_𝑘≠0, we have 𝑐=0.
Assume that 𝑏_𝑘=0. Then, taking the scalar product with 𝜉_𝑘 to both sides of (3.4) and using (2.28) and (3.1), we obtain4^𝑟_𝑖=1𝑔𝐴^∗𝜉_𝑖=𝑐−𝑋,𝐴^∗𝜉_𝑘𝑌−𝑔𝐴^∗𝜉_𝑖𝑌,𝐴^∗𝜉_𝑘𝑋𝜂_𝑖(𝑍)+2𝑔(𝐽𝑌,𝑍)𝑔𝑋,𝐽𝜉_𝑘−𝑔(𝐽𝑍,𝑋)𝑔𝑌,𝐽𝜉_𝑘𝑔(𝐽𝑋,𝑌)𝑔𝑍,𝐽𝜉_𝑘,∀𝑋,𝑌,𝑍∈Γ(𝑇𝑀).(3.9) Replace 𝑍 by 𝐽𝑁_𝑘 and 𝑌 by 𝜉_𝑘 in this equation and use (3.1), then we have 𝑐𝑔𝑋,𝐽𝜉_𝑘=0,∀𝑋∈Γ(𝑇𝑀),(3.10) because 𝜂_𝑖(𝐽𝑁_𝑘)=0 by (3.6) and 𝑔(𝐽𝜉_𝑘,𝐽𝑁_𝑘)=1. Replace 𝑋 by 𝐽𝑁_𝑘 in this equation, we get 𝑐=0.

Corollary 3.5. There exist no irrotational generic 𝑟-lightlike submanifolds 𝑀 of an indefinite cosymplectic space form 𝑀(𝑐) with 𝑐≠0.

Proposition 3.6. Let 𝑀 be an 𝑟-lightlike submanifold of an indefinite cosymplectic manifold 𝑀. Then the characteristic 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=∑^𝑟𝑖=1𝑎_𝑖𝜉_𝑖 [or 𝜁=∑^𝑟𝑖=1𝑏_𝑖𝑁_𝑖]. From this, we have 1=𝑔(𝜁,𝜁)=^𝑟_𝑖_,𝑗=1𝑎_𝑖𝑎_𝑗𝑔𝜉_𝑖,𝜉_𝑗=0or1=𝑔(𝜁,𝜁)=^𝑟_𝑖_,𝑗=1𝑏_𝑖𝑏_𝑗𝑔𝑁_𝑖,𝑁_𝑗=0.(3.11) It is a contradiction. Thus 𝜁 does not belong to Rad(𝑇𝑀) and ltr(𝑇𝑀).

4. Generic Lightlike Submanifolds

If the characteristic vector field 𝜁 is tangent to 𝑀, then, by Proposition 3.6, 𝜁 does not belong to Rad(𝑇𝑀). 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 also proved this result in his book [11] which Kang et al. [12] and Duggal and Sahin [5, 8] assumed in their papers. We also assumed this result in this paper. In this case, all of the functions 𝑎_𝑖, 𝑏_𝑖, and 𝑒_𝛼 on 𝑀, defined by (3.5), vanish identically.

Theorem 4.1. Let 𝑀 be a generic 𝑟-lightlike submanifold of an indefinite cosymplectic manifold 𝑀. Then 𝜁 is a parallel vector field on 𝑀 and 𝑆(𝑇𝑀). Furthermore 𝜁 is conjugate to any vector field on M with respect to ℎ and ℎ^∗𝑖. In particular, 𝜁 is an asymptotic vector field on 𝑀.

