Turk J Math
(2014) 38: 1050 – 1070
c
T
¨
UB
˙
ITAK
doi:10.3906/mat-1311-19
Turkish Journal of Mathematics
http://journals.tubitak.gov.tr/math/
Research Article
Pseudosymmetric lightlike hypersurfaces
Sema KAZAN, Bayram S¸AH
˙
IN
∗
Department of Mathematics, Faculty of Science and Arts,
˙
In¨on ¨u University, 44280, Malatya, Turkey
Received: 11.11.2013 • Accepted: 06.06.2014 • Published Online: 24.10.2014 • Printed: 21.11.2014
Abstract: We study lightlike hypersurfaces of a semi-Riemannian manifold satisfying pseudosymmetry conditions. We
give sufficient conditions for a lightlike hypersurface to be pseudosymmetric and show that there is a close relationship
of the pseudosymmetry condition of a lightlike hypersurface and its integrable screen distribution. We obtain that a
pseudosymmetric lightlike hypersurface is a semisymmetric lightlike hypersurface or totally geodesic under certain condi-
tions. Moreover, we give an example of pseudosymmetric lightlike hypersurfaces and investigate pseudoparallel lightlike
hypersurfaces. Furthermore, we introduce Ricci-pseudosymmetric lightlike hypersurfaces, obtain characterizations, and
give an example for such hypersurfaces.
Key words: Semisymmetric lightlike hyp ersurface, Ricci-semisymmetric lightlike hypersurface, pseudosymmetric light-
like hypersurface, pseudoparallel lightlike hypersurface
1. Introduction
Let (M, g) be a Riemannian manifold of dimension n and ∇ be the Levi-Civita connection. A Riemannian
manifold is called locally symmetric if ∇R = 0, where R is the Riemannian curvature tensor of M [6]. Locally
symmetric Riemannian manifolds are a generalization of manifolds of constant curvature. As a generalization of
locally symmetric Riemannian manifolds, semisymmetric Riemannian manifolds were defined by the condition
R · R = 0.
It is known that locally symmetric manifolds are semisymmetric manifolds but the converse is not true [28].
Such manifolds were investigated by Cartan and they were locally classified by Szabo [5].
The Riemannian manifold (M, g) is called a pseudosymmetric manifold if at every point of M the
following condition is satisfied: the tensor R · R and Q(g, R) are linearly dependent.
The manifold (M, g) is pseudosymmetric if only if R · R = LQ(g, R) on the set U = {x ∈ M | Q(g, R) =
0 at x}, where L is some function on U .
Pseudosymmetric manifolds were discovered during the study of totally umbilical submanifolds of semisym-
metric manifolds [1]. It is clear that every semisymmetric Riemannian manifold is a pseudosymmetric manifold
but the converse is not true.
On the other hand, lightlike hypersurfaces of a semi-Riemannian manifold were studied by Duggal and
Bejancu and they obtained a transversal bundle for such hypersurfaces to the overcome anomaly that occurred
due to degenerate metric. After their book [18], many authors studied lightlike hypersurfaces by using their
∗
Correspondence: bayram.sahin@inonu.edu.tr
2010 AMS Mathematics Subject Classification: 53C15, 53C40, 53C50.
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KAZAN and S¸AH
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IN/Turk J Math
approach. In [27], S¸ahin introduced the notion of semisymmetric lightlike hypersurfaces of a semi-Riemannian
manifold and obtained many new results. After S¸ahin’s paper, many authors have studied such surfaces in
various semi-Riemannian manifolds (see [20, 21, 22, 23, 24, 29]).
In this paper, we study a more general curvature condition for lightlike hypersurfaces: pseudosymmetry
conditions. We define a pseudosymmetric lightlike hypersurface, give an example, and obtain certain sufficient
conditions for such hyp ersurfaces in Section 3, after we establish the basic information needed for the rest
of the paper in Section 2. We also investigate sufficient conditions for a lightlike hypersurface (Einstein) to
be pseudosymmetric. In Section 4, we study lightlike hypersurfaces by imposing a pseudoparallel condition
and we observe that the situation is very different from the nondegenerate case. In Section 5, we check the
Ricci-pseudosymmetry conditions for a lightlike hypersurface and provide an example of such hypersurfaces.
