yari-r emann man foldunun l ghtl ke sotrop k altman foldu lightlike

Yar - Riemann Manifoldunun Lightlike sotropik Altmanifoldu
C.B.Ü. Fen Bilimleri Dergisi
2.2 (2006) 123 128
ISSN 1305-1385
C.B.U. Journal of Science
2.2 (2006) 123 128
YARI-R EMANN MAN FOLDUNUN
L GHTL KE SOTROP K ALTMAN FOLDU
Erol YA AR1
Mu la Üniversitesi, Ula Ali Koçman M.Y.O., 48600 Ula-Mu la, Türkiye.
Özet: Bu makalede, yar -Riemann manifoldunun isotropic altmanifoldu çal ld . Lightlike isotropic
altmanifoldun denklem yap lar verildi. Sonra ise M isotropic altmanifoldu total geodezik yapan artlar elde
edildi.
LIGHTLIKE ISOTROPIC SUBMANIFOLD OF
SEMI-RIEMANNIAN MANIFOLD
Abstract: In this paper, we study isotropic submanifold of semi-Riemannian manifold. We give the
structure equations of lightlike isotropic submanifold. Then we obtain that the condition on M lightlike
isotropic submanifold is totally geodesic.
Key Words and Phrases : Lightlike isotropic submanifolds, Totally geodesic, Gauss and Codazzi equations.
MOS Clasification : 53B15 53B30 53C05
--------------------------------------------------------------------------------------------------------1
Sorumlu Yazar
[email protected]
C.B.Ü. Fen Bil. Dergisi (2006)123 128, 2006 /Erol YA AR
1. Introduction
The theory of submanifolds of a Riemannian
or semi-Riemannian manifold is one of the
most important topics in differential
geometry. It is well known that the primary
difference between theory of lightlike
submanifold
and
semi-Riemannian
submanifold arises due to the fact that in the
first case, a part of the normal vector bundle
TM lies in the tangent bundle TM of the
~
submanifold M of M , whereas in the
second case TM TM ={0}. In 1992,
Duggal and Bejancu [3] introduced a half
lightlike submanifold M, of codimension 2
and found geometric conditions for the
induced connection on M to be metric
connection. In 2004, Erol K l ç and Bayram
ahin [5] studied coisotropic submanifold of
a semi-Riemannian manifold. In this study
they investigated the integrability condition
of the screen distribution and gave a
necessary and sufficient condition on Ricci
tensor of a coisotropic submanifold to be
symmetric.
In the present paper, we have proved that the
connection induced from semi-Riemannian
manifold of codimension (m+n) on lightlike
isotropic submanifold is metric. Besides we
obtain the structure equations of lightlike
isotropic submanifold and proved the
theorem on M in semi-Riemannian manifold
of constant curvature, of codimension (m+n).
2. Preliminaries
Let ( M , g ) be a real (2m+n)-dimensional
semi-Riemannian manifold of constant index
q such as m 1,1 q 2m, and M be an mdimensional submanifold of M . In case g
is degenerate on the tangent bundle TM of M.
We say that M is a lightlike submanifold of
~
M [2]. Throughout this paper we denote the
algebra of smooth function on M by F(M)
and the F(M) module of smooth section of a
vector bundle E over M by
(E). The
following range of induced is used :
124
i, j , k
1,..., r and
, ,
r 1,..., n .
For a degenerate tensor field g on M, there
exists a locally vector field
(TM ) ,
0 such as g ( X , ) 0 for any
X
(TM ) . Then each tangent space
Tx M we have
Tx M
u Tx M
g u, v
0, v Tx M
which is degenerate (m+1)-dimensional
subspace of Tx M . The radical (null)
subspace of Tx M , denoted by RadTx M , is
defined by
Tx M
Tx M
x
g
x
, Xx
0, X x Tx M .
The dimension of RadTx M
depends on
x M. The submanifold M of M is said to be
r-lightlike submanifold if the mapping
RadTM : x M RadTxM
define a smooth distribution on M of rank
r>0, where RadTM is called the radical
(null) distribution on M [2]. In this paper, we
study lightlike isotropic submanifold where
RadTM {0} and 1 r m .
Therefore, S (TM ) {0} . Thus we can write
(2.1)
TM M
TM
tr (TM )
(TM
ltr (TM )
S (TM ).
