Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 304647, 11 pages
doi:10.1155/2012/304647
Research Article
On Geometry of Submanifolds of (LCS)n -Manifolds
Mehmet Atceken
Department of Mathematics, Faculty of Arts and Sciences, Gaziosmanpasa University, 60250 Tokat, Turkey
Correspondence should be addressed to Mehmet Atceken, mehmet.atceken@gop.edu.tr
Received 13 September 2011; Revised 16 December 2011; Accepted 17 December 2011
Academic Editor: Attila Gilányi
Copyright q 2012 Mehmet Atceken. 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.
We show geometrical properties of a submanifold of a LCSn -manifold. The properties of the
induced structures on such a submanifold are also studied.
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 LCSn -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
An n-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
2
International Journal of Mathematics and Mathematical Sciences
type 0, 2 such that, for each point p ∈ M, the tensor gp : Tp M×Tp M → R is a nondegenerate
inner product of signature −, , . . . , , where Tp M denotes the tangent vector space of M at
p and R is the real number space. A nonzero vector v ∈ Tp M 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
gX, P 2.1
AX,
for any X ∈ ΓT M, is said to be a concircular vector field if
∇X A Y α gX, Y ωXAY ,
2.2
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.
Let M 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
gξ, ξ
−1.
2.3
Since ξ is a unit concircular vector field, it follows that there exists a nonzero 1-form η such
that, for
gX, ξ
2.4
ηX,
the equation of the following form holds:
∇X η Y α gX, Y ηXηY α / 0
2.5
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
∇X α
Xα
ρ being a certain scalar function given by ρ
φX
dαX
ρηX,
2.6
−ξα. If we put
1
∇X ξ,
α
2.7
X ηXξ,
2.8
then from 2.5 and 2.7 we have
φX
International Journal of Mathematics and Mathematical Sciences
3
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, LCSn -manifold. Especially, if we take α 1, then
we can obtain the LP-Sasakian structure of Matsumoto 4. In a LCSn -manifold n > 2, the
following relations hold:
ηξ
−1,
φξ
η φX
0,
n − 1 α2 − ρ ηX,
SX, ξ
α2 − ρ
∇X φ Y
Xρ
RX, Y Z
2.10
2.11
2.12
gY, Zξ − ηZY ,
2.13
α gX, Y ξ 2ηXηY ξ ηY X ,
2.9
ηY X − ηXY ,
α2 − ρ
Rξ, Y Z
gX, Y ηXηY ,
X ηXξ,
φ2 X
RX, Y ξ
g φX, φY
0,
dρX
βηX,
φRX, Y Z α2 − ρ gY, ZηX − gX, ZηY ξ,
2.14
2.15
2.16
for all X, Y, Z ∈ ΓT M.
3. Submanifolds of a (LCS)-Manifold
Let M be an isometrically immersed submanifold of a LCSn -manifold M with induced
metric g; we define the isometric immersion by i : M → M and denote by B the differential
gBX, BY , for all
of i. The induced Riemannian metric g on M by g satisfies gX, Y X, Y ∈ ΓT M.
⊥
p,
We denote the tangent and normal spaces of M at point p ∈ M by TM p and TM
⊥
respectively. Let {N1 , N2 , , . . . , Ns } be an orthonormal basis of the normal space TM p, where
s dimM − dimM, that is, s codimM.
For any X ∈ ΓT M, we can write
φBX
BψX s
υi XNi ,
3.1
i 1
φNi
BUi s
λij Nj ,
j 1
1 ≤ i ≤ s,
3.2
4
International Journal of Mathematics and Mathematical Sciences
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:
s
BV ξ
3.3
αi Ni ,
i 1
where V and αi are vector field and functions on M and M, respectively. From 3.1 and 3.2,
we can derive
υk X
g φBX, Nk
g BX, φNk
gBX, BUk g Ni , φNk
λik g φNi , Nk
λki ,
gUk , X,
3.4
that is, λik is symmetric and
αk
gξ, Nk 1 ≤ i, k ≤ s.
ηNk ,
3.5
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. Let M be an isometrically immersed submanifold of a LCS-manifold M. Then, the
following assertions are true:
I μ⊗V −
ψ2
s
υi ⊗ Ui ,
3.6
i 1
αj V
ψUj s
λji Ui ,
1 ≤ j ≤ s,
3.7
i 1
s
λkj λjp
δkp αk αp − υp Uk ,
1 ≤ k, p ≤ s,
3.8
j 1
gX, V where μ denotes the induced 1-form on M by η on M and given by μX
ηBX.
