HALF LIGHTLIKE SUBMANIFOLDS OF AN INDEFINITE TRANS-SASAKIAN MANIFOLD OF QUASI-CONSTANT CURVATURE

Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics.
2015.
May,
22(2):
113-125

- Received : October 07, 2014
- Accepted : February 04, 2015
- Published : May 31, 2015

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We study half lightlike submanifolds
M
of an indefinite trans-Sasakian manifold
of quasi-constant curvature subject to the condition that the 1-form
θ
and the vector field
ζ
, defined by (1.1), are identical with the 1-form
θ
and the vector field
ζ
of the indefinite trans-Sasakian structure {
J
,
θ
,
ζ
} of
.
M
is a lightlike submanifold of codimension 2 such that
rank
{
Rad
(
TM
)} = 1, where
Rad
(
TM
) is the radical distribution of
M
. It is a special case of an
r
-lightlike submanifold
[3]
such that
r
= 1. Its geometry is more general than that of lightlike hypersurfaces or coisotropic submanifolds which are lightlike submanifolds
M
of codimension 2 such that
rank
{
Rad
(
TM
)} = 2. Much of its theory will be immediately generalized in a formal way to arbitrary
r
-lightlike submanifolds. For this reason, we study half lightlike submanifolds.
B.Y. Chen and K. Yano
[2]
introduced the notion of a
Riemannian manifold of quasi-constant curvature
as a Riemannian manifold
endowed with the curvature tensor
satisfying the following form:
for any vector fields
X
,
Y
and
Z
of
, where
ℓ
and
ħ
are smooth functions,
ζ
is a smooth vector field and
θ
is a 1-form associated with
ζ
by
If
ħ
= 0, then
is a space of constant curvature
ℓ
.
J.A. Oubina
[10]
introduced the notion of a trans-Sasakian manifold of type (
α
,
β
). We say that a trans-Sasakian manifold
of type (
α
,
β
) is an
indefinite trans-Sasakian manifold
if
is a semi-Riemannian manifold. Indefinite Sasakian, Kenmotsu and cosymplectic manifolds are three important kinds of trans-Sasakian manifold such that
α
= 1,
β
= 0, and
α
= 0,
β
= 1, and
α
=
β
= 0, respectively.
In this paper, we study half lightlike submanifolds
M
of an indefinite trans-Sasakian manifold
of quasi-constant curvature subject to the condition that the 1-form
θ
and the vector field
ζ
, defined by (1.1), are identical with the 1-form
θ
and the vector field
ζ
of the indefinite trans-Sasakian structure {
J
,
ζ
,
θ
} of
. The paper contains several new results which are related to the induced structure on
M
.
M
,
g
) be a half lightlike submanifold, with the radical distribution
Rad
(
TM
), and screen and coscreen distributions
S
(
TM
) and
S
(
TM
^{⊥}
) respectively, of a semi-Riemannian manifold
. We follow Duggal and Jin
[4]
for notations and structure equations used in this article. Denote by
F
(
M
) the algebra of smooth functions on
M
, by Г(
E
) the
F
(
M
) module of smooth sections of a vector bundle
E
over
M
and by (* . *)
_{i}
the
i
-th equation of (* . *). We use the same notations for any others. For any null section
ξ
of
Rad
(
TM
) on a coordinate neighborhood
U
⊂
M
, there exists a uniquely defined null vector field
N
∈ Г(
S
(
TM
^{⊥}
)
^{⊥}
) satisfying
Denote by
ltr
(
TM
) the subbundle of
S
(
TM
^{⊥}
)
^{⊥}
locally spanned by
N
. Then we show that
S
(
TM
^{⊥}
)
^{⊥}
=
Rad
(
TM
) ⊕
ltr
(
TM
). Let
tr
(
TM
) =
S
(
TM
^{⊥}
)⊕
_{orth}
ltr
(
TM
). We call
N
,
ltr
(
TM
) and
tr
(
TM
) the
lightlike transversal vector field, lightlike transversal vector bundle
and
transversal vector bundle
of
M
with respect to the screen distribution
S
(
TM
) respectively. Let
be the Levi-Civita connection of
and
P
the projection morphism of
TM
on
