Some Properties of M-projective Curvature Tensor on Generalized Sasakian-Space-Forms

International Journal of Mathematics and Statistics Invention (IJMSI)
E-ISSN: 2321 – 4767 P-ISSN: 2321 - 4759
www.ijmsi.org Volume 4 Issue 4 || April. 2016 || PP-38-43
www.ijmsi.org 38 | Page
Some Properties of M-projective Curvature Tensor on
Generalized Sasakian-Space-Forms
D. G. Prakasha , B. S. Hadimani and Vasant Chavan
(Department of Mathematics, Karnatak University, Dharwad - 580 003, INDIA)
ABSTRACT : The object of this paper is to study the M -projective curvature tensor on generalized
Sasakian-space-forms. We study M -projectively semisymmetric, M -projectively pseudosymmetric and  -M -
projectively semisymmetric generalized Sasakian-space-form.
KEYWORDS - Generalized Sasakian-space-form, M -projective curvature tensor, semisymmetric,
pseudosymmetric.
I. INTRODUCTION
The notion of generalized Sasakian-space-forms was introduced and studied by Alegre et al., [1] with
several examples. A generalized Sasakian-space-form is an almost contact metric manifold ),,,,( gM 
whose curvature tensor is given by
}),(),({=),( 1
YZXgXZYgfZYXR  (1)
}),(2),(),({2
ZYXgXZYgYZXgf  
XZYYZXf )()()()({3
  })(),()(),(  XZYgYZXg  ,
where 321
,, fff are differentiable functions on M and ZYX ,, are vector fields on M . In such a case we
will write the manifold as ),,( 321
12
fffM
n 
. This kind of manifolds appears as a natural generalization of the
Sasakian-space-forms by taking:
4
3
=1
c
f and
4
1
== 32
c
ff , where c denotes constant  -sectional
curvature. The  -sectional curvature of generalized Sasakian-space-form ),,( 321
12
fffM
n 
is 21
3 ff  .
Moreover, cosymplectic space-form and Kenmotsu space-form are also considered as particular types of
generalized Sasakian-space-form. The generalized Sasakian-space-forms have also been studied in ([2], [3], [4],
[6], [7], [8], [9], [12], [14], [21], [20], [22]) and many others.
Apart from the conformal curvature tensor, the M -projective curvature tensor is another important
tensor from the differential geometric point of view. This curvature tensor bridges the gap between conformal
curvature tensor, conharmonic curvature tensor and concircular curvature tensor on one side and H -projective
curvature tensor on the other.
The M -projetive curvature tensor of Riemannian manifold ),(
12
gM
n 
was defined by Pokhariyal
and Mishra [19] as
YZXSXZYS
n
ZYXRZYX ),(),([
4
1
),(=),( M
],),(),( QYZXgQXZYg  (2)
where R and S are the curvature tensor and the Ricci tensor of M , respectively, and Q is the Ricci operator
defined as ),(=),( YQXgYXS . Some properties of this tensor in Sasakian and K a hler manifolds have
been studied earlier ([16], [17], [25]).
In the context of generalized Sasakian-space-forms, Kim [14] studied conformally flat and locally
symmetric generalized Sasakian-space-forms. De and Sarkar [6] studied some symmetric properties of
generalized Sasakian-space-forms with projective curvature tensor. In [20], Prakasha shown that every
generalized Sasakian-space-form is Weyl-pseudosymmetric. The symmetric properties of generalized Sasakian-
space-forms have also been studied in [12] with 2
W -curvature tensor. Also, Prakasha and Nagaraj [21] studied
quasi-conformally flat and quasi-confomally semisymmetric generalized Sasakian-space-forms. Recently Hui
and Prakasha [13] studied certain properties on the C-Bochner curvature tensor of generalized Sasakian-space-

