# Generic Lightlike Submanifolds of an Indefinite Cosymplectic Manifold

Authors: Dae Ho Jin; Jae Won Lee
Journal: Mathematical Problems in Engineering
 (2011)

---

## 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 submanifoldsM of an indefinite cosymplectic manifold M¯.


---

## Body

## 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 genericr-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 that(2.1)TM=Rad(TM)⊕orthS(TM),TM⊥=Rad(TM)⊕orthS(TM⊥),
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:(2.2)i,j,k,…∈{1,…,r},α,β,γ,…∈{r+1,…,n}.Lettr(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,…,Nr} 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 that(2.3)g¯(Ni,ξj)=δij,g¯(Ni,Nj)=0,
where {ξ1,…,ξr} is a lightlike basis of Γ(Rad(TM)). Then we have(2.4)TM¯=TM⊕tr⁡(TM)={Rad(TM)⊕tr⁡(TM)}⊕orthS(TM)={Rad(TM)⊕ltr(TM)}⊕orthS(TM)⊕orthS(TM⊥).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¯: (2.5){ξ1,…,ξr,N1,…,Nr,Fr+1,…,Fm,Er+1,…,En},
where the sets {Fr+1,…,Fm} and {Er+1,…,En} 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(2.6)∇¯XY=∇XY+∑i=1rhil(X,Y)Ni+∑α=r+1nhαs(X,Y)Eα,(2.7)∇¯XNi=-ANiX+∑j=1rτij(X)Nj+∑α=r+1nρiα(X)Eα,(2.8)∇¯XEα=-AEαX+∑i=1rϕαi(X)Ni+∑β=r+1nσαβ(X)Eβ,(2.9)∇XPY=∇X*PY+∑i=1rhi*(X,PY)ξi,(2.10)∇Xξi=-Aξi*X-∑j=1rτji(X)ξj,
for any X,Y∈Γ(TM), where ∇ and ∇* are induced linear connections on TM and S(TM), respectively, the bilinear forms hiℓ and hαs on M are called the local lightlike and screen second fundamental forms on TM, respectively, hi* are called the local radical second fundamental forms on S(TM). ANi, Aξi*, and AEα are linear operators on Γ(TM) and τij, ρiα, ϕαi and σαβ are 1-forms on TM. Since ∇¯ is torsion-free, ∇ is also torsion-free and both hiℓ and hαs are symmetric. From the fact hiℓ(X,Y)=g¯(∇¯XY,ξi), we know that hiℓ are independent of the choice of a screen distribution. We say that (2.11)h(X,Y)=∑i=1rhil(X,Y)Ni+∑α=r+1nhαs(X,Y)Eα
is the second fundamental tensor of M.The induced connection∇ on TM is not metric and satisfies(2.12)(∇Xg)(Y,Z)=∑i=1r{hil(X,Y)ηi(Z)+hil(X,Z)ηi(Y)},
for all X,Y∈Γ(TM), where ηis are the 1-forms such that(2.13)ηi(X)=g¯(X,Ni),∀X∈Γ(TM).