S
(
TM
). Then the local Gauss and Weingarten formulas of
M
and
S
(
TM
) are given respectively by
where ∇ and ∇* are induced 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 called the
shape operators
, and
τ
,
ρ
and
ϕ
are 1-forms on
TM
. From now and in the sequel, let
X
,
Y
,
Z
and
W
be the vector fields on
M
, unless otherwise specified.
Since the connection
on
is torsion-free, the induced connection ∇ on
M
is also torsion-free, and
B
and
D
are symmetric. The above three local second fundamental forms of
M
and
S
(
TM
) are related to their shape operators by
where
η
is a 1-form on
TM
such that
for any
X
∈ Г(
TM
). From (2.6), (2.7) and (2.8), we see that
B
and
D
satisfy
and
A_{N}
are
S
(
TM
)-valued, and
is self-adjoint on
TM
such that
Denote by
,
R
and
R
* the curvature tensors of the connections
, ∇ and ∇* respectively. Using the local Gauss-Weingarten formulas for
M
and
S
(
TM
), we have the Gauss equations for
M
and
S
(
TM
) such that
In the case
R
= 0, we say that
M
is
flat
.
is called an
indefinite trans-Sasakian manifold
[10]
if there exists a structure set
, where
J
is a tensor field of type (1, 1),
ζ
is a vector field which is called the
structure vector field
of
and
θ
is a 1-form such that
for any vector fields
X
and
Y
on
, where
ϵ
= 1 or −1 according as the vector field
ζ
is spacelike or timelike respectively. In this case, the set
is called an
indefinite trans-Sasakian structure of type
(
α
,
β
).
In the entire discussion of this paper, we may assume that
ζ
is unit spacelike,
i
,
e
.,
ϵ
= 1, without loss generality. From (3.1) and (3.2), we get
Let
M
be a half lightlike submanifold of an indefinite trans-Sasakian manifold
such that the structure vector field
ζ
of
is tangent to
M
. Călin
[1]
proved that if
ζ
is tangent to
M
, then it belongs to
S
(
TM
) which assume in this paper. It is known
[8]
that, for any half lightlike submanifold
M
of an indefinite trans-Sasakian manifold
,
J
(
Rad
(
TM
)),
J
(
ltr
(
TM
)) and
J
(
S
(
TM
^{⊥}
)) are subbundles of
S
(
TM
), of rank 1. Thus there exists a non-degenerate almost complex distribution
H_{o}
with respect to
J
,
i
.
e
.,
J
(
H_{o}
) =
H_{o}
, such that
S (TM ) = { J (Rad (TM )) ⊕ J (ltr (TM )) } ⊕_{orth} J (S (TM ^{⊥})) ⊕_{orth} H_{o} .
Denote by
H
the almost complex distribution with respect to
J
such that
H = Rad (TM ) ⊕_{orth} J (Rad (TM )) ⊕_{orth} H_{o} ,
TM = H ⊕ J (ltr (TM )) ⊕_{orth} J (S (TM ^{⊥})).
Consider two local null vector fields
U
and
V
, a local unit spacelike vector field
W
on
S
(
TM
), and their 1-forms
u
,
v
and
w
defined by
Let
S
be the projection morphism of
TM
on
H
and
F
the tensor field of type (1, 1) globally defined on
M
by
F
=
J
○
S
. Then
JX
is expressed as
Applying
J
to (3.6) and using (3.1) and (3.4), we have
Applying
to (3.4) ~ (3.6) by turns and using (2.1), (2.2), (2.3), (2.6) ~ (2.8), (2.10) and (3.4) ~ (3.6), we have
Substituting (3.6) into (3.3) and using (2.1), we see that
Applying
to
and using (3.1) and (3.3), we have
M
be a half lightlike submanifold of an indefinite trans-Sasakian manifold
of quasi-constant curvature. Comparing the tangential, lightlike transversal and co-screen components of the two equations (2.11) and (4.1), we get
Taking the scalar product with
N
to (2.12), we have
Substituting (4.1) into the last equation, we see that
Theorem 4.1.
Let M be a half lightlike submanifold of an indefinite trans-Sasakian manifold
of quasi-constant curvature. Then α is a constant, and
β = 0, ℓ = α ^{2}, ħ = 0.
Proof.
Applying ∇
_{Y}
to (3.16), we obtain
