Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 636782, 18 pages
doi:10.1155/2012/636782
Research Article
Lightlike Submanifolds of a Semi-Riemannian
Manifold of Quasi-Constant Curvature
D. H. Jin1 and J. W. Lee2
1
2
Department of Mathematics, Dongguk University, Kyongju 780-714, Republic of Korea
Department of Mathematics, Sogang University, Sinsu-dong, Mapo-gu, Seoul 121-742, Republic of Korea
Correspondence should be addressed to J. W. Lee, leejaewon@sogang.ac.kr
Received 19 January 2012; Revised 29 February 2012; Accepted 14 March 2012
Academic Editor: Chein-Shan Liu
Copyright q 2012 D. H. Jin and J. W. Lee. 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 study the geometry of lightlike submanifolds M, g, ST M, ST M⊥ of a semi-Riemannian
g of quasiconstant curvature subject to the following conditions: 1 the curvature
manifold M,
is tangent to M, 2 the screen distribution ST M of M is totally geodesic in
vector field ζ of M
M, and 3 the coscreen distribution ST M⊥ of M is a conformal Killing distribution.
1. Introduction
In the generalization from the theory of submanifolds in Riemannian to the theory of
submanifolds in semi-Riemannian manifolds, the induced metric on submanifolds 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 nondegenerate
case. In lightlike case, the standard text book definitions do not make sense, and one fails
to use the theory of nondegenerate geometry in the usual way. The primary difference
between the lightlike submanifolds and nondegenerate 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 nondegenerate
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 ndimensional spacetime manifold. Lightlike hypersurfaces are also studied in the theory of
electromagnetism 1. Thus, large number of applications but limited information available
2
Journal of Applied Mathematics
motivated us to do research on this subject matter. Kupeli 2 and Duggal and Bejancu 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.
In the study of Riemannian geometry, Chen and Yano 3 introduced the notion of a
g with the
Riemannian manifold of a quasiconstant curvature as a Riemannian manifold M,
satisfying the condition
curvature tensor R
g RX,
Y Z, W α g Y, Z
g X, W − g X, Z
g Y, W
β g X, WθY θZ − g X, ZθY θW
g Y, ZθXθW − g Y, WθXθZ ,
1.1
where α, β are scalar functions and θ is a 1-form
for any vector fields X, Y, Z, and W on M,
defined by
θX g X, ζ,
1.2
which called the curvature vector field. It is well known that
where ζ is a unit vector field on M
is of the form 1.1, then the manifold is conformally flat. If β 0,
if the curvature tensor R
then the manifold reduces to a space of constant curvature.
A nonflat Riemannian manifold of dimension n> 2 is defined to be a quasi-Einstein
manifold 4 if its Ricci tensor satisfies the condition
RicX,
Y a
g X, Y bφXφY ,
1.3
where a, b are scalar functions such that b /
0, and φ is a nonvanishing 1-form such that
g X, U φX for any vector field X, where U is a unit vector field. If b 0, then
the manifold reduces to an Einstein manifold. It can be easily seen that every Riemannian
manifold of quasiconstant curvature is a quasi-Einstein manifold.
The subject of this paper is to study the geometry of lightlike submanifolds of a
g of quasiconstant curvature. We prove two characterization
semi-Riemannian manifold M,
theorems for such a lightlike submanifold M, g, ST M, ST M⊥ as follows.
g of quasiTheorem 1.1. Let M be an r-lightlike submanifold of a semi-Riemannian manifold M,
constant curvature. If the curvature vector field ζ of M is tangent to M and ST M is totally geodesic
in M, then one has the following results:
1 if ST M⊥ is a Killing distribution, then the functions α and β, defined by 1.1, vanish
M, and the leaf M∗ of ST M are flat manifolds;
identically. Furthermore, M,
2 if ST M⊥ is a conformal Killing distribution, then the function β vanishes identically. Fur and M∗ are space of constant curvatures, and M is an Einstein manifold such
thermore, M
that Ric r/m − rg, where r is the induced scalar curvature of M.
Theorem 1.2. Let M be an irrotational r-lightlike submanifold of a semi-Riemannian manifold
g of quasiconstant curvature. If ζ is tangent to M, ST M is totally umbilical in M, and
M,
Journal of Applied Mathematics
3
ST M⊥ is a conformal Killing distribution with a nonconstant conformal factor, then the function
and M∗ are space of constant curvatures, and M is a totally
β vanishes identically. Moreover, M
umbilical Einstein manifold such that Ric c/m − rg, where c is the scalar quantity of M.
