Filomat 26:2 (2012), 277–287
DOI 10.2298/FIL1202277S
Published by Faculty of Sciences and Mathematics,
University of Ni
ˇ
s, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
Slant lightlike submanifolds of indefinite Sasakian manifolds
Bayram S
.
ahin
a
, Cumali Yıldırım
a
a
Department of Mathematics, Inonu University, 44280, Malatya, Turkey.
Abstract. In this paper, we define and study both slant lightlike submanifolds and screen slant lightlike
submanifolds of an indefinite Sasakian manifold. We provide non-trivial examples and obtain necessary
and suffi cient conditions for the existence of a slant lightlike submanifold.
1. Introduction
In [8], Duggal and Bejancu introduced the geometry of arbitrary lightlike submanifolds of semi-
Riemannian manifolds. Since then, many authors have studied the geometry of lightlike hypersurfaces
and lightlike submanifolds. 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 [13]. A Killing horizon is a lightlike hypersurface which is a local
isometry horizon with respect to 1− parameter group. Physically, a particle on local isometry horizon of
a 4− dimensional spacetime manifold may immediately be travelling at the speed of light along the single
null generator, but standing still to relative to its surroundings. Roughly speaking, a Killing horizon is a
lightlike hypersurface whose generating null vector can be normalized so as to coincide with one of the
Killing vector. The surface of a black hole is described in terms of Killing horizon. This relation has its
roots in Hawking’s area theorem which states that if matter satisfies the dominant energy condition, then
the area of the black hole can not decrease [15].
On the other hand, the theory of contact manifolds has its roots in differential equations, optics and
phase space of a dynamical system (for details see [1]. Recently, Fritelli at all [12] gave a self-contained
presentation of the null surface formulation of the Einstein based on the contact geometry of differential
equation. The essential idea of the null surface formulation is to start from family of co-dimension one
foliations of the spacetime manifold by hypersurfaces, fix the conformal structure of spacetime by requiring
these hypersurface to be null and formulate the Einstein equations in terms of these data.
Let
¯
M be a Sasakian manifold with almost contact structure (ϕ, η, V) and M a Riemannian manifold
isometrically immersed in
¯
M such that the structure vector field V is tangent to M. Then M is called
invariant if ϕ(TpM) = TpM, for every p ∈ M, where T
p
M denotes the tangent space to M at the point p.
M is called anti-invariant if ϕ(TpM) ⊂ T
p
M
⊥
for every p ∈ M, where T
p
M
⊥
denotes the normal space to
M at the point p. As a generalization of invariant and totally real submanifolds of almost contact metric
2010 Mathematics Subject Classification. Primary 53C15; Secondary 53C40; 53C50
Keywords. Degenerate metric, slant lightlike submanifold, Sasakian manifold
Received: 20 October 2009; Accepted: 1 August 2011
Communicated by Vladimir Dragovi
´
c
Email addresses: bayram.sahin@inonu.edu.tr ( Bayram S
.
ahin ), cumali.yildirim@inonu.edu.tr (Cumali Yıldırım )
B. S
.
ahin, C. Yıldırım / Filomat 26:2 (2012), 277–287 278
manifolds, following Chen’s definition [6], A. Lotta [16] and Cabrerizo et all [5] studied the geometry of
slant submanifolds of a Sasakian manifold
¯
M as a real submanifold such that the angle between ϕX and
T
x
M is constant for every vector X ∈ T
x
M and x ∈ M. The first author of this paper introduced lightlike slant
submanifolds of indefinite Kaehler manifolds in [18] and [19]. On the other hand, in [10], Duggal-S
.
ahin
studied various lightlike submanifolds of indefinite Sasakian manifolds. However, the concept of slant
lightlike submanifolds of indefinite Sasakian manifolds has not been studied as yet.
The objective of this paper is to introduce the notion of slant submanifolds of an indefinite Sasakian
manifolds. We study the existence problem and establish an interplay between slant lightlike submanifolds
and contact Cauchy Riemann (CR)-lightlike submanifolds [10].
