Science and Technology Journal,
Vol. 3
Issue: II
ISSN: 2321-3388
Some Results on α–Kenmotsu Manifolds
Rajesh Kumar
Department of Mathematics
Pachhunga University College, Aizawl–796001, Mizoram, India
E-mail: rajesh_mzu@yahoo.com
Abstract—The object of the paper is to study
-Kenmotsu manifolds which can be derived from almost contact
Riemannian manifolds satisfying certain conditions. We first examine the projective flat -Kenmotsu manifold and
give some interesting results. Next, it is shown that an -Kenmotsu manifold is a manifold of constant curvature in
both the case of Einstein -Kenmotsu manifold satisfying
as well as -Kenmotsu manifold satisfying
, where is the projective curvature tensor. It is also proved that if -Kenmotsu manifold satisfying
then the manifold is projectively flat.
Keywords: Almost contact metric manifold, -Kenmotsu manifold, Einstein manifold, scalar curvature and Projective
curvature.
INTRODUCTION
Let
be an -dimensional
-manifold admitting a
vector ield , a tensor ield , and a 1-form such that
,
and
.
,
there exists a metric tensor
where denotes the operator of covariant differentiation
with respect to the Riemannian connection , then
(2)
called an
satisfying
-Kenmotsu manifold [8, 9]. If
and
-Kenmotsu manifold is called a Kenmotsu manifold and
if
is constant then it is called a homothetic Kenmotsu
(4)
by many authors [ 10, 11, 12].
Further, on an -Kenmotsu manifold
, the following
relations hold
for any vector ield and , then
is called an almost
contact metric manifold with structure
[1].
In 1972 Kenmotsu [2] introduced a new class of almost
contact Riemannian manifolds which are now a days called
Kenmotsu manifolds. It is well known that odd dimensional
spheres admit Sasakian structures where as odd dimensional
hyperbolic spaces cannot admit Sasakian structure, but
have so-called Kenmotsu structure. Kenmotsu manifolds are
normal (noncontact) almost contact Riemannian manifolds.
Kenmotsu manifolds have been studied by many authors [3,
4, 5, 6, 7].
Also, if the manifold
,
is
, then a
manifold. Recently -Kenmotsu manifold have been studied
(3)
,
(7)
And
(1)
then such a manifold is called an almost contact manifold.
If in
(6)
,
satis ies the following relations
(5)
,
(9)
,
(10)
,
.
(8)
,
(11)
,
(12)
(13)
The projective curvature tensor on a Riemannian
manifold is given by [13]
(14)
Kumar
PROJECTIVELY FLAT –KENMOTSU
MANIFOLD
This shows that either
or
Let us suppose that in an -Kenmotsu manifold
,
.
Now, if
(15)
,
then from (3), we get
then it follows from (14) that
,
(16)
which is not possible.
Or
Therefore
which gives from (21)
(17)
Taking
and (13), we get
.
in (17) and then using (1), (4), (12)
.
Theorem 3. A projectively lat Einstein -Kenmotsu
manifold
is the manifold of constant curvature, provided
.
(18)
Thus, we have the theorem.
Theorem 1. A projective lat -Kenmotsu manifold is an
Einstein manifold.
From (18), we have
,
where
Thus, we have the following theorem.
AN EINSTEIN -KENMOTSU MANIFOLD
SATISFYING
In this section we assume that
(19)
.
.
From (14) and (20), we have
Contracting (19) with respect to , we have
.
It can be written as
Hence, we have the following result.
Theorem 2. The scalar curvature of a projectively lat
-Kenmotsu manifold
is constant.
(24)
Let an -Kenmotsu manifold is an Einstein manifold,
then its Ricci tensor is of the form
,
where
Putting
in (24) and using (8), we get
(20)
.
is a constant. Then (16) reduces to
Putting
(21)
(25)
in (25), we get
(26)
or
Again putting
.
Taking
(12), we get
(23)
in (25), we get
.
(22)
Now,
in (22) and then using (1), (4) and
180
(27)
Some Results on α–Kenmotsu Manifolds
In view of (23), we get
AN -KENMOTSU MANIFOLD SATISFYING
From (8) and (14), we have
.
Therefore,
(32)
Putting
.
in (32), we get
. (33)
From this it follows that
Again putting
in (32), we have
.
(34)
Now, we have
,
(28)
where
On putting
.
in (26), we get
Let
, then we have
.
,
Therefore,
(29)
be an orthonormal basis of the
Let
of
tangent space at any point. Then the sum for
yields
the relation (29) for
.
.
From this it follows that
(30)
Using (25) and (30), it follows from (28) that
,
.
(35)
where
(31)
.
Let
be an orthonormal basis of the
tangent space at any point. Then the sum for
of
the relation (35) for
yields
From (24) and (31), we get
,
.
Therefore, we can state.
(36)
In virtue of (34) and (36), we have
Theorem 4. If in an Einstein -Kenmotsu manifold the
hold, then the manifold is of
relation
.
constant curvature, provided
.
Putting
181
in (37), we get
(37)
Kumar
REFERENCES
(38)
Hence (37) becomes
1.
.
(39)
Using (32), (34) and (39) it follows from (35) that
2.
3.
.
4.
Hence, we have the following theorems
5.
Theorem 5. If in an -Kenmotsu manifold the relation
hold, then the manifold is of scalar
curvature
.
6.
7.
Theorem 6. If in an -Kenmotsu manifold the relation
hold, then the manifold is projectively lat.
8.
From theorem 1 and theorem 6, we conclude the
following theorem.
9.
Theorem 7. An
-Kenmotsu manifold satisfying
is an Einstein manifold of negative
curvature
10.
11.
ACKNOWLEDGEMENT
12.
The author is thankful to the referee for his valuable
suggestions towards the improvement of this paper.
13.
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