According to the decomposition (2.1), we
choose the field of frames { 1 ,..., m } and
{N1 ,..., N m , Wm 1 ,..., Wn } on M and M ,
respectively.
Example 2.1 Suppose that ( M , g ) be a
surface of R25 given by equations
x3
cos x1 , x 4
sin x1 , x 5 x 2 .
We choose a set of vectors { 1 , 2 , u1 , u2 }
given by
1
u1
so
TM
2
5
,
sin x1
that
2
1
sin x1
3
cos x1 4 ,
, u2 cos x1 1 4
RadTM TM sp{ 1 , 2 } ,
1
3
sp{ 1 , u1 , u2 }. Therefore M is an
Yar - Riemann Manifoldunun Lightlike sotropik Altmanifoldu
isotropic lightlike submanifold. Construct
two null vectors
N1 12 { 1 5 } and
{ 1 sin x1 3 cos x1 4 }
such as g ( N i , i )
{1, 2} and
ij for i, j
N2
ltr (TM )
Let W
1
2
cos x1
sin x1
3
4
be a spacelike
sp{W } . Thus
i
,W
Tx M
such
that g ( i , u ) 0, g (W ,W ) 0, u Tx M .
Above relation implies that
i
TM and
hence i RadTx M . Thus, locally there
exists a lightlike vector fields on M, it is also
denoted by i such as
g ( i ,Y )
0, X
(TM ), Y
(TM )
Consequently, the m-dimensional radical
distribution RadTM of lightlike isotropic
submanifld M of M is locally spanned by
i . We choose such a non-degenerate
distribution on the screen transversal vector
bundle S (TM ) of M. Thus we have the
following orthogonal direct decomposition
(2.2)
TM = Rad(TM) S(TM ).
Thus we choose W as a unit vector field and
put g (W , W )
where
1.
Theorem 2.2 Let ( M , g , S (TM )) be an
isotropic submanifold of ( M , g ). Suppose
that U be a coordinate neighborhood of M
and { 1 ,..., m } be a basis of (TM u ) . Then
there exist smooth {N1 ,..., N m } of TM M
such that g ( N i ,
j
)
ij
and
g ( N i , N j ) 0, g ( N i ,W ) 0,
for any i, j {1,..., m},
{m 1,..., n} and
W
( S (TM )) . Suppose that be the
Y
X
h s ( X , Y ),
Y
(2.4)
N
s
AN X
X
L
N
X
N,
(2.5)
s
AW X
XW
for any X , Y
(TM ), N
and
W
( S (TM ))
{ X Y , AN X , AW X }
X
{ 1 , 2 , N1 , N 2 , W } is a basis of R25 along M.
From (2.1), there exist
X
X
sp {N1 , N 2 }.
vector such that S (TM )
g( i , X )
~
Levi-Civita connection on M . According to
(2.1) we have
(2.3)
{h s ( X , Y ),
belong
L
W
s
X
to
respectively.
W
(ltr (TM ))
X
where
and
N , s XW , L X N , L XW }
(TM ) and
(tr (TM ))
Here
is
~
connection on M but
the
metric
s
and
are linear
connections on M and tr(TM) respectively.
