Advances in Pure Mathematics, 2011, 1, 378-383
doi:10.4236/apm.2011.16067 Published Online November 2011 (http://www.SciRP.org/journal/apm)
Geodesic Lightlike Submanifolds of Indefinite
Sasakian Manifolds*
Junhong Dong1, Ximin Liu2
1Department of Mat hem at i cs, South China University of Technology, Guangzhou, China
2School of Mathematical Sciences, Dalian University of Technology, Dalian, Chi na
Received July 23, 201 1; revised August 15, 2011; accepted August 25, 2011
Abstract
In this paper, we study geodesic contact CR-lightlike submanifolds and geodesic screen CR-lightlike (SCR)
submanifolds of indefinite Sasakian manifolds. Some necessary and sufficient conditions for totally geodesic,
mixed geodesic,
D
-geodesic and -geodesic contact CR-lightlike submanifolds and SCR submanifolds
are obtained. D
Keywords: CR-Lightlike Submanifolds, Sasakian Manifolds, Totally Geodesic Submanifolds
1. Introduction
A submanifold
M
of a se mi -R ie ma n ni an ma ni f ol d
M
is called lightlike submanifold if the induced metric on
M
is degenerate. The general theory of a lightlike sub-
manifold has been developed by Kupeli [1] and Bejancu-
Duggal [2].
The geometry of CR-lightlike submanifolds of inde-
finite Kaehler manifolds was studied by Guggal and
Bejancu [2]. The geodesic CR-lightlike submanifolds in
indefinite Kaehler manifolds were studied by Sahin and
Günes [3,4].
Lightlike submanifold of indefinite Sasakian mani-
folds can be defined according to the behavior of the
almost contact structure, and contact CR-lightlike sub-
manifolds and screen CR-lightlike (SCR) submanifolds
of indefinite Sasakian manifolds were studied by Duggal
and Sahin in [5]. The study of the geometry of subma-
nifolds of indefinite Sasakian manifolds has been de-
veloped by [6] and others.
In this paper, geodesic contact CR-lightlike submani-
folds and geodesic screen CR-lightlike (SCR) submani-
folds of indefinite Sasakian manifolds are considered.
Some necessary and sufficient conditions for totally
geodesic, mixed geodesic, D-geodesic and D

-geo-
desic contact CR-lightlike submanifolds and SCR sub-
manifolds are obtained.
2. Preliminaries
A submanifold m
M
immersed in a semi-Riemannian
manifold ()
mn
,
M
g
 is called a lightlike submanifold if
it admits a degenerate metric
g
induced from
g
whose radical distribution is of rank where RadTM r
1rm

, , where =RadTMTMTM 



=| ,=0,
xx
xM
TMuTMguvvTM



.
Let


STM
TM
be a screen distribution which is semi-
Riemannian complementary distribution of in
, i.e. RadTM
TM =(RadTMS TM

. As is a
nondegenerate vector subbundle of

STM
|
M
TM , we put
 
.MS TM|=S T
M
TM


We consider a nondegenerate vector subbundle of


STM
RadTM, which is a complementary vector bundle of
in TM

. Since, for any local basis {}
i

of
, there exists a local frame of sections
with value in the orthogonal complement of
RadTM {}
i
N


STM

such that


=,
ii
gN ij

 and

,N

=0
ij , there
exists a lightlike, transversal vector bundle
gN


ltrTM
locally spanned by . Let be the comple-
mentary (but not orthogonal) vector bundle to TM in
{}
i
N

trTM

|
M
TM .
Then





=,trTMltrTMSTM  




=.TMSTMRadTMltrTMS TM  

*This work is supporte d by NSFC (10931005).

J. H. DONG ET AL.379
Now, let  be the levi-Civita connection on
M
, we
have
  

,= ,,
,, ,
X, X
X
gYZgYZ gYZ
XYZ TM

 (2.1)
 
=,, ,
XX
YYhXYXYTM , (2.2)

 
=,
,
t
XVX
VAXVX TM
VtrTM


,
(2.3)
where and

,
XV
YAX



,,
t
X
hXY V

belong to


TM and ,
 
TMtr
respectively. Using the projectors
and
 
:ltrTM STM
 

:str TMltr TM

,
from [1], we have



=,,,,
ls
XX
YYh XYhXYXYTM ,
(2.4)



 
=,,,
ls
XNX
NAXNDXNN ltrTM
(2.5)


 
=,,
sl
XWX
WAX WDXWW STM

 .