2. Lightlike Submanifolds
Let M, g be an m-dimensional lightlike submanifold of an m n-dimensional semi g . We follow Duggal and Bejancu 1 for notations and results
Riemannian manifold M,
used in this paper. The radical distribution RadT M T M ∩ T M⊥ is a vector subbundle of
the tangent bundle T M and the normal bundle T M⊥ , of rank r 1 ≤ r ≤ min{m, n}. Then, in
general, there exist two complementary nondegenerate distributions ST M and ST M⊥ of
RadT M in T M and T M⊥ , respectively, called the screen and coscreen distribution on M, and
we have the following decompositions:
T M⊥ RadT M⊕orth S T M⊥ ,
T M RadT M⊕orth ST M;
2.1
where the symbol ⊕orth denotes the orthogonal direct sum. We denote such a lightlike
submanifold by M M, g, ST M, ST M⊥ . Let trT M and ltrT M be complementary
|M and T M⊥ in ST M⊥ , respectively, and
but not orthogonal vector bundles to T M in T M
let {Ni } be a lightlike basis of ΓltrT M|U consisting of smooth sections of ST M⊥|U , where
U is a coordinate neighborhood of M, such that
g Ni , ξj δij ,
g Ni , Nj 0,
2.2
where {ξ1 , . . . , ξr } is a lightlike basis of ΓRadT M. Then,
T M ⊕ trT M {RadT M ⊕ trT M}⊕orth ST M
TM
{RadT M ⊕ ltrT M}⊕orth ST M⊕orth S T M⊥ .
2.3
is
We say that a lightlike submanifold M, g, ST M, ST M⊥ of M
1 r-lightlike submanifold if 1 ≤ r < min{m, n},
2 coisotropic submanifold if 1 ≤ r n < m,
3 isotropic submanifold if 1 ≤ r m < n,
4 totally lightlike submanifold if 1 ≤ r m n.
The above three classes 2∼4 are particular cases of the class 1 as follows: ST M⊥ {0}, ST M {0}, and ST M ST M⊥ {0}, respectively.
Example 2.1. Consider in R42 the 1-lightlike submanifold M given by equations
1 x √ x1 x2 ,
2
3
2 1
1
2
x log 1 x − x
,
2
4
2.4
4
Journal of Applied Mathematics
then we have T M span{U1 , U2 } and T M⊥ {H1 , H2 }, where we set
2 2 √ √ 1
2
1
1
2
U1 2 1 x − x
∂x 1 x − x
∂x3 2 x1 − x2 ∂x4 ,
2 2 √ √ U2 2 1 x1 − x2
∂x2 1 x1 − x2
∂x3 2 x1 − x2 ∂x4 ,
√
H1 ∂x1 ∂x2 2∂x3 ,
2 2 √ H2 2 1 x2 − x1
∂x2 2 x1 − x2 ∂x3 1 x1 − x2
∂x4 .
2.5
It follows that RadT M is a distribution on M of rank 1 spanned by ξ H1 . Choose ST M
and ST M⊥ spanned by U2 and H2 where are timelike and spacelike, respectively. Finally,
the lightlike transversal vector bundle
1
1
1
ltrT M Span N ∂x1 ∂x2 √ ∂x3
2
2
2
2.6
and the transversal vector bundle
trT M Span{N, H2 }
2.7
are obtained.
be the Levi-Civita connection of M
and P the projection morphism of ΓT M on
Let ∇
ΓST M with respect to the decomposition 2.1. For an r-lightlike submanifold, the local
Gauss-Weingartan formulas are given by
X Y ∇X Y ∇
r
n
h
i X, Y Ni hsα X, Y Wα ,
i1
X Ni −AN X ∇
i
r
n
τij XNj ρiα XWα ,
j1
X Wα −AW X ∇
α
2.8
αr1
r
2.9
αr1
φαi XNi i1
∇X P Y ∇∗X P Y n
θαβ XWβ ,
2.10
βr1
r
h∗i X, P Y ξi ,
2.11
i1
∇X ξi −A∗ξi X −
r
τji Xξj ,
2.12
j1
for any X, Y ∈ ΓT M, where ∇ and ∇∗ are induced linear connections on T M and ST M,
respectively, the bilinear forms h
i and hsα on M are called the local lightlike second fundamental
form and local screen second fundamental form on T M, respectively, and h∗i is called the local
radical second fundamental form on ST M. ANi , A∗ξi , and AWα are linear operators on ΓT M,
and τij , ρiα , φαi , and θαβ are 1-forms on T M.
Journal of Applied Mathematics
5
is torsion-free, ∇ is also torsion-free and both h
and hsα are symmetric. From
Since ∇
i
X Y, ξi , we know that h
are independent of the choice of a screen
the fact that h
i X, Y g ∇
i
distribution. Note that h
i , τij , and ρiα depend on the section ξ ∈ ΓRadT M|U . Indeed, take
ξi rj1 aij ξj , then we have dtrτij dtr
τij 5.
The induced connection ∇ on T M is not metric and satisfies
r h
i X, Y ηi Z h
i X, Zηi Y ,
∇X g Y, Z 2.13
i1
where ηi is the 1-form such that
ηi X g X, Ni ,
∀X ∈ ΓT M, i ∈ {1, . . . , r}.
