IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 12, Issue 4 Ver. II (Jul. - Aug.2016), PP 39-43
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DOI: 10.9790/5728-1204023943 www.iosrjournals.org 39 | Page
Ricci Solitons on Para-Sasakian Manifolds
Balachandra S. Hadimani and D. G. Prakasha
Department of Mathematics, Karnatak University, Dharwad - 580 003, INDIA.
Abstract: The present paper aims at studying a Ricci solitons on para-Sasakian manifolds. We give the results
on Ricci solitons in para-Sasakian manifold satisfying the conditions
0=),( RXS
,
0=),( PXR
and
0=),( HXR
,where
P
,
H
are pseudo-projective and quasi-conharmonic curvature tensors, respectively.
MSC(2000): 53C21; 53C44; 53C25.
Keywords and phrases: Para-Sasakian manifold; Ricci Solitons; Pseudo-projective; Quasi-conharmonic
curvature tensor.
I. Introduction
A Ricci soliton is a natural generalization of Einstein metric. A Ricci soliton
),,(
Vg
is defined on a
pseudo-Riemannian manifold
),( gM
by
0,=),(2),(2),()(£ YXgYXSYXg
V
(1.1)
where
g
V
£
denotes the Lie derivative of Riemannian metric
g
along a vector feild
V
,
is a constant, and
X
,
Y
are arbitrary vector fields on
M
. A Ricci soliton is said to be shrinking, steady, and expanding
according as
is negative, zero, and positive, respectively. Theoretical physicists have also been taking
interest in the equation of Ricci soliton in relation with string theory, and the fact that equation (1.1) is a special
case of Einstein field equations. The of Ricci soliton in Riemannian Geometry was introduced [1] as self-similar
solutions of the Ricci flow. Recent progress on Riemannian Ricci solitons may be found in [2]. Also, Ricci
solitons have been studied extensively in the context of pseudo-Riemannian Geometry; we may refer to [3, 4, 5,
6, 7, 8] and references therein.
In 1976, Sato [9] introduced the notion of almost paracontact structure
),,(
on a differentiable
manifolds. This structure is an analogue of the almost contact structure. An almost contact manifold is always
odd-dimensional but an almost paracontact manifold could be of even dimension as well. Takahashi [10]
defined almost contact manifolds (in particular, Sasakian manifolds) equipped with an associated Pseudo-
Riemannian metric.
In 1977, Adati and Matsumoto defined Para-Sasakian and special Para-Sasakian manifolds [11], which
are special classes of an almost paracontact manifold. Also, The geometry of these manifolds is extensively
studied by [12, 13, 14, 15, 16, 17] and many others. In 1985, Kaneyuki and Williams [18] defined the notion of
almost para contact structure on pseudo-Riemannian manifold of dimension
1)(2 n
. Later, Zamkovoy [19]
showed that any almost para contact structure admits a pseudo-Riemannian metric with signature
)1,( nn
.
In the present paper, we study Ricci solitons on para-Sasakian manifolds. The paper is organised as
follows: section
2
is devoted to preliminaries on para-Sasakian manifolds. In section
3
,
4
and
5
we study
para-Sasakian Ricci solitons satisfying
0=),( RXS
,
0=),( PXR
and
0=),( HXR
, where
P
and
H
are pseudo-projective and quasi-conharmonic curvature tensors, respectively.
II. Preliminaries
An almost paracontact structure on a manifold
M
of dimension
n
is a triplet
),,(
consisting
of a (1,1)-tensor field
, a vector field
, a 1-form
satisfying:
0,=1,=)(,=
2
I
(2.1)
1,=)(0,= nrank
(2.2)
where
I
denotes the identity transformation. A pseudo-Riemannian metric
g
on
M
is compatible with the
almost paracontact structure
),,(
if
).()(),(=),( YXYXgYXg
(2.3)
Ricci Solitons on Para-Sasakian Manifolds
DOI: 10.9790/5728-1204023943 www.iosrjournals.org 40 | Page
In such case,
),,,( g
is called an almost paracontact metric structure. By (2.1)-(2.3), it is clear that,
)(=),( XXg
for any compatible metric. Any almost paracontact structure admits compatible metrics. The
fundamental 2-form
of an almost paracontact structure
),,,( g
is defined by
),(= YXg
, for all
tangent vector fields X, Y. If
d=
, then the manifold
),,,,( gM
is called a paracontact metric
manifold associated to the metric
g
.
