In this paper, we study two types of 1lightlike submanifolds, named by lightlike hypersurface and half lightlike submanifold, of an indefinite Sasakian manifold admitting nonmetric
θ
connections. We prove that there exist no such two types of 1lightlike submanifolds of an indefinite Sasakian manifold.
1. INTRODUCTION
A linear(affine) connection
on a semiRiemannian manifold (
,
) is called a
nonmetric
θ

connection
if, for any vector felds
X
,
Y
and
Z
on
, it satisfes
where
θ
is a 1form, associated with a nonvanishing smooth vector field
ζ
by
Two special cases are important for both the mathematical study and the applications to physics: (1) A nonmetric
θ
connection
on
is called a
semisymmetric nonmetric connection
if its torsion tensor
satisfes
The notion of semisymmetric nonmetric connections on a Riemannian manifold was introduced by Ageshe and Chafle
[1]
and later studied by many authors. The lightlike version of Riemannian manifolds with semisymmetric nonmetric connections have been studied by some authors
[11
,
12
,
13
,
14
,
17]
.
(2) A nonmetric
θ
connection
is called a
quartersymmetric nonconnection
if its torsion tensor
satisfies
where
ϕ
is a (1, 1)type tensor field. The quartersymmetric nonmetric connection was introduced by S. Golad
[8]
, and then, studied by many authors
[2
,
3
,
16]
.
The theory of lightlike submanifolds is an important topic of research in differential geometry due to its application in mathematical physics. The study of such notion was initiated by Duggal and Bejancu
[4]
and later studied by many authors
[6
,
7]
. Although now we have lightlike version of a large variety of Riemannian submanifolds, the geometry of lightlike submanifolds of indefinite Sasakian manifolds admitting nonmetric
θ
connections has not been introduced as yet.
In this paper, we study the geometry of two type of lightlike submanifolds, named by lightlike hypersurface and half lightlike submanifold, of an indefinite Sasakian manifold
admitting nonmetric
θ
connection, in which the 1form
θ
and its associated vector field
ζ
, defined by (1.1), is identical with the structure 1form
θ
and its associated vector field
ζ
, respectively, of the indefinite Sasakian structure (
J
,
ζ
,
θ
,
). We prove the following result:
•
There exist no such two types of
1
lightlike submanifolds of an indefinite Sasakian manifold
admitting nonmetric θconnections
.
From these results we deduce to the following result:
•
There exist no such two types of
1
lightlike submanifolds of an indefinite Sasakian manifold
admitting either semisymmetric nonmetric connection or quartersymmetric nonmetric connection.
2. NONEXISTENCE THEOREM FOR LIGHTLIKE HYPERSURFACES
An odddimensional semiRiemannian manifold (
,
) is said to be an
indefinite Sasakian manifold
(
[9]
~
[10]
) if there exists a structure set {
J
,
ζ
,
θ
,
}, where
J
is a (1, 1)type tensor field,
ζ
is a vector field which is called the
structure vector field
of
and
θ
is a 1form such that, for any vector fields
X
and
Y
on
,
holds, where
∊
= 1 or ‒1 according as
ζ
is spacelike or timelike respectively.
In this case, we show that
Jζ
= 0 and
θ
○
J
= 0. The structure set {
J
,
ζ
,
θ
,
} is called an
indefinite Sasakian structure
of
. From (2.1) and (2.2), we get
In the entire discussion of this article, we shall assume that the structure vector field
ζ
of
to be unit spacelike,
i
.
e
.,
∊
= 1, without loss of generality.
