In this paper, we study screen quasiconformal irrotational half lightlike submanifolds
M
of a semiRiemannian space form
(
c
) admitting a semisymmetric nonmetric connection, whose structure vector field
ζ
is tangent to
M
. The main result is a classification theorem for such Einstein half lightlike submanifolds of a Lorentzian space form admitting a semisymmetric nonmetric connection.
1. INTRODUCTION
The theory of lightlike submanifolds is indeed important for both the geometry of submanifolds to mathematics and its applications to physics. The study of such notion was initiated by Duggal and Bejancu
[3]
and later studied by many authors (see upto date in
[4
,
5]
). The notion of a semisymmetric nonmetric connection on a Riemannian manifold was introduced by Ageshe and Cha°e
[1]
. Although now we have lightlike version of a large variety of Riemannian submanifolds, the geometry of lightlike submanifolds of semiRiemannian manifolds admitting semisymmetric nonmetric connections has been few known. Recently Yasar, Cöken and Yücesan
[15]
and Jin
[6
,
7]
studied lightlike hypersurfaces in a semiRiemannian manifold admitting a semisymmetric nonmetric connection. Jin
[10]
and JinLee
[11]
studied general lightlike submanifolds and half lightlike submanifolds of a semiRiemannian manifold with a semisymmetric nonmetric connection.
The objective of this paper is to study screen quasiconformal irrotational half lightlike submanifolds
M
of a semiRiemannian space form
(
c
) admitting a semisymmetric nonmetric connection, whose structure vector field
ζ
of
(
c
) is tangent to
M
but it does not belongs to
S
(
TM
). The reason for this geometric restriction on
M
is due to the fact that such a class admits an integrable screen distribution and a symmetric induced Ricci tensor of
M
. Our main result is a classification theorem for such Einstein half lightlike submanifolds
M
of a Lorentzian space form admitting a semisymmetric nonmetric connection.
2. SEMISYMMETRIC NONMETRIC CONNECTION
Let (
,
) be a semiRiemannian manifold. A connection
on
is called a
semisymmetric nonmetric connection
[1]
if
and its torsion tensor
satisfy
for any vector fields
X
,
Y
and
Z
on
, where 𝜋 is a 1form associated with a nonvanishing vector field
ζ
, which is called the
structure vector field
, by
𝜋(X ) = (X, ζ ).
A submanifold (
M
,
g
) of codimension 2 is called
half lightlike submanifold
if the radical distribution
Rad
(
TM
) =
TM
∩
TM
^{⊥}
is a vector subbundle of the tangent bundle
TM
and the normal bundle
TM
^{⊥}
of
M
, with rank 1. In this case, there exists complementary nondegenerate distributions
S
(
TM
) and
S
(
TM
^{⊥}
) of
Rad
(
TM
) in
TM
and
TM
^{⊥}
respectively, which are called the
screen
and
coscreen distributions
on
M
respectively, 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
, by 𝚪(
E
) the
F
(
M
) module of smooth sections of a vector bundle
E
over
M
and by (2.3)
_{i}
the
i
th equation of (2.3). We use same notations for any others. Choose
L
∈ 𝚪(
S
(
TM
^{⊥}
)) as a spacelike unit vector field, without loss of generality, i.e.,
(
L
,
L
) = 1. We call
L
the
canonical normal vector field
of
M
. Consider the orthogonal complementary vector bundle
S
(
TM
)
^{⊥}
to
S
(
TM
) in
T
. Certainly
Rad
(
TM
) and
S
(
TM
^{⊥}
) are vector subbundles of
S
(
TM
)
^{⊥}
. As
S
(
TM
^{⊥}
) is nondegenerate, we have
S(TM)^{⊥} = S(TM^{⊥}) ⊕_{orth} S(TM^{⊥})^{⊥},
where
S
(
TM
^{⊥}
)
^{⊥}
is the orthogonal complementary to
S
(
TM
^{⊥}
in
S
(
TM
)
^{⊥}
. It is wellknown
[3]
that, for any null section
ξ
of
Rad
(
TM
) on a coordinate neighborhood 𝒰 ⊂
M
, there exists a uniquely defined lightlike vector bundle
ltr
(
TM
) and a null vector field
N
of
ltr
(
TM
) on 𝒰 satisfying
(ξ,N) = 1, (N,N) = (N,X) = (N,L) = 0, ∀X ∈ 𝚪(S(TM)).
