Advances in Pure Mathematics, 2012, 2, 285-290
http://dx.doi.org/10.4236/apm.2012.24037 Published Online July 2012 (http://www.SciRP.org/journal/apm)
Totally Umbilical Screen Transversal Lightlike
Submanifolds of Semi-Riemannian
Product Manifolds
S. M. Khursheed Haider, Advin, Mamta Thakur
Department of Mathematics, Jamia Millia Islamia, New Delhi, India
Email: smkhaider@rediffmail.com, {advin.maseih, mthakur09}@gmail.com
Received February 17, 2012; revised March 5, 2012; accepted March 12, 2012
ABSTRACT
We study totally umbilical screen transversal lightlike submanifolds immersed in a semi-Riemannian product manifold
and obtain necessary and sufficient conditions for induced connection
on a totally umbilical radical screen transver-
sal lightlike submanifold to be metric connection. We prove a theorem which classifies totally umbilical ST-anti-in-
variant lightlike submanifold immersed in a semi-Riemannian product manifold.
Keywords: Semi-Riemannian Product Manifolds; Lightlike Submanifolds; Totally Umbilical Radical ST-Lightlike
Submanifolds; Totally Umbilical ST-Anti-Invariant Lightlike Submanifolds
1. Introduction
It is well known that the geometry of lightlike submani-
folds of semi-Riemannian manifolds is different from the
geometry of submanifolds immersed in a Riemannian
manifold since the normal vector bundle of lightlike
submanifolds intersect with tangent bundle making it
more interesting to study. The general theory of lightlike
submanifolds of a semi-Riemannian manifold has been
developed by Duggal-Bejancu [1] and Kupeli [2]. Totally
umbilical CR-submanifolds of a Kaehler manifold with
Riemannian metric were studied by Bejancu [3], Desh-
mukh and Husain [4] and many more whereas, totally
umbilical lightlike submanifolds of semi-Riemannian
manifolds of constant curvature was investigated by
Duggal-Jin [5] and totally umbilical CR-lightlike sub-
manifolds of an indefinite Kaehler manifold were studied
by Duggal-Bejancu [1] and Gogna et al. [6]. In [7], B.
Sahin initiated the study of transversal lightlike sub-
manifolds of an indefinite Kaehler manifold and investi-
gated the existence of such lightlike submanifolds in an
indefinite space form. These submanifolds in Sasakian
setting were studied by Yildirim and Sahin [8]. As a gen-
eralization of real null curves of indefinite Kaehler
manifolds, B. Sahin [9] introduced the notion of screen
transversal lightlike submanifolds and obtained many
interesting results. In this paper, we study totally umbili-
cal screen transversal lightlike submanifolds of semi-
Riemannian product manifolds.
This paper is arranged as follows. In Sections 2 and 3,
we give the basic concepts on lightlike submanifolds and
semi-Riemannian product manifolds needed for this pa-
per. In Section 4, we study the integrability of distribu-
tions involved in the definition of totally umbilical radi-
cal screen transversal lightlike submanifolds and obtain
necessary and sufficient conditions for induced connec-
tion
on totally umbilical radical screen transversal
lightlike submanifolds to be metric connection. In Sec-
tion 5, we prove a theorem which shows that the induced
connection
on a totally umbilical ST-anti-invariant
lightlike submanifold is a metric connection under some
conditions. We also prove a theorem which classifies
totally umbilical ST-anti-invariant lightlike submanifold
immersed in a semi-Riemannian product manifold.
2. Preliminaries
We follow [1] for the notation and fundamental equation
for lightlike submanifolds used in this paper. A sub-
manifold Mm immersed in a semi-Riemannian manifold
,
mn g
M
is called a lightlike submanifold if it is a
lightlike manifold with respect to the metric g induced
from
g
and radical distribution RadTM is of rank r,
where 1 r m. Let
STM

TMRadTMS TM
be a screen distribution
which is a semi-Riemannian complementary distribution
of RadTM in TM, i.e.,
STMConsider a screen transversal vector bundle
,
which is a semi-Riemannian complementary vector bun-
C
opyright © 2012 SciRes. APM
S. M. K. HAIDER ET AL.
286
i
STM

.
dle of RadTM in . Since for any local basis
TM
of
RadTM, there exists a local null frame i of sections
with values in the orthogonal complement of