2.14
But the connection ∇∗ on ST M is metric. The above three local second fundamental forms
of M and ST M are related to their shape operators by
r
h
i X, Y g A∗ξi X, Y − h
k X, ξi ηk Y ,
h
i X, P Y g A∗ξi X, P Y ,
k1
α hsα X, Y gAWα X, Y −
2.15
g A∗ξi X, Nj 0,
r
2.16
2.17
φαi Xηi Y ,
i1
α hsα X, P Y gAWα X, P Y ,
h∗i X, P Y gANi X, P Y ,
g AWα X, Ni α ρiα X,
ηj ANi X ηi ANj X 0,
2.18
2.19
and β θαβ −α θβα , where X, Y ∈ ΓT M. From 2.19, we know that the operators ANi are
shape operators related to h∗i for each i, called the radical shape operators on ST M. From
2.16, we know that the operators A∗ξi are ΓST M valued. Replace Y by ξj in 2.15, then
we have h
i X, ξj h
j X, ξi 0 for all X ∈ ΓT M. It follows that
h
i X, ξi 0,
h
i ξj , ξk 0.
2.20
Also, replace X by ξj in 2.15 and use 2.20, then we have
h
i X, ξj g X, A∗ξi ξj ,
A∗ξi ξj A∗ξj ξi 0,
A∗ξi ξi 0.
2.21
Thus ξi is an eigenvector field of A∗ξi corresponding to the eigenvalue 0. For an r-lightlike
submanifold, replace Y by ξi in 2.17, then we have
hsα X, ξi −α φαi X.
2.22
6
Journal of Applied Mathematics
From 2.15∼2.18, we show that the operators A∗ξi and AWα are not self-adjoint on
ΓT M but self-adjoint on ΓST M.
Theorem 2.2. Let M, g, ST M, ST M⊥ be an r-lightlike submanifold of a semi-Riemannian
g , then the following assertions are equivalent:
manifold M,
i A∗ξi are self-adjoint on ΓT M with respect to g, for all i,
ii h
i satisfy h
i X, ξj 0 for all X ∈ ΓT M, i and j,
iii A∗ξi ξj 0 for all i and j, that is, the image of RadT M with respect to A∗ξi for each i is a
trivial vector bundle,
iv h
i X, Y gA∗ξi X, Y for all X, Y ∈ ΓT M and i, that is, A∗ξi is a shape operator on
M, related by the second fundamental form h
i .
Proof. From 2.15 and the fact that h
i are symmetric, we have
r h
k X, ξi ηk Y − h
k Y, ξi ηk X .
g A∗ξi X, Y − g X, A∗ξi Y 2.23
j1
i⇔ii. If h
i X, ξj 0 for all X ∈ ΓT M, i and j, then we have gA∗ξi X, Y for all X, Y ∈ ΓT M, that is, A∗ξi are self-adjoint on ΓT M with respect to g.
Conversely, if A∗ξi are self-adjoint on ΓT M with respect to g, then we have
gA∗ξi Y, X
h
k X, ξi ηk Y h
k Y, ξi ηk X,
2.24
for all X, Y ∈ ΓT M. Replace Y by ξj in this equation and use the second equation of 2.20,
then we have h
j X, ξi 0 for all X ∈ ΓT M, i and j.
ii⇔iii. Since ST M is nondegenerate, from the first equation of 2.21, we have
h
i X, ξj 0 ⇔ A∗ξi ξj 0, for all i and j.
ii⇔iv. From 2.16, we have h
i X, Y gA∗ξi X, Y ⇔ h
j X, ξi 0 for any X, Y ∈
ΓT M and for all i and j.
Corollary 2.3. Let M, g, ST M, ST M⊥ be a 1-lightlike submanifold of a semi-Riemannian
g , then the operators A∗ are self-adjoint on ΓT M with respect to g.
manifold M,
ξi
Definition 2.4. An r-lightlike submanifold M, g, ST M, ST M⊥ of a semi-Riemannian
X ξi ∈ ΓT M for any X ∈ ΓT M and i.
g is said to be irrotational if ∇
manifold M,
the above definition is equivalent to h
X, ξi For an r-lightlike submanifold M of M,
j
0 and hsα X, ξi 0 for any X ∈ ΓT M. In this case, A∗ξi are self-adjoint on ΓT M with respect
to g, for all i.
