TWO CHARACTERIZATION THEOREMS FOR HALF LIGHTLIKE SUBMANIFOLDS OF AN INDEFINITE KENMOTSU MANIFOLD

Journal of the Korean Society of Mathematical Education Series B The Pure and Applied Mathematics.
2014.
Jan,
21(1):
1-10

- Received : February 22, 2013
- Accepted : January 15, 2014
- Published : January 30, 2014

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In this paper, we study the curvature of locally symmetric or semi-symmetric half lightlike submanifolds
M
of an indefinite Kenmotsu manifold
, whose structure vector field is tangent to
M
. After that, we study the existence of the totally geodesic screen distribution of half lightlike submanifolds of indefinite Kenmotsu manifolds with parallel co-screen distribution subject to the conditions: (1)
M
is locally symmetric, or (2) the lightlike transversal connection is flat.
half lightlike and coisotropic submanifolds
[3]
. Half lightlike submanifold is a special case of
r
-lightlike submanifold such that
r
= 1 and its geometry is more general form than that of coisotrophic submanifold. Much of the works on half lightlike submanifolds will be immediately generalized in a formal way to general r-lightlike submanifolds of arbitrary codimension
n
and arbitrary rank
r
.
In the theory of Sasakian manifolds, the following result is well-known
[9]
:
If a Sasakian manifold is locally symmetric, then it is of constant positive curvature
1. In 1971, K. Kenmotsu proved the following result
[8]
:
If a Kenmotsu manifold is locally symmetric, then it is of constant negative curvature
–1.
In this paper, we study the curvature of locally symmetric or semi-symmetric half lightlike submanifolds of an indefinite Kenmotsu manifold
, whose structure vector fild is tangent to
M
. After that, we study the existence of the totally geodesic screen distribution of half lightlike submanifolds of indefinite Kenmotsu manifolds with parallel co-screen distribution subject such that either
M
is locally symmetric or the lightlike transversal connection is flat. We prove the following results:
Theorem 1.1.
Let M be a half lightlike submanifold of an indefinite Kenmotsu manifold
,
whose structure vector field is tangent to M. If M is locally symmetric or semi-symmetric, then M is a space of constant negative curvature –1. In this case, the induced connection on M is a torsion-free metric connection and the lightlike transversal connection is flat.
Theorem 1.2.
Let M be a half lightlike submanifold of an indefinite Kenmotsu manifold
with parallel co-screen distribution. If either M is locally symmetric or the lightlike transversal connection is flat, then the screen distribution S(TM) of M is never totally geodesic in M.
is said to be an
indefinite Kenmotsu manifold
[7
,
8
,
10]
if there exist a structure set
, where
J
is a
(1, 1)-type tensor field, 𝜁 is a vector field and
θ
is a 1-form such that
for any vector fields
X, Y
on
, where
is the Levi-Civita connection of
.
A submanifold (
M, g
) of a semi-Riemannian manifold
of codimension 2 is called
a half lightlike submanifold
if the radical distribution
Rad(TM)
=
TM
∩
TM
^{⊥}
of
M
is a vector subbundle of the tangent bundle
TM
and the normal bundle
TM
^{⊥}
of rank 1. Then there exist complementary non-degenerate distributions
S(TM)
and
S
(
TM
^{⊥}
) of
Rad(TM)
in
TM
and
TM
^{⊥}
respectively, which are called the
screen
and
co-screen distributions
on
M
, such that
where ⊕
_{orth}
denotes the orthogonal direct sum. We denote such a half lightlike submanifold by
M = (M, g, S(TM))
. Denote by
F(M)
the algebra of smooth functions on
M
and by 𝚪(
E
) the
F(M)
module of smooth sections of a vector bundle
E
over
M
. Choose
L
∈ 𝚪(
S
(
TM
^{⊥}
)) as a unit vector field with
. In this paper we may assume that
, without loss of generality. Consider the orthogonal complementary distribution
S(TM)
^{⊥}
to
S(TM)
in
. For any null section 𝜉 of
Rad(TM)
, certainly 𝜉 and
L
belong to 𝚪(
S(TM)
^{⊥}
). Thus we have
S(TM)
^{⊥}
=
S
(
TM
^{⊥}
) ⊕
_{orth}
S
(
TM
^{⊥}
)
^{⊥}
,
where
S
(
TM
^{⊥}
)
^{⊥}
is the orthogonal complementary to
S
(
TM
^{⊥}
) in
S
(
TM
)
^{⊥}
. For any null section 𝜉 of
