Journal of mathematics and computer science
On a Pseudo Projective
14 (2015), 309-314
Recurrent Sasakian Manifolds
1
2
3
A. Singh , R. Kumar Pandey , A. Prakash , S. Khare
4
1
Department of Mathematics, B.B.D. University, Lucknow-226004, Uttar Pradesh, India
Department of Mathematics, B.B.D. University, Lucknow-226004, Uttar Pradesh, India
3
Department of Mathematics, N.I.T., Kurukshetra -136119, Haryana, India.
4
Department of Mathematics, B.B.D. University, Lucknow-226004, Uttar Pradesh, India
[email protected]
2
[email protected]
[email protected]
[email protected]
Article history:
Received June 2014
Accepted November 2014
Available online January 2015
Abstract
The object of the present paper is to study the pseudo projective
Keywords: Pseudo projective
Einstein manifold.
recurrent Sasakian manifolds.
symmetric manifold, pseudo projective
recurrent manifold,
1. Introduction
The notion of local symmetry of a Riemannian manifold has been studied by many authors in several
ways to a different extent. As a weaker version of local symmetry, in 1977, Takahashi [9] introduced
the notion of locally
symmetric Sasakian manifold and obtained their several interesting results.
The properties of pseudo projective curvature tensor is studied by many geometers [17], [18], [19],
[22] and obtained their some interesting results.
In this paper we shown that pseudo projective
recurrent Sasakian manifold is an Einstein
manifold and in a pseudo projective
recurrent Sasakian manifold, the characteristic vector field
and the vector field associated to the
form are co-directional. Finally, we proved that a
three dimensional locally pseudo – projective
recurrent Sasakian manifold is of constant
curvature.
2. Preliminaries
Let
be an almost contact Riemannian manifold, where is a
tensor field, is
the structure vector field, is a
form and is the Riemannian metric.It is well known that the
structure
satisfy
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A. Singh, R. K. Pandey, A. Prakash , S. Khare / J. Math. Computer Sci. 14 (2015), 309-314
(1)
(2)
(3)
(4)
(5)
(6)
for all vector fields
then
where denotes the operator of covariant differentiation with respect to
is called a Sasakian manifold [1].
Sasakian manifolds have been studied by many authors such as De, Shaikh and Biswas [3],
Takahashi [9], Tanno [15] and many others.
In a Sasakian manifold the following relations hold: [1]
(7)
(8)
(9)
(10)
for all vector fields
where
curvature tensor of the manifold.
is the Ricci tensor of type
and
is the Riemannian
A Sasakian manifold is said to be an Einstein manifold if the Ricci tensor is of the form
where λ is a constant.
Definition 2.1. A Sasakian manifold is said to be a locally
symmetric manifold if [9]
(
for all vector fields
)
(11)
orthogonal to
Definition 2.2. A Sasakian manifold is said to be a locally pseudo projective
̃)
if
((
)
for all vector fields
̃
̃
̃
(13)
[
are constants such that
,b
]
][
[
̃
recurrent Sasakian manifold if
where ̃ is a pseudo projective curvature tensor given by [17]
for arbitrary vector fields
and
(12)
orthogonal to
Definition 2.3. A Sasakian manifold is said to be pseudo projective
there exists a non-zero
form such that
where
symmetric manifold
If
[
]
and
(14)
Then (14) takes of the form
]
.
where is the projective curvature tensor [21]. Hence the Projective curvature is a particular
case of the tensor ̃ . For the reason ̃ is called Pseudo projective curvature tensor, where is the
Riemann curvature tensor is Ricci tensor and is the scalar curvature.
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A. Singh, R. K. Pandey, A. Prakash , S. Khare / J. Math. Computer Sci. 14 (2015), 309-314
If the 1-form
manifold.
vanishes, then the manifold reduces to a locally pseudo projective
3. Pseudo projective
symmetric
recurrent Sasakian manifold
In this section we consider a Sasakian manifold which is pseudo projective
manifold. Then by virtue of (1) and (3), we get
(
̃)
((
̃)
recurrent Sasakian
̃
)
(15)
from which it follows that
((
̃)
Let
manifold. Then putting
)
((
̃)
(̃
)
)
(16)
be an orthonormal basis of the tangent space at any point of the
in (16) and taking summation over
we get
(
[
)
]
(17)
Replacing Z by ξ in (17) and using (2) and (9), we get
(
[
)]
(18)
Now we have
using (5), (6) and (9) in the above relation, it follows that
(19)
In view of (18) and (19), we get
(
[
Replacing
by
)]
(20)
and using (3), (4) and (10) in (20), we get
(21)
for all
Hence, we can state the following theorem:
Theorem 3.1 A Pseudo projective
recurrent Sasakian manifold
is an Einstein manifold.
Now from (15), we have
̃)
(
Using (14) in (22), we get
((
̃)
(
[
[
]
]
)[
]
[
]
]
From (23) and the Bianchi identity, we get
[
(
(
]
)[
[
]
)[
]
(22)
)
[
(
̃
)
[
]
]
[
]
311
(23)
A. Singh, R. K. Pandey, A. Prakash , S. Khare / J. Math. Computer Sci. 14 (2015), 309-314
[
]
[
(
)[
]
By virtue of (8), we obtain from (24) that
]
[
[
[
(24)
]
]
]
[
(
]
)[
[
]
]
[
(
]
)[
[
]
]
[
(
Putting
]
)[
[
]
]
in (25) and taking summation over
(25)
we get
(26)
for all vector fields
Replacing
for any vector field
i.e.,
by in (26), we get
where
From (27),
being the vector field associated to the
(27)
form
we can state the following theorem:
Theorem 3.2 In a Pseudo projective
Sasakian manifold
the characteristic vector
field and the vector field associated to the
form are co-directional and the
form is given
by (27).
4. On a
Manifold
On a
dimensional Locally Pseudo Projective
Recurrent Sasakian
dimensional Sasakian Manifold Ricci tensor and curvature tensor has the following form
(
)
(
)
)[
(
(
(28)
]
)[
]
(29)
Taking covariant differentiation of (29), we get
[
(
)[
]
Taking
orthogonal to and using (5) and (6), we get
312
(30)
A. Singh, R. K. Pandey, A. Prakash , S. Khare / J. Math. Computer Sci. 14 (2015), 309-314
[
]
)[
(
]
(31)
from (31) it follows that
[
orthogonal to and using (1) and (2) in (32), we get
Now, taking
[
[
Using (28)in (34) and then taking
Now, applying
]
orthogonal to
(34)
we get
][
]
(35)
to the both side of (35), we get
[
[
[
orthogonal to
]
] is a scalar, since
[
]
(36)
]
]
[
, we get
̃
Putting
in (38), where
of the manifold and taking summation over
̃
[
][
[
̃
where
]
̃)
̃)
(
Using (13), (33), (1) in (36), we obtain
taking
(33)
we get
̃)
[
(
(32)
]
Differentiating covariantly (14) with respect to W
(
]
]
]
[
(37)
(38)
is an orthonormal basis of the tangent space at any point
we obtain
[
]
is a non-zero
form. Then, by Schur’s theorem will be
a constant on the manifold Hence, we can state the following theorem:
Theorem 4.1 On a
dimensional locally Pseudo-projective
projective curvature tensor is of the form of constant curvature.
recurrent Sasakian manifold, pseudo-
Acknowledgements. The authors are thankful to Professor S. Ahmad Ali Dean School of Applied
Sciences, B.B.D. University, Lucknow, providing suggestions for the improvement of this paper.
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