Moreover, we show that a Ricci-pseudosymmetric lightlike hypersurface is totally geodesic under certain geo-
metric conditions.
2. Preliminaries
In this section, we give a review on manifolds with pseudosymmetry type and lightlike hypersurfaces.
Let (M, g) be a connected n-dimensional, n ≥ 3, semi-Riemannian manifold of class C
∞
. For a (0, k)-
tensor field T on M , k ≥ 1, we define the (0, k + 2)-tensors R · T and Q(g , T ) by
(R · T )(X
1
, ..., X
k
; X, Y ) = (
˜
R(X, Y ) · T )(X
1
, ..., X
k
)
= −T(
˜
R(X, Y )X
1
, X
2
, ..., X
k
)
− ... − T (X
1
, ..., X
k−1
,
˜
R(X, Y )X
k
), (2.1)
and
Q(g, T )(X
1
, ..., X
k
; X, Y ) = ((X ∧ Y ) · T )(X
1
, ..., X
k
)
= −T((X ∧ Y )X
1
, X
2
, ..., X
k
)
− ... − T (X
1
, ..., X
k−1
, (X ∧ Y )X
k
), (2.2)
respectively, for X
1
, ..., X
k
, X, Y ∈ Γ(T M), where
˜
R is the curvature tensor field of M and R is the
Riemannian–Christoffel tensor field given by R(X
1
, X
2
, X
3
, X
4
) = g(
˜
R(X
1
, X
2
)X
3
, X
4
), and the endomor-
phisms are defined by
˜
R(X, Y )Z = [∇
X
, ∇
Y
]Z − ∇
[X,Y ]
Z,
(X ∧ Y )Z = g(Y, Z)X − g(X, Z)Y . Curvature conditions, involving the form R · T = 0, are called curvature
conditions of semisymmetric type [7]. A semi-Riemannian manifold (M, g) is then said to be semisymmetric
if it satisfies the condition R ·R = 0. It is well known that the class of semisymmetric manifolds includes the set
of locally symmetric manifolds (∇R = 0) as a proper subset [2]; here, we suppose that (M, g) is a Riemmanian
manifold. If M satisfies the condition
∇R = 0,
then M is called a locally symmetric manifold. A semi-Riemannian manifold (M, g) is said to be a pseudosym-
metric manifold if at every point of M the tensor R · R and Q(g, R) are linearly dependent. This is equivalent
to the fact that the equality
R · R = L
R
Q(g, R) (2.3)
holds on U
R
= {x ∈ M : Q(g, R) = 0} for some function L
R
on U
R
[10].
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KAZAN and S¸AH
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On the other hand, (M, g) is said to be a Ricci-pseudosymmetric manifold if at every point of M the
tensor R · S and Q(g, S) are linearly dependent. This is equivalent to the fact that the equality
R · S = L
S
Q(g, S), (2.4)
holds the set U
S
= {x ∈ M : Q(g, S) = 0} for some function L
S
on U
S
, where S is the Ricci tensor
[9]. For pseudosymmetry, Ricci-pseudosymmetry, and pseudosymmetry type curvature conditions, see also
[3, 11, 10, 9, 12, 14, 16, 17, 15, 13, 26].