Proof. Replace 𝑌 by 𝜁 to (2.6) and use (3.3) and 𝜁∈Γ(𝑇𝑀), we get ∇_𝑋𝜁+^𝑟_𝑗=1ℎ^ℓ𝑗(𝑋,𝜁)𝑁_𝑗+^𝑛_𝛽_=𝑟+1ℎ^𝑠𝛽(𝑋,𝜁)𝐸_𝛽=0,∀𝑋∈Γ(𝑇𝑀).(4.1) Taking the scalar product with 𝜉_𝑖 and 𝐸_𝛼 in this equation by turns, we have ∇_𝑋𝜁=0,ℎ^ℓ𝑖(𝑋,𝜁)=0,ℎ^𝑠𝛼(𝑋,𝜁)=0.(4.2) Thus 𝜁 is parallel on 𝑀 and conjugate to any vector field on 𝑀 with respect to ℎ. Replace 𝑃𝑌 by 𝜁 to (2.9) and use (4.2) and 𝜁∈Γ(𝑆(𝑇𝑀)), we have ∇^∗𝑋𝜁+^𝑟_𝑗=1ℎ^∗𝑗(𝑋,𝜁)𝜉_𝑗=0,∀𝑋∈Γ(𝑇𝑀).(4.3) Taking the scalar product with 𝑁_𝑖 to this equation we have ∇^∗𝑋𝜁=0,ℎ^∗𝑖(𝑋,𝜁)=0,∀𝑋∈Γ(𝑇𝑀).(4.4) Thus 𝜁 is also parallel on 𝑆(𝑇𝑀) and conjugate to any vector field on 𝑀 with respect to ℎ^∗. Thus we have our assertions.

Definition 4.2. An 𝑟-lightlike submanifold 𝑀 of 𝑀 is said to be totally umbilical [13] if there is a smooth vector field ℋ∈Γ(tr(𝑇𝑀)) such that ℎ(𝑋,𝑌)=ℋ𝑔(𝑋,𝑌),∀𝑋,𝑌∈Γ(𝑇𝑀).(4.5) In case ℋ=0, we say that 𝑀 is totally geodesic.
It is easy to see that 𝑀 is totally umbilical if and only if, on each coordinate neighborhood 𝒰, there exist smooth functions 𝒜_𝑖 and ℬ_𝛼 such thatℎ^ℓ𝑖(𝑋,𝑌)=𝒜_𝑖𝑔(𝑋,𝑌),ℎ^𝑠𝛼(𝑋,𝑌)=ℬ_𝛼𝑔(𝑋,𝑌),∀𝑋,𝑌∈Γ(𝑇𝑀).(4.6)

Theorem 4.3. Let 𝑀 be a totally umbilical generic 𝑟-lightlike submanifold of an indefinite cosymplectic manifold 𝑀. Then 𝑀 is totally geodesic.

Proof. From (4.2) and (4.6), we obtain 𝒜_𝑖𝑔(𝑋,𝜁)=0,ℬ_𝛼𝑔(𝑋,𝜁)=0.(4.7) Replace 𝑋 by 𝜁 to this equations and use 𝑔(𝜁,𝜁)=1, we have 𝒜_𝑖=0 for all 𝑖 and ℬ_𝛼=0 for all 𝛼. Thus 𝑀 is totally geodesic.

Definition 4.4. A screen distribution 𝑆(𝑇𝑀) of 𝑀 is said to be totally umbilical [13] in 𝑀 if, for each locally second fundamental form ℎ^∗𝑖, there exist smooth functions 𝒞_𝑖 on any coordinate neighborhood 𝒰 in 𝑀 such that ℎ^∗𝑖(𝑋,𝑃𝑌)=𝒞_𝑖𝑔(𝑋,𝑌),∀𝑋,𝑌∈Γ(𝑇𝑀).(4.8) In case 𝒞_𝑖=0 for all 𝑖, we say that 𝑆(𝑇𝑀) is totally geodesic in 𝑀.
Due to (2.18) and (4.8), we know that 𝑆(𝑇𝑀) is totally umbilical in 𝑀 if and only if each shape operators 𝐴_𝑁𝑖 of 𝑆(𝑇𝑀) satisfies𝑔𝐴_𝑁𝑖𝑋,𝑃𝑌=𝒞_𝑖𝑔(𝑋,𝑃𝑌),∀𝑋,𝑌∈Γ(𝑇𝑀),(4.9) for some smooth functions 𝒞_𝑖 on 𝒰⊆𝑀.

Theorem 4.5. Let 𝑀 be a generic 𝑟-lightlike submanifold of an indefinite cosymplectic manifold 𝑀 such that 𝑆(𝑇𝑀) is totally umbilical in 𝑀. Then 𝑆(𝑇𝑀) is totally geodesic in 𝑀.