Some Properties of M-projective Curvature…
forms. As a motivation of this, here we plan to study certain symmetric properties on generalized Sasakian-
space-form with M -projective curvature tensor.
The present paper is organised as follows: After preliminaries in section 3, we study M -projectively
semisymmetric and M -projectively pseudosymmetric generalized Sasakian-space-forms. Also in section 4 we
study  -M -projectively semisymmetric generalized Sasakian-space-form.
II. PRELIMINARIES
An odd-dimensional Riemannian manifold ),( gM is said to be an almost contact metric manifold
[5] if there exist on M a (1,1) tensor field  , a vector field  (called the structure vector field) and a 1-form
 such that 1=)( ,  )(=)(
2
XXX  and )()(),(=),( YXYXgYXg   , for any vector
fields X , Y on M . In particular, in an almost contact metric manifold we also have 0= and 0=  .
Such a manifold is said to be a contact metric manifold if =d , where ),(=),( YXgYX  is
called the fundamental 2-form of M . If , in addition,  is a killing vector field, then M is said to be a K -
contact manifold. It is well-known that a contact metric manifold is K -contact manifold if and only if
XX
  = , for any vector field X on M . On the other hand, the almost contact metric structure of M is
said to be normal if  ),(2=),](,[ YXdYX  , for any X , Y , where ],[  denotes the Nijenhuis
torsion of  . A normal contact metric manifold is called Sasakian manifold. An almost contact metric manifold
is Sasakian if and only if XYYXgYX
)(),(=)(   , for any X , Y .
In addition to the relation (1), for a 1)(2 n -dimensional 1)>(n generalized Sasakian-space-form
),,( 321
12
fffM
n 
the following relations also hold [1]:
)()()1)(2(3),()3(2=),( 32321
YXfnfYXgffnfYXS  , (3)
 )()1)(2(3)3(2= 32321
XfnfXffnfQX  , (4)
))()()((=),( 31
YXXYffYXR   , (5)
)),()()((=),( 31
 YXgXYffYXR  , (6)
)()(2=),( 31
XffnXS   , (7)
321
461)(22= nfnffnnr  , (8)
where R , S and r are respectively denotes the curvature tensor of type (1,3) , Ricci tensor of type (0,2)
and scalar curvature of the space-form.
In view of (3)–(6), it can be easily construct that in a 1)(2 n -dimensional 1)>(n generalized
Sasakian-space-form ),,( 321
12
fffM
n 
, the M -projective curvature tensor satisfies the following conditions:
})()(),({
4
)1)(2(3
=),(
32
 YXYXg
n
fnf
YX 

M , (9)
})()({
4
)1)(2(3
=),(
32
YXXY
n
fnf
YX  

M , (10)
)},(),()(),({
4
)1)(2(3
=)),((
32
YZXgXZYg
n
fnf
ZYX  

M (11)
for all vector fields ZYX ,, on ),,( 321
12
fffM
n 
.
III.  -PROJECTIVELY SEMISYMMETRIC AND  - PROJECTIVELY
PSEUDOSYMMETRIC GENERALIZED SASAKIAN-SPACE-FORMS
A Riemannain manifold M is called locally symmetric if its curvature tensor R is parallel, that is,
0=R , where  denotes the Levi-Civita connection. As a proper generalization of locally symmetric
manifolds the notion of semisymmetric manifolds was defined by
)(,,,,0,=),)(),(( MWVUYXWVURYXR  (12)

and studied by many authors(e.g.,[15, 18, 24]). A complete intrinsic classification of these spaces was given by
Z. I. Szabo[23].
A generalized Sasakian space-form ),,( 321
12
fffM
n 
is said to be M -projectively symmetric if
0=M and it is called M -projectively semisymmetric if,
0.=),)(),(( WVUYXR M (13)
From (13), we have
WVUXRWVUXR ),),((),(),(  MM  (14)
0.=),(),()),(,( WXRVUWVXRU  MM 
In the view of (6), the above equation becomes
WVXUWVUXgXWVUgff ),()()),(,()),(,()[( 31
MMM  
WUVXgWXUVWVUXg ),(),(),()(),(),(  MMM 
0.=]),(),(),()(  VUWXgXVUW MM  (15)
Putting =V in (15) and using (9), (10) and (11), we have
XWUgXWU
n
fnf
WXUff ),()()({
4
1)(23
),()[(
32
31