Proof. For any X ∈ ΓT M, by using 2.10, 3.1, and 3.2, we have
φ2 BX
φBψX s
υi XφNi
i 1
Bψ 2 X s
υj ψX Nj j 1
BX ηBXξ
Bψ 2 X s
j 1
s
i 1
υj ψX Nj s
i 1
⎧
⎨
υi X BUi ⎩
υi XBUi ⎫
⎬
s
λij Nj
j 1
s
i 1
υi X
,
⎭
s
λij Nj .
j 1
3.9
International Journal of Mathematics and Mathematical Sciences
5
Also considering 3.3, we arrive at
s
BX μXBV μX
υj ψX Nj s
Bψ 2 X αi Ni
i 1
j 1
s
s
υi XBUi
i 1
υi X
i 1
3.10
s
λij Nj .
j 1
From the tangential components of 3.10, we conclude that
X μXV
s
ψ2X υi XUi ,
3.11
i 1
which is equivalent to 3.6. On the other hand, with the normal components of 3.10, we
have
s
s
αi Ni
μX
i 1
υj ψX Nj s
j 1
s
υi X
i 1
3.12
λij Nj ,
j 1
which implies that
μXαk
υk ψX s
υi Xλik ,
3.13
i 1
that is,
gX, V αk
g ψX, Uk s
gX, Ui λik .
3.14
i 1
This proves 3.7. In order to prove 3.8, taking 2.10 and 3.2, into account we have
φ 2 Nk
φBUk s
λjk φNj ,
j 1
Nk ηNk ξ
Nk αk BV αk
s
αi Ni
BψUk BψUk i 1
s
υi Uk Ni i 1
j 1
s
s
υi Uk Ni i 1
s
s
λjk
j 1
s
λjt Nt .
t 1
λjk BUj λjt Nt ,
t 1
λjk BUj
j 1
s
3.15
6
International Journal of Mathematics and Mathematical Sciences
Taking the product of 3.15 with Np , 1 ≤ p ≤ s, we reach
s
δkp αk αp − υp Uk ,
λjk λjp
3.16
j 1
which gives us 3.8.
Lemma 3.2. Let M be an isometrically immersed submanifold of a LCS-manifold M. Then, the
following assertions are true:
s
ψV αi Ui
s
υp V 0,
i 1
αi λip
1 ≤ p ≤ s,
0,
3.17
i 1
μV s
−1 −
3.18
α2i ,
i 1
g ψX, ψY
s
gX, Y μXμY υi Xυi Y ,
3.19
i 1
for any X, Y ∈ ΓT M.
0 and 3.3, we have
Proof. Making use of φξ
φBV s
αi φNi
s
BψV i 1
υi V Ni i 1
0
s
BψV s
αi
i 1
υi V Ni B
i 1
s
⎧
⎨
s
⎩
j 1
BUi λij Nj
s
αi Ui i 1
⎫
⎬
⎛
αi ⎝
i 1
s
,
⎭
⎞
3.20
λij Nj ⎠.
j 1
From the tangential and normal components of this last equation, respectively, we get
ψV s
αi Ui
0,
υp V i 1
s
αi λip
0.
3.21
i 1
Again, taking into account that ξ is time-like vector and 3.3, we reach
⎛
g ⎝BV s
i 1
αi Ni , BV s
⎞
αj Nj ⎠
gV, V −
j 1
−1
s
αj αi g Ni , Nj ,
i,j 1
μV s
i 1
α2i .
3.22
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7
Finally, we conclude that
g ψX, ψY
g BψX, BψY
⎛
s
g ⎝φBX −
υi XNi , φBY −
i 1
⎞
υj Y Nj ⎠
j 1
s
gBX, BY ηBXηBY s
3.23
υi Xυi Y i 1
s
gX, Y μXμY υi Xυi Y .
i 1
This proves our assertions.
Now, we suppose that {N1 , N2 , , . . . , Ns } and {N 1 , N 2 , . . . , N s } are two orthonormal
⊥
p at p ∈ M and set
bases of TM
s
Ni
1 ≤ i ≤ s,
kji Ni ,
3.24
j 1
s
δij . So, we mean that the basis with another basis
by means of gN i , N j p 1 kip kpj
transition matrix kij is an orthogonal matrix. From 3.24 we have
s
3.25
kjp N p .
Nj
p 1
Taking 3.24 into account, 3.1, 3.2, and 3.3 are, respectively, written in the following
way:
s
BψX φBX
υk XN k ,
3.26
k 1
φN p
BUp s
1 ≤ p ≤ s,
λpt N t ,
3.27
t 1
ξ
BV s
3.28
α N ,
1
where
υp X
s
kip υi X,
s
U
i 1
ki Ui ,
s
λpt
s
kip λij λit ,
ij
3.29
i 1
λpt
λtp ,
α
ki αi .
i 1
3.30
8
International Journal of Mathematics and Mathematical Sciences
Furthermore, because λij is symmetric, from 3.30, we can derive that under the suitable
transformation 3.24 λij reduce to λij
λi δij , where λi are eigenvalues of matrix λij . So,
again 3.27 and 3.8 can be, respectively, written in the following way:
φN
υp Uk
BU λ N ,
3.31
δkp αk αp − λp λk δkj ,
2
which implies that υk Uk 1 − α2k − λk and υk Uj −αk αj for k / j.
Now, let M be an isometrically immersed submanifold of a LCS-manifold M. If
φBTM 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,
φBX
BψX,
3.32
λij Nj ,
3.33
s
φNi
j 1
ξ
BV s
3.34
αi Ni
i 1
for any X ∈ ΓT M.