Proof. As 𝑆(𝑇𝑀) is totally umbilical in 𝑀. Replace 𝑌 by 𝜁 to (4.8) and use (4.4), we have 𝒞_𝑖𝑔(𝑋,𝜁)=0 for all 𝑋∈Γ(𝑇𝑀). Replace 𝑋 by 𝜁 to this equation and use the fact 𝑔(𝜁,𝜁)=1, we obtain 𝒞_𝑖=0 for all 𝑖.
From (3.6), the screen distribution 𝑆(𝑇𝑀) splits as follows:𝑆(𝑇𝑀)={𝐽(Rad(𝑇𝑀))⊕𝐽(ltr(𝑇𝑀))}⊕_orth𝐽𝑆𝑇𝑀^⟂⊕_orth𝐻_𝑜,(4.10) where 𝐻_𝑜 is a non-degenerate almost complex distribution 𝐻_𝑜 on 𝑀 with respect to 𝐽, that is, 𝐽(𝐻_𝑜)=𝐻_𝑜. Thus the general decompositions of 𝑇𝑀 and 𝑇𝑀 in (2.1) and (2.4) reduce, respectively, to 𝑇𝑀=𝐻⊕𝐻^,𝑇𝑀=𝐻⊕𝐻^⊕tr(𝑇𝑀),(4.11) where 𝐻 and 𝐻^ are 2𝑟- and 𝑟-lightlike distributions on 𝑀 such that 𝐻𝐻′==R𝐽ad(lt(r𝑇(𝑀𝑇𝑀)⊕))⊕_orth𝐽_orth(Rad(𝑇𝑀))⊕_orth𝐻_𝑜,𝐽𝑆𝑇𝑀^⟂.(4.12) In this case, 𝐻 is an almost complex distribution of 𝑀 with respect to 𝐽. Consider the local null vector fields 𝑈_𝑖 and 𝑉_𝑖 on 𝑆(𝑇𝑀) and the local nonnull vector field 𝑊_𝛼 on 𝑆(𝑇𝑀) defined respectively by 𝑈_𝑖=−𝐽𝑁_𝑖,𝑉_𝑖=−𝐽𝜉_𝑖,𝑊_𝛼=−𝐽𝐸_𝛼.