We now recall the main notions and formulas for lightlike hypersurfaces. For the geometry of lightlike
hypersurfaces, we refer to [18] and [19]. Let (M, g) be a hypersurface of a (m + 2)-dimensional, with g = ¯g
|M
,
m > 0, semi-Riemannian manifold (
¯
M, ¯g) of index q ≥ 1 and T
x
M be the tangent space of M at x. Then
T
⊥
x
M = {V
x
∈ T
x
¯
M : ¯g
x
(V
x
, W
x
) = 0, ∀W
x
∈ T
x
M},
and
RadT
x
M = T
x
M ∩ T
⊥
x
M,
whose dimensional is the nullity degree of g . Then M is called a lightlike hypersurface of
¯
M if RadT
x
M = {0},
where RadT M is called radical distribution [18]. We also recall that the nullity degree of g for a lightlike
hypersurface of M is 1. The complementary vector bundle S(T M) of RadT M in T M is called the screen
bundle of M . It is known that any screen bundle is nondegenerate. Thus, we have
T M = RadTM ⊥ S(T M), (2.5)
where ⊥ denotes the orthogonal direct sum. On the other hand, there exists a unique vector bundle tr(T M )
of rank 1 over M , such that for any nonzero section ξ of T M
⊥
on a coordinate neighborhood U ⊂ M , there
exists a unique section N of tr(T M ) on U such that
¯g(ξ, N) = 1, ¯g(N, N ) = ¯g(N, X) = 0, ∀X ∈ Γ(S(T M|
U
)). (2.6)
It follows from (2.6) that tr(T M) is a lightlike vector bundle such that tr(T M)
x
∩ T
x
M = {0} for any x ∈ M
. Thus, from (2.5) and (2.6), we have
T
¯
M|
M
= S(T M) ⊕ (T M
⊥
⊕ tr(T M)) (2.7)
= TM ⊕ tr(T M). (2.8)
Here, the complementary (nonorthogonal) vector bundle tr(T M ) to the tangent bundle T M in T
¯
M|
M
is called
the lightlike transversal bundle of M with respect to screen distribution S(T M ).
Suppose that ∇ and
¯
∇ are the Levi-Civita connections of the M lightlike hypersurface and
¯
M semi-
Riemannian manifold, respectively. According to (2.8), we have
¯
∇
X
Y = ∇
X
Y + h(X, Y ) (2.9)
¯
∇
X
N = −A
N
X + ∇
t
X
N, (2.10)
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for any X, Y ∈ Γ(T M), N ∈ Γ(tr(T M)), where ∇
X
Y, A
N
X ∈ Γ(T M) and h(X, Y ), ∇
t
X
N ∈ Γ(tr(T M)). If
we set B(X, Y ) = g(h(X, Y ), ξ) and τ(X) = ¯g(∇
t
X
N, ξ), then from (2.9) and (2.10), we have
¯
∇
X
Y = ∇
X
Y + B(X, Y )N (2.11)
¯
∇
X
N = −A
N
X + τ (X)N , (2.12)
for any X, Y ∈ Γ(T M), N ∈ Γ(tr(T M)). A
N
and B are called the shape operator and the second
fundamental form of the lightlike hypersurface M , respectively.
Let P be the projection of Γ(T M ) on Γ(S(T M)). Then we have
∇
X
P Y = ∇
∗
X
P Y + C(X, P Y )ξ (2.13)
∇
X
ξ = −A
∗
ξ
X + τ (X)ξ, (2.14)
for any X, Y ∈ Γ(T M), where ∇
∗
X
P Y, A
∗
ξ
X ∈ Γ(S(T M)) and C is a 1-form on U defined by
C(X, P Y ) = ¯g(∇
X
P Y, N). (2.15)
C, A
∗
ξ
X , and ∇
∗
are called the local second fundamental form, the local shape operator, and the induced
connection on S(T M ), respectively. Then we have the following assertions:
g(A
N
Y, P W ) = C(Y, P W ), g(A
N
Y, N) = 0, B(X, ξ) = 0, (2.16)
g(A
∗
ξ
X, P Y ) = B(X, P Y ), g(A
∗
ξ
X, N) = 0, (2.17)
for X, Y, W ∈ Γ(TM), ξ ∈ Γ(T M
⊥
), and N ∈ Γ(tr(T M)).
Now let M be a lightlike hypersurface of a semi-Euclidean space R
(n+2)
q
. The Gauss equation of M is
then given by
R(X, Y )Z = B(Y, Z)A
N
X − B(X, Z)A
N
Y, (2.18)
for any X, Y, Z ∈ Γ(TM ) and N ∈ Γ(tr(T M )), where R is curvature tensor field of M .
Let M be a lightlike hypersurface of semi-Euclidean (m + 2)-space. Then the Ricci tensor Ric of M is
given by
Ric(X, Y ) = −Σ
m
i=1
ε
i
{B(X, Y )C(W
i
, W
i
) − g(A
∗
ξ
Y, A
N
X)}, ε
i
= ±1 (2.19)
for any X, Y ∈ Γ(T M), N ∈ Γ( tr(T M)) and {W
m
i=1
} is an orthonormal basis of S(T M).