But the connection ∇* on S(TM) is metric. The above three local second fundamental forms are related to their shape operators by(2.14)hil(X,Y)=g(Aξi*X,Y)-∑k=1rhkl(X,ξi)ηk(Y),(2.15)hil(X,PY)=g(Aξi*X,PY),g¯(Aξi*X,Nj)=0,(2.16)ϵαhαs(X,Y)=g(AEαX,Y)-∑i=1rϕαi(X)ηi(Y),(2.17)ϵαhαs(X,PY)=g(AEαX,PY),g¯(AEαX,Ni)=ϵαρiα(X),(2.18)hi*(X,PY)=g(ANiX,PY),ηj(ANiX)+ηi(ANjX)=0,(2.19)ϵβσαβ=-ϵασβα,∀X,Y∈Γ(TM),
where ϵα=g¯(Eα,Eα)(=±1) is the sign of the vector field Eα. From (2.18), we know that each ANi is shape operator related to the local second fundamental form hi* on S(TM). Replace Y by ξj in (2.14), we have(2.20)hil(X,ξj)+hjl(X,ξi)=0,
for all X∈Γ(TM). It follows(2.21)hil(X,ξi)=0,hil(ξj,ξk)=0.
Also, replace X by ξj in (2.14) and use (2.21), we have(2.22)hil(X,ξj)=g(X,Aξi*ξj),Aξi*ξj+Aξj*ξi=0,Aξi*ξi=0.
For an r-lightlike submanifold, replace Y by ξi in (2.16), we have(2.23)hαs(X,ξi)=-ϵαϕαi(X),∀X∈Γ(TM).
From (2.6), (2.10), and (2.23), for all X∈Γ(TM), we have(2.24)∇¯Xξi=-Aξi*X-∑j=1rτji(X)ξj-∑α=r+1nϵαϕαi(X)Eα+∑j=1rhjl(X,ξi)Nj.Definition 2.1.
A lightlike submanifoldM of a semi-Riemannian manifold (M¯,g¯) is said to be irrotational if ∇¯Xξi∈Γ(TM) for any X∈Γ(TM) and ξi∈Γ(Rad(TM)) for all i.Note 1.
For anr-lightlike M, the above definition is equivalent to
(2.25)hjl(X,ξi)=0,hαs(X,ξi)=ϕαi(X)=0,∀X∈Γ(TM).Denote byR¯ and R the curvature tensors of ∇¯ and ∇, respectively. Using the local Gauss-Weingarten formulas for M, we obtain(2.26)R¯(X,Y)Z=R(X,Y)Z+∑i=1r{hil(X,Z)ANiY-hil(Y,Z)ANiX}+∑α=r1n{hαs(X,Z)AEαY-hαs(Y,Z)AEαX}+∑i=1r{∑i=1r(∇Xhil)(Y,Z)-(∇Yhil)(X,Z)+∑j=1r[τji(X)hjl(Y,Z)-τji(Y)hjl(X,Z)]+∑α=r+1n[ϕαi(X)hαs(Y,Z)-ϕαi(Y)hαs(X,Z)]}Ni+∑α=r+1n{∑i=1r[ρiα(X)hil(Y,Z)-ρiα(Y)hαs(X,Z)](∇Xhαs)(Y,Z)-(∇Yhαs)(X,Z)+∑i=1r[ρiα(X)hil(Y,Z)-ρiα(Y)hαs(X,Z)]+∑β=r+1n[σβα(X)hβs(Y,Z)-σβα(Y)hβs(X,Z)]}Eα,
for all X,Y,Z∈Γ(TM). Assume that M is irrotational. Replace Z by ξk in (2.26) and use (2.10), (2.15), (2.17), and (2.25), then we have(2.27)R¯(X,Y)ξk=R(X,Y)ξk+∑i=1r{g(Aξi*Y,Aξk*X)-g(Aξi*X,Aξk*Y)}Ni+∑α=r+1nϵα{g(AEαY,Aξk*X)-g(AEαX,Aξk*Y)}Eα.
Using (2.27) and the fact R(X,Y)Z∈Γ(TM) for X,Y,Z∈Γ(TM), we get(2.28)g¯(R¯(X,Y)Z,ξk)=-g¯(R¯(X,Y)ξk,Z)=-g(R(X,Y)ξk,Z)+∑i=1r{g(Aξi*X,Aξk*Y)-g(Aξi*Y,Aξk*X)}ηi(Z)=g(R(X,Y)Z,ξk)+∑i=1r{g(Aξi*X,Aξk*Y)-g(Aξi*Y,Aξk*X)}ηi(Z)=∑i=1r{g(Aξi*X,Aξk*Y)-g(Aξi*Y,Aξk*X)}ηi(Z),∀X,Y,Z∈Γ(TM).