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
[11]
. Then
T
is decomposed as
In the entire discussion of this article, we shall assume that the structure vector field
ζ
of
to be spacelike unit tangent vector field of
M
. In the sequel, we take
X, Y, Z, W
∈ 𝚪(
TM
), unless otherwise specified. Let
P
be the projection morphism of
TM
on
S
(
TM
) with respect to the decomposition (2.3)
_{1}
. Then the local Gauss and Weingartan 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
, which are called the
shape operators
, and 𝜏, 𝜌 and 𝜙 are 1forms on
TM
. We say that
h
(
X, Y
) =
B
(
X, Y
)
N
+
D
(
X, Y
)
L
is the
global second fundamental form tensor
of
M
. Using (2.1), (2.2) and (2.5), we have
and
B
and
D
are symmetric on
TM
, where
T
is the torsion tensor with respect to the induced connection 𝛁 and 𝜂 is a 1form on
TM
such that
𝜂(X) = (X, N )
From the facts
B
(
X, Y
) =
(
_{X}
Y
,
ξ
) and
D
(
X, Y
) =
(
_{X}
Y
,
L
), we know that
B
and
D
are independent of the choice of
S
(
TM
) and satisfy
The above three local second fundamental forms
M
and
S
(
TM
) are related to their shape operators by
where
f
is the smooth function given by
f
= 𝜋(
N
). From (2.12) and (2.13), we show that
is
S
(
TM
)valued selfadjoint and satisfies
Denote by
,
R
and
R
* the curvature tensors of the semisymmetric nonmetric connection
of
, the induced connection 𝛁 on
M
and the induced connection 𝛁* on
S
(
TM
). Using the GaussWeingarten formulas for
M
and
S
(
TM
), we obtain the GaussCodazzi equations for
M
and
S
(
TM
) :
A complete simply connected semiRiemannian manifold
of constant curvature
c
is called a
semiRiemannian space form
and denote it by
(
c
). For any
X, Y, Z
∈ 𝚪(
T
), the curvature tensor
of
(
c
) is given by
Taking the scalar product with
ξ
and
L
to (2.22), we get
From this results and (2.17), for all
X, Y, Z
∈ 𝚪(
TM
), we obtain
3. CHARACTERIZATION THEOREMS
Definition.
A half lightlike submanifold
M
of a semiRiemannian manifold
is said to be
irrotational
[12]
if
_{X}
ξ
∈ 𝚪(
TM
) for any
X
∈ 𝚪(
TM
).
From (2.5) and (2.12), we show that the above definition is equivalent to the condition:
D
(
X
,
ξ
) = 0 = 𝜙(
X
) for all
X
∈ 𝚪(
TM
).
Lemma 1
(
[8
,
11]
).
Let M be an irrotational half lightlike submanifold of a semiRiemannian manifold
admitting a semisymmetric nonmetric connection such that the structure vector field
ζ
of
is tangent to M. Then
ζ
is conjugate to any vector field X on M, i.e.,
ζ
satisfies h
(
X
,
ζ
) = 0.
Note that
h
(
X
,
ζ
) = 0 is equivalent to the following two equations:
Definition.
A half lightlike submanifold
M
of a semiRiemannian manifold
admitting a semisymmetric nonmetric connection is called
screen quasiconformal
[9
,
13]
if the second fundamental forms
B
and
C
satisfy
where 𝜑 is a nonvanishing function on a coordinate neighborhood 𝒰 in
M
.
Due to (2.13) and (2.15), we show that
M
is screen quasiconformal if and only if the shape operators
A_{N}
and
are related by
We quote the following results for irrotational screen quasiconformal half lightlike submanifold due to Jin
[9]
:
Theorem 3.1.
Let M be an irrotational screen quasiconformal half lightlike submanifolds M of a semiRiemannian space form
(
c
)
admitting a semisymmetric nonmetric connection. If the structure vector field
ζ
is tangent to M but it does not belong to S
(
TM
),
then we have c
= 1.