Journal of Applied Mathematics
7
We need the following Gauss-Codazzi equations for a full set of these equations see
R, and R∗ the curvature tensors of the Levi1, chapter 5 for M and ST M. Denote by R,
of M,
the induced connection ∇ of M, and the induced connection ∇∗
Civita connection ∇
on ST M, respectively:
g RX,
Y Z, P W gRX, Y Z, P W
r h
i X, Zh∗i Y, P W − h
i Y, Zh∗i X, P W
i1
n
2.25
α {hsα X, Zhsα Y, P W − hsα Y, Zhsα X, P W},
αr1
α g RX,
Y Z, Wα ∇X hsα Y, Z − ∇Y hsα X, Z
r h
i Y, Zρiα X − h
i X, Zρiα Y i1
n
2.26
hsβ Y, Zθβα X − hsβ X, Zθβα Y ,
βr1
g RX,
Y Z, Ni g RX, Y Z, Ni r h
j X, Zηi ANj Y − h
j Y, Zηi ANj X
j1
n
2.27
α hsα X, Zρiα Y − hsα Y, Zρiα X ,
αr1
g RX,
Y ξi , Nj g RX, Y ξi , Nj
r h
k X, ξi ηj ANk Y − h
k Y, ξi ηj ANk X
k1
n
ρjα Xφαi Y − ρjα Y φαi X
αr1
g A∗ξi X, ANj Y − g A∗ξi Y, ANj X − 2dτji X, Y r h
k X, ξi ηj ANk Y − h
k Y, ξi ηj ANk X
2.28
k1
r
τjk Xτki Y − τjk Y τki X
k1
n
ρjα Xφαi Y − ρjα Y φαi X ,
αr1
g RX, Y P Z, P W gR∗ X, Y P Z, P W
r h∗i X, P Zh
i Y, P W − h∗i Y, P Zh
i X, P W ,
i1
2.29
8
Journal of Applied Mathematics
gRX, Y P Z, Ni ∇X h∗i Y, P Z − ∇Y h∗i X, P Z
r h∗j X, P Zτij Y − h∗j Y, P Zτij X .
2.30
j1
is given by
The Ricci tensor of M
RicX,
Y trace Z −→ RZ,
XY ,
,
∀X, Y ∈ Γ T M
2.31
Let dim M
m n. Locally, Ric
is given by
for any X,Y ∈ ΓT M.
i , XY, Ei ,
i g RE
mn
Y RicX,
2.32
i1
If dimM
> 2 and
where {E1 , . . . , Emn } is an orthonormal frame field of T M.
κ
Ric
g ,
κ
is a constant,
2.33
is an Einstein manifold. If dimM
2, any M
is Einstein, but κ
then M
in 2.33 is
not necessarily constant. The scalar curvature r is defined by
r mn
i , Ei .
i RicE
2.34
i1
is Einstein if and only if
Putting 2.33 in 2.34 implies that M
Ric
r
g .
mn
2.35
3. The Tangential Curvature Vector Field
Let R0,2 denote the induced Ricci tensor of type 0, 2 on M, given by
R0,2 X, Y trace{Z −→ RZ, XY },
.
∀X, Y ∈ Γ T M
3.1
Consider an induced quasiorthonormal frame field
{ξ1 , . . . , ξr ; N1 , . . . , Nr ; Xr1 , . . . , Xm ; Wr1 , . . . , Wn },
3.2
Journal of Applied Mathematics
9
where {Ni , Wα } is a basis of ΓtrT M|U on a coordinate neighborhood U of M such that
Ni ∈ ΓltrT M|U and Wα ∈ ΓST M⊥ |U . By using 2.29 and 3.1, we obtain the following
local expression for the Ricci tensor:
RicX,
Y n
r
a , XY, Wa g Rξ
i , XY, Ni
a g RW
ar1
i1
m
b , XY, Xb
b g RX
r
i , XY, ξi ,
g RN
i1
br1
R0,2 X, Y m
3.3
a gRXa , XY, Xa ar1
r
g Rξi , XY, Ni .
3.4
i1
Substituting 2.25 and 2.27 in 3.3 and using 2.15∼2.18 and 3.4, we obtain
R0,2 X, Y RicX,
Y r
n
h
i X, Y tr ANi hsα X, Y tr AWα
i1
−
αr1
r
n
g ANi X, A∗ξi Y −
α gAWα X, AWα Y i1
−
αr1
r
h
j ξi , Y ηi ANj X i,j1
−
n
r 3.5
ρiα Xφαi Y i1 αr1
n
r
α , XY, Wα − g Rξ
i , Y X, Ni ,
α g RW
αr1
i1
for any X, Y ∈ ΓT M. This shows that R0,2 is not symmetric. A tensor field R0,2 of M, given
by 3.1, is called its induced Ricci tensor if it is symmetric. From now and in the sequel, a
symmetric R0,2 tensor will be denoted by Ric.
Using 2.28, 3.5, and the first Bianchi identity, we obtain
R0,2 X, Y − R0,2 Y, X r g A∗ξi X, ANi Y − g A∗ξi Y, ANi X s
i1
r h
j X, ξi ηi ANj Y − h
j Y, ξi ηi ANj Y
i, j1
r n
ρiα Xφαi Y − ρiα Y φαi X
3.6
i1 αr1
−
r
g RX,
Y ξi , Ni .
i1
From this equation and 2.28, we have
R0,2 X, Y − R0,2 Y, X 2d tr τij X, Y .