## 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 satisfying(3.1)J2X=-X+θ(X)ζ,Jζ=0,θ∘J=0,θ(ζ)=1,g¯(ζ,ζ)=ϵ,g¯(JX,JY)=g¯(X,Y)-ϵθ(X)θ(Y),θ(X)=ϵg¯(ζ,X),dθ(X,Y)=g¯(JX,Y),ϵ=±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 NJ+dθ⊗ζ=0, where NJ is the Nijenhuis tensor field of J. A normal contact metric manifold is called a cosymplectic [9, 10] for which we have(3.2)∇¯Xθ=0,∇¯XJ=0,
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(3.3)∇¯Xζ=0.An indefinite cosymplectic manifoldM¯ 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 by(3.4)R¯(X,Y)Z=c4{g¯(Y,Z)X-g¯(X,Z)Y+θ(X)θ(Z)Y-θ(Y)θ(Z)X+g¯(X,Z)θ(Y)ζ-g¯(Y,Z)θ(X)ζ+g¯(JY,Z)JX+g¯(JZ,X)JY-2g¯(JX,Y)JZ},
for any vector fields X,Y, and Z in M¯.LetM 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 by(3.5)ζ=Pζ+∑i=1raiξi+∑i=1rbiNi+∑α=r+1neαEα,
where ai=θ(Ni), bi=θ(ξi) and eα=ϵαθ(Eα) are smooth functions on M¯.Note 2.
AlthoughS(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 thatM is generic lightlike submanifold of M¯ if there exists a screen distribution S(TM) of M such that
(3.6)J(S(TM)⊥)⊂S(TM).Proposition 3.2 (see [3]).
LetM be a lightlike hypersurface of an indefinite cosymplectic manifold M¯.  Then M is a generic lightlike submanifold of M¯.Proposition 3.3 (see [4]).
LetM 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.
LetM be an irrotational generic r-lightlike submanifold of an indefinite cosymplectic space form M¯(c). Then one has c=0.Proof.
Assume thatbk≠0 in (3.5). Note that (3.1) implies g¯(JX,ξk)=-g¯(X,Jξk) for all X∈Γ(TM). Then, taking the scalar product with ξk to (3.4), and using (2.28), we get
(3.7)4∑i=1r{g(Aξi*X,Aξk*Y)-g(Aξi*Y,Aξk*X)}ηi(Z)=c{bkg(X,Z)θ(Y)-bkg(Y,Z)θ(X)-g¯(JY,Z)g(X,Jξk)-g¯(JZ,X)g(Y,Jξk)+2g¯(JX,Y)g(Z,Jξk)},∀X,Y,Z∈Γ(TM).
Replace Z by Jξk and Y by ξk in the equation and use (3.1), then we have
(3.8)bk2cg(X,Jξk)=0,∀X∈Γ(TM),
because ηi(Jξk)=0 by (3.6). Replacing X by Jξk in this equation, we obtain bk4c=0. Since bk≠0, we have c=0.
Assume thatbk=0. Then, taking the scalar product with ξk to both sides of (3.4) and using (2.28) and (3.1), we obtain(3.9)4∑i=1r{g(Aξi*X,Aξk*Y)-g(Aξi*Y,Aξk*X)}ηi(Z)=c{-g¯(JY,Z)g(X,Jξk)-g¯(JZ,X)g(Y,Jξk)+2g¯(JX,Y)g(Z,Jξk)},∀X,Y,Z∈Γ(TM).
Replace Z by JNk and Y by ξk in this equation and use (3.1), then we have
(3.10)cg(X,Jξk)=0,∀X∈Γ(TM),
because ηi(JNk)=0 by (3.6) and g(Jξk,JNk)=1. Replace X by JNk in this equation, we get c=0.Corollary 3.5.
There exist no irrotational genericr-lightlike submanifolds M of an indefinite cosymplectic space form M¯(c) with c≠0.Proposition 3.6.
LetM 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 thatζ belongs to Rad(TM) (or ltr(TM)). Then (3.1) deduces to ζ=∑i=1raiξi [or ζ=∑i=1rbiNi]. From this, we have
(3.11)1=g¯(ζ,ζ)=∑i,j=1raiajg¯(ξi,ξj)=0[or1=g¯(ζ,ζ)=∑i,j=1rbibjg¯(Ni,Nj)=0].
It is a contradiction. Thus ζ does not belong to Rad(TM) and ltr(TM).