,,
,
X
(2.6)
Denote the projection of TM to by ,
we have the decomposition

STM P

=
XX
PYPYhX PY

 (2.7)
=
t
X
AX




 (2.8)
for any








,, ,X YTMRadTMNltrTM   .
From the above equations we have




,,= ,,
l
g
hXY gAXY


 (2.9)




,,= ,
N,
g
hXPYN gAXPY
 (2.10)


,,=0, =0.
(2.11)
l
gh XA

 

Definition 2.1 A (2n + 1)-dimensional Semi-Rieman-
nian manifold


,
M
g is called a contact metric mani-
fold if there is a tensor field (1,1)

, a vector field ,
called the characteristic vector field, and its dual 1-form
V

such that

,= ,,,=gXY gXYXYgVV,

 
(2.12)
 


2=,,=

,
X
XXVgXV X    (2.13)
 
,=,, ,dXY gXYXYTM  

, (2.14)
where =1

.
From the above definiton, it follows that

=0,=0,=1.V  V (2.15)
The (, ,, )Vg

is called a contact metric structure of
M
. If =0Nd V



, we say that
M
has a normal
contact structure, where N

is the Nijenhuis tensor
field of

. A normal contact metric manifold is called a
Sasakian m a ni fo ld for which we have
=
XVX.

 (2.16)




=,
XYgXYV YX
 .  (2.17)
Let





,, ,MgSTMSTM
 be a lightlike
submanifold of


,
M
g. For any vector field
X
tangent
to
M
, we put
=,
X
PX QX

 (2.18)
where and are the tangential and the
transversal parts of
PX QX
X

, respectively.
Let’s suppose is a spacelike vector field so that V
=1

, it’s similar when is a timelike vector field. V
3. Geodesic Invariant L ightlike
Submanifolds
Definition 3.1 Let




,, ,MgSTM STM
 be a
lightlike submanifold, tangent to the structure vector fi eld
,V


VSTM, immersed in an indefinite Sasakian
manifold


,
M
g, we say that
M
is an invariant subma -
nifolds of
M
if the following conditions are satisfied

  
=, =RadTMRadTMS TMSTM  . (3.1)
From (2.16), (2.17), (2.18) and (2.4) we hav e




,=,=0, = =
ls XX
hXVhXVVVPX, (3.2)



 


,= ,= ,,,
l
hXYhXYh XYXYTM    .
(3.3)
From (3.1) and (2.12) we have





=, =ltr TMltr TMSTMSTM 


.

(3.4)
Theorem 3.1 Let




,, ,MgSTM STM
 be an
invariant lightlike submanifold of an indefinite Sasakian
manifold
M
, then
M
is totally geodesic if and only if
and
l
h
s
h of
M
are parallel.
Proof. Suppose is parallel, for any
l
h


,,
X
YZ TM , we have


  

,= ,,
,=0.
lll
XX
lX
hYVhYV hYV
hY V


X
By (3.2), we h a v e

 
,= ,=0
ll
X
hYV hYV,

J. H. DONG ET AL. 380
so

,=
lX
hY Y00.. That is to say

,=
l
hYPX

,=0.hYPX
In a similar way, we can get Thus,
s
M
is totally geodesic.
Conversely, if since

,= ,=0
ls
hXY hXY,

  

,= ,,
,=0,
l
XX X
lX
hYZhYZ hYZ
hY Z


ll

  

,= ,,
,=0,
sss
XX
sX
hYZhYZhYZ
hY Z


X
so and
l
h
s
h are parallel, which completes the proof.
4. Geodesic Contact CR-Lightlike
Submanifolds
Definition 4.1 Let be a
lightlike submanifold, tangent to the structure vector
field , immersed in an indefinite Sasakian manifold

 
,, ,MgSTM STM


V

,

M
g. We say that
M
is a contact CR-lightlike sub-
manifold of
M
if the following conditions are satisfied
[(A)] is a distribution on RadTM
M
such that
(RadTadTM )={0}.MR


0
D [(B)] There exist vector bundles and over D
M
such that
  
0
=,S TMRadTMDDV 
 
00 12
=, =DDDLL,



where 0 is non-degenerate and , 2 is
a vector subbundle of ST . So we have the
decomposition
D

1=Lltr TM

L

M

 
0
=,=TMDDV DRadTMRadTMD 
  .
If we denote , then we have
ˆ=
DDV
ˆˆ
=,=
TMDDD D 
 ˆ
.
Definition 4.2 A contact CR-lightlike submanifold of
an indefinite Sasakian manifold is called -geodesic
contact CR-lightlike submanifold if its second funda-
mental form satisfied
ˆ
D
h


,=0hXY , for any

ˆ
,

X
YD .
Definition 4.3 A contact CR-lightlike submanifold of
an indefinite Sasakian manifold is called mixed geodesic
contact CR-lightlike submanifold if its second funda-
mental form satisfied for any h