3.7
10
Journal of Applied Mathematics
g ,
Theorem 3.1 see5. Let M be a lightlike submanifold of a semi-Riemannian manifold M,
0,2
is a symmetric Ricci tensor Ric if and only if each 1-form trτij is closed,
then the tensor field R
that is, dtrτij 0, on any U ⊂ M.
Note 1. Suppose that the tensor R0,2 is symmetric Ricci tensor Ric, then the 1-form trτij is closed by Theorem 3.1. Thus, there exist a smooth function f on U such that trτij df.
Consequently, we get trτij X Xf. If we take ξi rj1 αij ξj , it follows that trτij X τij X 0 for any X ∈
tr
τij X Xln Δ. Setting Δ expf in this equation, we get tr
ΓT M|U . We call the pair {ξi , Ni }i on U such that the corresponding 1-form trτij vanishes
the canonical null pair of M.
For the rest of this paper, let M be a lightlike submanifold of a semi-Riemannian
of quasiconstant curvature. We may assume that the curvature vector field ζ
manifold M
is a unit spacelike tangent vector field of M and dim M
> 4,
of M
RicX,
Y n m − 1α β gX, Y n m − 2βθXθY ,
i , Y X, Ni αgX, Y βθXθY ,
g Rξ
3.9
α , Y X, Wα αgX, Y βθXθY ,
α g RW
3.10
3.8
for all X, Y ∈ ΓT M. Substituting 3.8∼3.10 into 3.5, we have
R0,2 X, Y m − 1α β gX, Y m − 2βθXθY r
n
h
i X, Y tr ANi hsα X, Y tr AWα
i1
αr1
r
n
− g ANi X, A∗ξi Y −
α gAWα X, AWα Y i1
−
r
3.11
αr1
n
r h
j ξi , Y ηi ANj X ρiα Xφαi Y .
i,j1
i1 αr1
Definition 3.2. We say that the screen distribution ST M of M is totally umbilical 1 in M if,
on any coordinate neighborhood U ⊂ M, there is a smooth function γi such that ANi X γi P X
for any X ∈ ΓT M, or equivalently,
h∗i X, P Y γi gX, Y ,
∀X, Y ∈ ΓT M.
3.12
In case γi 0 on U, we say that ST M is totally geodesic in M.
is said to be a conformal Killing vector field 6 if LX g −2δg for
A vector field X on M
any smooth function δ, where LX denotes the Lie derivative with respect to X, that is,
LX g Y, Z X g Y, Z − g X, Y , Z − g Y, X, Z,
.
∀X, Y, Z ∈ Γ T M
3.13
Journal of Applied Mathematics
11
is called
In particular, if δ 0, then X is called a Killing vector field 7. A distribution G on M
if each vector field belonging to G is a
a conformal Killing resp., Killing distribution on M
conformal Killing resp., Killing vector field on M. If the coscreen distribution ST M⊥ is a
Killing distribution, using 2.10 and 2.17, we have
r
X Wα , Y −gAW X, Y φαi Xηi Y −α hsα X, Y .
g ∇
α
3.14
i1
Therefore, since hsα are symmetric, we obtain
LWα g Y, Z −2α hsα X, Y .
3.15
g , then the
Theorem 3.3. Let M be an r-lightlike submanifold of a semi-Riemannian manifold M,
coscreen distribution ST M⊥ is a conformal Killing (resp., Killing) distribution if and only if there
exists a smooth function δα such that
hsα X, Y α δα gX, Y ,
{resp. hsα X, Y 0, }
∀X, Y ∈ ΓT M.
3.16
Theorem 3.4. Let M be an irrotational r-lightlike submanifold of a semi-Riemannian manifold
g of quasiconstant curvature. If the curvature vector field ζ of M
is tangent to M, ST M
M,
⊥
is totally umbilical in M, and ST M is a conformal Killing distribution, then the tensor field R0,2
is an induced symmetric Ricci tensor of M.
Proof. From 2.17∼2.20, 2.22, 3.16, and 3.11, we have
hsα X, Y α δα gX, Y ,
AW α X δ α P X φαi X 0,
r
α ρiα Xξi ,
3.17
i1
0,2
R
X, Y m − 1α β m − r −
n
α δα2
1
αr1
m − 2β θXθY r
m − r − 1 γi g A∗ξi X, Y ,
r
n δα ρiα ξi gX, Y αr1 i1
3.18
∀X, Y ∈ ΓT M.
i1
Using 3.17, we show that R0,2 is symmetric.