Let (
M
,
g
) be a lightlike hypersurface, with a screen distribution
S
(
TM
), of an indefinite Sasakian manifold (
,
). We follow Duggal and Bejancu
[4]
for notations and structure equations used in this section. For any null section
ξ
of
TM
^{⊥}
on a coordinate neighborhood 𝒰 ⊂
M
, there exists a unique null section
N
of a unique vector bundle
tr
(
TM
) in
S
(
TM
)
^{⊥}
satisfying
We call
tr
(
TM
) and
N
the
transversal vector bundle
and the
null transversal vector field
of
M
with respect to the screen distribution respectively. Let
P
be the projection morphism of
TM
on
S
(
TM
). Then the local Gauss and Weingartan formulas of
M
and
S
(
TM
) are given respectively by
for any
X
,
Y
∈ Γ(
TM
), where the symbols ∇ and ∇* are the induced linear connections on
TM
and
S
(
TM
) respectively,
B
and
C
are the local second fundamental forms on
TM
and
S
(
TM
) respectively,
A_{N}
and
are the shape operators on
TM
and
S
(
TM
) respectively and
τ
and
σ
are 1forms on
TM
.
The induced connection ∇ on
M
is not metric and satisfies
for any
X
,
Y
,
Z
∈ Γ(
TM
), where
η
is a 1form on
TM
such that
From the fact that
, we know that
B
is independent of the choice of the screen distribution
S
(
TM
), and satisfies
From this result and (2.4), for all
X
∈ Γ(
TM
) we obtain
Now we set
a
=
θ
(
N
) and
b
=
θ
(
ξ
). For all
X
,
Y
∈ Γ(
TM
), the above second fundamental forms
B
and
C
are related to their shape operators by
Now we quote the following result by Jin
[9]
:
Lemma 1.
Let M be a lightlike hypersurface of an indefinite almost contact metric manifold
.
Then J
(
TM
^{⊥}
)
and J
(
tr
(
TM
))
are subbundles of S
(
TM
),
of rank
1.
Theorem 2.1.
There exist no lightlike hypersurfaces of an indefinite Sasakian manifold admitting a nonmetric θconnection.
Proof
. Now we consider two vector fields
V
and
U
on
S
(
TM
) such that
For any
X
∈ Γ(
TM
), the action
JX
of
X
by
J
is expressed as
where
FX
is the tangential component of
JX
and
u
is a 1form given by
Applying
_{X}
to (2.13)
_{1}
and using (2.2), (2.4), (2.10) and (2.14), we have
On the other hand, taking
Y
=
V
to (2.11) and using (2.15), we have
From the last two equations, we obtain
bu
(
X
) = 0 for all
X
∈ Γ(
TM
). Taking
X
=
V
to this result and using (2.1), we get
b
= 0. This implies that
ζ
is tangent to
M
. Replacing
Y
by
ζ
to (2.4) and using (2.3), we obtain
Applying ∇
_{X}
to
g
(
ζ
,
ζ
) = 1 and using (2.8), we get
Substituting (2.16)
_{1}
into the last equation and using (2.15) and (2.16)
_{2}
, we have
θ
= 0 on
TM
. It is a contradiction as
θ
(
ζ
) = 1. Thus there exist no lightlike hypersurfaces of an indefinite Sasakian manifold admitting a nonmetric
θ
connection.
Corollary 1.
There exist no lightlike hypersurfaces of an indefinite Sasakian manifold admitting a semisymmetric nonmetric connection or a quartersymmetric nonmetric connection.
3. NONEXISTENCE THEOREM FOR HALF LIGHTLIKE SUBMANIFOLDS
Let (
M
,
g
) be a half lightlike submanifold, with a screen distribution
S
(
TM
) and the radical distribution
Rad
(
TM
), of an indefinite Sasakian manifold (
,
). We follow Duggal and Jin
[5]
for notations and structure equations used in this section. For any null section
ξ
of
Rad
(
TM
), there exists a uniquely defined lightlike vector bundle
ltr
(
TM
) and a null vector field
N
of
ltr
(
TM
) satisfying
We call
N
,
ltr
(
TM
) and
tr
(
TM
) =
S
(
TM
^{⊥}
)⊕
_{orth}
ltr
(
TM
)
the lightlike transversal vector field, lightlike transversal vector bundle
and
transversal vector bundle
of
M
with respect to
S
(
TM
) respectively. In this case, the local Gauss and Weingartan formulas of
M
and
S
(
TM
) are given by
for all
X
,
Y
∈ Γ(
TM
), where ∇ and ∇* are induced linear connections on
TM
and
S
(
TM
) respectively,
B
and
D
are called the
local second fundamental forms
of
M
,
C
is called the
local second fundamental form
on
S
(
TM
).