Section 2 includes basic information on the lightlike geometry as needed in this paper. In section 3,
we introduce the concept of slant lightlike submanifolds and give a non-trivial example. We show that,
contrary to the Riemannian case, the geometry of slant lightlike and screen slant lightlike submanifolds is
very different from the Riemannian case. We prove a characterization theorem and show that co-isotropic
contact CR-lightlike submanifolds are slant lightlike submanifolds. Because, a slant lightlike submanifold of
an indefinite Sasakian manifold do not contain invariant and screen real submanifolds, finally, in section 4,
we introduce screen slant lightlike submanifoldds of indefinite Sasakian manifold and give an example of
such submanifolds.
2. Preliminaries
An odd dimensional semi-Riemannian manifold (
¯
M,
¯
1) is called a contact metric manifold [4] if there
exists a (1, 1) tensor field ϕ, a vector field V, called the characteristic vector field, and its 1-form η satisfying
¯
1(ϕ X, ϕ Y) =
¯
1(X, Y) − ϵ η(X)η(Y),
¯
1(V, V) = ϵ, (1.1)
ϕ
2
(X) = −X + η (X) V,
¯
1(X, V) = η(X),
d η(X, Y) =
¯
1(X, ϕ Y), ∀X, Y ∈ Γ(T
¯
M),
where ϵ = 1 or −1. It follows that ϕV = 0, η ◦ ϕ = 0, η(V) = ϵ.
Then (ϕ, V, η,
¯
1) is called contact metric structure of
¯
M. We say that M has a normal contact structure if
N
ϕ
+ d η ⊗ξ = 0, where N
ϕ
is the Nijenhuis tensor field of ϕ [4]. A normal contact metric manifold is called
a Sasakian manifold [20] for which we have
¯
∇
X
V = ϕX, (1.2)
(
¯
∇
X
ϕ)Y = −
¯
1(X, Y)V + ϵ η(Y)X. (1.3)
We follow [8] for the notation and formulas used in this paper. A submanifold (M
m
, 1) immersed in
a semi-Riemannian manifold (
¯
M
m+n
,
¯
1) is called a lightlike submanifold if the metric 1 induced from
¯
1 is
degenerate and the radical distribution Rad(TM) is of rank r, where 1 ≤ r ≤ m. Let S(TM) be a screen
distribution which is a semi-Riemannian complementary distribution of Rad(TM) in TM, i.e.,
TM = Rad (TM) ⊥ S(TM).
Consider a screen transversal vector bundle S(TM
⊥
), which is a semi-Riemannian complementary vector
bundle of Rad(TM) in TM
⊥
. Since, for any local basis {ξ
i
} of Rad(TM), there exists a local null frame {N
i
}
of sections with values in the orthogonal complement of S(TM
⊥
) in [S(TM)]
⊥
such that
¯
1(ξ
i
, N
j
) = δ
ij
, it
follows that there exists a lightlike transversal vector bundle ltr(TM) locally spanned by {N
i
} [8, page 144].
Let tr(TM) be complementary (but not orthogonal) vector bundle to TM in T
¯
M|
M
. Then,
tr(TM) = ltr(TM) ⊥ S(TM
⊥
),
T
¯
M|
M
= S(TM) ⊥ [Rad(TM) ⊕ ltr(TM)] ⊥ S(TM
⊥
).
Although S(TM) is not unique, it is canonically isomorphic to the factor vector bundle TM/Rad TM [14].
Following result is important to this paper.
B. S
.
ahin, C. Yıldırım / Filomat 26:2 (2012), 277–287 279
Proposition 2.1 ([8])
The lightlike second fundamental forms of a lightlike submanifold
M
do not depend
on
S(TM), S(TM
⊥
)
and
ltr(TM)
.
Throughout this paper, we will discuss the dependence (or otherwise) of the results on induced object(s)
and refer [8] for their transformation equations.