4. Proof of Theorem 1.1
As h∗i 0, we get g RX, Y P Z, Ni 0 by 2.30. From 2.27 and 3.16, we have
n
δα gX, P Zρiα Y − gY, P Zρiα X .
g RX,
Y P Z, Ni αr1
4.1
12
Journal of Applied Mathematics
By Theorems 3.1 and 3.4, we get dτ 0 on T M. Thus, we have g RX,
Y ξi , Ni 0 due to
2.28. From the above results, we deduce the following equation:
n
δα gX, P Zρiα Y − gY, P Zρiα X .
g RX,
Y Z, Ni 4.2
αr1
Taking X ξi and Z X to 4.2 and then comparing with 3.9, we have
βθXθY − α n
δα ρiα ξi gX, Y ,
∀X, Y ∈ ΓT M.
4.3
αr1
Case 1. If ST M⊥ is a Killing distribution, that is, δα 0, then we have
βθXθY −αgX, Y ,
∀X, Y ∈ ΓT M.
4.4
Substituting 4.3 into 1.1 and using 2.25 and the facts g RX,
Y Z, ξi 0 and
g RX, Y Z, Ni 0 due to 1.1, we have
RX, Y Z −α gY, ZX − gX, ZY ,
∀X, Y, Z ∈ ΓT M.
4.5
Thus, M is a space of constant curvature −α. Taking X Y ζ to 4.3, we have β −α.
Substituting 4.3 into 3.18 with δα γi 0, we have
RicX, Y 0,
∀X, Y ∈ ΓT M.
4.6
On the other hand, substituting 4.5 and gRξi , Y X, Ni 0 into 3.4, we have
RicX, Y − m − 1αgX, Y ,
∀X, Y ∈ ΓT M.
4.7
and M are flat
From the last two equations, we get α 0 as m > 1. Thus, β 0, and M
∗
manifolds by 1.1 and 4.5. From this result and 2.29, we show that M is also flat.
Case 2. If ST M⊥ is a conformal Killing distribution, assume that β /
0. Taking X Y ζ to
4.3, we have β −{α nαr1 δα ρiα ξi }. From this and 4.3, we show that
gX, Y θXθY ,
∀X, Y ∈ ΓT M.
4.8
Substituting 4.8 into 1.1 and using 2.25 with h∗i 0 and 3.16, we have
gRX, Y Z, W α 2β n
α δα2
αr1
gY, ZgX, W − gX, ZgY, W ,
4.9
Journal of Applied Mathematics
13
for all X, Y, Z, W ∈ ΓT M. Substituting 4.8 into 3.18 with γi 0, we have
RicX, Y m − r − 1 α β n
gX, Y ,
α δα2
αr1
∀X, Y ∈ ΓT M,
4.10
by the fact that nαr1 δα ρiα ξi −α β. On the other hand, from 2.27, 3.9, and 4.3, we
have gRξi , Y X, Ni 0. Substituting this result and 4.9 into 3.4, we have
RicX, Y m − r − 1 α 2β n
α δα2
αr1
gX, Y ,
∀X, Y ∈ ΓT M.
4.11
The last two equations imply β 0 as m − r > 1. It is a contradiction. Thus, β 0 and
is a space of constant curvature α. From 2.29 and 4.9, we show that M∗ is a space of
M
constant curvature α nαr1 α δα2 . But M is not a space of constant curvature by 3.173 .
n
Let κ m − r − 1α αr1 α δα2 , then the last two equations reduce to
R0,2 X, Y RicX, Y κgX, Y ,
∀X, Y ∈ ΓT M.
4.12
Thus M is an Einstein manifold. The scalar quantity r of M 8, obtained from R0,2 by the
method of 2.34, is given by
r
r
m
R0,2 ξi , ξi a R0,2 Xa , Xa .
i1
4.13
ar1
Since M is an Einstein manifold satisfying 4.12, we obtain
r
m
r κ gξi , ξi κ
a gXa , Xa κm − r.
i1
4.14
ar1
Thus, we have
RicX, Y r
gX, Y ,
m−r
4.15
which provides a geometric interpretation of half lightlike Einstein submanifold the same as
in Riemannian case as we have shown that the constant κ r/m − r.
14
Journal of Applied Mathematics
5. Proof of Theorem 1.2
Assume that ζ is tangent to M, ST M is totally umbilical, and ST M⊥ is a conformal Killing
vector field. Using 1.1, 2.26 reduces to
∇X hsα Y, Z − ∇Y hsα X, Z r h
i X, Zρiα Y − h
i Y, Zρiα X
i1
n hsβ X, Zθβα Y − hsβ Y, Zθβα X ,
5.1
βr1
for all X, Y, Z ∈ ΓT M. Replacing W by N to 1.1, we have
g RX,
Y Z, Ni αηi X ei βθX gY, Z
− αηi Y ei βθY gX, Z β θY ηi X − θXηi Y θZ,
5.2
for all X, Y, Z ∈ ΓT M and where ei θNi . Applying ∇X to 3.12 and using 2.13, we
have
r
∇X h∗i Y, P Z X γi gY, P Z γi h
j X, P Zηj Y ,
5.3
j1
for all X, Y, Z ∈ ΓT M. Substituting this equation into 2.30, we obtain
⎧
⎧
⎫
⎫
r
r
⎨ ⎨ ⎬
⎬
g RX, Y P Z, Ni X γi − γj τij X gY, P Z − Y γi − γj τij Y gX, P Z
⎩
⎩
⎭
⎭
j1
j1
γi
r
j1
h
j X, P Zηj Y − γi
r
h
j Y, P Zηj X,
∀X, Y, Z ∈ ΓT M.