N
STM
in 
such that
ST
M
,gN
ij ij
, it follows that
there exists a lightlike transversal vector bundle ltr(TM)
locally spanned by
[[1]; pg-144]. Let tr(TM) be
complementary (but not orthogonal) vector bundle to TM
in
i
N
M
TM . Then

tr TMltr

TM

,
S TM
 
M
TMMSTM

STM

STM
. S TMRadT



,M STM
ltr


M
TM
ST
STM
Following are four subcases of a lightlike submanifold
,,MgST .
Case 1: r-lightlike if r < min{m, n}.
Case 2: Co-isotropic if r = n < m; = 0.
Case 3: Isotropic if r = m < n;
= 0.
Case 4: Totally lightlike if r = n = m; = 0 =
.
The Gauss and Weingarten formulae are
 
XY TM,,XY
XX
YYh (2.1)
and
 
,
X
U

,
t
XU
UAX
X
TM U
 
 tr TM (2.2)
where and

,
XU
YAX
,,
t
Y
hX t
X belong to
and , respectively, and
U
TM


tr TM
are linear connection on M and on the vector bundle
, respectively. Moreover, we have
TM
tr

l
YYhX
h
,Y,
s
Y X
XX (2.3)
,
ls
DX
X
N
XN
NAX N (2.4)
,
sl
DX
X
W
XW
WAX 
 
,,
W

(2.5)
X
YTMNltr TM  and
TM

WS .
Denote the projection of TM on by P. Then, by
using (2.1), (2.3)-(2.5) and the fact that
STM
is a metric
connection, we obtain








,,, ,,,
,, ,
sl
W
sW.
g
hXYW gYDXWgAXY
gDXNWgNAX

,,
,
t
XX
h




X PY
(2.6)
From the decomposition of the tangent bundle of a
lightlike submanifold, we have
XX
PY PY
AX

 

,
(2.7)
for
X
YTM and
RadTM
 .
In general, the induced connection on M is not a
metric connection whereas
is a metric connection
on
,, ,MgSTMSTM
be a lightlike submani- Let
folds of
,
M
g. For any vector field X tangent to M, we
put
F
XfX X (2.8)
where fX and
X
are the tangential and transversal
parts of FX respectively. For


VtrTM
F
VBVCV
π:
(2.9)
where BV and CV are the tangential and transversal parts
of FV respectively.
3. Semi-Riemannian Product Manifolds
Let (M1, g1) and (M2, g2) be two m1 and m2-dimensional
semi-Riemannian manifolds with constant indices q1 > 0
and q2 > 0 respectively. Let 12 1
M
MM
12
:
, and
2
M
MM
be the projections which are given
by
π,
,
x
yx
x
yy
for any
and
12
,
x
yMM . We denote the product manifold by
12
,
M
MMg , where
 
12
,π,π,
g
XYgX YgX Y

 

for any
,YTM
2
ππ

2
X
, where denotes the differen-
tial mapping. Then we have ,
ππ0

,
 
π and
I

where I is the iden-
tity map of
12
M
M . Thus

,
M
g is a (m1 + m2)-
dimensional semi-Riemannian manifold with constant
index (q1 + q2). The Riemannian product manifold
,
M
12
MMg is characterized by M1 and M2 which
are totally geodesic submanifolds of
M
.
πF
Now, if we put
then we can easily see
that F2 = I and

,,
g
FX YgXFY (3.1)
for any
,
X
YTM , where F is called almost Rie-
mannian product structure on 12
M
M. If we denote the
Levi-Civita connection on
M
by , then
0
XFY (3.2)
for any
,
X
YTM , that is, F is parallel with respect
to .
4. Totally Umbilical Radical ST-Lightlike
Submanifolds
In this section, we study totally umbilical radical ST-
lightlike submanifolds of a semi-Riemannian product
manifold. We first recall the following definitions from
[9].
Definition 4.1. A r-lightlike submanifold M of a semi-
Riemannian product manifold M is said to be a screen
transversal (ST) lightlike submanifold of
M
if there
Copyright © 2012 SciRes. APM
S. M. K. HAIDER ET AL. 287
exists a screen transversal bundle such that

STM

.STM

STM

FRadTM
,, 0,gZWX
0
s
H
Definition 4.2. A ST-lightlike submanifold M of a
semi-Riemannian product manifold M is said to be a
radical ST-lightlike submanifold if is invariant
with respect to F.
We also need the following definition of totally um-
bilical lightlike submanifolds of a semi-Riemannian
manifold.
Definition 4.3. [5] A lightlike submanifold (M, g) of a
semi-Riemannian manifold
,
M
g is called totally
umbilical in
M
, if there is a smooth transversal vector
field
H
tr TM