A_{N}
,
and
A_{L}
are linear operators on
TM
and
τ
,
ρ
,
ϕ
and
σ
are 1forms on
TM
.
Using (1.1) and (3.1), for all
X
,
Y
,
Z
∈ Γ(
TM
) we have
From the facts
and
, we know that
B
and
D
are independent of the choice of
S
(
TM
) and satisfy
From this result and (3.1), for all
X
∈ Γ(
TM
) we obtain
We set
b
=
θ
(
ξ
),
a
=
θ
(
N
) and
e
=
θ
(
L
). For any
X
,
Y
∈ Γ(
TM
), the above three local second fundamental forms are related to their shape operators by
Now we quote the following result by Jin
[10]
:
Lemma 2.
Let M be a half lightlike submanifold of an indefinite almost contact metric manifold
.
Then the distributions J
(
TM
^{⊥}
),
J
(
tr
(
TM
))
and J
(
S
(
TM
^{⊥}
))
are vector subbundles of S
(
TM
),
of rank
1.
Theorem 3.1.
There exist no half lightlike submanifolds of an indefinite Sasakian manifold admitting a nonmetric θconnection.
Proof
. Now we consider three vector fields
V
,
U
and
W
on
S
(
TM
) such that
For any
X
∈ Γ(
TM
), the action
JX
of
X
by
J
is expressed as
where
FX
is the tangential component of
JX
and
u
and
w
are 1forms given by
Applying
to (3.12)
_{1}
and using (2.2), (3.1), (3.8) and (3.13), we have
On the other hand, taking
Y
=
V
to (3.9) and using (3.14)
_{1}
, we have
From this and (3.16)
_{1}
, we obtain
bu
(
X
) = 0 for any
X
∈ Γ(
TM
). Thus we get
b
= 0. It follow that
and
τ
=
σ
. Applying
to (3.12)
_{3}
and using (2.2), (3.1), (3.3), (3.12) and (3.13), we have
On the other hand, taking
Y
=
W
to (3.10), we have
From this and (3.18)
_{2}
, we obtain
ew
(
X
) = 0 for any
X
∈ Γ(
TM
). Thus we get
e
= 0. As
b
=
e
= 0, the structure vector field
ζ
is tangent to
M
.
Applying
to (3.13) and using (3.1) ~ (3.3), (3.12) and (3.13), we have
(∇_{X}F)Y = u(Y)A_{N}X + w(Y)A_{L}X − B(X, Y)U − D(X, Y)W + g(X, Y)ζ − θ(Y)X,
(∇_{X}u)Y = −u(Y)τ (X) − w(Y)ϕ(X) − B(X, FY),
(∇_{X}w)(Y) = −u(Y)ρ(X) − D(X, FY).
On the other hand, applying ∇
_{X}
to
u
(
Y
) =
g
(
Y
,
V
) and
w
(
Y
) =
g
(
Y
,
W
) by turns and using (3.6), (3.15), (3.16)
_{1}
, (3.17), (3.18)
_{1}
and
θ
○
J
= 0, we have
(∇_{X}u)(Y) = − u(Y)τ(X) − w(Y)ϕ(X) − B(X, FY) − θ(Y)u(X) ,
(∇_{X}w)(Y) = − u(Y)ρ(X) − D(X, FY) − θ(Y)w(X).
From the last four equations, for all
X
,
Y
∈ Γ(
TM
) we obtain
Taking
X
=
U
and
Y
=
ζ
to the first equation, or taking
X
=
W
and
Y
=
ζ
to the second equation, we have 1 = 0. It is a contradiction. Thus there exist no half lightlike submanifolds of an indefinite Sasakian manifold admitting a nonmetric
θ
connection.
Corollary 2.
There exist no half lightlike submanifolds of an indefinite Sasakian manifold admitting a semisymmetric nonmetric connection or a quartersymmetric nonmetric connection.
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