j1
5.4
Substituting this equation and 5.2 into 2.27 and using θξi 0, we obtain
⎧
⎫
r
n
⎨ ⎬
δα ρiα X gY, Z
X γi − γj τij X − αηi X − ei βθX −
⎩
⎭
j1
αr1
⎧
⎫
r
n
⎨ ⎬
δα ρiα Y gX, Z
− Y γi − γj τij Y − αηi Y − ei βθY −
⎩
⎭
j1
αr1
⎧
⎫
r
r
⎨
⎬
h
j Y, P Zηj X − h
j X, P Zηj Y γi
⎩ j1
⎭
j1
β θY ηi X − θXηi Y θZ,
∀X, Y, Z ∈ ΓT M.
5.5
Journal of Applied Mathematics
15
Replacing Y by ξi to this and using 2.201 and the fact θξi 0, we have
⎧
⎫
r
n
⎨ ⎬
δα ρiα ξi gX, Y − βθXθY ,
γi h
i X, Y ξi γi − γj τij ξi − α −
⎩
⎭
j1
αr1
5.6
for all X, Y ∈ ΓT M. Differentiating 3.16 and using 5.1, we have
r
r
δα ηi X − α ρiα X h
i Y, Z −
δα ηi Y − α ρiα Y h
i X, Z
i1
i1
⎧
⎨
n
⎩
βr1
Xδα α
⎫
⎬
β δβ θβα X gY, Z
⎭
⎧
⎫
n
⎨
⎬
Y δα α
−
β δβ θβα Y gX, Z.
⎩
⎭
βr1
5.7
Replacing Y by ξi in the last equation and using 2.201 , we obtain
⎧
⎫
n
⎨
⎬
β δβ θβα ξi gX, Z.
δα − α ρiα ξi hi X, Z ξi δα α
⎩
⎭
βr1
5.8
0. Thus, we have
As the conformal factor δα is nonconstant, we show that δα − α ρiα ξi /
h
i X, Y σi gX, Y ,
∀X, Y ∈ ΓT M,
5.9
where σi {ξi δα α nβr1 β δβ θβα ξi }δα − α ρiα ξi −1 . From 3.171 and 5.9, we
r show that the second fundamental form tensor h, given by hX, Y i1 hi X, Y Ni n
s
h
X,
Y
W
,
satisfies
α
αr1 α
hX, Y HgX, Y ,
∀X, Y ∈ ΓT M.
5.10
Thus, M is totally umbilical 5. Substituting 5.9 into 5.6, we have
⎧
⎫
r
n
⎨ ⎬
ξi γi − γj τij ξi − γi σi − α −
δα ρiα ξi gX, Y βθXθY ,
⎩
⎭
j1
αr1
5.11
for all X, Y ∈ ΓT M. Taking X Y ζ to this equation, we have
r
n
β ξi γi − γj τij ξi − γi σi − α −
δα ρiα ξi .
j1
αr1
5.12
16
Journal of Applied Mathematics
Assume that β /
0, then we have
gX, Y θXθY ,
∀X, Y ∈ ΓT M.
5.13
Substituting 5.13 into 1.1 and using 2.25, 3.12, 3.171 , and 5.9, we have
gRX, Y Z, W
r
n
2
α 2β σi γi α δα gY, ZgX, W − gX, ZgY, W ,
i1
5.14
αr1
for all X, Y, Z, W ∈ ΓT M. Substituting 5.9 and 5.13 into 3.18, we have
RicX, Y r
n
σi γi α δα2
m − 1 α β m − r − 1
r
n i1
αr1
5.15
δα ρiα ξi gX, Y .
αr1 i1
On the other hand, substituting 5.14 and the fact that
g Rξi , Y X, Ni αβ
n
α δα ρiα ξi gX, Y 5.16
αr1
into 3.4, we have
RicX, Y r
n
2
σi γi α δα
m − 1α 2m − 1β m − r − 1
r
n δα ρiα ξi gX, Y .
i1
αr1
5.17
αr1 i1
Comparing 5.15 and 5.17, we obtain m − 1β 0. As m > 1, we have β 0, which is a
contradiction. Thus, we have β 0. Consequently, by 1.1, 2.29, and 5.14, we show that
2
and M∗ are spaces of constant curvatures α and α 2 r σi γi n
M
αr1 α δα , respectively.
i1
Let
r
r
n
n 2
σi γi α δα δα ρiα ξi ,
m − 1α m − r − 1
κ
i1
αr1
5.18
αr1 i1
then 5.15 and 5.17 reduce to
R0,2 X, Y RicX, Y κgX, Y ,
∀X, Y ∈ ΓT M.