## 4. Generic Lightlike Submanifolds

If the characteristic vector fieldζ 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 ifζis tangent toM, then it belongs toS(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 ai, bi, and eα on M¯, defined by (3.5), vanish identically.Theorem 4.1.
LetM 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 hi*. In particular, ζ is an asymptotic vector field on M.Proof.
ReplaceY by ζ to (2.6) and use (3.3) and ζ∈Γ(TM), we get
(4.1)∇Xζ+∑j=1rhjl(X,ζ)Nj+∑β=r+1nhβs(X,ζ)Eβ=0,∀X∈Γ(TM).
Taking the scalar product with ξi and Eα in this equation by turns, we have
(4.2)∇Xζ=0,hil(X,ζ)=0,hαs(X,ζ)=0.
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
(4.3)∇X*ζ+∑j=1rhj*(X,ζ)ξj=0,∀X∈Γ(TM).
Taking the scalar product with Ni to this equation we have
(4.4)∇X*ζ=0,hi*(X,ζ)=0,∀X∈Γ(TM).
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.
Anr-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
(4.5)h(X,Y)=Hg(X,Y),∀X,Y∈Γ(TM).
In case ℋ=0, we say that M is totally geodesic.
It is easy to see thatM is totally umbilical if and only if, on each coordinate neighborhood 𝒰, there exist smooth functions 𝒜i and ℬα such that(4.6)hil(X,Y)=Aig(X,Y),hαs(X,Y)=Bαg(X,Y),∀X,Y∈Γ(TM).Theorem 4.3.
LetM 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
(4.7)Aig(X,ζ)=0,Bαg(X,ζ)=0.
Replace X by ζ to this equations and use g(ζ,ζ)=1, we have 𝒜i=0 for all i and ℬα=0 for all α. Thus M is totally geodesic.Definition 4.4.
A screen distributionS(TM) of M is said to be totally umbilical [13] in M if, for each locally second fundamental form hi*, there exist smooth functions 𝒞i on any coordinate neighborhood 𝒰 in M such that
(4.8)hi*(X,PY)=Cig(X,Y),∀X,Y∈Γ(TM).
In case 𝒞i=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 ANi of S(TM) satisfies(4.9)g(ANiX,PY)=Cig(X,PY),∀X,Y∈Γ(TM),
for some smooth functions 𝒞i on 𝒰⊆M.Theorem 4.5.
LetM 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.
AsS(TM) is totally umbilical in M. Replace Y by ζ to (4.8) and use (4.4), we have 𝒞ig(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 𝒞i=0 for all i.
From (3.6), the screen distribution S(TM) splits as follows:(4.10)S(TM)={J(Rad(TM))⊕J(ltr(TM))}⊕orthJ(S(TM⊥))⊕orthHo,
where Ho is a non-degenerate almost complex distribution Ho on M with respect to J, that is, J(Ho)=Ho. Thus the general decompositions of TM and TM¯ in (2.1) and (2.4) reduce, respectively, to
(4.11)TM=H⊕H′,TM¯=H⊕H′⊕tr⁡(TM),
where H and H′ are 2r- and r-lightlike distributions on M such that
(4.12)H=Rad(TM)⊕orthJ(Rad(TM))⊕orthHo,H′=J(ltr(TM))⊕orthJ(S(TM⊥)).
In this case, H is an almost complex distribution of M with respect to J. Consider the local null vector fields Ui and Vi on S(TM) and the local nonnull vector field Wα on S(TM) defined respectively by
(4.13)Ui=-JNi,Vi=-Jξi,Wα=-JEα.
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:
(4.14)X=SX+∑i=1rui(X)Ui+∑α=r+1mwα(X)Wα,JX=FX+∑i=1rui(X)Ni+∑α=r+1mwα(X)Eα,
where ui, vi, and wα are 1-forms locally defined on M by
(4.15)ui(X)=g(X,Vi),vi(X)=g(X,Ui),wi(X)=ϵαg(X,Eα),