In case the paracontact metric structure is normal. The structure is called para-Sasakian. Equivalently,
a paracontact metric structure
),,,( g
is para-Sasakian if An almost paracontact metric structure becomes
a paracontact metric structure if
XYYXgY
X
)(),(=)(
(2.4)
for any vector fields
)(, TMYX
, where
is Levi-Civita connection of
g
.
From (2.4), it follows that
.= X
X
(2.5)
Also, in an
n
-dimensional para-Sasakian manifold, the following relations hold:
),(),()(),(=)),(( XZYgYZXgZYXR
(2.6)
,)()(=),( XYYXYXR
(2.7)
,)(),(=),( XYYXgYXR
(2.8)
),(1)(=),( XnXS
(2.9)
)(,, TMZYX
. Here R is Riemannian curvature tensor and
S
is Ricci tensor defined by
),(=),( YQXgYXS
, where
Q
is Ricci operator.
Let
),,,( gM
be an
n
-dimension para-Sasakian manifold and let
),,(
g
be a Ricci soliton
on
M
. Then the relation (1.1) implies
0=),(2),(2),)(£( YXgYXSYXg
or
),(2),)(£(=),(2 YXgYXgYXS
(2.10)
for any
X
,
Y
)(M
.
On a para-Sasakian manifold M, from (2.5) and the skew-symmetric property of
, we obtain
0.=),(),(=),)(£(
YX
XgYgYXg
(2.11)
By plugging (2.11) in (2.10), we have
).,(=),( YXgYXS
(2.12)
Also, By Putting
=Y
in (2.12) we have
)(=),( XXS
. (2.13)
By virtue of (2.9), we obtain from (2.13) that
)1( n
. (2.14)
Thus, we can state the following:
Theorem 1. A para-Sasakian Ricci soliton in an n-dimentsional (n>1) para-Sasakian manifold is expanding.
III. Ricci Soliton in a Para-Sasakian Manifold Satisfying
0.=),( RXS
Consider a para-Sasakian manifold
),( gM
n
, satifying the condition
0.=),( RXS
(3.1)
By definition we have
WVURXWVURXS
S
),()((=),)(),((
WVUXRWVURX
S
),)((),()(=
,))(,())(,( WXVURWVXUR
SS
(3.2)
where the endomorphisim
YX
S
is defined by
Ricci Solitons on Para-Sasakian Manifolds
DOI: 10.9790/5728-1204023943 www.iosrjournals.org 41 | Page
.),(),(=)( YZXSXZYSZYX
S
(3.3)
In the view of (3.3) in (3.2), we get
WVURXS ),)(),((
WVRUXSWVXRUSWVURXSXWVURS ),(),(),(),()),((,()),(,(=
.),(),(),(),(),(),(),(),(
VURWXSXVURWSWURVXSWXURVS
(3.4)
Using (2.12) and (2.13), we obatin
WVRUXgWVXRUWVURXgXWVUR ),(),(),()()),(,()),(([
0.=]),(),(),()(),(),(),()(
VURWXgXVURWWURVXgWXURV
(3.5)
By taking an inner product with
and by Using (2.6), (2.7) and (2.8) in (3.5), we get
),(),(),(),()),(,([ WUgVXgWVgUXgWVURXg
0.=)}](),()(),(){(2 VWUgUWVgX
(3.6)
Putting
i
eUX ==
in (3.6) and summing over
ni 1,2,...,=
, we get
0.=)]()(2),(3)(),([ WVWVgnWVS
(3.7)
Putting
==WV
in (3.7) and by virtue of (2.12) and (2.13), we have
0.=1)}({ n
(3.8)
Implies either
0=
or
1)(= n
.
Therefore, we can state the following theorem.
Theorem 2. A Ricci soliton in a para-Sasakian manifold
1)>(n
satisfying
0=),( RXS
is either steady
or shrinking.
IV. Ricci Soliton in a Para-Sasakian Manifolds Satisfying
0),( PXR
.
Pseudo-projective curvature tensor
P
is defined by B. Prasad [20].