5.19
Journal of Applied Mathematics
17
Thus, M is an Einstein manifold. The scalar quantity c of M is given by
c
r
R0,2 ξi , ξi i1
r
m
a R0,2 Xa , Xa ar1
κgξi , ξi κ
i1
m
5.20
a gXa , Xa κm − r.
ar1
Thus, we have
RicX, Y c
gX, Y .
m−r
5.21
Example 5.1. Let M, g be a lightlike hypersurface of an indefinite Kenmotsu manifold
M equipped with a screen distribution ST M, then there exist an almost contact metric
structure J, ζ, ϑ, g on M, where J is a 1, 1-type tensor field, ζ is a vector field, ϑ is a 1-form,
and g is the semi-Riemannian metric on M such that
J 2 X −X ϑXζ,
ϑX gζ, X,
∇X ζ −X ϑXζ,
Jζ 0,
ϑ ◦ J 0,
ϑζ 1,
gJX, JY gX, Y − ϑXϑY ,
∇X J Y −gJX, Y ζ ϑY JX,
5.22
for any vector fields X, Y on M, where ∇ is the Levi-Civita connection of M. Using the
local second fundamental forms B and C of M and ST M, respectively, and the projection
morphism P of M on ST M, the curvature tensors R, R, and R∗ of the connections ∇, ∇,
and ∇∗ on M, M, and ST M, respectively, are given by see 9
g RX, Y Z, P W gRX, Y Z, P W
BX, ZCY, P W − BY, ZCX, P W,
gRX, Y P Z, P W gR∗ X, Y P Z, P W
5.23
CX, P ZBY, P W − CY, P ZBX, P W,
for any X, Y, Z, W ∈ ΓT M. In case the ambient manifold M is a space form Mc of constant
J-holomorphic sectional curvature c, R is given by see 10
RX, Y Z gX, ZY − gY, ZX.
5.24
Assume that M is almost screen conformal, that is,
CX, P Y ϕBX, P Y ηXϑY ,
5.25
18
Journal of Applied Mathematics
where ϕ is a nonvanishing function on a neighborhood U in M, and ζ is tangent to M, then,
by the method in Section 2 of 9, we have
BX, Y ρ gX, Y − ϑXϑY ,
5.26
where ρ is a nonvanishing function on a neighborhood U. Then the leaf M∗ of ST M is a
semi-Riemannian manifold of quasiconstant curvature such that α −1 2ϕρ2 , β −2ϕρ2 ,
and θ ϑ in 1.1.
Acknowledgment
The authors are thankful to the referee for making various constructive suggestions and corrections towards improving the final version of this paper.
References
1 K. L. Duggal and A. Bejancu, Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications, vol.
364 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands,
1996.
2 D. N. Kupeli, Singular Semi-Riemannian Geometry, vol. 366 of Mathematics and Its Applications, Kluwer
Academic Publishers, Dordrecht, The Netherlands, 1996.
3 B.-Y. Chen and K. Yano, “Hypersurfaces of a conformally flat space,” Tensor, vol. 26, pp. 318–322, 1972.
4 M. C. Chaki and R. K. Maity, “On quasi Einstein manifolds,” Publicationes Mathematicae Debrecen, vol.
57, no. 3-4, pp. 297–306, 2000.
5 K. L. Duggal and D. H. Jin, “Totally umbilical lightlike submanifolds,” Kodai Mathematical Journal, vol.
26, no. 1, pp. 49–68, 2003.
6 D. H. Jin, “Geometry of screen conformal real half lightlike submanifolds,” Bulletin of the Korean Mathematical Society, vol. 47, no. 4, pp. 701–714, 2010.
7 D. H. Jin, “Einstein half lightlike submanifolds with a Killing co-screen distribution,” Honam Mathematical Journal, vol. 30, no. 3, pp. 487–504, 2008.
8 K. L. Duggal, “On scalar curvature in lightlike geometry,” Journal of Geometry and Physics, vol. 57, pp.
473–481, 2007.
9 D. H. Jin, “Screen conformal lightlike real hypersurfaces of an indefinite complex space form,” Bulletin
of the Korean Mathematical Society, vol. 47, no. 2, pp. 341–353, 2010.
10 K. Kenmotsu, “A class of almost contact Riemannian manifolds,” The Tohoku Mathematical Journal. Second Series, vol. 21, pp. 93–103, 1972.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014