# On Geometry of Submanifolds of(LCS)n-Manifolds

Authors: Mehmet Atceken
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

We show geometrical properties of a submanifold of a(LCS)n-manifold. The properties of the induced structures on such a submanifold are also studied.


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## Body

## 1. Introduction

The geometry of manifolds endowed with geometrical structures has been intensively studied, and several important results have been published. In this paper, we deal with manifolds having a Lorentzian concircular structure ((LCS)n-manifold) [1–3] (see Section 2 for detail).The study of the Lorentzian almost paracontact manifold was initiated by Matsumoto in [4]. Later on, several authors studied the Lorentzian almost paracontact manifolds and their different classes including [1, 4, 5]. Recently, the notion of the Lorentzian concircular structure manifolds was introduced in (briefly (LCS)-manifolds) with an example, which generalizes the notion of the LP-Sasakian manifolds introduced by Matsumoto in [4].Papers related to this issue are very few in the literature so far. But the geometry of submanifolds of a (LCS)-manifold is rich and interesting. So, in the present paper we introduce the concept of submanifolds of a (LCS)-manifold and investigate the fundamental properties of such submanifolds. We obtain the necessary and sufficient conditions for a submanifold of (LCS)-manifold to be invariant. In this case, the induced structures on submanifold by the structure on ambient space are classified. I think that the results will contribute to geometry.


## 2. Preliminaries

Ann-dimensional Lorentzian manifold M is a smooth connected paracompact Hausdorff manifold with a Lorentzian metric g, that is, M admits a smooth symmetric tensor field g of type (0,2) such that, for each point p∈M, the tensor gp:TpM×TpM→ℝ is a nondegenerate inner product of signature (-,+,…,+), where TpM denotes the tangent vector space of M at p and ℝ is the real number space. A nonzero vector v∈TpM is said to be timelike (resp., non-spacelike, null, and spacelike) if it satisfies gp(v,v)<0 (resp., ≤0, =0, and >0) [6].Definition 2.1.
In a Lorentzian manifold(M¯,g¯), a vector field P defined by
(2.1)g¯(X,P)=A(X),
for any X∈Γ(TM¯), is said to be a concircular vector field if
(2.2)(∇¯XA)(Y)=α{g¯(X,Y)+ω(X)A(Y)},
where α is a nonzero scalar and ω is a closed 1-form and ∇¯ denotes the operator of covariant differentiation with respect to the Lorentzian metric g.LetM be an n-dimensional Lorentzian manifold admitting a unit time-like concircular vector field ξ, called the characteristic vector field of the manifold. Then, we have
(2.3)g̅(ξ,ξ)=-1.
Since ξ is a unit concircular vector field, it follows that there exists a nonzero 1-form η such that, for
(2.4)g̅(X,ξ)=η(X),
the equation of the following form holds:
(2.5)(∇̅Xη)(Y)=α{g̅(X,Y)+η(X)η(Y)}(α≠0)
for all vector fields X, Y, where ∇¯ denotes the operator of covariant differentiation with respect to the Lorentzian metric g and α is a nonzero scalar function satisfying
(2.6)∇̅Xα=(Xα)=dα(X)=ρη(X),ρ being a certain scalar function given by ρ=-(ξα). If we put
(2.7)ϕX=1α∇̅Xξ,
then from (2.5) and (2.7) we have
(2.8)ϕX=X+η(X)ξ,
from which, it follows that ϕ is a symmetric (1,1) tensor and called the structure tensor of the manifold. Thus, the Lorentzian manifold M together with the unit time-like concircular vector field ξ, its associated 1-form η, and an (1,1) tensor field ϕ is said to be a Lorentzian concircular structure manifold (briefly, (LCS)n-manifold). Especially, if we take α=1, then we can obtain the LP-Sasakian structure of Matsumoto [4]. In a (LCS)n-manifold (n>2), the following relations hold:
(2.9)η(ξ)=-1,ϕξ=0,η(ϕX)=0,g̅(ϕX,ϕY)=g̅(X,Y)+η(X)η(Y),(2.10)ϕ2X=X+η(X)ξ,(2.11)S(X,ξ)=(n-1)(α2-ρ)η(X),(2.12)R(X,Y)ξ=(α2-ρ)[η(Y)X-η(X)Y],(2.13)R(ξ,Y)Z=(α2-ρ)[g(Y,Z)ξ-η(Z)Y],(2.14)(∇̅Xϕ)Y=α{g(X,Y)ξ+2η(X)η(Y)ξ+η(Y)X},(2.15)(Xρ)=dρ(X)=βη(X),(2.16)R(X,Y)Z=ϕR(X,Y)Z+(α2-ρ){g(Y,Z)η(X)-g(X,Z)η(Y)}ξ,
for all X,Y,Z∈Γ(TM).