Let
be the Ricci curvature tensor of
and
R
^{(0, 2)}
the induced Ricci type tensor on
M
given respectively by
(X, Y) = trace{Z → (Z,X)Y}, ∀X, Y ∈ 𝚪(T),
R(0, 2)(X, Y) = trace{Z → R(Z,X)Y}, ∀X, Y ∈ 𝚪(TM).
Consider a quasiorthonormal frame field {
ξ
,
W_{a}
} on
M
, where
Rad
(
TM
) =
Span
{
ξ
} and
S
(
TM
) =
Span
{
W_{a}
} and let
E
= {
ξ
,
N
,
W_{a}
} be the corresponding frame field on
. Using this quasiorthonormal frame field, we obtain
R^{(0, 2)}(X, Y) = (X, Y) + B(X, Y)tr A_{N} + D(X, Y)trA_{L}  g(A_{N}X,Y)  g(A_{L}X, A_{L}Y) + 𝜌(X)𝜙(Y)  ((ξ, Y)X, N)  ((L, X)Y, L),
This shows that
R
^{(0, 2)}
is not symmetric. The tensor field
R
^{(0, 2)}
is called its
induced Ricci tensor
[4
,
5]
, denoted by
Ric
, of
M
if it is symmetric. It is known
[11]
that
R
^{(0, 2)}
is an induced Ricci tensor of
M
if and only if the 1form 𝜏 is closed, i.e.,
d
𝜏 = 0, for any coordinate neighborhood 𝒰 ⊂
M
.
Remark 1.
If
R
^{(0, 2)}
is symmetric, then there exists a null pair {
ξ
,
N
} such that the corresponding 1form 𝜏 satisfies 𝜏 = 0
[11]
, which called a
canonical null pair
of
M
. Although
S
(
TM
) is not unique, it is canonically isomorphic to the factor vector bundle
S
(
TM
)
^{#}
=
TM
/
Rad
(
TM
)
[12]
. This implies that all screen distribution are mutually isomorphic. For this reason, in case
d
𝜏 = 0 we consider only lightlike hypersurfaces
M
endow with the canonical null pair.
We say that
M
is an
Einstein manifold
if the Ricci tensor of
M
satisfies
It is wellknown that if dim
M
> 2, then
κ
is a constant. For dim
M
= 2, any manifold
M
is Einstein but
κ
is not necessarily constant.
In case the ambient manifold
is a space form
(
c
),
R
^{(0, 2)}
is given by
Taking the scalar product with
ξ
to (2.17) and using (2.22), we have
Definition.
A vector field
X
on
is said to be
conformal Killing
[8]
if
_{x} = 2𝛿
for any nonvanishing smooth function 𝛿, where
denotes the Lie derivative on
, that is, for all
Y, Z
∈ 𝚪(
T
),
(_{x})(Y,Z) = X((Y,Z))  ([X,Y],Z)  (Y,[X,Z]).
In particular, if 𝛿 = 0, then
X
is called a
Killing vector field
on
.
Theorem 3.2
(
[8
,
11]
).
Let M be a half lightlike submanifold of
admitting a semisymmetric nonmetric connection. If the canonical normal vector field L is conformal Killing, then L is a Killing vector field.
Proof
. Using (2.1) and (2.2), for any
X, Y, Z
∈ 𝚪(
T
), we have
(_{x})(Y,Z) = (_{Y}X,Z) + (Y, _{Z}X)  2𝜋(X)(Y,Z).
As
L
is conformal Killing, we have
(
_{X}L, Y
) = 
D
(
X, Y
) by (2.9) and (2.16). This implies (
_{L}
)(
X, Y
) = 2
D
(
X, Y
) for any
X, Y
∈ 𝚪(
TM
). Thus we have
D(X, Y) = 𝛿g(X, Y ), ∀X, Y ∈ 𝚪(TM).
Taking
X
=
Y
=
ζ
to this and using (3.1)
_{2}
, we get 𝛿 = 0 and
L
is Killing. □
Theorem 3.3
(
[11]
).