Using this equation, (2.3)
_{2}
, (3.12) and (3.16), we have
Replacing
Z
by
ζ
to (2.11) and then, taking the scalar product with
ζ
and using (3.17) and the fact that ,
ḡ
(
(
X
,
Y
)
ζ
,
ζ
) = 0, we have
g (R (X , Y ) ζ , ζ ) = α { u (X )g (A_{N} Y , ζ ) − u (Y )g (A_{N} X , ζ )}.
Taking the scalar product with
ζ
to (4.5) and using (3.17), we have
β (α − 1)ḡ (X ,JY ) = 0.
Taking
X
=
U
and
Y
=
ξ
to this equation, we obtain
β
(
α
− 1) = 0.
Applying ∇
_{X}
to (3.8)
_{1}
:
B
(
Y
,
U
) =
C
(
Y
,
V
), we have
Using (3.8), (3.9), (3.10), (3.17) and (3.18), the last equation is reduced to
Substituting this equation into (4.2) such that
Z
=
U
and using (3.8), we get
Comparing this equation with (4.4) such that
PZ
=
V
, we obtain
(ℓ − α ^{2} + β ^{2}){u (Y )η (X ) − u (X )η (Y )} = 2αβ {u (Y )v (X ) − u (X )v (Y )}.
Taking
X
=
ξ
and
Y
=
U
, and then,
X
=
V
and
Y
=
U
to this, we have
ℓ
=
α
^{2}
−
β
^{2}
and
αβ
= 0. From the facts that
αβ
= 0 and
β
(
α
− 1) = 0, we obtain
β
= 0,
i
.
e
.,
ℓ = α ^{2} − β ^{2}, β = 0.
Applying ∇
_{Y}
to (3.17)
_{1}
and using (3.13) and (3.16) ~ (3.18), we have
Substituting this into (4.2) such that
Z
=
ζ
, we have
(Xα )u (Y ) = (Yα )u (X ).
Replacing
Y
by
U
to this equation, we obtain
Applying
to
and using (2.1) and (2.2) we have
(∇_{X}η )(Y ) = − g (A_{N}X , Y) + τ (X )η (Y ).
Applying ∇
_{Y}
to (3.18) and using (3.14), (3.16) and (3.18), we have
Substituting this into (4.4) such that
PZ
=
ζ
and using (4.5), we get
ħ {θ (X )η (Y ) − θ (Y )η (X )} = (X α )v (Y ) − (Y α )v (X ).
Taking
X
=
ξ
and
Y
=
ζ
, and then,
X
=
U
and
Y
=
V
to this, we obtain
ħ = 0, Uα = 0.
As
Uα
= 0, from (4.6), we see that
α
is a constant. ☐
Corollary 1.
Let
be an indefinite trans-Sasakian manifold, of type (α, β), of quasi-constant curvature with a half lightlike submanifold. Then
is an indefinite α-Sasakian manifold of constant positive curvature α^{2}.
Theorem 4.2.
Let M be a half lightlike submanifolds of an indefinite trans-Sasakian manifold
of quasi-constant curvature. If one of the followings ;
is satisfied, then
is a flat manifold with an indefinite cosymplectic structure. In case (1), M is also flat.
Proof.
Denote
λ
,
μ
,
σ
and
δ
by the 1-forms such that
λ (X ) = B (X , U ) = C (X , V ), σ (X ) = D (X , W ),
μ (X ) = B (X , W ) = D (X , V ), δ (X ) = B (X , V ).
(1) If
F
is parallel, then, as
β
= 0, from (3.12) we have
Replacing
Y
by
ξ
and using (2.9) and (3.5), we obtain
ϕ
(
X
)
W
= 0. From this result, we see that
ϕ
= 0. Taking the scalar product with
U
to (4.7), we get
u (Y )v (A_{N}X ) + w (Y )v (A_{L}X ) − αθ (Y )v (X ) = 0.
Taking
Y
=
W
and
Y
=
ζ
to this equation by turns, we get
From (4.8)
_{2}
, we get
α
= 0. By Theorem 4.1,
ℓ
= 0 and
is flat manifold with an indefinite cosymplectic structure. Taking
Y
=
U
to (4.7), we have
due to (4.8)
_{1}
. Taking the scalar product with
N
,
V
and
W
to (4.7) by turns and using (2.7), (2.8), (3.8) and (4.8)
_{1}
, we have
Taking
Y
=
W
to the first equation, we obtain
ρ
= 0. As
ρ
= 0, from (2.8) we see that
A_{L}X
belongs to
S
(
TM
). As
and
A_{L}X
belong to
S
(
TM
) and
S
(
TM
) is non-degenerate, from the last two equations, we have
Taking the scalar product with
V
to the second equation, we see that
μ (X ) = B (X , W ) = D (X , V ) = 0,
As
ℓ
=
ħ
= 0, substituting (4.9) and (4.10) into (4.1), we get
Thus
M
is also flat.
(2) If
U
is parallel with respect to ∇, then, from (3.6) and (3.9), we have
J (A_{N}X ) − u (A_{N}X )N − w (A_{N}X) L + τ (X )U + ρ (X )W − αη (X )ζ = 0.
Taking the scalar product with
ζ
,
V
and
W
by turns, we get