(4.13) Denote by 𝑆 the projection morphism of 𝑇𝑀 on 𝐻 with respect to the decomposition (4.11). Then any vector field 𝑋 on 𝑀 is expressed as follows: 𝑋=𝑆𝑋+^𝑟_𝑖=1𝑢_𝑖(𝑋)𝑈_𝑖+^𝑚_𝛼_=𝑟+1𝐽𝑋=𝐹𝑋+_𝑟𝑤_𝛼(𝑋)𝑊_𝛼,_𝑖=1𝑢_𝑖(𝑋)𝑁_𝑖+^𝑚_𝛼_=𝑟+1𝑤_𝛼(𝑋)𝐸_𝛼,(4.14) where 𝑢_𝑖, 𝑣_𝑖, and 𝑤_𝛼 are 1-forms locally defined on 𝑀 by 𝑢_𝑖(𝑋)=𝑔𝑋,𝑉_𝑖,𝑣_𝑖(𝑋)=𝑔𝑋,𝑈_𝑖,𝑤_𝑖(𝑋)=𝜖_𝛼𝑔𝑋,𝐸_𝛼,(4.15) and 𝐹 is a tensor field of (1,1)-type globally defined on 𝑀 by 𝐹𝑋=𝐽𝑆𝑋,∀𝑋∈Γ(𝑇𝑀).(4.16) Apply 𝐽 to (2.6), (2.7), (2.8), and (2.24) and use (4.13) and (4.14), we have ℎℎ^ℓ𝑗𝑋,𝑈_𝑖=ℎ^∗𝑖𝑋,𝑉_𝑗,ℎ^∗𝑖𝑋,𝑊_𝛼=𝜖_𝛼ℎ^𝑠𝛼𝑋,𝑈_𝑖,(4.17)_ℓ𝑗ℎ𝑋,𝑉_𝑖=ℎ^ℓ𝑖𝑋,𝑉_𝑗,ℎ^ℓ𝑖𝑋,𝑊_𝛼=𝜖_𝛼ℎ^𝑠𝛼𝑋,𝑉_𝑖,(4.18)_ℓ𝑖∇𝑋,𝑊_𝛼=𝜖_𝛼ℎ^𝑠𝛼𝑋,𝑉_𝑖,𝜖_𝛽ℎ^𝑠𝛽𝑋,𝑊_𝛼=𝜖_𝛼ℎ^𝑠𝛼𝑋,𝑊_𝛽,(4.19)_𝑋𝑈_𝑖=𝐹𝐴_𝑁𝑖𝑋+^𝑟_𝑗=1𝜏_𝑖𝑗(𝑋)𝑈_𝑗+^𝑟_𝛼_=𝑟+1∇𝜌_𝑖𝛼(𝑋)𝑊_𝛼,(4.20)_𝑋𝑉_𝑖=𝐹𝐴^∗𝜉_𝑖𝑋−^𝑟_𝑗=1𝜏_𝑗𝑖(𝑋)𝑉_𝑗−^𝑟_𝛼_=𝑟+1𝜖_𝛼𝜙_𝛼𝑖(𝑋)𝑊_𝛼+^𝑟_𝑗=1∇ℎ^ℓ𝑗𝑋,𝜉_𝑖𝑈_𝑗,(4.21)_𝑋𝑊_𝛼=𝐹𝐴_𝐸𝛼𝑋+^𝑟_𝑖=1𝜙_𝛼𝑖(𝑋)𝑈_𝑖+^𝑛_𝛽_=𝑟+1∇𝜎_𝛼𝛽(𝑋)𝑊_𝛽,(4.22)_𝑋𝐹(𝑌)=^𝑟_𝑖=1𝑢_𝑖(𝑌)𝐴_𝑁𝑖𝑋+^𝑛_𝛼_=𝑟+1𝑤_𝛼(𝑌)𝐴_𝐸𝛼𝑋−^𝑟_𝑖=1ℎ^ℓ𝑖(𝑋,𝑌)𝑈_𝑖−^𝑛_𝛼_=𝑟+1ℎ^𝑠𝛼(𝑋,𝑌)𝑊_𝛼.(4.23)