,
of M, called the transversal cur-
vature vector of M, such that for all
X
YTM ,
,,XYHhXY g
It is known that M is totally umbilical if and only if on
each co-ordinate neighborhood U, there exists smooth
vector fields

l
H
ltr TM and

HSTM

s
such that

 
,,
ss
gXYH
,

,
and ,
l
hXY gXY
DXW

,,
0
ll
XYHh

(4.1)
for any
X
Y 
TM
and WS


.TM
In respect of the integrability of the distributions in-
volved in the definition of totally umbilical radical
ST-lightlike submanifolds immersed in a semi-Rieman-
nian product manifold, we have:
Theorem 4.4. Let M be a totally umbilical radical
ST-lightlike submanifold of a Semi-Riemannian product
manifold. Then the screen distribution S(TM) is always
integrable.
Proof. From (2.3) and (3.2), a direct calculation shows
that


,,


, ,
ss
,
g
X YNhY FXFNghX FY

(4.2)
for
,
X
YS


ltr TMTM and . Using (4.1)
in (4.2), we get
N

0N,,gXY ,
from which our assertion follows.
Theorem 4.5. Let M be a totally umbilical radical
ST-lightlike submanifold of a semi-Riemannian product
manifold. Then the distribution RadTM is always inte-
grable.
Proof. For

,
Z
W RadTM and
X
STM ,
from (2.3) and (3.2) we get




, ,
,,.
,, s
s
g
ZWXg hZFXFW
g
hW

FX FZ (4.3)
Taking account of (4.1) in (4.3), we obtain
which proves our assertion.
The necessary and sufficient conditions under which
is given by the following result.
Theorem 4.6. Let M be a totally umbilical radical
ST-lightlike submanifold of a semi-Riemannian product
manifold
,0hXY
if and only if
s
M
. Then
H
has no components in
F
RadTM for any
,.XY STM
X
Proof. Using (2.3) and (3.2), for any ,YSTM ,
we obtain
 
 
,,
,,.
ls
X
ls
X
FYhXFYhX FY
F
YFhXY FhXY
 
  (4.4)
Taking inner product of (4.4) with FN for any
NltrTM 2
and using the fact that
F
I
, we get

,, ,.
sX
g
hX FYFNgY N (4.5)
From (2.7), (4.1) and (4.5), we have


,, ,,.
s
XFYgH FNghXYN
g
(4.6)
Thus, our assertion follows from (4.6).
It is known that the induced connection on a lightlike
submanifold immersed in a semi-Riemannian manifold is
not a metric connection. In view of this, it is interesting
to see under what condition the induced connection on a
totally umbilical radical ST-lightlike submanifold is a
metric connection. The following theorem gives the
geometric conditions for the induced connection to be a
metric connection.
Theorem 4.7. Let M be a totally umbilical radical
ST-lightlike submanifold of a semi-Riemannian product
manifold
M
. Then the induced connection
on M is
a metric connection if and only if for
0
F
AX
RadTM
 .
TM ,
X
Proof. For
X
TM

RadTM

, , from (3.2)
we have
.
XX
FF



 
,
,,.
sl
FX XX
ss
AXF fFhX
Bh XCh X
(4.7)
Using (2.3), (2.5), (2.8), (2.9) and (4.1) in (4.7), we
obtain
 



XF
Taking tangential components of the above equation
and then using (4.1), we arrive at
f
AX
,
which proves our assertion.
Corollary 4.8. Let M be a totally umbilical radical
ST-lightlike submanifold of a semi-Riemannian product
manifold . Then the distribution RadTM is parallel if
M
Copyright © 2012 SciRes. APM
S. M. K. HAIDER ET AL.
288
and only if for any .
21F
Proof. From (3.2), for any we
obtain
0
12
,

12
,

A

RadTM

RadTM
11
22
.FF



Using (2.3), (2.5), (2.8), (2.9) and (4.1) in the above
equation, we get


1
212
12
,
,,
l
s
Fh

21 1
12 2
12
s
s
AFf
Bh Ch
 
F






21
12
Af

(4.8)
Considering the tangential components of (4.8) and using
(4.1), we arrive at
F

,
from which our assertion follows.
Lemma 4.9. Let M be a totally umbilical ST-lightlike
submanifold of a semi-Riemannian product manifold
M
. Then

,
s
W
A
XgHWX
for any


X
STM


TM
 and WS .
Proof. For
,
X
YTM

, from (2.6) and (4.1), we
have


,.
s
,,
W
g
AXY gXYgH W

(4.9)
If
X
RadTM


, then from (4.9) we infer that
Moreover, if
0.
W
AX
X
STM , then due to
non-degeneracy of , we have