Rad(TM)
on a coordinate neighborhood
U
⊂
M
, there exists a uniquely defined null vector field
N
∈ 𝚪(
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. Therefore
T
is decomposed as
Let
P
be 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 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
and
ϕ
are 1-forms on
TM
.
Since
is torsion-free, ∇ is also torsion-free and both
B
and
D
are symmetric. From the facts
, we know that
B
and
D
are independent of the choice of
S(TM)
and satisfy
The induced connection ∇ of
M
is not metric and satisfies
for all
X, Y, Z
∈ 𝚪(
TM
), where 𝜂 is a 1-form on
TM
such that
But the connection ∇* on
S(TM)
is metric. The above three local second fundamental forms are related to their shape operators by
In case
C
= 0 on any coordinate neighborhood
U
, we say that
S(TM)
is
totally geodesic
in
M
. From (2.10), we show that
S(TM)
is totally geodesic in
M
if and only if
S(TM)
is a parallel distribution on
M
, i.e.,
∇
_{X}Y
∈ 𝚪(
S(TM))
, ∀
X
∈ 𝚪(
TM
) and
Y
∈ 𝚪(
S(TM)
).
In the sequel, we let
X, Y, Z, U
, … be the vector fields of
M
, unless otherwise specified. Denote by
and
R
the curvature tensors of
and ∇ respectively. Using (2.7)~(2.11), we have the Gauss-Codazzi equations for
M
and
S(TM)
:
A half lightlike submanifold
M
= (
M, g
, ∇) equipped with a degenerate metric
g
and a linear connection ∇ is said to be of constant curvature
c
if there exists a constant
c
such that the curvature tensor
R
of ∇ satisfies
For any
X
∈ 𝚪(
TM
), let
where
Q
is the projection morphism of
on 𝚪(
ltr(TM)
) with respect to (2.6). Then ∇
^{𝓁}
is a linear connection on the lightlike transversal vector bundle
ltr(TM)
of
M
. We say that ∇
^{𝓁}
is the
lightlike transversal connection
of
M
. We define the curvature tensor
R
^{𝓁}
on
ltr(TM)
by
If
R
^{𝓁}
vanishes identically, then the transversal connection is said to be
flat
.
From (2.8) and the definition of ∇
^{𝓁}
, we get
for all
X
∈ 𝚪(
TM
). Substituting this equation into the right side of (2.24), we get
R
^{𝓁}
(
X, Y
)
N
= 2dτ(
X, Y
)
N
.
From this result we deduce the following theorem:
Theorem 2.1
(
[6]
).
Let M be a half lightlike submanifold of a semi-Riemannian manifold
.
Then the lightlike transversal connection of M is flat, if and only if the 1-form
τ
is closed, i.e., dτ
= 0,
on any
𝒰 ⊂
M
.
Note 1.
We know that dτ is independent of the choice of the section 𝜉 on
Rad(TM)
, where τ is given by
. In fact, if we take
and
, it follows that
. If we take the exterior derivative d on the last equation, then we have
.
M
. It is well known
[1]
that if 𝛇 is tangent to
M
, then it belongs to
S(TM)
. Replacing
Y
by 𝛇 to (2.7) and using (2.2), we have
Substituting (3.1)
_{1}
into
R(X, Y)
𝛇 = ∇
_{X}
∇
_{Y}
𝛇−∇
_{Y}
∇
_{X}
𝛇−∇
_{[X, Y]}
𝛇 and using (2.19), (3.1) and the fact that ∇ is torsion-free, we have
Taking the scalar product with 𝛇 to this and using the fact
and (2.1), we show that
θ
is closed, i.e.,
dθ
= 0 on
TM
. Thus we obtain
Applying
to
θ(Y)
= g(Y, 𝛇) and using (2.2), (2.5) and
, we have
Case 1. Assume that
M
is locally symmetric, i.e., ∇
R
= 0. Applying ∇
_{Z}
to (3.2) and using the first equation of (3.1)[denote by (3.1)
_{1}
], (3.2) and (3.3), we have
Thus
M
is a space of constant curvature −1. Applying ∇
_{U}
to (3.4), we have
(∇_{Ug})(X,Z)Y = (∇_{Ug})(Y,Z)X
.
Taking
Z = Y
= 𝜉 to this and using (2.12)
_{1}
and (2.13), we get
B
= 0. Thus ∇ is a torsion-free metric connection on
M
by (2.13). As
B
= 0, we have
by (2.15). From (2.22), we get
R(X, Y)
𝜉 = −2dτ(X,Y )𝜉. On the other hand, replacing
Z
by 𝜉 to (3.4), we have
R(X, Y)
𝜉 = 0. These two results imply
d
τ = 0. Thus the lightlike transversal connection ∇
^{𝓁}
is flat.
Case 2. Assume that
M
is semi-symmetric, i.e.,
R(X, Y)R
= 0. Applying ∇
_{Z}
to (3.2) and using (3.1)
_{1}
, (3.2) and (3.3), we have
Substituting (3.5) into
(R(U,Z)R)(X, Y)
𝜉 = 0 and using (3.1)
_{1}
, we have
Replacing
U
by 𝛇 to (3.6) and using (
∇_{𝛇}R
)(
X, Y
)𝛇 = 0 due to (3.2) and (3.5), we have (
∇_{Z}R
)(
X, Y
)𝛇 = 0. From this and (3.5), we show that
Thus
M
is a space of constant negative curvature −1. Replacing
U
by 𝜉 to (3.6) and using (2.12)