Theorem 4.6. Let 𝑀 be a generic 𝑟-lightlike submanifold of an indefinite cosymplectic manifold 𝑀. Then 𝐻 is integrable if and only if ℎ(𝑋,𝐹𝑌)=ℎ(𝐹𝑋,𝑌),∀𝑋,𝑌∈Γ(𝐻).(4.24) Moreover, if 𝑀 is totally umbilical, then 𝐻 is a parallel distribution on M.

Proof. Take 𝑋,𝑌∈Γ(𝐻). Then we have 𝐹𝑌=𝐽𝑌∈Γ(𝐻). Apply ∇_𝑋 to 𝐹𝑌=𝐽𝑌 and use (2.6), (3.2), (4.13), and (4.14), we have ℎ∇^ℓ𝑖(𝑋,𝐹𝑌)=𝑔∇_𝑋𝑌,𝑉_𝑖,ℎ^𝑠𝛼(𝑋,𝐹𝑌)=𝜖_𝛼𝑔∇_𝑋𝑌,𝑊_𝛼,(4.25)_𝑋𝐹(𝑌)=−^𝑟_𝑖=1ℎ^ℓ𝑖(𝑋,𝑌)𝑈_𝑖−^𝑚_𝛼_=𝑟+1ℎ^𝑠𝛼(𝑋,𝑌)𝑊_𝛼.(4.26) By directed calculations from two equations of (4.25), we have ℎ(𝑋,𝐹𝑌)−ℎ(𝐹𝑋,𝑌)=^𝑟_𝑖=1𝑔[𝑋,𝑌],𝑉_𝑖𝑁_𝑖+^𝑚_𝛼_=𝑟+1𝜖_𝛼𝑔[𝑋,𝑌],𝑊_𝛼𝐸_𝛼.(4.27) If 𝐻 is an integrable distribution on 𝑀, then we have [𝑋,𝑌]∈Γ(𝐻) for any 𝑋,𝑌∈Γ(𝐻). This implies 𝑔([𝑋,𝑌],𝑉_𝑖)=𝑔([𝑋,𝑌],𝑊_𝛼)=0 for all 𝑖 and 𝛼. Therefore we obtain ℎ(𝑋,𝐹𝑌)=ℎ(𝐹𝑋,𝑌) for all 𝑋,𝑌∈Γ(𝐻). Conversely if ℎ(𝑋,𝐹𝑌)=ℎ(𝐹𝑋,𝑌) for all 𝑋,𝑌∈Γ(𝐻), then we have 𝑔([𝑋,𝑌],𝑉_𝑖)=𝑔([𝑋,𝑌],𝑊_𝛼)=0 for all 𝑖 and 𝛼. This implies [𝑋,𝑌]∈Γ(𝐻) for all 𝑋,𝑌∈Γ(𝐻). Thus 𝐻 is an integrable distribution of 𝑀.
If 𝑀 is totally umbilical, from Theorem 4.3 and (4.25), we have𝑔∇_𝑋𝑌,𝑉_𝑖=𝑔∇_𝑋𝑌,𝑊_𝛼=0,∀𝑖,𝛼.(4.28) This implies ∇_𝑋𝑌∈Γ(𝐻) for all 𝑋,𝑌∈Γ(𝐻). Thus 𝐻 is a parallel distribution on 𝑀.

Theorem 4.7. Let 𝑀 be a generic 𝑟-lightlike submanifold of an indefinite cosymplectic manifold 𝑀. Then 𝐹 is parallel on 𝐻 with respect to ∇ if and only if 𝐻 is a parallel distribution on𝑀.

Proof. Assume that 𝐹 is parallel on 𝐻 with respect to ∇. For any 𝑋,𝑌∈Γ(𝐻), we have (∇_𝑋𝐹)𝑌=0. Taking the scalar product with 𝑉_𝑘 and 𝑊_𝛽 to (4.26) with (∇_𝑋𝐹)𝑌=0, we have ℎ^ℓ𝑘(𝑋,𝑌)=0 and ℎ^𝑠𝛽(𝑋,𝑌)=0 for all 𝑋,𝑌∈Γ(𝐻) and for each 𝑘 and 𝛽, respectively. From (4.25), we have 𝑔(∇_𝑋𝑌,𝑉_𝑖)=0 and 𝑔(∇_𝑋𝑌,𝑊_𝛼)=0. This implies ∇_𝑋𝑌∈Γ(𝐻) for all 𝑋,𝑌∈Γ(𝐻). Thus 𝐻 is a parallel distribution on 𝑀.
Conversely, if 𝐻 is parallel on 𝑀, from (4.25) we haveℎ^ℓ𝑖(𝑋,𝐹𝑌)=0,ℎ^𝑠𝛼(𝑋,𝐹𝑌)=0,∀𝑋,𝑌∈Γ(𝐻).(4.29) For any 𝑌∈Γ(𝐻), we show that 𝐹²𝑌=𝐽²𝑌=−𝑌+𝜃(𝑌)𝜁. Replace 𝑌 by 𝐹𝑌 to the equations and use (4.2), we have ℎ^ℓ𝑖(𝑋,𝑌)=0 and ℎ^𝑠𝛼(𝑋,𝑌)=0 for any 𝑋,𝑌∈Γ(𝐻). Thus 𝐹 is parallel on 𝐻 with respect to ∇ by (4.25).

Theorem 4.8. Let 𝑀 be a generic 𝑟-lightlike submanifold of an indefinite cosymplectic manifold 𝑀. If F is parallel on 𝑇𝑀 with respect to ∇, then 𝐻 is a parallel distribution on 𝑀 and 𝑀 is locally a product manifold 𝑀^𝑟×𝑀^𝑛−𝑟×𝑀^𝑚−𝑛, where 𝑀^𝑟, 𝑀^𝑛−𝑟, and 𝑀^𝑚−𝑛 are leafs of 𝐽(ltr(𝑇𝑀)), 𝐽(𝑆(𝑇𝑀^⟂)) and 𝐻, respectively.