αη (X ) = 0, τ = 0, ρ = 0,
respectively. Taking
X
=
ξ
to the first result, we have
α
= 0. As
α
= 0, we see that
ℓ
= 0 and
is a flat manifold with an indefinite cosymplectic structure.
(3) If
V
is parallel with respect to ∇, then, from (3.6) and (3.10), we have
Taking the scalar product with
U
and
W
by turns, we get
τ
= 0 and
ϕ
= 0, respectively. Applying
J
to the last equation and using (3.1) and (3.17)
_{1}
, we have
Taking the scalar product with
U
to this equation, we get
Replacing
X
by
ζ
to this equation and using (3.17)
_{1}
, we get
α = αu (U ) = − B (U , ζ ) = 0.
Thus
ℓ
= 0 and
is a flat manifold with an indefinite cosymplectic structure.
(4) If
W
is parallel with respect to ∇, then, from (3.6) and (3.11), we get
J (A_{L}X ) − u (A_{L}X )N − w (A_{L}X )L + ϕ (X )U = 0.
Taking the scalar product with
V
and
U
by turns, we have
ϕ = 0, ρ = 0.
respectively. Applying
J
to the last equation and using (3.1), (3.17)
_{2}
, we have
A_{L}X = −αw (X )ζ + μ (X )U + σ (X )W .
Taking the scalar product with
U
to this, we have
D
(
X
,
U
) = 0 and
C (X ,W ) = 0.
Applying ∇
_{X}
to
C
(
Y
,
W
) = 0 and using (3.10) and
ϕ
=
β
= 0, we have
(∇_{X}C )(Y , W ) = − g (A_{N} Y , F (A_{L}X )).
Taking
PZ
=
W
to (4.4) and using the last two equations, we obtain
g (A_{N}X , F (A_{L}Y )) − g (A_{N} Y , F (A_{L}X )) = ℓ {w (Y )η (X ) − w (X )η (Y )}
as
ρ
= 0. Taking
X
=
ξ
and
Y
=
W
to this and using the facts that
F
(
A_{L}W
) = 0 and
A_{L}ξ
= 0, we obtain
ℓ
= 0. As
ℓ
= 0, we see that
α
= 0 and
is a flat manifold with an indefinite cosymplectic structure. ☐
Definition.
The structure tensor field
F
on
M
is said to be recurrent
[9]
if there exists a 1-form
on
M
such that
Theorem 5.1.
Let M be a half lightlike submanifold of an indefinite trans-Sasakian manifold
of quasi-constant curvature. If F is recurrent, then it is parallel,
and M are flat, and the transversal connection of M is flat.
Proof.
As
F
is recurrent, from (3.12) and the fact that
β
= 0, we get
Replacing
Y
by
ξ
to this and using (2.10), (3.1), (3.4), (3.5) and the fact that
Fξ
= −
V
, we get
. Taking the scalar product with
U
, we get
. Thus
F
is parallel with respect to ∇. From Theorem 4.3, we see that
and
M
are flat, and the transversal connection of
M
is flat. ☐
Definition.
The structure tensor field
F
of
M
is said to be
Lie recurrent
[9]
if there exists a 1-form 𝜗 on
M
such that
where
L_{X}F
denotes the Lie derivative on
M
of
F
with respect to
X
, that is,
The structure tensor field
F
is called
Lie parallel
if 𝜗 = 0.
Theorem 5.2.
Let M be a half lightlike submanifold of an indefinite trans-Sasakian manifold
of quasi-constant curvature. If F is Lie recurrent, then it is Lie parallel, and
is a flat manifold with indefinite cosymplectic structure.
Proof.
As
F
is Lie recurrent, from (3.12), (5.1) and (5.2) we get
Replacing
Y
by
ξ
to (5.3) and using (2.6), (3.4) and
Fξ
= −
V
, we have
Taking the scalar product with
V
,
W
and
ζ
to this by turns, we get
On the other hand, taking
Y
=
V
to (5.3) and using (3.4), we have
𝜗(X )ξ = − B (X , V )U − D (X , V )W − ∇_{ξ}X + F ∇_{V} X + αu (X )ζ .
Applying
F
and using (3.7), (5.5) and
FU
=
FW
=
Fζ
= 0, we get
𝜗(X )V = ∇_{V} X + F ∇_{ξ}X + ϕ (X )W .
Comparing this with (5.4), we get 𝜗 = 0. Therefore
F
is Lie parallel.
Taking
X
=
U
to (5.3) and using (3.7), (3.8), (3.9) and (3.18), we get
Taking the scalar product with
ζ
to this equation and using (3.18), we get
αv
(
Y
) = 0. Taking
Y
=
V
to this result, we have
α
= 0. Therefore,
ℓ
= 0 and
is a flat manifold with indefinite cosymplectic structure. ☐