,=0hXZ,

ˆ
X
D and

Z
D  .
Definition 4.4 A contact CR-lightlike submanifold of
an indefinite Sasakian manifold is called -geodesic
contact CR-lightlike submanifold if its second funda-
mental form satisfied for any
D
h

,=0hZU ,

,
Z
UD

 .
Theorem 4.1 Let
M
be a contact CR-lightlike
submanifold of an indefinite Sasakian manifold
M
.
Then
M
is totally geodesic if and only if






,=,,
l
w
g
YAXgYDXW, XY

 has no compo-
nents in 1
L

,



spanVYTM  or
X
has no
components in 1
L

.
Proof. We know that
M
is totally geodesic if and
only if


=0,YhX , for any

,
X
Y TM. By the
definition of the second fundamental form,


,=0YhX
is equivalent to





,Y

, =0W,,Y
W =0,ghX
 
STM 
ghX 

,, for any . RadTM  
From (2.4) and (2.7) we have









   

= ,
, ,
,
X
XX
X
X
X
ghXg Y
gY Y
gY
gY ggXYV
gY YgX

 

,,
=,
=,
=,
=,
Y


YX




 



 


(4.1)
and




  



 

,, =,
=, ,
=,
=, ,
=,, ,.
sX
X
X
sl
WX
l
W
ghXY WgYW
XgYWgY W
gY W
g
YAXWDXW
gYAXgYDXW





(4.2)
Thus, from (4.1) and (4.2), the proof is completed.
Theorem 4.2 Let
M
be a contact CR-lightlike sub-
manifold of an indefinite Sasakian manifold
M
. Then
M
is mixed geodesic if and only if Y
A
X

has no
components in 2
RadT LM


.
Proof. By the definition,
M
is mixed geodesic if and
only if








,,=0, ,,
ˆ
, .
ghXYghXY W
xDYD


 
=0.
Then we have










  





,,= ,
=,
=,
=, ,,
=,(),
=,(),
=,
X
XX
X
X
X
Y
Y
ghXYg Y
gY Y
gY
gY ggXYV
gY YgX
gAXYgX
gAX



 

YX
 
 
 




 




J. H. DONG ET AL.381
and
 

 


  


,, =,
=,
=,
=,, ,
=,
=,.
X
XX
X
X
X
Y
ghXY WgYW
gYW YW
gYW
g
YW ggXYVYXW
gYW
gAXW

 

 





 


Thus, the proof of the theorem is complete.
Theorem 4.3 Let
M
be a contact CR-lightlike sub-
manifold of an indefinite Sasakian manifold
M
. Then
M
is -geodesic if and only if
ˆ
D

2XRadTM L
   

XY
,

has no components
in .

ˆ
,,LXY D  
2
Proof.
M
is -geodesic if and only if
ˆ
D




,,=0, ,,=0
ls
ghXYghXYW 

, for any


ˆ
,,
X
YD RadTM    and .


WSTM



Then we have
  
 


 




,,= ,
=,
=,
=, ,,
=,
=,
=,
X
XX
X
X
X
X
X
ghXYg Y
gY Y
gY
gY ggXYVYX
gY
gY
gY

 


 







 



and

 




  


,, =,
=,
=,
=,, ,
=,
=,.
sX
XX
X
X
X
X
ghXY WgYW
gYW YW
gYW
g
YW ggXYVYXW
gYW
gYW
 

 





 


Thus the assertions of the theorem follows.
Theorem 4.4 Let
M
be a contact CR-lightlike sub-
manifold of an indefinite Sasakian manifold
M
. Then
M
is -geodesic if and only if DW
A
X,
A
X

 have
no components in




2 , MXLRadT D
 Y

.
Proof.
M
is -geodesic if and, only if D








,,=0, ,,=0
ls
ghXYghXY W  , for any




,,
X
YD RadTM 

  and



WSTM

 .
So we have







,,=,= ,
=,
XX
ghXYg YgY
gAXY





and







,, =, =,
=,.
XX
W
g
hXY WgYWgYW
gAXY

Thus the assertions of the theorem follows.
5. Geodesic Contact SCR-Lightlike
Submanifolds
Definition 5.1 Let




,, ,MgSTM STM
 be a
lightlike submanifold, tangent to the structure vector
field , immersed in an indefinite Sasakian manifold V


,
M
g. We say that
M
is a contact SCR-lightlike sub-
manifold of
M
if the following conditions are satisfied
[(A)] There exist real non-null distributions and
D
D

, such that


 

=,
=0,
STMD DVDSTM
DD



 

,

where D

is the orthogonal complementary to DV

in


MST . [(B)]



TM=, D  =, =RadTMRadTMltr TMltr  D.
Hence we have the decomposition
=, =TMDD VDDRadTM