Besides, we define the F(M)-bilinear
mappings
(2.6)
L
X
: (ltr (TM ))
(ltr (TM ));
L
( LV )
DX L ( LV ),
s
( SV )
DX s ( SV ),
X
(2.7)
s
X
: ( S (TM ))
( S (TM ));
X
(2.8)
D L : (TM )
L
D ( X , SV )
( S (TM ))
D
L
X
(ltr (TM ))
( SV ),
and
(2.9)
D s : (TM )
s
(ltr (TM ))
( S (TM ))
s
D ( X , LV ) D X (LV )
for any x
(TM ), V
(tr (TM )). Since
{ i , N j } are locally lightlike sections on
U
M , we define symmetric F ( M ) -
bilinear form D s and 1-forms pij ,
i
,
and V i on U by
Ds ( X ,Y )
g (h s ( X , Y ),W ),
Pij ( X )
i
L
g(
X
N i , i ),
g ( D s ( X , N i ),W ),
g(
s
X
W ,W )
125
C.B.Ü. Fen Bil. Dergisi (2006)123 128, 2006 /Erol YA AR
and
X ( g (Y , Z )) g (
V i g ( D L ( X ,W ), i )
(TM ). It follows that
for any X , Y
h s ( X , Y ) D s ( X , Y )W ,
L
X
Ni
D ( X , Ni )
s
X
i
W
D s ( X , Z ) g (W , Y )
m 1
W ,
(
Hence (2.3), (2.4) and (2.5) become
(2.10)
n
D s ( X , Y )W ,
g )(Y , Z )
and R the curvature tensor of
and
respectively. Then by straightforward
calculation and using (2.10), (2.11), (2.12),
(2.13), (2.14) and (2.15) we obtain
(2.17)
R( X , Y )Z
(2.11)
{D s ( X , Y ) AW Y
R( X , Y ) Z
m 1
m n
Ni
ANi X
n
Pij ( X ) N j
j 1
i
( X )W ,
m 1
m n
XW
AW X
( X )W
for any X , Y
(TM ). We call D the
screen second fundamental form of M with
respect to tr(TM). Both ANi and AW are
linear operators on (TM ) . We will see by
(2.15) that the first one is RadTM-valued,
called the shape operations of M. Since i
j
D )(Y , Z )W
X
are lightlike vector fields, from
(2.10)-(2.12) we obtain
(2.13)
X
W )
s
D ( X , Z )(
Y
W )
s
Y
D )( X , Z )W }
m
n
{D s (Y , Z )V i ( X ) N i
m 1
s
D s (Y , Z )(
s
(
n
V i ( X ) Ni
i 1
D s (Y , Z ) AW X
(
(2.12)
i 1
D s ( X , Z )V i (Y ) N i },
m 1
(2.18)
m
n
{R s ( X , Y )W
R ( X , Y )W
i 1
m 1
s
D (Y , AW X )W
D s ( X , AW Y )W
V i (Y )
V i (X )
(
i
( X )W
A)(W , X ) (
Y
X
i
(Y )W
A)(W , Y )
V i ( X ) ANi Y V i (Y ) ANi X
s
D ( X , i ) 0,
(
(2.14)
X
D L )(Y , W ) (
Y
D L )( X , W )},
(2.19)
D L ( X , i ) 0,
m
n
{R L ( X , Y ) N i
R( X , Y ) Ni
(2.15)
i 1
g ( ANi X , i )
g ( AW X , i ).
Further, taking in to account that
is a
metric connection and by using (2.10) we
obtain
0 ( X g)(Y, Z)
X(g(Y, Z)) g( XY, Z) g(Y,
X
Z)
i
(Y )V i ( X ) N i
i
(Y ) AW X
,
m 1
( X )V i (Y ) N i
i
(Y ) AW X
i
(
Y
A)( N i , X ) (
(
X
D s )(Y , N i ) (
X
A)( N i , Y )
Y
D s )( X , N i )
D s (Y , ANi X ) D s ( X , ANi Y )
for any X , Y , Z
126
(TM ) . Denote by R
n
m 1
and
X
for any X , Y , Z
D ( X ,W ) V i Ni .
X
Z)
m 1
W
XY
X
n
L
XY
Y , Z ) g (Y ,
D s ( X , Y ) g (W , Z )
Pij ( X ) N j ,
s
X
n
(TM ),
Yar - Riemann Manifoldunun Lightlike sotropik Altmanifoldu
W ,W
Ni
( S (TM ))
and
m
{D s ( X , AW Y )W
(ltr (TM )). Consider the Riemannian
curvature of type (0,4) of
and by using
(2.17)-(2.19) and the definition of curvature
tensor, we derive the following structure
equations :
(2.20)
m
n
i 1
m 1
R( X , Y , Z , Ni )
i
{g ( R ( X , Y ) Z , N i )
(Y ) D s ( X , Z )
i
( X ) D s (Y , Z )},
n
R s ( X , Y )W
i 1
m 1
s
D (Y , AW X )W
V i (X )
V i (Y )
(
(
Y
i
( X )W
X
(
Y
D L )(Y , W )},
(2.25)
m
n
{
i 1
(2.21)
m
n
i1
(Y )V i ( X ) N i
i
{g(( Y A)(W , X)
m1
(
V i (X)g(ANiY, Ni )},
(2.22)
m
n
R( X , Y , Ni , Ni ' )
{g ((
i 1
'
(
'
i
(Y )
i
A)( N i ' , X )
m 1
'
A)( N i , Y ), N i )
Y
X
i
(X )
i
(Y )
( X )}.