Lemma 3.3. Let M be an invariant submanifold of a LCS-manifold M. Then, the following
assertions are true:
ψ2
I μ ⊗ V,
αi V
s
δkj αk αj −
s
λki λij
0,
ψV
1
s
α2i ,
0,
αi λij
0,
3.36
i 2
i 1
−υV 3.35
0,
g ψX, ψY
gX, Y μXμY ,
3.37
i 1
for any X, Y ∈ ΓT M.
Proof. The proof is obvious. Therefore, we omit it.
Theorem 3.4. Let M be an invariant submanifold of a (LCS)-manifold M. One of the following cases
occurs.
1 If ξ is normal to M, then the induced structure ψ, g on M is an almost product
Riemannian structure whenever ψ is nontrivial.
2 If ξ is tangent to M, then the induced structure ψ, V, μ, g on M is a Lorentzian
concircular structure.
International Journal of Mathematics and Mathematical Sciences
9
Proof. 1 If ξ is normal to the submanifold, then the vector field V
0. From 3.35 and
3.37, we have ψ 2 I, gψX, ψY gX, Y , that is, ψ, g is an almost product Riemannian
structure whenever ψ is nontrivial.
2 If ξ is tangent to the submanifold i.e., V / 0, αi 0, then we have μX gX, V ,
ψ 2 I μ ⊗ V , ψV 0, μV −1, that is, ψ, V, μ, g is a Lorentzian concircular structure.
Theorem 3.5. Let M 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 Theorem 3.4 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
μ2 X s
υi2 X
0,
3.38
i 1
that is, μX 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
s
υi XUi
0,
3.39
i 1
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. Let M be an isometrically immersed submanifold of (LCS)-manifold M. Then, M
⊥
p, at every point p ∈ M, admits an
is invariant submanifold if and only if the normal space TM
orthonormal basis consisting of the eigenvectors of the matrix φ.
Proof. Let us suppose that M is invariant.
1 When ξ 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.
s
0 is an
Now, we suppose that a vector ω1 , ω2 , . . . , ωs satisfying
i 2 αi ωi
eigenvector and its eigenvalue is λ. Then, we have
s
λij ωj
j 1
λωi ,
1 ≤ i ≤ s,
3.40
10
International Journal of Mathematics and Mathematical Sciences
which implies that
s
s
λki λji ωj
λ
i,j 1
λki ωi ,
1 ≤ k ≤ s,
3.41
i 1
from which
s
s
λ2 ωk .
λki λji ωj
j 1
3.42
i 1
On the other hand, from 3.36 we get
s
⎛
s
δkj − αk αj ωj
j 1
⎝
i 1
⎞
s
λkj λij ⎠ωj
λ2 ωk ,
3.43
j 1
that is, ωk λ2 ωk , which is equivalent to λ2 1.
Consequently, if by a suitable transformation of the orthonormal basis {N1 ,
⊥
p, the matrix λij becomes a diagonal matrix, then the diagonal compoN2 , . . . , Ns } of TM
nents λ1 , λ2 , . . . , λs satisfy relations
λs1
···
λ22
λ2s−1 ,
λs
0.
3.44
⊥
p, then, from
In this case, if we denote by {N 1 , N 2 , . . . , N s } another orthonormal basis of TM
3.31, we have φN
λ N , 1 ≤ ≤ s. So, N , 1 ≤ ≤ s, are eigenvectors of the matrix-φ
and N s ξ.
2 When ξ is tangent to M, since αi 0, 1 ≤ i ≤ s, from 3.36, we have
s
δkj
3.45
λki λij .
i 1
If we denote by {ω1 , ω2 , . . . , ωs } an eigenvector of matrix λij and by λ its eigenvalue, then
we have
s
λji ωj
λωi ,
1 ≤ i ≤ s.
3.46
j 1
So, we obtain
s
s
λki λji ωj
i,j 1
λ
λki ωi ,
i 1
1 ≤ k ≤ s,
3.47
International Journal of Mathematics and Mathematical Sciences
11
λ2 ωk , which implies that λ2
1. Since the eigenvalues of λij are ±1, by
that is, ωk
⊥
p, {N1 , N2 , . . . , Ns } to become
a suitable transformation of the orthonormal basis of TM
{N 1 , N 2 , . . . , N s }, then N 1 , N 2 , . . . , N s are eigenvectors of matrix-φ.
⊥
p consists of eigenvecConversely, if the orthonormal basis {N 1 , N 2 , . . . , N s } of TM
2
tors of matrix-φ and these eigenvalues λ1 , λ2 , . . . , λs satisfy λ1 λ22 , . . . , λ2s−1 1 and λ2s ±1
λN and we conclude that U
0, 1 ≤
≤ s, and so M is
or 0, then we have φN
invariant.
Acknowledgment
The author would like to express my gratitude to the referees for valuable comments and
suggestions.
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3 A. A. Shaikh and K. K. Baishya, “On concircular structure spacetimes II,” American Journal of Applied
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