_{1}
, (3.7) and (∇
_{Z}R
)(
X, Y
)𝛇 = 0, we have
B(Y,Z)X = B(X,Z)Y.
Replacing
Y
by 𝜉 to this and using (2.12)
_{1}
, we get
B
= 0. Thus, by (2.13), ∇ is a torsion-free metric connection on
M
. Using (2.22), (3.7) and the method of Case 1, we see that the lightlike transversal connection is flat. □
T
, the vector field 𝛇 is decomposed as
where
W
is a smooth vector field on
M
and
m = θ
(𝜉) and
n = θ(L)
are smooth functions. Substituting (4.1) in (2.2) and using (2.8) and (2.9), we have
Substituting (4.3) and (4.4) into the following two equations
[X, Y]m = X(Y m) − Y (Xm)
,
[X, Y ]n = X(Y n) − Y (Xn)
,
and using (2.19), (2.20), (2.21), (4.1), (4.3), (4.4), we have respectively
Substituting (4.2) into
R(X, Y)W = ∇_{X}∇_{Y}W − ∇_{Y}∇_{X}W − ∇_{[X, Y]}W
and using (2.19)~(2.21), (4.2)~(4.5) and the fact ∇ is torsion-free, we have
Taking the scalar product with 𝛇 to (4.6) and using (2.1), we show that the structure 1-form
θ
is closed, i.e.,
dθ
= 0 on
TM
.
Assume that
S(TM)
is totally geodesic in
M
. In this case, 𝛇 is not tangent to
M
and
l
=
θ(N)
≠ 0. In fact, if 𝛇 is tangent to M or
l
= 0, then
. Applying
to
and using (2.2) and (2.8), we have
η(X)
= 0 for all
X
∈ 𝚪(
TM
). It is a contradiction as η(𝜉) = 1. Thus 𝛇 is not tangent to
M
and
l
≠ 0. As 𝛇 is not tangent to
M
, we see that (
m, n
) ≠ (0, 0). As
S
(
TM
^{⊥}
) is a parallel distribution, we have
A_{L}
=
ϕ
= 0 due to (2.9). From (2.17) and (2.18), we also have
D
=
ρ
= 0.
Substituting (2.19)~(2.21) into (4.6) and using (4.5), we get
Applying
to
θ(Y)
=
g
(
Y
, 𝛇) and using (2.2) and (2.6), we have
Case 1. Assume
M
is locally symmetric. Applying ∇
_{Z}
to (4.7), we have
R(X, Y)∇_{Z}W = (∇_{Z}θ)(X)Y − (∇_{Z}θ)(Y)X.
Substituting (4.2) and (4.8) in this equation and using (4.7), we obtain
Replacing
Z
by 𝜉 to (4.9) and using (2.12)
_{1}
, we have
R(X, Y)
𝜉 = 0. Comparing the
Rad(TM)
-components of this and (2.22), we have
d
τ = 0. Thus by Theorem 2.1 the lightlike transversal connection is flat. From (2.19), (2.20) and (4.9), we have
Replacing
Y
by 𝜉 to (4.10) and using (2.12)
_{1}
, we get
From (4.9) and (4.11), we show that
R
= 0. From this and (4.7), we have
θ(X)Y = θ(Y )X
.
Replacing
Y
by 𝜉 to this equation and using
X = PX + η(X)𝜉
, we have
mPX
=
g(X,W)
𝜉.
As the left term of this equation belongs to
S(TM)
and the right term belongs to
Rad(TM)
, we have
mPX
= 0 and
g(X,W)
𝜉 = 0 for all
X
∈ 𝚪(
TM
). Thus
m
= 0 and
g(X,W)
= 0 for all
X
∈ 𝚪(
TM
). This imply
W
=
l
𝜉 and
From this and the fact
, we show that
n
^{2}
= 1.
It is known
[6]
that, for any half lightlike submanifold of an indefinite almost contact metric manifold
,
J(Rad(TM))
,
J(ltr(TM))
and
J(S(TM^{⊥}))
are vector subbundles of
S(TM)
of rank 1 respectively. Applying
to
and using (2.1), (2.3), (2.8) and (2.9), we have
Replacing
X
by
J
𝜉 to (4.13) and using (2.1)
_{6}
, we have
n
= 0. It is a contradiction as
n
^{2}
= 1. Thus
S(TM)
is not totally geodesic in
M
.
Case 2. Assume that the transversal connection is flat. We have
d
τ = 0. Substituting (4.1) into (4.6) with
dθ
= 0 and using (2.19)~(2.21) and (4.5), we have
Taking the scalar product with
W
to this and using the facts
θ(X) − m𝜂(X) = g(X,W)
and
, we have
θ(Y)𝜂(X) − θ(X)𝜂(Y)
= 0.
Replacing
Y
by 𝜉 to this equation, we have
g(X,W)
= 0 for all
X
∈ 𝚪(
TM
). This implies
W
=
l
𝜉. Thus 𝛇 is decomposed as
From the fact
and (4.14), we show that 2
lm
= 1 − n
^{2}
. Applying
to (4.14) and using (2.2), (2.8), (2.9) and (2.11), we have
Taking the scalar product with 𝜉,
N
and
L
to this result by turns, we get
respectively. From (2.15) and (4.15), we have
Applying
to
and using (2.1), (2.3), (2.8), (2.9) and the fact
S(TM)
is non-degenerate, we have
Taking the scalar product with
J
𝜉 to this and using (2.1)
_{6}
, we have n(1 −
ml
) = −
lmn
. This implies
n
= 0. As (
m, n
) ≠ (0, 0) and n = 0, we have
m
≠ 0 and 2
lm
= 1. Consequently we get
JL
= 0 by (4.17). It is a contradiction as
Thus
S(TM)
is not totally geodesic in
M
.
Corollary 1.
Let M be a half lightlike submanifold of an indefinite Kenmotsu manifold
.
Then the structure 1-form θ, given by (2.1), is closed on TM
.