Let M be a half lightlike submanifold of a semiRiemannian manifold
admitting a semisymmetric metric connection. Then the following assertions are equivalent :

(1)The screen distribution S(TM)is an integrable distribution.

(2)C is symmetric, i.e., C(X, Y) =C(Y, X)for all X, Y∈ 𝚪(S(TM)).

(3)The shape operator ANis selfadjoint with respect to g, i.e.,
g(A_{N}X, Y ) = g(X,A_{N} Y ), ∀X, Y ∈ 𝚪(S(TM )).
Remark 2.
Just as in the wellknown case of locally product Riemannian or semiRiemannian manifolds
[3
,
4
,
5
,
14]
, if
S
(
TM
) is an integrable distribution, then
M
is locally a product manifold
C
×
M
* where
C
is a null curve tangent to
Rad
(
TM
) and
M
* is a leaf of the integrable distribution
S
(
TM
).
Theorem 3.4.
Let M be a screen quasiconformal irrotational Einstein half lightlike submanifold of a Lorentzian space form
(
c
)
with a semisymmetric nonmetric connection
.
If
ζ
is tangent to M but it does not belong to S
(
TM
),
the canonical normal vector field is conformal Killing and the mean curvature of M is constant, then M is locally a product manifold M
=
C
×
M
_{1}
×
M
_{2}
,
where C is a null curve,
M
_{1}
is an Euclidean space and M
_{2}
is a totally umbilical Riemannian space
.
Proof
. As
L
is Killing, we get
D
= 𝜙 = 0 and
g
(
A_{L}X, Y
) = 0 for any
X, Y
∈ 𝚪(
TM
). From (3.3), (3.5) and the fact
is selfadjoint, we show that
R
^{(0, 2)}
is a symmetric induced Ricci tensor
Ric
and
S
(
TM
) is an integrable distribution. As
g
(
ζ
,
X
) =
B
(
ζ
,
X
) = 0 and
S
(
TM
) is nondegenerate, we have
Using (2.13), (3.3), (3.4) and the fact
c
= 1, from (3.5) we have
for all
X, Y
∈ 𝚪(
TM
) due to
c
= 1, where 𝛼 =
tr

fm
𝜑
^{1}
. Taking
X
=
Y
=
ζ
to (3.8) and using (3.7), we have 𝜅 =
m
. Thus (3.8) becomes
As
is Lorentzian manifold,
S
(
TM
) is a Riemannian. Since
ξ
is an eigenvector field of
corresponding to the eigenvalue 0 due to (2.16) and
is
S
(
TM
)valued real selfadjoint operator,
have
m
real orthonormal eigenvector fields in
S
(
TM
) and is diagonalizable. Consider a frame field of eigenvectors {
ξ
,
E
_{1}
, . . . ,
E_{m}
} of
such that {
E
_{1}
, . . . ,
E_{m}
} is an orthonormal frame field of
S
(
TM
) and
E_{i}
= 𝜆
_{i}E_{i}
. Put
X
=
Y
=
E_{i}
in (3.9), each eigenvalue 𝜆
_{i}
is a solution of the equation
x^{2}  𝛼x = 0.
As this equation has at most two distinct solutions 0 and 𝛼, there exists
p
∈ {0, 1, . . . ,
m
} such that 𝜆
_{1}
= . . . = 𝜆
_{p}
= 0 and 𝜆
_{p+1}
= . . . = 𝜆
_{m}
= 𝛼(≠ 0), by renumbering if necessary. As
tr
= 0
p
+ (
m

p
)𝛼, we have
(m  p  1)𝛼 = fm𝜑^{1}.
Consider four distributions
D
_{o}
,
D
_{𝛼}
,
and
on
S
(
TM
) given by

Do= {X∈ 𝚪(TM) X= 0},=Do∩S(TM),

D𝛼= {U∈ 𝚪(TM) U= 𝛼PU},=D𝛼∩S(TM).