 M
}].)(),()(),()()()(2  WXUgUWXgWUX  (16)
Thus, we obtain either 0=31
ff  or
 )()(2)()(),({
4
1)(23
=),(
32
WXXWUXWUg
n
fnf
WXU 

M
}.)(),()(),(  WXUgUWXg  (17)
Taking the inner product on both sides of relation (17) with V , we get
),()()(),(),({
4
1)(23
=),),((
32
VXgWUVXgWUg
n
fnf
VWXUg 

M
)()(),()()()(2 VUWXgVWX  
)}.()(),( VWXUg  (18)
Let }{ i
e is an orthonormal basis of the tangent space at each point of the manifold and taking summation over
i , 11,2,...,2= ni in (17), we have
),,(
12
=),( WXg
n
r
WXS

(19)
plugging =W in (19) and using (7) and (8), we obtain
0.=)(]3)2[(1 23
Xffn  (20)
In this case, since 0)( X , the relation (20) implies that
.
21
3
=
2
3
n
f
f

(21)
Now, with the help of (21), the equation (17) reduces to
0.=),( WXUM (22)
That is ),,( 321
12
fffM
n 
is M -projectively flat. Hence we conclude the following:
Theorem 3.1 A 1)(2 n -dimension 1)>(n M -projectively semisymmetric generalized Saskian-
space-form is M -projectively flat (then
n
f
f
21
3
=
2
3

) or 31
= ff .
Corollary 3.2 A 1)(2 n -dimension 1)>(n generalized Saskian-space-form is M -projectively
semisymmetric if and only if M -projectively flat (
n
f
f
21
3
=
2
3

).

Next, for a )(0, k -tensor field T on M , 1k , and a symmetric (0,2) -tensor field A on M , we
define the 2)(0, k -tensor fields TR  and ),( TAQ by
),;,...,)(.( 1
YXXXTR K
)),(,,...,(...),...,,),((= 1121 kkk
XYXRXXTXXXYXRT 
 (23)
and
),;,...,)(,( 1
YXXXTAQ K
))(,,...,(...),...,,)((= 1121 kAkkA
XYXXXTXXXYXT  
(24)
respectively, where YX A
 is the endomorphism given by
.),(),(=)( YZXAXZYAZYX A
 (25)
A Riemannian manifold M is said to be pseudosymmetric (in the sense of R. Deszcz [10, 11]) if
),(= RgQLRR R
 (26)
holds on }0
1)(
|{= xatG
nn
r
RMxU R


 , where G is the (0,4) -tensor defined by
),)((=),,,( 43214321
XXXXgXXXXG  and R
L is some smooth function on R
U . A Riemannian
manifold M is said to be M -projectively pseudosymmetric if
),,;,,)(,(=),)(),(( YXWVUgQLWVUYXR MM M
 (27)
holds on the set 0}:{=  MM
MxU at x , where M
L is some function on M
U and M is the M -
projective curvature tensor.
Let ),,( 321
12
fffM
n 
be a 1)(2 n -dimensional 1)>(n M -projectively pseudosymmetric
generalized Sasakian-space-form. Then from (25) and (27), we have
].),)()[((=),)(),(( WVUYLWVUYR MM M
  (28)
If ),,( 321
12
fffM
n 
be a 1)(2 n -dimensional 1)>(n generalized Sasakian-space-form, from
(6) and (25), we get
.))((=),( 31
YXffYXR   (29)
In view of (28) in (29), it is easy to see that
).(= 31
ffL M
(30)
Hence, by taking account of previous calculations and discussion, we conclude the following:
Theorem 3.3 Let ),,( 321
12
fffM
n 
be a 1)(2 n -dimensional 1)>(n generalized Sasakian-
space-form. If ),,( 321
12
fffM
n 
is M -projectively pseudosymmetric then ),,( 321
12
fffM
n 
is either M -
projectively flat, in which case
n
f
f
21
3
=
2
3

or 31
= ffL M
holds on ),,( 321
12
fffM
n 
.
But M
L need not be zero, in general and hence there exists M -projectively pseudosymmetric
manifolds which are not M -projectively semisymmetric. Thus the class of M -projectively pseudosymmetric
manifolds is a natural extension of the class of M -projective semisymmetric manifolds. Thus, if 0M
L then
it is easy to see that ),()(= 31
MM gQffR  , which implies that the pseudosymmetric function
31
= ffL M
. Therefore, we able to state the following result:
Theorem 3.4 Every generalized Sasakian-space-form is M -projectively pseudosymmetric of the form
),()(= 31
MM gQffR  .