STM

,
s
W
A
XgHWX
t
,
which proves the assertion.
For the induced connection of a totally umbilical
radical ST-lightlike submanifold in semi-Riemannian
product manifolds to be a metric connection on
tr TM,
we have:
Theorem 4.10. Let M be a totally umbilical radical
ST-lightlike submanifold of a semi-Riemannian product
manifold
M
. Then is a metric connection on
if and only if X has no component in
t

tr TMsN
for any

X
TM 

,WXTM 
 and


TM

TM
.Nltr

S


NltrTM
Proof. For and
, using (2.2), (2.5), (2.9) and (3.2), we
get

12
,
,
ts
XFN

,,
l
X
g
NWgAXFN
BW CW CW


DX FN


ltr TM
(4.10)
where , 1 and

RadTMCWBW
2
CW
 . Using (4.1) and (4.10), we obtain,


Considering lemma 4.9, we get

12
,
ts
XFNX
,, ,.
g
NWgAXCWgFNCW 

2
,,, .
ts
XX
g
NW gFNCW (4.11)
Thus our assertion follows from (4.11) and Theorem
2.3 page 159 of [1].
Theorem 4.11. Let M be a totally umbilical radical
ST-lightlike submanifold of a semi-Riemannian product
manifold
M
. Then
12
21FF
AA

,.RadTM



,.RadTM


for all 12
Proof. For any 12 , using product
structure on
M
, we get
11
22
,FF



from which we have

121
2121212
,, ,
ls s
F
FFhFhAF

 
 
(4.12)
where we have used (2.3), (2.5) and (3.2). Interchanging
1
and 2
in (4.12) and then subtracting the resulting
equation from (4.12), we obtain
12 212
21 11221
.
ss
FF
F
FAA FF
 
 

(4.13)
Taking inner product of (4.13) with
,XSTM
we get
 
12 12
21 21
,, ,.
FF
g
FXgFXg AAX
 
 
 
(4.14)
Now, from (2.3) and (4.1), a direct calculation shows
that
 
12
21
,0, ,0.gFXgFX


  (4.15)
Using (4.15) in (4.14), we get
12
21
,0.
FF
gAA X


(4.16)
Thus our assertion follows from (4.16) together with
non-degeneracy of
STM .
5. Totally Umbilical ST-Anti-Invariant
Lightlike Submanifolds
In this section, we study totally umbilical ST-anti-in-
variant lightlike submanifolds immersed in a semi-Rie-
mannian product manifold. First we recall the following
definition from [9].
Definition 5.1. [9] A ST-lightlike submanifold M of a
semi-Riemannian product Manifold
M
is said to be a
ST-anti-invariant lightlike submanifold of M if
STM
is screen transversal with respect to F, i.e.,

FSTM STM
.
Copyright © 2012 SciRes. APM
S. M. K. HAIDER ET AL. 289
The necessary and sufficient conditions for the in-
duced connection on a totally umbilical ST-anti-
invariant lightlike submanifold M to be a metric connec-
tion is given by the following result.
Theorem 5.2. Let M be a totally umbilical ST-anti-
invariant lightlike submanifold of a semi-Riemannian
product manifold
M
. Then the induced connection
on M is a metric connection if and only if s
X
F
has
no component in

F
STM for all ,

X
TM

,,
s
Bh X

.

RadTM

Proof. Using (2.3), (2.5), (2.8), (2.9), (3.2) and (4.1),
we arrive at


,.
sl
FX X
s
AX FChX
Ch X


 

YSTM
(5.1)
Taking inner product of (5.1) with FY for
and then using (4.1), we obtain

s

,,,
XX
g
FFYg Y

which proves the assertion.
Theorem 5.3. Let M be a totally umbilical ST-anti-
invariant lightlike submanifold of a semi-Riemannian
product manifold
M
. Then RadTM is parallel if and
only if 12
s
F


has no component in
F
STM

RadTM
for
all .
12
Proof. From (2.3), (2.5), (2.8), (2.9), (3.2) and (4.1),
we have
,


2 12
,,
s
Bh

21 1
12 21
12
,
sl
F
s
AF Ch
Ch
 
 




RadTM
1
12 2
for any 12 . Using (4.1) in the above
equation, we get
,

21
s
F
AF





YSTM
. (5.2)
Taking inner product of (5.2) with FY for
, we obtain

s

11
22
,,,
g
FFYg Y


from which our assertion follows.
Theorem 5.4. Let M be a totally umbilical ST-anti-
invariant lightlike submanifold of a semi-Riemannian
product manifold
M
. Then
s
H
has no component in