]),(),([
1
]),(),([),(=),( YZXgXZYgb
n
a
n
r
YZXSXZYSbZYXaRZYXP
(4.1)
where
0, ba
are constants.
Taking
=Z
in (4.1) and by using (2.7) and (2.12) in (4.1), we have
].)()([
1
=),( XYYXb
n
a
n
r
baYXP
(4.2)
Similarly using (2.6), (2.12) in (4.1), we get
)].(),()(),([
1
=)),(( XZYgYZXgb
n
a
n
r
baZYXP
(4.3)
By using the condition
0=),( PXR
, we have
0.=),(),()),(,(),),((),(),( WXRVUPWVXRUPWVUXRPWVUPXR
(4.4)
In the view of (2.8) in (4.4), we obtain
WVPUXgWVXPUWVUPXgXWVUP ),(),(),()()),(,()),((
0.=),(),(),()(),(),(),()(
VUPWXgXVUPWWUPVXgWXUPV
(4.5)
Taking inner product with
in (4.5), we get
)),((),()),(()()),(,()()),(( WVPUXgWVXPUWVUPXgXWVUP
0.=)),((),()),(()()),((),(),(()(
VUPWXgXVUPWWUPVXgWXUPV
(4.6)
By virtue of (4.2), (4.3) in (4.6), we have
Ricci Solitons on Para-Sasakian Manifolds
DOI: 10.9790/5728-1204023943 www.iosrjournals.org 42 | Page
0.=)],(),(),(),([
1
)),(,( WVgUXgXVgWUgb
n
a
n
r
baWVUPXg
(4.7)
Using (4.1) in (4.7), we obtain
0.=)],(),(),(),([)),(,( WVgUXgVXgWUgaWVURXag
(4.8)
Putting
i
eUX ==
in (4.8) and summing over
,1,2,...,= ni
we get
0.=),(1)(),( WVgnWVS
(4.9)
Putting
==WV
in (4.9) and by virtue of (2.12), we obtain
1).(= n
(4.10)
It implies
is positive for every
1)>(n
. Hence we state the following theorem.
Theorem 3. A Ricci soliton in a para-Sasakian manifold
1)>(n
satisfying
0=),( PXR
is expanding.
V. Ricci Soliton in a Para-Sasakian Manifold Satisfying
0),( HXR
.
Quasi-conharmonic curvature tensor
H
)3( n
of the type
(1,3)
is defined [21].
]),(),([]),(),([),(=),( QYZXgQXZYgcYZXSXZYSbZYXaRZYXH
],),(),([
2
2
YZXgXZYgcb
n
a
n
r
(5.1)
where
a
,
b
and
c
are constants such that
0,, cba
.
Taking
=Z
in (5.1) and by using (2.7) and (2.12) in (5.1), we have
].)()([
2
2
)(=),( XYYXcb
n
a
n
r
cbaYXH
(5.2)
Similarly using (2.6), (2.12) in (5.1), we get
)].(),()(),([
2
2
)(=)),(( XZYgYZXgcb
n
a
n
r
cbaZYXH
(5.3)
We assume the condition
0=),( HXR
, then we have
0.=),(),()),(,(),),((),(),( WXRVUHWVXRUHWVUXRHWVUHXR
(5.4)
In the view of (2.8) in (5.4), we obtain
WVHUXgWVXHUWVUHXgXWVUH ),(),(),()()),(,()),((
0.=),(),(),()(),(),(),()(
VUHWXgXVUHWWUHVXgWXUHV
(5.5)
Taking inner product with
in (5.5), we get
)),((),()),(()()),(,()()),(( WVHUXgWVXHUWVUHXgXWVUH
0.=)),((),()),(()()),((),()),()(
VUHWXgXVUHWWUHVXgWXUHV
(5.6)
By virtue of (5.2), (5.3) in (5.6), we have
0.=)],(),(),(),([
2
2
)()),(,( UXgWVgVXgWUgcb
n
a
n
r
cbaWVUHXg
(5.7)
Using (5.1) in (5.7), we obtain
Ricci Solitons on Para-Sasakian Manifolds
DOI: 10.9790/5728-1204023943 www.iosrjournals.org 43 | Page
0.=)],(),(),(),([)),(,( UXgWVgVXgWUgaWVURXag
(5.8)
Putting
i
eUX ==
in (5.8) and summing over
,1,2,...,= ni
we get
0.),(1)(),( WVgnWVS
(5.9)
Putting
==WV
in (4.9) and by virtue of (2.12), we obtain
1).(= n
(5.10)
It implies
is positive for every
1)>(n
. Hence we state the following theorem.