Proof. Assume that 𝐹 is parallel on 𝑇𝑀 with respect to ∇. Then 𝐹 is parallel on 𝐻 with respect to ∇. By Theorem 4.7, 𝐻 is a parallel distribution on 𝑀. Apply the operator 𝐹 to (4.23) with (∇_𝑋𝐹)𝑌=0, we have ^𝑟_𝑖=1𝑢_𝑖(𝑌)𝐹𝐴_𝑁𝑖𝑋+^𝑛_𝛼_=𝑟+1𝑒_𝛼(𝑌)𝐹𝐴_𝐸𝛼𝑋=0,∀𝑋,𝑌∈Γ(𝑇𝑀),(4.30) due to 𝐹𝑈_𝑖=𝐹𝑊_𝛼=0 for all 𝑖 and 𝛼. Replace 𝑌 by 𝑈_𝑘 and 𝑊_𝛽 to this equation by turns and use (4.15), we have 𝐹(𝐴_𝑁𝑖𝑋)=0 and 𝐹(𝐴_𝐸𝛼𝑋)=0. Taking the scalar product with 𝑊_𝛽 and 𝑁_𝑘 to (4.23) with (∇_𝑋𝐹)𝑌=0 by turns, we have ℎ^𝑠𝛼(X,𝑌)=^𝑟_𝑖=1𝑢_𝑖(𝑌)𝑤_𝛼𝐴_𝑁𝑖𝑋+^𝑚_𝛽_=𝑟+1𝑤_𝛽(𝑌)𝑤_𝛼𝐴^𝐸_𝛽𝑋,(4.31)_𝑟_𝑖=1𝑢_𝑖(𝑌)𝑔𝐴_𝑁𝑖𝑋,𝑁_𝑘+^𝑚_𝛼_=𝑟+1𝑤_𝛼(𝑌)𝑔𝐴_𝐸𝛼𝑋,𝑁_𝑘=0,(4.32) for all 𝑋,𝑌∈Γ(𝑇𝑀). Replace 𝑌 by 𝜉_𝑗 to (4.31), we get 𝜙_𝛼𝑗(𝑋)=0 due to (2.23). Also replace 𝑌 by 𝑊_𝛽 to (4.32), we have 𝜌_𝑘𝛽(𝑋)=0 due to (2.17). From this results, (4.11) and (4.14), we get ∇_𝑋𝑈_𝑖=^𝑟_𝑗=1𝜏_𝑖𝑗(𝑋)𝑈_𝑗,∇_𝑋𝑊_𝛼=^𝑚_𝛽_=𝑟+1𝜎_𝛼𝛽(𝑋)𝑊_𝛽,∀𝑋∈Γ(𝑇𝑀).(4.33) 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=𝑀^𝑟×𝑀^𝑛−𝑟×𝑀^𝑚−𝑛, where 𝑀^𝑟,𝑀^𝑛−𝑟, and 𝑀^𝑚−𝑛 are some leafs of 𝐽(ltr(𝑇𝑀)), 𝐽(𝑆(𝑇𝑀^⟂)) and 𝐻, respectively.