IV.  -M -PROJECTIVELY SEMISYMMETRIC GENERALIZED SASAKIAN-SPACE-
FORM
Definition 4.1 For a 1)(2 n -dimensional 1)>(n generalized Sasakian-space-form is said to be
 -M -projectively semisymmetric if it satisfies the condition 0=),( YXM .
Thus, we get
0,=),(),(=)),(( ZYXZYXZYX MMM   (31)
for all vector fields )(,, MZYX  .
Now, by virtue of (2), we have
].),(),(),(),([
4
1
),(=),( QYZXgQXZYgYZXSXZYS
n
ZYXRZYX  M (32)
Taking use of (1), (3) and (4) in (32), we obtain
YXZYZXgfYZXgXZYg
n
ff
ZYX  )()(),({}),(),({
2
)(3
=),( 2
32


M
})(),(2),(2)()(),(  ZYXgZYXgXYZXZYg 
}.)(),()(),({3
 XZYgYZXgf  (33)
Similarly,
YZXgfYZXgXZYg
n
ff
ZYX ),({}),(),({
2
)(3
=),( 2
32
 

M
ZYXgXZYgXZYgYZXg ),(2)(),(),()(),(  
}.)()()()({})(),(2 3
XZYYYXfZYXg   (34)
Substituting (33) and (34) in (31), we obtain
}),(),(),(),({
2
3
3
32
YZXgZYgYZXgXZYgf
n
ff
 







 )(),()()()()(){( 31
XZYgXYZYYZff 
0.=})(),(  YZXg (35)
Setting =Y in (35), we get
0.=})(),({
2
3
3
32
XZZXgf
n
ff
 







(36)
In this case, since 0)(),(  XZZXg  , the relation (36) implies that
.
21
3
=
2
3
n
f
f

(37)
Hence we state the following:
Theorem 4.2 For a 1)(2 n -dimensional 1)>(n  -M -projectively semisymmetric generalized
Sasakian-space-form ),,( 321
12
fffM
n 
,
n
f
f
21
3
=
2
3

holds.
In a recent paper [6], De and Sarkar have proved the following:
Theorem 4.3 [6] A 1)(2 n -dimensional 1)>(n generalized ),,( 321
12
fffM
n 
is projectively
flat if and only if
n
f
f
21
3
=
2
3

.
Suppose,
n
f
f
21
3
=
2
3

holds. Therefore 0=M and hence 0=),( YXM .

Thus in view of Theorem (4.2), we can state the following:
Theorem 4.4 A 1)(2 n -dimensional 1)>(n generalized Sasakian-space-form
),,( 321
12
fffM
n 
is  -M - projectively semisymmetric if and only if
n
f
f
21
3
=
2
3

.
From the Theorem 7.2 of the [7], we note that a 1)(2 n -dimensional 1)>(n generalized Sasakian-
space-form is Riccisymmetric if and only if
n
f
f
21
3
=
2
3

.
By virtue of Theorem (4.4) we can state the following :
Theorem 4.5 A 1)(2 n -dimensional 1)>(n generalized Sasakian-space-form
),,( 321
12
fffM
n 
is  -M -projectively semisymmetric if and only if it is Ricci semisymmetric.
By continuing Theorems (4.2), (4.3), (4.5), we can state the following:
Corollary 4.6 Let ),,( 321
12
fffM
n 
be a 1)(2 n -dimensional 1)>(n generalized Sasakian-
space-form. Then the following statements are equivalent.
1. ),,( 321
12
fffM
n 
is  -M -projectively semisymmetric.
2. ),,( 321
12
fffM
n 
is projectively flat.
3. ),,( 321
12
fffM
n 
is Ricci semisymmetric.
4.
n
f
f
21
3
=
2
3

holds ),,( 321
12
fffM
n 
.
REFERENCES
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