and F is a tensor field of (1,1)-type globally defined on M by
(4.16)FX=JSX,∀X∈Γ(TM).
Apply J to (2.6), (2.7), (2.8), and (2.24) and use (4.13) and (4.14), we have
(4.17)hjl(X,Ui)=hi*(X,Vj),hi*(X,Wα)=ϵαhαs(X,Ui),(4.18)hjl(X,Vi)=hil(X,Vj),hil(X,Wα)=ϵαhαs(X,Vi),(4.19)hil(X,Wα)=ϵαhαs(X,Vi),ϵβhβs(X,Wα)=ϵαhαs(X,Wβ),(4.20)∇XUi=F(ANiX)+∑j=1rτij(X)Uj+∑α=r+1rρiα(X)Wα,(4.21)∇XVi=F(Aξi*X)-∑j=1rτji(X)Vj-∑α=r+1rϵαϕαi(X)Wα+∑j=1rhjl(X,ξi)Uj,(4.22)∇XWα=F(AEαX)+∑i=1rϕαi(X)Ui+∑β=r+1nσαβ(X)Wβ,(4.23)(∇XF)(Y)=∑i=1rui(Y)ANiX+∑α=r+1nwα(Y)AEαX-∑i=1rhil(X,Y)Ui-∑α=r+1nhαs(X,Y)Wα.Theorem 4.6.
LetM be a generic r-lightlike submanifold of an indefinite cosymplectic manifold M¯. Then H is integrable if and only if
(4.24)h(X,FY)=h(FX,Y),∀X,Y∈Γ(H).
Moreover, if M is totally umbilical, then H is a parallel distribution on M.Proof.
TakeX,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
(4.25)hil(X,FY)=g(∇XY,Vi),hαs(X,FY)=ϵαg(∇XY,Wα),(4.26)(∇XF)(Y)=-∑i=1rhil(X,Y)Ui-∑α=r+1mhαs(X,Y)Wα.
By directed calculations from two equations of (4.25), we have
(4.27)h(X,FY)-h(FX,Y)=∑i=1rg([X,Y],Vi)Ni+∑α=r+1mϵαg([X,Y],Wα)Eα.
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],Vi)=g([X,Y],Wα)=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],Vi)=g([X,Y],Wα)=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.
IfM is totally umbilical, from Theorem 4.3 and (4.25), we have(4.28)g(∇XY,Vi)=g(∇XY,Wα)=0,∀i,α.
This implies ∇XY∈Γ(H) for all X,Y∈Γ(H). Thus H is a parallel distribution on M.Theorem 4.7.
LetM 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 thatF is parallel on H with respect to ∇. For any X,Y∈Γ(H), we have (∇XF)Y=0. Taking the scalar product with Vk and Wβ to (4.26) with (∇XF)Y=0, we have hkℓ(X,Y)=0 and hβs(X,Y)=0 for all X,Y∈Γ(H) and for each k and β, respectively. From (4.25), we have g(∇XY,Vi)=0 and g(∇XY,Wα)=0. This implies ∇XY∈Γ(H) for all X,Y∈Γ(H). Thus H is a parallel distribution on M.
Conversely, ifH is parallel on M, from (4.25) we have(4.29)hil(X,FY)=0,hαs(X,FY)=0,∀X,Y∈Γ(H).
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 hiℓ(X,Y)=0 and hαs(X,Y)=0 for any X,Y∈Γ(H). Thus F is parallel on H with respect to ∇ by (4.25).Theorem 4.8.
LetM 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 Mr×Mn-r×Mm-n,  where Mr, Mn-r, and Mm-n are leafs of J(ltr(TM)),  J(S(TM⊥)) and H, respectively.Proof.
Assume thatF 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
(4.30)∑i=1rui(Y)F(ANiX)+∑α=r+1neα(Y)F(AEαX)=0,∀X,Y∈Γ(TM),
due to FUi=FWα=0 for all i and α. Replace Y by Uk and Wβ to this equation by turns and use (4.15), we have F(ANiX)=0 and F(AEαX)=0. Taking the scalar product with Wβ and Nk to (4.23) with (∇¯XF)Y=0 by turns, we have
(4.31)hαs(X,Y)=∑i=1rui(Y)wα(ANiX)+∑β=r+1mwβ(Y)wα(AEβX),(4.32)∑i=1rui(Y)g(ANiX,Nk)+∑α=r+1mwα(Y)g(AEαX,Nk)=0,
for all X,Y∈Γ(TM). Replace Y by ξj to (4.31), we get ϕαj(X)=0 due to (2.23). Also replace Y by Wβ to (4.32), we have ρkβ(X)=0 due to (2.17). From this results, (4.11) and (4.14), we get
(4.33)∇XUi=∑j=1rτij(X)Uj,∇XWα=∑β=r+1mσαβ(X)Wβ,∀X∈Γ(TM).