Theorem 4.9. Let 𝑀 be a generic 𝑟-lightlike submanifold of an indefinite cosymplectic manifold 𝑀. One has the following assertions.(i)If each 𝑉_𝑖 is parallel with respect to ∇, then 𝜏_𝑖𝑗=𝜙_𝛼𝑖=ℎ^ℓ(𝑋,𝜉_𝑖)=0. In this case 𝑀 is irrotational. Moreover, one has𝐴^∗𝜉_𝑖𝑋=^𝑟_𝑗=1𝑢_𝑗𝐴^∗𝜉_𝑖𝑋𝑈_𝑗+^𝑚_𝛼_=𝑟+1𝑤_𝛼𝐴^∗𝜉_𝑖𝑋𝑊_𝛼,∀𝑋∈Γ(𝑇𝑀).(4.34)(ii)If each 𝑈_𝑖 is parallel with respect to ∇, then 𝜏_𝑖𝑗=𝜌_𝑖𝛼=0 and𝐴_𝑁𝑖𝑋=^r_𝑗=1𝑢_𝑗𝐴_𝑁𝑖𝑋𝑈_𝑗+^𝑚_𝛼_=𝑟+1𝑤_𝛼𝐴_𝑁𝑖𝑋𝑊_𝛼,∀𝑋∈Γ(𝑇𝑀).(4.35)(iii)If each 𝑊_𝛼 is parallel with respect to ∇, then 𝜙_𝛼𝑖=0 and𝐴_𝐸𝛼𝑋=^𝑟_𝑖=1𝑢_𝑖𝐴_𝐸𝛼𝑋𝑈_𝑖+^𝑚_𝛽_=𝑟+1𝑤_𝛽𝐴_𝐸𝛼𝑋𝑊_𝛽,∀𝑋∈Γ(𝑇𝑀).(4.36) Moreover, if all of 𝑉_𝑖, 𝑈_𝑖, and 𝑊_𝛼 are parallel on TM with respect to ∇, then 𝑆(𝑇𝑀) is totally geodesic in 𝑀 and 𝜏_𝑖𝑗=𝜙_𝛼𝑖=𝜌_𝑖𝛼=0 on Γ(𝑇𝑀). In this case, each null transversal vector fields 𝑁_𝑖 of 𝑀 is a constant on 𝑀.

Proof. If 𝑉_𝑖 is parallel with respect to ∇, then, taking the scalar product with 𝑈_𝑘, 𝑊_𝛽, and 𝑉_𝑘 to (4.21) by turns, we have 𝜏_𝑘𝑖(𝑋)=0, 𝜙_𝛽𝑖(𝑋)=0 and ℎ^ℓ𝑘(𝑋,𝜉_𝑖)=0, respectively. Thus 𝑀 is irrotational. We have 𝐹(𝐴^∗𝜉_𝑖𝑋)=0 for all 𝑋∈Γ(𝑇𝑀). From this result and (4.14), we obtain 𝐽𝐴^∗𝜉_𝑖𝑋=^𝑟_𝑗=1𝑢_𝑗𝐴^∗𝜉_𝑖𝑋𝑁_𝑗+^𝑚_𝛼_=𝑟+1𝑤_𝛼𝐴^∗𝜉_𝑖𝑋𝐸_𝛼.(4.37) Apply 𝐽 to this equation and use 𝜃(𝐴^∗𝜉𝑋_𝑖)=0, we obtain (i). In a similar way, by using (4.13), (4.14), (4.20), and (4.22), 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 (4.17)-1, we have𝑢_𝑗𝐴_𝑁𝑖𝑋=𝑣_𝑖𝐴^∗𝜉_𝑗𝑋=𝑔𝐴^∗𝜉_𝑗𝑋,𝑈_𝑖=0,∀𝑋∈Γ(𝑇𝑀).(4.38) Also, substituting the equation of (iii) into (4.17)-2, we have 𝑤_𝛼𝐴_𝑁𝑖𝑋=𝜖_𝛼𝑣_𝑖𝐴_𝐸𝛼𝑋=𝑔𝐴_𝐸𝛼𝑋,𝑈_𝑖=0,∀𝑋∈Γ(𝑇𝑀).(4.39) From the last two equations and the equation of (ii), we see that 𝐴_𝑁𝑖𝑋=0 for all 𝑋∈Γ(𝑇𝑀). From this and (2.18) we see that 𝑆(𝑇𝑀) is totally geodesic in 𝑀 and all 1-forms 𝜏_𝑖𝑗, 𝜙_𝛼𝑖, and 𝜌_𝑖𝛼, defined by (2.7) and (2.8), vanish identically. Using the results and (2.7), we show that 𝑁 is a constant on 𝑀.

Theorem 4.10. Let 𝑀 be a totally umbilical generic 𝑟-lightlike submanifold of an indefinite cosymplectic manifold 𝑀 such that 𝑆(𝑇𝑀) is totally umbilical. Then M is locally a product manifold 𝑀^𝑟×𝑀^𝑠×𝑀^𝑡, where 𝑀^𝑟, 𝑀^𝑠, and 𝑀^𝑡 are some leafs of Rad(𝑇𝑀), 𝐻^⟂𝑜=Span{𝑉_𝑖,𝑈_𝑖,𝑊_𝛼} and 𝐻_𝑜, respectively, and 𝑠=𝑛+𝑟,𝑡=𝑚−𝑛−2𝑟.