Theorem 2.3 Let M be lightlike isotropic
submanifold of an (2m+n)-dimensional semiRiemannian manifold of constant curvature
( M (c), g ), and of codimension (m+n). Then
the curvature tensor of M and M (c) related
to the following equations :
(2.23)
m
n
{D s (Y , Z ) AW X
R( X , Y ) Z
i 1
D s ( X , Z ) AW Y
m 1
D s ( X , Z )V i (Y ) N i
D s (Y , Z )V i ( X ) N i
D s (Y , Z )(
(
s
X
(
XW )
D )(Y , Z )W },
(2.24)
D s ( X , Z )(
Y
W )
s
Y D )( X , Z )W
i
( X )V i (Y ) N i
,
m 1
i
(Y ) AW X
( X ) AW Y
i
s
A)( N i , Y ) (
Y
A)( N i , X ) D ( X , ANi Y )
D (Y , ANi X )} (
Y
D s )( X , N i ) (
X
s
( X A)(W ,Y), Ni ) V i (Y)g(ANi X, Ni )
A)(W , Y )
D L )( X , W )
R L ( X , Y ) Ni
R(X,Y,W , Ni )
(Y )W
A)(W , X ) V i (Y ) ANi X
V i ( X ) ANi Y
(
X
i
X
D s )(Y , N i )
Proof. By using the definition of constant
curvature M (c) and (2.17), (2.18) and (2.19)
we obtain (2.23), (2.24) and (2.25).
Definition 2.4. A lightlike isotropic
submanifold (M,g) of a semi-Riemannian
manifold ( M , g ) is said to be totally
umbilical in M if there is a smooth vector
field H s such as
(2.26)
hs ( X ,Y )
H s g ( X , Y ), X , Y
(TM ).
Corollary
2.5.
Lightlike
isotropic
submanifold of ( M , g ) is totally geodesic if
M is totally umbilical , i.e.,
h s ( X , Y ) 0, X , Y
(TM ).
Then we have
Theorem 2.6. Let ( M , g ) be a isotropic
submanifold of ( M , g )
of codimension
(m+n) if M is totally umbilical in M then
(2.27)
R( X , Y )Z
R( X , Y ) Z , X , Y , Z
(TM ).
Proof. By using (2.17) and Corollary 2.3 we
get (2.27).
Corollary 2.7. Under the hypothesis of
theorem we have
127
C.B.Ü. Fen Bil. Dergisi (2006)123 128, 2006 /Erol YA AR
(2.28)
R ( X , Y )W
s
R ( X , Y )W
(
X
(
Y
A)(W , X )
A)(W , Y ),
(2.29)
R( X , Y ) Ni
(
Y
A)( N i , X ) (
X
A)( N i , Y ).
References
[1] Bejancu, A., Null Hypersurfaces in SemiEuclidean Space, Saitama Math. J. Vol. 14, 25-40,
(1996).
[2] Duggal, K., and Bejancu, A., Lightlike
Submanifold of Semi-Riemannian Manifold and
Applications, Kluver Academic Pub., (1996).
Geli Tarihi: 13/10/2005
128
[3] Duggal, K. and Jin, D.H., Half Lightlike
Submanifolds of Codimension 2, Math. J.
Toyama Uni., 22, 121-161, (1999)
[4] Duggal, K.L. and Bejancu A., Lightlike
Submanifold of Semi-Riemannian Manifold
Applications, Acta Appl. Math 38, 197-215,
(1995).
[5] K l ç, E., ahin, B., H.B. Karada , Güne , R.,
Coisotropic Submanifold of a Semi-Riemannian
Manifold, Turk J. Math., 28, 335-352, (2004).
[6] Katsuno, K., Null Hypersurfaces in Lorentzian
Manifolds I, Math. Proc. Camb. Phil. Soc. 88,
175-182, (1980).
[7] O Neill, B., Semi-Riemannian Geometry with
Applications to Relativity, Academic Press,
London, (1983).
Kabul Tarihi: 30/01/2006