## 3. Submanifolds of a (LCS)-Manifold

LetM be an isometrically immersed submanifold of a (LCS)n-manifold M¯ with induced metric g¯; we define the isometric immersion by i:M→M¯ and denote by B the differential of i. The induced Riemannian metric g on M by g¯ satisfies g(X,Y)=g¯(BX,BY), for all X,Y∈Γ(TM).We denote the tangent and normal spaces ofM at point p∈M by TM(p) and TM⊥(p), respectively. Let {N1,N2,,…,Ns} be an orthonormal basis of the normal space TM⊥(p), where s=dim(M¯)-dim(M), that is, s=codim(M).For anyX∈Γ(TM), we can write(3.1)ϕBX=BψX+∑i=1sυi(X)Ni,(3.2)ϕNi=BUi+∑j=1sλijNj,1≤i≤s,
where ψ,υi,Ui, and λij denote induced (1-1)-tensor, 1-forms, vector fields and functions on M, respectively. The vector field ξ on (LCS)-manifold M¯ can be written as follows:(3.3)ξ=BV+∑i=1sαiNi,
where V and αi are vector field and functions on M and M¯, respectively. From (3.1) and (3.2), we can derive(3.4)υk(X)=g¯(ϕBX,Nk)=g¯(BX,ϕNk)=g¯(BX,BUk)=g(Uk,X),λik=g¯(ϕNi,Nk)=g¯(Ni,ϕNk)=λki,
that is, λik is symmetric and(3.5)αk=g¯(ξ,Nk)=η(Nk),1≤i,k≤s.
Here, we note that the induced (1-1)-tensor field ψ is also symmetric because ϕ is symmetric.Next, we will the following Lemmas for later use.Lemma 3.1.
LetM be an isometrically immersed submanifold of a (LCS)-manifold M¯. Then, the following assertions are true:
(3.6)ψ2=I+μ⊗V-∑i=1sυi⊗Ui,(3.7)αjV=ψUj+∑i=1sλjiUi,1≤j≤s,(3.8)∑j=1sλkjλjp=δkp+αkαp-υp(Uk),1≤k,p≤s,
where μ denotes the induced 1-form on M by η on M¯ and given by μ(X)=g(X,V)=η(BX).Proof.
For anyX∈Γ(TM), by using (2.10), (3.1), and (3.2), we have
(3.9)ϕ2BX=ϕBψX+∑i=1sυi(X)ϕNi=Bψ2X+∑j=1sυj(ψX)Nj+∑i=1sυi(X){BUi+∑j=1sλijNj},BX+η(BX)ξ=Bψ2X+∑j=1sυj(ψX)Nj+∑i=1sυi(X)BUi+∑i=1sυi(X)∑j=1sλijNj.
Also considering (3.3), we arrive at
(3.10)BX+μ(X)BV+μ(X)∑i=1sαiNi=Bψ2X+∑j=1sυj(ψX)Nj+∑i=1sυi(X)BUi+∑i=1sυi(X)∑j=1sλijNj.
From the tangential components of (3.10), we conclude that
(3.11)X+μ(X)V=ψ2X+∑i=1sυi(X)Ui,
which is equivalent to (3.6). On the other hand, with the normal components of (3.10), we have
(3.12)μ(X)∑i=1sαiNi=∑j=1sυj(ψX)Nj+∑i=1sυi(X)∑j=1sλijNj,
which implies that
(3.13)μ(X)αk=υk(ψX)+∑i=1sυi(X)λik,
that is,
(3.14)g(X,V)αk=g(ψX,Uk)+∑i=1sg(X,Ui)λik.
This proves (3.7). In order to prove (3.8), taking (2.10) and (3.2), into account we have
(3.15)ϕ2Nk=ϕBUk+∑j=1sλjkϕNj,Nk+η(Nk)ξ=BψUk+∑i=1sυi(Uk)Ni+∑j=1sλjk{BUj+∑t=1sλjtNt},Nk+αkBV+αk∑i=1sαiNi=BψUk+∑i=1sυi(Uk)Ni+∑j=1sλjkBUj+∑j=1sλjk∑t=1sλjtNt.
Taking the product of (3.15) with Np, 1≤p≤s, we reach
(3.16)∑j=1sλjkλjp=δkp+αkαp-υp(Uk),
which gives us (3.8).Lemma 3.2.
LetM be an isometrically immersed submanifold of a (LCS)-manifold M¯. Then, the following assertions are true:
(3.17)ψV+∑i=1sαiUi=0,υp(V)+∑i=1sαiλip=0,1≤p≤s,(3.18)μ(V)=-1-∑i=1sαi2,(3.19)g(ψX,ψY)=g(X,Y)+μ(X)μ(Y)+∑i=1sυi(X)υi(Y),
for any X,Y∈Γ(TM).Proof.
Making use ofϕξ=0 and (3.3), we have
(3.20)ϕBV+∑i=1sαiϕNi=BψV+∑i=1sυi(V)Ni+∑i=1sαi{BUi+∑j=1sλijNj},0=BψV+∑i=1sυi(V)Ni+B∑i=1sαiUi+∑i=1sαi(∑j=1sλijNj).