F
ltr TM.
Proof. For


,
X
YSTM , using (2.3), (2.5) and
(3.2) we get

,
,,
s
X FY

sl
FX X
l
X
A YFYD
F
YFhXY

 FhXY

RadTM
(5.3)
Taking inner product of (5.3) with and
then using (4.1) we obtain

from which we have our assertion.
Theorem 5.5. Let M be a totally umbilical ST-anti-
invariant lightlike submanifold of a semi-Riemannian
product manifold

,0
s
H F
,gXYg .
M
. Then if and only if 0
l
H
s
X
F
X
has no component in

F
STM for all
X
STM

 
,
,,
sl
FX XX
ss
AX FXXChXX
Bh XXCh XX


.
Proof. Using (2.3), (2.5), (2.8), (2.9), (3.2) and (4.1)
we get
(5.4)
X
for any STM
 
,,.
sls
XX
. From screen transversal parts
of (5.4), we arrive at
F
XXCh XXChXX

F
Taking inner product of the above equation with
for
RadTM
 and using (2.8), (4.1) we get


,,,,
sl
X
gFXFgXXgH


which proves our assertion.
The following theorem classifies totally umbilical ST-
anti-invariant lightlike submanifold immersed in a semi-
Riemannian product manifold.
Theorem 5.6. Let M be a totally umbilical ST-anti-
invariant lightlike submanifold of a semi-Riemannian
product manifold
M
. Then either Hs has no components
in
dim 1STM

.
F
STM or
Proof. Taking inner product of the tangential compo-
nents of (5.4) with
Z
STM and using (3.1) and
(2.9), we get


,,,
s
FX
g
AXZghXXFZ (5.5)
X
for any STM . On the other hand, by virtue of
(2.6) we have

,,,
s
FX
g
AXZ ghXZFX (5.6)
Combining (5.5) and (5.6), we get


,, ,,
ss
g
hXXFZghXZFX
Using (4.1) in the above equation, we obtain


,, ,,.
ss
g
XXgH FZgXZgH FX (5.7)
Interchanging X and Z in (5.7) and rearranging the
terms, we get




,
,,.
,
ss
gXZ
g
HFX gHFZ
gZZ
 (5.8)
From (5.7) and (5.8), we conclude that




2
,
,,.
,,
ss
gXZ
g
HFX gHFX
gXXgZZ
 (5.9)
Thus our assertion follows from (5.9).
Copyright © 2012 SciRes. APM
S. M. K. HAIDER ET AL.
Copyright © 2012 SciRes. APM
290
REFERENCES
[1] K. L. Duggal and A. Bejancu, “Lightlike Submanifolds of
Semi-Riemannian Manifolds and Applications,” Klu-
werAcademic Publishers, Dordrecht, 1996.
[2] D. N. Kupeli, “Singular Semi-Riemannian Geometry,”
Kluwer Academic Publishers, Dordrecht, 1996.
[3] A. Bejancu, “Umbilical CR-Submanifolds of a Kaehler
Manifold,” Rendiconti di Matematica, Vol. 13, 1980, pp.
431-466.
[4] S. Deshmukh and S. I. Husain, “Totally Umbilical CR-
Submanifolds of a Kaehler Manifold,” Kodai Mathe-
matical Journal, Vol. 9, No. 3, 1986, pp. 425-429.
doi:10.2996/kmj/1138037271
[5] K. L. Duggal and D. H. Jin, “Totally Umbilical Lightlike
Submanifolds,” Kodai Mathematical Journal, Vol. 26,
No. 1, 2003, pp. 49-68.
doi:10.2996/kmj/1050496648
[6] M. Gogna, R. Kumar and R. K. Nagaich, “Totally Um-
bilical CR-lightlike Submanifolds of Indefinite Kaehler
Manifolds,” Bulletin of Mathematical Analysis and Ap-
plications, Vol. 2, No. 4, 2010, pp. 54-61.
[7] B. Sahin, “Transversal Lightlike Submanifolds of Indefi-
nite Kaehler Manifolds,” Analele Universitaii de Vest,
Timisoara Seria Matematica-Informatica, Vol. XLIV, No.
1, 2006, pp. 119-145.
[8] C. Yildirim and B. Sahin, “Transversal Lightlike Sub-
manifolds of Indefinite Sasakian Manifolds,” Turkish
Journal of Mathematics, Vol. 33, 2009, pp. 1-23.
[9] B. Sahin, “Screen Transversal Lightlike Submanifolds of
Indefinite Kaehler Manifolds,” Chaos, Solitons and Frac-
tals, Vol. 38, No. 5, 2008, pp. 1439-1448.
doi:10.1016/j.chaos.2008.04.008