Followings are four subcases of a lightlike submanifold (M, 1, S(TM), S(TM
⊥
).
Case 1: r - lightlike if r < min{m, n};
Case 2: Co - isotropic if r = n < m; S(T M
⊥
) = {0};
Case 3: Isotropic if r = m < n; S(TM) = {0};
Case 4: Totally lightlike if r = m = n; S(TM) = {0} = S(TM
⊥
).
The Gauss and Weingarten formulas are:
¯
∇
X
Y = ∇
X
Y + h(X, Y), ∀X, Y ∈ Γ(TM), (1.4)
¯
∇
X
V = −A
V
X + ∇
t
X
V, ∀X ∈ Γ(TM), V ∈ Γ(tr(TM)), (1.5)
where {∇
X
Y, A
V
X} and {h(X, Y), ∇
t
X
V} belong to Γ(TM) and Γ(ltr(TM)), respectively. ∇ and ∇
t
are linear
connections on M and on the vector bundle ltr(TM), respectively. The second fundamental form h is a
symmetric F(M)-bilinear form on Γ(TM) with values in Γ(tr(TM)) and the shape operator A
V
is a linear
endomorphism of Γ(TM). Then we have
¯
∇
X
Y = ∇
X
Y + h
l
(X, Y) + h
s
(X, Y), (1.6)
¯
∇
X
N = −A
N
X + ∇
l
X
(N) + D
s
(X, N), (1.7)
¯
∇
X
W = −A
W
X + ∇
s
X
(W) + D
l
(X, W), ∀X, Y ∈ Γ(T M), (1.8)
N ∈ Γ(ltr(TM)) and W ∈ Γ(S(T M
⊥
)). Denote the projection of TM on S(TM) by
¯
P. Then, by using (1.4),
(1.6)-(1.8) and taking account that
¯
∇ is a metric connection we obtain
¯
1(h
s
(X, Y), W) +
¯
1(Y, D
l
(X, W)) = 1(A
W
X, Y), (1.9)
¯
1(D
s
(X, N), W) =
¯
1(N, A
W
X). (1.10)
We set
∇
X
¯
PY = ∇
∗
X
¯
PY + h
∗
(X,
¯
PY), (1.11)
∇
X
ξ = −A
∗
ξ
X + ∇
∗
t
X
ξ, (1.12)
for X, Y ∈ Γ(TM) and ξ ∈ Γ(Rad(TM)). By using above equations we obtain
¯
1(h
l
(X,
¯
PY), ξ) = 1(A
∗
ξ
X,
¯
PY), (1.13)
¯
1(h
∗
(X,
¯
PY), N) = 1(A
N
X,
¯
PY), (1.14)
¯
1(h
l
(X, ξ), ξ) = 0 , A
∗
ξ
ξ = 0. (1.15)
In general, the induced connection ∇ on M is not metric connection. Since
¯
∇ is a metric connection, by
using (1.6) we get
(∇
X
1)(Y, Z) =
¯
1(h
l
(X, Y), Z) +
¯
1(h
l
(X, Z), Y). (1.16)
However, it is important to note that ∇
⋆
is a metric connection on S(T M). From now on, we briefly denote
(M, 1, S(TM), S(TM
⊥
)) by M in this paper.
Definition 2.1 ([10]) Let (M, 1, S(TM), S(TM
⊥
)) be a lightlike submanifold tangent to the structure vector field
V immersed in an indefinite Sasakian manifold (
¯
M,
¯
1). We say that M is a contact CR - lightlike submanifold
of
¯
M if the following conditions are satisfied:
B. S
.
ahin, C. Yıldırım / Filomat 26:2 (2012), 277–287 280
(A) Rad TM
is a distribution on
M
such that
Rad TM ∩ ϕ(RadTM) = {0},
(B)
There exist vector bundles
D
0
and
D
′
over
M
such that
S(TM) = {ϕ(RadTM) ⊕ D
′
} ⊥ D
o
⊥ V,
ϕD
o
= D
o
, ϕ(D
′
) = L
1
⊥ ltr(TM),
where
D
0
is a non-degenerate distribution on
M
and
L
1
is a vector sub bundle of
S(TM
⊥
).