Clearly we show that
D_{o}
∩
D
_{𝛼}
=
Rad
(
TM
),
∩
= {0} as 𝛼 ≠ 0 and
=
PD_{o}
,
=
D
_{𝛼}
In the sequel, we take the vector fields
X, Y
∈ 𝚪(
D_{o}
),
U, V
∈ 𝚪(D𝛼) and
Z, W
∈ 𝚪(
TM
). Denote
X
*=
PX, Y
* =
PY, U
* =
PU
and V *=
PV
. Then
X
*,
Y
* ∈ 𝚪(
) and
U
*,
V
* ∈ 𝚪(
). Since
X
* and
U
* are eigenvector fields of the real selfadjoint operator
corresponding to the different eigenvalues 0 and 𝛼 respectively,
X
* ⊥
U
* and
g
(
X,U
) =
g
(
X
*,
U
*) = 0, that is,
D_{o}
⊥
_{g}
D
_{𝛼}
. Also, since
B
(
X,U
) =
g
(
X,U
) = 0, we show that
D
_{𝛼}
⊥
_{B}
D_{o}
. Since {
E_{i}
}
_{1≤i≤p}
and {
E_{a}
}
_{p+1≤a≤m}
are vector fields of
and
respectively and
and
are mutually orthogonal,
and
are nondegenerate distributions of rank
p
and rank (
m

p
) respectively. Thus
S
(
TM
) is decomposed as
S
(
TM
) =
⊕
_{orth}
.
From (3.9), we get
(
 𝛼
P
) = 0. Let
W
∈
Im
. Then there exists
Z
∈ 𝚪(
TM
) such that
W
=
Z
. Then (
 𝛼
P
)
W
= 0 and
W
∈ 𝚪(
D
_{𝛼}
). Thus
Im
⊂ 𝚪(
D
_{𝛼}
). By duality, we have
Im
(
 𝛼
P
) ⊂ 𝚪(
D_{o}
).
Applying 𝛁
_{X}
to
B
(
Y,U
) = 0 and using (2.13) and
Y
= 0, we obtain
(𝛁_{X}B )(Y,U ) = g(𝛁_{X}Y,U ).
Substituting this into (3.6) and using (2.11) and
X
=
Y
= 0, we get
g([X, Y], U) = 0.
As
Im
⊂ 𝚪(
D
_{𝛼}
) and
D
_{𝛼}
is nondegenerate, we get
[
X, Y
] = 0. Thus [
X, Y
] ∈ 𝚪(
D_{o}
) and
D_{o}
is integrable. This result implies [
X
*,
Y
*] ∈ 𝚪(
D_{o}
). On the other hand, since
S
(
TM
) is integrable, [
X
*,
Y
*] ∈ 𝚪(
S
(
TM
)). Thus [
X
*,
Y
*] ∈ 𝚪(
). Thus
is also an integrable distribution.
Applying 𝛁
_{V}
to
B
(
U, Y
) = 0 and using
Y
= 0 and
U
= 𝛼
PU
, we get
(𝛁_{V} B)(U, Y ) = 𝛼g(𝛁_{V} Y,U ).
Substituting this into (3.6) and using the fact 𝛼 ≠ 0, we obtain
g(𝛁_{V} Y,U ) = g(V, 𝛁_{U}Y ).
Applying 𝛁
_{V}
to
g
(
Y,U
) = 0 and using (2.10), we have
𝜋(Y )g(U, V )  B(V,U )η(Y )  g(𝛁_{V} Y,U ) = g(Y,𝛁_{V} U ).
Taking the skewsymmetric part of this equation and using (2.11), we have
g([V,U], Y) = 0, ∀ Y ∈ 𝚪(D_{o}) and U, V ∈ 𝚪(D_{𝛼}).
From this, we get
g
([
V
*,
U
*],
Y
*) = 0 for all
Y
* ∈ 𝚪(
) and
U
*,
V
* ∈ 𝚪(
). As
and
are mutually orthogonal nondegenerate distributions, we show that [
V
*,
U
*] ∈ 𝚪(
). Thus
is also an integrable distribution.
Applying 𝛁
_{U}
to
B
(
X, Y
) = 0 and 𝛁
_{X}
to
B
(
U, Y
) = 0, we have
(𝛁_{U}B )(X, Y ) = 0, (𝛁_{X}B )(U, Y ) = 𝛼g(𝛁_{X}Y,U ).