Theorem 4 . A Ricci soliton in a para-Sasakian manifold
1)>(n
satisfying
0=),( HXR
is expanding.
References
[1] R. S. Hamilton, The Ricci flow on surfaces, Mathematics and general relativity (Santa Cruz, CA, 1986), Contemp. Math., vol. 71,
Amer. Math. Soc., Providence, RI, 1988, pp. 237–262.
[2] H. D. Cao, Recent progress on Ricci solitons, arXiv:0908:2006v1, Adv. Lect. Math. (ALM), 11 (2009), 1-38.
[3] C. L. Bejan and M. Crasmareanu, Second order parallel tensors and Ricci solitons in 3-dimensional normal paracontact geometry,
Ann. Glob. Anal. Geom., DOI 10.1007/s10455-014-9414-4.
[4] C. S. Bagewadi, Gurupadavva Ingalahalli, and S. R. Ashoka, A study on Ricci solitons in Kenmotsu manifolds, ISRN Geometry,
Hindawi Publishing Corporation., Volume 2013, Artcle ID 412593, 6 pages.
[5] G. Calvaruso and D. Perrone, Geometry of H-paracontact metric manifolds, arXiv:1307.7662v1.2013.
[6] G. Calvaruso and A. Perrone, Ricci solitons in three-dimensional paracontact geometry, arXiv:1407.3458v1.2014.
[7] G. Calvaruso and A. Zaeim, A complete classification of Ricci and Yamabe solitons on non-reductive homogeneous 4-spaces, J.
Geom. Phys., 80 (2014), 15-25.
[8] R. Pina and K. Tenenblat, On solutions of the Ricci curvature and the Einstein equation, Israel J. Math., (171) (2009), 61-76.
[9] I. Sato, On a structure similar to the almost contact structure I., Tensor N.S., 30,219-224, 1976.
[10] T. Takahashi, Sasakian manifold with pseudo-Riemannian metric, Tohoku Math. J., 21(2) (1969), 644-653.
[11] T. Adati and K. Matsumoto, On conformally recurrent and conformally symmetric P-Sasakian manifolds, TRU Math., 13 (1977),
25-32.
[12] T. Adati and T. Miyazawa, On P-Sasakian manifolds satisfying certain conditions, Tensor (N.S.), 33 (1979), 173-178.
[13] U. C. De, G. Pathak, On P-Sasakian manifolds satisfying certain conditions, J. Indian Acad. Math. 16 (1994), 72–77.
[14] K. Matsumoto, S. Ianus, I. Mihai, On a P-Sasakian manifolds which admit certain tensor fields, Publ. Math. Debrecen 33 (1986),
61–65.
[15] D. A. Patil, C. S. Bagewadi and D. G. Prakasha, On generalized
–recurrent para-Sasakian manifolds, J. Int. Acad. Physical
Sci.,15(1), (2011), 1-10.
[16] D. G. Prakasha, Torseforming vector fields in a 3-dimensional para-Sasakian manifold, Vasile Alecsandri, University of Bacau,
Faculty of science, Scientific studies and research series Mathematics and Informatics., 20(2), (2010), 61-66.
[17] D. Tarafdar, U.C. De, On a type of P-Sasakian Manifold, Extr. Math. 8(1) (1993), 31–36.
[18] S. Kaneyuki and F. L. Williams, Almost paracontact and parahodge structures on manifolds, Nagoya Math. J., 99 (1985), 173-187.
[19] S. Zamkovoy, Canonical connections on paracontact manifolds, Ann Glob Anal Geom., 36(1) (2009), 37-60.
[20] B. Prasad, On pseudo projective curvature tensor on a Riemannian manifold, Bull.Cal.Math.Soc.., 94 (3)(2002), 163-166.
[21] H. C. Lal, Vivek Kumar S and B. Prasad, Quasi conharmonic curvature tensor on a Riemannain manifold, J. Progressive Sci.
04(02), 151-156, 2013.