Thus, we have the following decomposition
TM = D ⊕V ⊕ D
′
, (1.17)
where
D = Rad T M ⊥ ϕ(Rad TM) ⊥ D
o
. (1.18)
A contact CR-lightlike submanifold is proper if D
o
, {0} and L
1
, {0}.
3. Slant lightlike submanifolds
We start with the following lemmas which will be useful for later results.
Lemma 3.1.
Let
M
be an
r−
lightlike submanifold of an indefinite Sasakian manifold
¯
M
of index
2q.
Suppose
that
ϕRadTM
is a distribution on
M
such that
RadTM ∩ ϕRadTM = {0}
. Then
ϕltr(TM)
is subbundle of the
screen distribution
S(TM)
and
ϕRadTM ∩ ϕltr(TM) = {0}
Proof. Since by hypothesis ϕRadTM is a distribution on M such that ϕRadTM ∩ RadTM = {0}, we have
ϕRadTM ⊂ S(TM). Now we claim that ltr(TM) is not invariant with respect to ϕ. Let us suppose that
ltr(TM) is invariant with respect to ϕ. Choose ξ ∈ Γ(RadTM) and N ∈ Γ(ltr(TM)) such that
¯
1(N, ξ) = 1.
Then from the decomposition of a lightlike submanifold, we have 1 =
¯
1(ξ, N) =
¯
1(ϕξ, ϕN) = 0 due to
ϕξ ∈ Γ(S(TM)) and ϕN ∈ Γ(ltr(TM)). This is a contradiction, so ltr(TM) is not invariant with respect to ϕ.
Also ϕN does not belong to S(TM
⊥
), since S(TM
⊥
) is orthogonal to S(TM),
¯
1(ϕN, ϕξ) must be zero, but
we have
¯
1(ϕN, ϕξ) =
¯
1(N, ξ) , 0 for some ξ ∈ Γ(RadTM), this is again a contradiction. Thus we conclude
ϕltr(TM) is a distribution on M. Moreover,ϕN does not belong to RadTM. Indeed, if ϕN ∈ Γ(RadTM), we
would have ϕ
2
N = −N ∈ Γ(ϕRadT M ), but this is impossible. Similarly, ϕ N does not belong to ϕRadTM.
Thus we conclude that ϕltr(TM) ⊂ S(TM) and ϕRadTM ∩ ϕltr(TM) = {0}.
Lemma 3.2.
Under the hypothesis of Lemma 3.1 and the spacelike characteristic vector field, if
r = q,
then
any complementary distribution to
ϕ(RadTM) ⊕ ϕltr(TM)
in
S(TM)
is Riemannian
.
Proof. Let dim(
¯
M) = m + n and dim(M) = m. Lemma 3.1 implies that ϕltr(TM) ⊕ ϕRadTM ⊂ S(TM). We
denote the complementary distribution to ϕltr(TM) ⊕ ϕRadTM in S(TM) by D
′
. Then we have a local quasi
orthonormal field of frames on
¯
M along M
{ξ
i
, N
i
, ϕξ
i
, ϕN
i
, X
α
, W
a
}, i ∈ {1, ..., r}, α ∈ {3r + 1, ..., m}, a ∈ {r + 1, ..., n},
where {ξ
i
}and {N
i
}are lightlike basis of RadTM and ltr(TM), respectively and ϕξ
i
, ϕN
i
, {X
α
}and {W
a
}are or-
thonormal basis of S(TM) and S(TM
⊥
), respectively. From the basis {ξ
1
, ..., ξ
r
, ϕξ
1
, ..., ϕξ
r
, ϕN
1
, ..., ϕN
r
, N
1
, ..., N
r
}
of ltr(TM) ⊕RadTM ⊕ϕRadTM ⊕ϕltr(TM), we can construct an orthonormal basis {U
1
, ..., U
2r
, V
1
, ..., V
2r
} as
B. S
.
ahin, C. Yıldırım / Filomat 26:2 (2012), 277–287 281
follows
U
1
=
1
√
2
(ξ
1
+ N
1
) U
2
=
1
√
2
(ξ
1
− N
1
)
U
3
=
1
√
2
(ξ
2
+ N
2
) U
4
=
1
√
2
(ξ
2
− N
2
)
... ...