Substituting this two equations into (3.6), we have 𝛼
g
(𝛁
_{X}Y, U
) = 0. As
g(𝛁_{X}Y, U ) = B(𝛁_{X}Y, U ) = 𝛼g(𝛁_{X}Y, U ) = 0
and
Im
⊂ 𝚪(
D
_{𝛼}
) and
D
_{𝛼}
is nondegenerate, we get
𝛁
_{X}Y
= 0. This implies 𝛁
_{X}Y
∈ 𝚪(
D_{o}
). Thus
D_{o}
is an autoparallel distribution on
S
(
TM
). This implies that 𝛁
_{X *}
Y
* ∈ 𝚪(
D_{o}
) for any
X
*,
Y
* ∈ 𝚪(
). As
C
(
X
*,
Y
*) = 𝜑
B
(
X
*,
Y
*) +
η
(
X
*)𝜋(
Y
*) = 0, we have 𝛁
_{X *}
Y
* = 𝛁*
_{X *}
Y
* ∈ 𝚪(
S
(
TM
)). Thus 𝛁
_{X *}
Y
* ∈ 𝚪(
) and
is also an autoparallel distribution.
As
ζ
= 0,
ζ
belongs to
D_{o}
. Thus 𝜋(
U
) = 0 for any
U
∈ 𝚪(
D
_{𝛼}
). Applying 𝛁
_{X}
to
g
(
U, Y
) = 0 and using (2.10) and the fact
D_{o}
is autoparallel, we get
g
(𝛁
_{X}U, Y
) = 0. This implies 𝛁
_{X}U
∈ 𝚪(
D
_{𝛼}
).
Assume that the mean curvature vector field
of
M
is constant. Then 𝛼 is a constant. Applying 𝛁
_{X}
to
B
(
U, V
) = 𝛼
g
(
U, V
) and 𝛁
_{U}
to
B
(
X, V
) = 0 and using the fact 𝛼 is constant, we have
(𝛁_{X}B )(U, V ) = 0, (𝛁_{U}B )(X, V ) = 𝛼g(𝛁_{U}X, V ).
Substituting this two equations into (3.6) and using
D_{o}
⊥
_{B}
D
_{𝛼}
, we have
g(𝛁_{U}X, V ) = 𝜋(X )g(U, V ).
Applying 𝛁
_{U}
to
g
(
X, V
) = 0 and using (2.10), we obtain
g(X,𝛁_{U}V ) = 0.
From this, we get
g
(
X
*, 𝛁
_{U *}
V
*) = 0 for all
X
* ∈ 𝚪(
) and
U
*,
V
* ∈ 𝚪(
). As
and
are mutually orthogonal nondegenerate distributions, we show that 𝛁
_{U *}
V
* ∈ 𝚪(
). Thus
is autoparallel distribution.
Since the leaf
M
* of
S
(
TM
) is a Riemannian manifold and
S
(
TM
) =
⊕
_{orth}
, where
and
are autoparallel distributions of
M
*, by the decomposition theorem of de Rham
[2]
we have
M
* =
M
_{1}
×
M
_{2}
, where
M
_{1}
is a totally geodesic leaf of
and
M
_{2}
is a totally umbilical leaf of
. Consider the frame field of eigenvectors {
ξ
,
E
_{1}
, . . . ,
E_{m}
} of
such that {
E_{i}
}
_{i}
is an orthonormal frame field of
S
(
TM
), then
B
(
E_{i}, E_{j}
) =
C
(
E_{i}, E_{j}
) = 0 for 1≤
i
<
j
≤
m
and
B
(
E_{i}, E_{i}
) =
C
(
E_{i}, E_{i}
) = 0 for 1 ≤
i
≤
m
 1. From (2.17) and (2.20), we have
(
(
E_{i}, E_{j}
)
E_{j} , E_{i}
) =
g
(
R
*(
E_{i}, E_{j}
)
E_{j}, E_{i}
) = 0. Thus the sectional curvature
K
of
M
_{2}
is given by
Thus
M
is a locally product
C
×
M
_{1}
×
M
_{2}
, where
C
is a null curve,
M
_{1}
is an Euclidean space and
M
_{2}
is a totally umbilical Riemannian space. □
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Chafle M.R.
1992
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