From the tangential and normal components of this last equation, respectively, we get
(3.21)ψV+∑i=1sαiUi=0,υp(V)+∑i=1sαiλip=0.
Again, taking into account that ξ is time-like vector and (3.3), we reach
(3.22)g¯(BV+∑i=1sαiNi,BV+∑j=1sαjNj)=g(V,V)-∑i,j=1sαjαig¯(Ni,Nj),-1=μ(V)+∑i=1sαi2.
Finally, we conclude that
(3.23)g(ψX,ψY)=g¯(BψX,BψY)=g¯(ϕBX-∑i=1sυi(X)Ni,ϕBY-∑j=1sυj(Y)Nj)=g¯(BX,BY)+η(BX)η(BY)+∑i=1sυi(X)υi(Y)=g(X,Y)+μ(X)μ(Y)+∑i=1sυi(X)υi(Y).
This proves our assertions.Now, we suppose that{N1,N2,,…,Ns} and {N¯1,N¯2,…,N¯s} are two orthonormal bases of TM⊥(p) at p∈M and set(3.24)N¯i=∑j=1skjiNi,1≤i≤s,
by means of g(N¯i,N¯j)=∑p=1skipkpj=δij. So, we mean that the basis with another basis transition matrix (kij) is an orthogonal matrix. From (3.24) we have(3.25)Nj=∑p=1skjpN¯p.
Taking (3.24) into account, (3.1), (3.2), and (3.3) are, respectively, written in the following way:(3.26)ϕBX=BψX+∑k=1sυ¯k(X)N¯k,(3.27)ϕN¯p=BU¯p+∑t=1sλ¯ptN¯t,1≤p≤s,(3.28)ξ=BV+∑l=1sα¯lN¯l,
where(3.29)υ¯p(X)=∑i=1skipυi(X),U¯l=∑i=1skilUi,(3.30)λ¯pt=∑ijskipλijλit,λ¯pt=λ¯tp,α¯l=∑i=1skilαi.
Furthermore, because λij is symmetric, from (3.30), we can derive that under the suitable transformation (3.24) λij reduce to λ¯ij=λiδij, where λi are eigenvalues of matrix (λij). So, again (3.27) and (3.8) can be, respectively, written in the following way:(3.31)ϕN¯l=BU¯l+λlN¯l,υ¯p(U¯k)=δkp+α¯kα¯p-λ¯pλ¯kδkj,
which implies that υk¯(Uk¯)=1-α¯k2-λ¯k2 and υ¯k(U¯j)=-α¯kα¯j for k≠j.Now, letM be an isometrically immersed submanifold of a (LCS)-manifold M¯. If ϕ(BTM(p))⊂TM(p) for any point p∈M, then M is said to be an invariant submanifold of M¯. In this case, (3.1), (3.2), and (3.3) become, respectively,(3.32)ϕBX=BψX,(3.33)ϕNi=∑j=1sλijNj,(3.34)ξ=BV+∑i=1sαiNi
for any X∈Γ(TM).Lemma 3.3.
LetM be an invariant submanifold of a (LCS)-manifold M¯. Then, the following assertions are true:
(3.35)ψ2=I+μ⊗V,αiV=0,(3.36)δkj+αkαj-∑i=1sλkiλij=0,ψV=0,∑i=2sαiλij=0,(3.37)-υ(V)=1+∑i=1sαi2,g(ψX,ψY)=g(X,Y)+μ(X)μ(Y),
for any X,Y∈Γ(TM).Proof.
The proof is obvious. Therefore, we omit it.Theorem 3.4.
LetM be an invariant submanifold of a (LCS)-manifold M¯. One of the following cases occurs.(1)
Ifξ is normal to M, then the induced structure (ψ,g) on M is an almost product Riemannian structure whenever ψ is nontrivial.(2)
Ifξ is tangent to M, then the induced structure (ψ,V,μ,g) on M is a Lorentzian concircular structure.Proof.
(1) Ifξ is normal to the submanifold, then the vector field V=0. From (3.35) and (3.37), we have ψ2=I, g(ψX,ψY)=g(X,Y), that is, (ψ,g) is an almost product Riemannian structure whenever ψ is nontrivial.
(2) Ifξ is tangent to the submanifold (i.e., V≠0, αi=0), then we have μ(X)=g(X,V), ψ2=I+μ⊗V, ψV=0, μ(V)=-1, that is, (ψ,V,μ,g) is a Lorentzian concircular structure.Theorem 3.5.
LetM be a submanifold of a (LCS)-manifold M¯. The submanifold M of a (LCS)-manifold M¯ is invariant if and only if the induced structure (ψ,g) on M is an almost product Riemannian structure whenever ψ is nontrivial or the induced structure (ψ,V,μ,g) on M is a Lorentzian concircular structure.Proof.