... ...
U
2r−1
=
1
√
2
(ξ
r
+ N
r
) U
2r
=
1
√
2
(ξ
r
− N
r
)
V
1
=
1
√
2
(ϕξ
1
+ ϕN
1
) V
2
=
1
√
2
(ϕξ
1
− ϕN
1
)
V
3
=
1
√
2
(ϕξ
2
+ ϕN
2
) V
4
=
1
√
2
(ϕξ
2
− ϕN
2
)
... ...
... ...
V
2r−1
=
1
√
2
(ϕξ
r
+ ϕN
r
) V
2r
=
1
√
2
(ϕξ
r
− ϕN
r
).
Hence, Span{ξ
i
, N
i
, ϕξ
i
, ϕN
i
} is a non-degenerate space of constant index 2r. Thus we conclude that
RadTM ⊕ ϕRadTM ⊕ltr(TM) ⊕ ϕltr(TM) is non-degenerate and of constant index 2r on
¯
M. Since
index(T
¯
M) = index(RadTM ⊕ltr(TM)) + index(ϕRadTM ⊕ϕltr(TM)
+ index(D
′
⊥ S(TM
⊥
)),
we have 2q = 2r + index(D
′
⊥ S(TM
⊥
)). Thus, if r = q, then D
′
⊥ S(TM
⊥
) is Riemannian, i.e., index(D
′
⊥
(S(TM)
⊥
)) = 0. Hence D
′
is Riemannian.
Remark 3.2. As mentioned in the introduction, the purpose of this paper is to introduce the notion of slant
lightlike submanifolds. To define this notion, one needs to consider angle between two vector fields. As we
can see from section 2, a lightlike submanifold has two (radical and screen) distributions: The radical distri-
bution is totally lightlike and therefore it is not possible to define angle between two vector fields of radical
distribution. On the other hand, the screen distribution is non-degenerate. Thus one way to define slant
notion is to choose a Riemannian screen distribution on lightlike submanifold, for which we use Lemma 3.2.
Proposition 3.1.
There exist no lightlike submanifolds of an indefinite almost contact manifold
¯
M
such
that the structure vector field
V
is belong to
RadTM
or
ltr(TM)
.
Proof. Suppose that M is a lightlike submanifold and V ∈ Γ(RadTM). Then there exist a vector field
W ∈ Γ(ltr(TM)) 1(N, V) = 1 , 0. On the other hand from (1) we have
1(ϕN, ϕV) = 1(V, N) −η(V)η(N).
Since V is null and ϕV = 0, we obtain
1(V, N) = 0.
This is a contradiction which proves our assertion.
From now on, we suppose that the structure vector field V is tangent to M. Then proposition 3.1 implies
that V ∈ Γ(S(TM)). In this paper we assume that V is spacelike.
Definition 3.1. Let M be a q− lightlike submanifold of an indefinite Sasakian manifold
¯
M of index 2q. Then
we say that M is a slant lightlike submanifold of
¯
M if the following conditions are satisfied:
(A) RadTM is a distribution on M such that
ϕRadTM ∩ RadTM = {0}. (3.1)
(B) For each non-zero vector field tangent to D at x ∈ U ⊂ M, the angle θ(X) between ϕX and the vector
space D
x
is constant, that is, it is independent of the choice of x ∈ U ⊂ M and X ∈ D
x
, where D is
complementary distribution to ϕRadTM ⊕ ϕltr(TM) in the screen distribution S(TM).