Proof. By Theorem 4.6, 𝐻 is a parallel distribution 𝑀. Thus, for all 𝑋,𝑌∈Γ(𝐻_𝑜), we have ∇_𝑋𝑌∈Γ(𝐻). From (2.9) and (4.26), we have ℎ^∗𝑖(𝑋,𝐹𝑌)==𝑔𝑔∇𝐹_𝑋𝐹∇𝑌_𝑋,𝑁_𝑖=𝑔∇_𝑋𝐹𝑌+𝐹∇_𝑋𝑌,𝑁_𝑖𝑌,𝑁_𝑖=−𝑔∇_𝑋𝑌,𝐽𝑁_𝑖=𝑔∇_𝑋𝑌,𝑈_𝑖,(4.40) due to 𝐹𝑌∈Γ(𝐻_𝑜). If 𝑆(𝑇𝑀) is totally umbilical in 𝑀, then we have ℎ^∗𝑖=0 due to Theorem 4.5. By (2.9) and (4.40), we get 𝑔∇_𝑋𝑌,𝑁_𝑖=0,𝑔∇_𝑋𝑌,𝑈_𝑖=0,∀𝑋∈Γ(𝑇𝑀),∀𝑌∈Γ𝐻_𝑜.(4.41) These results and (4.25) imply ∇_𝑋𝑌∈Γ(𝐻_𝑜) for all 𝑋,𝑌∈Γ(𝐻_𝑜). Thus 𝐻_𝑜 is a parallel distribution on 𝑆(𝑇𝑀) and 𝑇𝑀=𝐻_𝑜⊕_orth𝐻^⟂𝑜, where 𝐻^⟂𝑜=Span{𝜉_𝑖,𝑉_𝑖,𝑈_𝑖,𝑊_𝛼}. By Theorems 4.3 and 4.5, we have ℎ^ℓ𝑖=ℎ^𝑠𝛼=𝐴_𝑁𝑖=𝜙_𝛼𝑖=0 and 𝐴_𝐸𝛼𝑋=∑^𝑟𝑖=1𝜌_𝑖𝛼(𝑋)𝜉_𝑖. Thus (2.10) and (4.20)~(4.22) deduce, respectively, to ∇_𝑋𝑈_𝑖=^𝑟_𝑗=1𝜏_𝑖𝑗(𝑋)𝑈_𝑗+^𝑚_𝛼_=𝑟+1∇_𝑋𝜌_𝑖𝛼(𝑋)𝑊_𝛼,𝑉_𝑖=−^𝑟_𝑗=1𝜏_𝑗𝑖(𝑋)𝑉_𝑗+^𝑟_𝑗=1∇_𝑋ℎ^ℓ𝑗𝑋,𝜉_𝑖𝑈_𝑗,𝑊_𝛼=−^𝑟_𝑖=1𝜌_𝑖𝛼(𝑋)𝑉_𝑗+^𝑚_𝛽_=𝑟+1∇_𝑋𝜎_𝛼𝛽(𝑋)𝑊_𝛽,𝜉=−^𝑟_𝑗=1𝜏_𝑗𝑖(𝑋)𝜉_𝑗,∀𝑋∈Γ𝐻^⟂𝑜.(4.42) Thus 𝐻^⟂𝑜 is also a parallel distribution on 𝑀. Thus we have 𝑀=𝑀^𝑟×𝑀^𝑠×𝑀^𝑡, where 𝑀^𝑟, 𝑀^𝑠, and 𝑀^𝑡 are some leafs of Rad(𝑇𝑀), 𝐻^⟂𝑜=Span{𝑉_𝑖,𝑈_𝑖,𝑊_𝛼} and 𝐻_𝑜, respectively, and 𝑠=𝑛+𝑟,𝑡=𝑚−𝑛−2𝑟.

Acknowledgment

The authors are thankful to the referee for making various constructive suggestions and corrections towards improving the final version of this paper.