locally symmetric
;
semi-symmetric manifold
;
half lightlike submanifolds
;
indefinite Kenmotsu manifold

1. INTRODUCTION

The theory of lightlike submanifolds is an important topic of research in differential geometry due to its application in mathematical physics, especially in the electromagnetic field theory. The study of such notion was initiated by Duggal and Bejancu
[2]
and later studied by many authors (see up-to date results in two books
[4
,
5]
). The class of lightlike submanifolds of codimension 2 is compose of two classes by virtue of the rank of its radical distribution, which are called the
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2. HALF LIGHTLIKE SUBMANIFOLDS

An odd dimensional semi-Riemannian manifold
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3. PROOF OF THEOREM 1.1

Assume that 𝛇 is tangent to
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4. PROOF OF THEOREM 1.2

From the decomposition (2.6) of
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Călin C.
1998
Contributions to geometry of CR-submanifold. Thesis
University of Iasi

Duggal K.L.
,
Bejancu A.
1996
Lightlike Submanifolds of Semi-Riemannian Manifolds and Applications
Kluwer Acad. Publishers
Dordrecht

Duggal K.L.
,
Jin D.H.
1999
Half-Lightlike Submanifolds of Codimension 2
Math. J. Toyama Univ.
22
121 -
161

Duggal K.L.
,
Jin D.H.
2007
Null Curves and Hypersurfaces of Semi-Riemannian Manifolds
World Scientific

Duggal K.L.
,
Sahin B.
2010
Differential geometry of lightlike submanifolds. Frontiers in Mathematics
Birkhäuser

Jin D.H.
2011
Half lightlike submanifolds of an indefinite Sasakian manifold
J. Korean Soc. Math. Edu. Ser. B: Pure Appl. Math.
18
(2)
173 -
183

Jin D.H.
2012
The curvatures of lightlike hypersurfaces in an indefinite Kenmotsu manifold
Balkan J. of Geo. and Its Appl.
17
(1)
49 -
57

Kenmotsu K.
1972
A class of almost contact Riemannian manifolds
Tôhoku Math. J.
21
93 -
103

Okumura M.
1962
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Tôhoku Math. J.
14
(2)
135 -
145
** DOI : 10.2748/tmj/1178244168**

Shankar Gupta R.
,
Sharfuddin A.
2010
Lightlike submanifolds of indefinite Kenmotsu manifold
Int. J. Contemp. Math. Sciences
5
(10)
475 -
496

Citing 'TWO CHARACTERIZATION THEOREMS FOR HALF LIGHTLIKE SUBMANIFOLDS OF AN INDEFINITE KENMOTSU MANIFOLD
'

@article{ SHGHCX_2014_v21n1_1}
,title={TWO CHARACTERIZATION THEOREMS FOR HALF LIGHTLIKE SUBMANIFOLDS OF AN INDEFINITE KENMOTSU MANIFOLD}
,volume={1}
, url={http://dx.doi.org/10.7468/jksmeb.2014.21.1.1}, DOI={10.7468/jksmeb.2014.21.1.1}
, number= {1}
, 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={2014}
, month={Jan}