Let M be a lightlike hypersurface of a semi-Euclidean space. We say that M is a semisymmetric if the
following condition is satisfied:
(R(X, Y ) · R)(X
1
, X
2
, X
3
, X
4
) = 0 (2.20)
for any X, Y, X
1
, X
2
, X
3
, X
4
∈ Γ(T M) [27]. Additionally, a lightlike hypersurface M is called a Ricci
semisymmetric lightlike hypersurface if the following condition is satisfied:
(R(X, Y ) · Ric)(X
1
, X
2
) = 0 (2.21)
for any X, Y, X
1
, X
2
∈ Γ(T M ) [27].
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3. Pseudosymmetric lightlike hypersurfaces in semi-Euclidean spaces
In this section, we consider pseudosymmetric lightlike hypersurfaces in a semi-Euclidean space. We give a
nontrivial example, obtain certain sufficient conditions for lightlike hypersurfaces to be pseudosymmetric, and
show that under certain conditions a pseudosymmetric lightlike hypersurface is totally geodesic. We also relate
the pseudosymmetry condition of the leaves of integrable screen distribution with the pseudosymmetry condition
of lightlike hypersurfaces.
Definition 3.1 Let M be a lightlike hypersurface of a semi-Euclidean space. We say that M is a pseudosym-
metric lightlike hypersurface if the tensors of R · R and Q(g, R) are linearly dependent at ∀p ∈ M . This is
equivalent to R · R = L
R
Q(g, R) on U
R
= {p ∈ M|Q(g, R) = 0}, where L
R
is some function on U
R
.
First of all, we give a nontrivial example of pseudosymmetric lightlike hypersurface in R
4
1
.
Example 3.2 Let M be a hypersurface in R
4
1
given by
x
1
= u
1
sec u
3
, x
2
= u
1
cos(u
2
+ u
3
) , x
3
= u
1
sin(u
2
+ u
3
) , x
4
= u
1
tan u
3
,
where R
4
1
is semi-Euclidean space of signature (−, +, +, +) with respect to canonical basis
{∂x
1
, ∂x
2
, ∂x
3
, ∂x
4
}
and u
1
= 0; u
3
, u
2
+ u
3
∈ (0,
π
2
). Then T M is spanned by
Z
1
= sec u
3
∂x
1
+ cos(u
2
+ u
3
)∂x
2
+ sin(u
2
+ u
3
)∂x
3
+ tan u
3
∂x
4
Z
2
= −u
1
sin(u
2
+ u
3
)∂x
2
+ u
1
cos(u
2
+ u
3
)∂x
3
Z
3
= u
1
sec u
3
tan u
3
∂x
1
− u
1
sin(u
2
+ u
3
)∂x
2
+ u
1
cos(u
2
+ u
3
)∂x
3
+ u
1
sec
2
u
3
∂x
4
.
Thus, the induced metric tensor of M is given by
∂s
2
= 0∂u
2
1
+ u
2
1
(∂u
2
2
+ ∂u
2
∂u
3
+ (1 + sec
2
u
3
)∂u
2
3
)
= u
2
1
(∂u
2
2
+ ∂u
2
∂u
3
+ (1 + sec
2
u
3
)∂u
2
3
).
Hence, M is a warped product lightlike hypersurface with RadTM = Span{Z
1
} and S(T M) = Span{Z
2
, Z
3
}.
Then the lightlike transversal vector bundle of M is spanned by
N = −
1
2
(sec u
3
∂x
1
− cos(u
2
+ u
3
)∂x
2
− sin(u
2
+ u
3
)∂x
3
+ tan u
3
∂x
4
).
By direct computations, we then get η(Z
2
) = 0, η(Z
3
) = 0 and η([Z
2
, Z
3
]) = 0. Thus, S(T M) is integrable.
Now, by using the Gauss formula, we obtain
B(Z
2
, Z
2
) = −u
1
, B(Z
2
, Z
3
) = −u
1
, B(Z
3
, Z
3
) = −u
1
− u
1
sec
2
u
3
.
On the other hand, from the Weingarten formula (2.12), we obtain
A
N
Z
2
= −
1
2u
1
Z
2
, A
N
Z
3
= −
1
u
1
Z
2
+
1
2u
1
Z
3
.
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