From Theorem3.4 we know that the necessary is obvious.
Conversely, we suppose that the induced structure(ψ,g) is an almost product Riemannian structure. Then, from (3.19), we have(3.38)μ2(X)+∑i=1sυi2(X)=0,
that is, μ(X)=υi(X)=0, 1≤i≤s. So from (3.1) and (3.3) we can derive that the submanifold M is invariant and ξ is normal to M.
Now, we suppose that the induced structure(ψ,V,μ,g) is a Lorentzian concircular structure. Then, from (3.6), we get(3.39)∑i=1sυi(X)Ui=0,
which implies that υi(X)=0, 1≤i≤s. From (3.7), by a direct calculation, we derive αi=0, 1≤i≤s. So from (3.1) and (3.3), we conclude that M is invariant submanifold and ξ is tangent to M.Theorem 3.6.
LetM be an isometrically immersed submanifold of (LCS)-manifold M¯. Then, M is invariant submanifold if and only if the normal space TM⊥(p), at every point p∈M, admits an orthonormal basis consisting of the eigenvectors of the matrix (ϕ).Proof.
Let us suppose thatM is invariant.
(1) Whenξ is normal to M, at p∈M we consider an s-dimensional vector space W and investigate the eigenvalues of the matrix (λij)s×s. From (3.36) and (3.37), it is easy to see that the vector (α1,α2,…,αs) of the vector space W is a unit eigenvector of the matrix (λij)s×s and its eigenvalue is equal to 0.
Now, we suppose that a vector(ω1,ω2,…,ωs) satisfying ∑i=2sαiωi=0 is an eigenvector and its eigenvalue is λ. Then, we have(3.40)∑j=1sλijωj=λωi,1≤i≤s,
which implies that
(3.41)∑i,j=1sλkiλjiωj=λ∑i=1sλkiωi,1≤k≤s,
from which
(3.42)∑j=1s(∑i=1sλkiλji)ωj=λ2ωk.
On the other hand, from (3.36) we get
(3.43)∑j=1s(δkj-αkαj)ωj=∑i=1s(∑j=1sλkjλij)ωj=λ2ωk,
that is, ωk=λ2ωk, which is equivalent to λ2=1.
Consequently, if by a suitable transformation of the orthonormal basis{N1,N2,…,Ns} of TM⊥(p), the matrix λij becomes a diagonal matrix, then the diagonal components λ1,λ2,…,λs satisfy relations(3.44)λ1s=λ22=⋯=λs-12,λs=0.
In this case, if we denote by {N¯1,N¯2,…,N¯s} another orthonormal basis of TM⊥(p), then, from (3.31), we have ϕN¯ℓ=λℓN¯ℓ, 1≤ℓ≤s. So, N¯ℓ, 1≤ℓ≤s, are eigenvectors of the matrix-(ϕ) and N¯s=ξ.
(2) Whenξ is tangent to M, since αi=0, 1≤i≤s, from (3.36), we have(3.45)δkj=∑i=1sλkiλij.
If we denote by {ω1,ω2,…,ωs} an eigenvector of matrix (λij) and by λ its eigenvalue, then we have
(3.46)∑j=1sλjiωj=λωi,1≤i≤s.
So, we obtain
(3.47)∑i,j=1sλkiλjiωj=λ∑i=1sλkiωi,1≤k≤s,
that is, ωk=λ2ωk, which implies that λ2=1. Since the eigenvalues of (λij) are ±1, by a suitable transformation of the orthonormal basis of TM⊥(p), {N1,N2,…,Ns} to become {N¯1,N¯2,…,N¯s}, then N¯1,N¯2,…,N¯s are eigenvectors of matrix-(ϕ).
Conversely, if the orthonormal basis{N¯1,N¯2,…,N¯s} of TM⊥(p) consists of eigenvectors of matrix-(ϕ) and these eigenvalues λ1,λ2,…,λs satisfy λ12=λ22,…,λs-12=1 and λs2=±1 or 0, then we have ϕN¯ℓ=λN¯ℓ and we conclude that U¯ℓ=0, 1≤ℓ≤s, and so M is invariant.



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*Source: 304647-2012-02-22.xml*