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=Mr×Mn-r×Mm-n, where Mr,Mn-r, and Mm-n are some leafs of J(ltr(TM)), J(S(TM⊥)) and H, respectively.Theorem 4.9.
LetM be a generic r-lightlike submanifold of an indefinite cosymplectic manifold M¯. One has the following assertions.(i)
If eachVi is parallel with respect to ∇, then τij=ϕαi=hℓ(X,ξi)=0.   In this case M is irrotational. Moreover, one has(4.34)Aξi*X=∑j=1ruj(Aξi*X)Uj+∑α=r+1mwα(Aξi*X)Wα,∀X∈Γ(TM).(ii)
If eachUi is parallel with respect to ∇, then τij=ρiα=0 and(4.35)ANiX=∑j=1ruj(ANiX)Uj+∑α=r+1mwα(ANiX)Wα,∀X∈Γ(TM).(iii)
If eachWα is parallel with respect to ∇, then ϕαi=0 and(4.36)AEαX=∑i=1rui(AEαX)Ui+∑β=r+1mwβ(AEαX)Wβ,∀X∈Γ(TM).
Moreover, if all of Vi, Ui, and Wα are parallel on TM with respect to ∇, then S(TM) is totally geodesic in M and τij=ϕαi=ρiα=0 on  Γ(TM). In this case, each null transversal vector fields Ni of  M is a constant on M.Proof.
IfVi is parallel with respect to ∇, then, taking the scalar product with Uk, Wβ, and Vk to (4.21) by turns, we have τki(X)=0, ϕβi(X)=0 and hkℓ(X,ξi)=0, respectively. Thus M is irrotational. We have F(Aξi*X)=0 for all X∈Γ(TM). From this result and (4.14), we obtain
(4.37)J(Aξi*X)=∑j=1ruj(Aξi*X)Nj+∑α=r+1mwα(Aξi*X)Eα.
Apply J to this equation and use θ(Aξ*Xi)=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 ofV, U and W are parallel on TM with respect to ∇. Substituting the equation of (i) into (4.17)-1, we have(4.38)uj(ANiX)=vi(Aξj*X)=g(Aξj*X,Ui)=0,∀X∈Γ(TM).
Also, substituting the equation of (iii) into (4.17)-2, we have
(4.39)wα(ANiX)=ϵαvi(AEαX)=g(AEαX,Ui)=0,∀X∈Γ(TM).
From the last two equations and the equation of (ii), we see that ANiX=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 τij, ϕαi, and ρiα, 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.
LetM 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 Mr×Ms×Mt, where Mr,  Ms, and  Mt are some leafs of Rad(TM),  Ho⊥=Span{Vi,Ui,Wα} and Ho, respectively, and s=n+r,t=m-n-2r.Proof.
By Theorem4.6, H is a parallel distribution M. Thus, for all X,Y∈Γ(Ho), we have ∇XY∈Γ(H). From (2.9) and (4.26), we have
(4.40)hi*(X,FY)=g(∇XFY,Ni)=g((∇XF)Y+F(∇XY),Ni)=g(F(∇XY),Ni)=-g(∇XY,JNi)=g(∇XY,Ui),
due to FY∈Γ(Ho). If S(TM) is totally umbilical in M, then we have hi*=0 due to Theorem 4.5. By (2.9) and (4.40), we get
(4.41)g(∇XY,Ni)=0,g(∇XY,Ui)=0,∀X∈Γ(TM),∀Y∈Γ(Ho).
These results and (4.25) imply ∇XY∈Γ(Ho) for all X,Y∈Γ(Ho). Thus Ho is a parallel distribution on S(TM) and TM=Ho⊕orthHo⊥, where Ho⊥=Span{ξi,Vi,Ui,Wα}. By Theorems 4.3 and 4.5, we have hiℓ=hαs=ANi=ϕαi=0 and AEαX=∑i=1rρiα(X)ξi. Thus (2.10) and (4.20)~(4.22) deduce, respectively, to
(4.42)∇XUi=∑j=1rτij(X)Uj+∑α=r+1mρiα(X)Wα,∇XVi=-∑j=1rτji(X)Vj+∑j=1rhjl(X,ξi)Uj,∇XWα=-∑i=1rρiα(X)Vj+∑β=r+1mσαβ(X)Wβ,∇Xξ=-∑j=1rτji(X)ξj,∀X∈Γ(Ho⊥).
Thus Ho⊥ is also a parallel distribution on M. Thus we have M=Mr×Ms×Mt, where Mr, Ms, and Mt are some leafs of Rad(TM), Ho⊥=Span{Vi,Ui,Wα} and Ho, respectively, and s=n+r,t=m-n-2r.



---

*Source: 610986-2011-11-03.xml*