1. INTRODUCTION

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-Bejancu
[3]
and later studied by many authors (see two books
[5
,
6]
). Half lightlike submanifold
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2. HALF LIGHTLIKE SUBMANIFOLD

Let (
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3. INDEFINITE TRANS-SASAKIAN MANIFOLDS

An odd-dimensional semi-Riemannian manifold
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4. MANIFOLD OF QUASI-CONSTANT CURVATURE

Let
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- (1)F is parallel with respect to the connection∇,
- (2)U is parallel with respect to the connection∇,
- (3)V is parallel with respect to the connection∇,and
- (4)W is parallel with respect to the connection∇

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5. RECURRENT HALF LIGHTLIKE SUBMANIFOLDS

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Citing 'HALF LIGHTLIKE SUBMANIFOLDS OF AN INDEFINITE TRANS-SASAKIAN MANIFOLD OF QUASI-CONSTANT CURVATURE
'

@article{ SHGHCX_2015_v22n2_113}
,title={HALF LIGHTLIKE SUBMANIFOLDS OF AN INDEFINITE TRANS-SASAKIAN MANIFOLD OF QUASI-CONSTANT CURVATURE}
,volume={2}
, url={http://dx.doi.org/10.7468/jksmeb.2015.22.2.113}, DOI={10.7468/jksmeb.2015.22.2.113}
, number= {2}
, journal={Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics}
, publisher={Korean Society of Mathematical Education}
, author={JIN, DAE HO}
, year={2015}
, month={May}