 .
Let us denote ˆ=DDV

.
Definition 5.2 A contact SCR-lightlike submanifold of
an indefinite Sasakian man ifold is called mixed geodesic
contact SCR-lightlike submanifold if its second fun-
damental form satisfied for any h

,=0hXY ,


X
D and


DY

 .
Theorem 5.1 Let
M
be a contact SCR-lightlike
submanifold of an indefinite Sasakian manifold
M
.
Then
M
is totally g eodesic if and only if













,=,=0, , ,
, .
W
LgXYLgXYXYTM
RadTMWS TM




 
Proof. We know
M
is totally geodesic if and only if








,,=0, ,,=0.
ˆ
, .
ghXYghXY W
XDYD




J. H. DONG ET AL. 382
From (2.1) and Lie derivative we obtain



 

  



 





 




 
,,= ,
=, ,
=,,,
=,,,,
=,,,,,,
=,,
=, ,,.
X
Y
Y
ghXYg Y
XgYgY X
gY X gYX
gYXgXYgX Y
g
YXgXYgXYgX
Lg XYgX
LgXYghXY


















Hence we have
   
2,,= ,.
g
hXYLg XY

 
In a similar way, we can get
   
2,,= ,
W,
g
hXYWLgXY
thus the proof is completed.
Theorem 5.2 Let
M
be a contact SCR-lightlike sub-
manifold of an indefinite Sasakian manifold
M
. Then
M
is mixed geodesic if and only if



ˆ
,
s
XY
YDAXD



 , for any



ˆ,
X
DY D

 .
Proof. For any




 
ˆ, ,
,
XDYD
RadTMWSTM



 
 
denote by
=, =,
X
PX QXWBW CW


 

where





, , PXD QXDBWD 

 

and .


CWSTMD 

 
If
M
is mixed geodesic, then

,=Y=0hXY Y
 
XX
 
WSTM

 . From the definition, there
exists such that =.WY

Thus we
have

0= =
=
=.
XXXX
t
WX X
tt
WW XX
WY WY
AX WY
PA XQA XBWCWY




 
 
X
=0
From the definition of the and C, we know that
. So we have
Q
=
t
WX
QA XCW





ˆ
,
XW
WDAX   
tD
. From =WY

and
(2.13), we have =WY

, thus the proof is completed.
Theorem 5.3 Let
M
be a contact SCR-lightlike
submanifold of an indefinite Sasakian manifold
M
.
Then D

defines a totally geodesic foliatio n if and only
if


sZ,hX

and


,
s
hXN  has no components in







,   DX ZD

, D .
Proof. From the definition, we have that D

is a
totally geodesic foliation if and only if


XYD

 ,
for any


,
X
YD

 , which is equivalent to



 
,=,=0,
,.
XX
gYZgYN
ZDNltrTM


Then we have




  

 




,=,=,
=,
=,
=, ,
=,
=,,
XX X
XX
X
X
X
s
g
YZ gYZgYZ
gY ZYZ
gYZ
g
YZgXZVZX
gYZ
gYhXZ
 





 

 


and






 





,= ,
=,
=,
=,
=,
=,
=,,.
X
X
XX
X
X
X
X
s
gYNgYN
gYNYN
gYN
,
g
YgXYV YXN
gYN
gYN
gYhXN










 



Thus the assertion is proved.
6. References
[1] D. N. Kupeli, “Singular Semi-Riemannian Geometry,”
Kluwer, Dordrecht, 1996.
[2] K. L. Duggal and A. Bejancu, “Lightlike Submanifolds of
Semi-Riemannian Manifolds and Applications,” Kluwer
Academic, Dordrecht, 1996.
[3] B. Sahin, “Transversal Lightlike Submanifolds of Indefi-
nite Kaehler Manifolds,” Analele Universitaii de Vest,
Timisoara Seria Matematica—Informatica, Vol. 44, No.
1, 2006, pp. 119-145.
[4] B. Sahin and R. Günes, “Geodesic CR-Lightlike Sub-
manifolds,” Contributions to Algebra and Geometry, Vol.
42, No. 2, 2001, pp. 583-594.

J. H. DONG ET AL. 383
[5] K. L. Duggal and B. Sahin, “Lightlike Submanifolds of
Indefinite Sasakian Manifolds,” International Journal of
Mathematics and Mathematical Sciences, Article ID
57585, 2007, 21 Pages.
[6] K. L. Duggal and B. Sahin, “Generalized Cauchy-Rieman
Lightlike Submanifolds of Indefinite Sasakian Mani-
folds,” Acta Mathematica Hungarica, Vol. 122, No. 1-2,
2009, pp. 45-58. doi:10.1007/s10474-008-7221-8