Article
Kyungpook Mathematical Journal 2017; 57(2): 309-318
Published online June 23, 2017
Copyright © Kyungpook Mathematical Journal.
Almost Ricci Soliton and Gradient Almost Ricci Soliton on 3-dimensional f -Kenmotsu Manifolds
Pradip Majhi
Department of Mathematics, University of North Bengal, Raja Rammohunpur, Darjeeling, Pin-$734013$, West Bengal, India
Received: May 8, 2016; Accepted: December 29, 2016
Abstract
The object of the present paper is to study almost Ricci solitons and gradient almost Ricci solitons in 3-dimensional
Keywords: $f$-Kenmotsu manifold, Ricci soliton, gradient Ricci soliton, almost Ricci soliton, gradient almost Ricci soliton
1. Introduction
The study of almost Ricci soliton was introduced by Pigola et. al. [18], where essentially they modified the definition of Ricci soliton by adding the condition on the parameter
where
where ∇2
Moreover, if the vector field
In the direction to understand the geometry of almost Ricci soliton, Barros and Ribeiro Jr. proved in [2] that a compact gradient almost Ricci soliton with nontrivial conformal vector field is isometric to a Euclidean sphere. In the same paper they proved an integral formula for compact case, which was used to prove several rigidity results, for more details see [2].
The existence of Ricci almost soliton has been confirmed by Pigola et. al. [18] on some certain class of warped product manifolds. Some characterization of Ricci almost soliton on a compact Riemannian manifold can be found in ([1], [2], [3]). It is interesting to note that if the potential vector field
In [6], authors studied Ricci solitons and gradient Ricci solitons on 3-dimensional normal almost contact metric manifolds. In [10] authors studied compact Ricci soliton. Beside these, A. Ghosh [12] studied
The present paper is organized as follows:
After preliminaries, in section 3 we study almost Ricci soliton in 3-dimensional
2. Preliminaries
Let
For further use, we recall the following definitions ([4], [11], [19]). The manifold
normal if the almost complex structure defined on the product manifold
M ×ℝ is integrable (equivalently [φ ,φ ] + 2dη ⊗ξ = 0),almost cosymplectic if
dη = 0 andd Φ = 0,cosymplectic if it is normal and almost cosymplectic (equivalently, ∇
φ = 0, ∇ being covariant differentiation with respect to the Levi-Civita connection).
The manifold
is cosymplectic (respectively, almost cosymplectic).
Olszak and Rosca [16] studied normal locally conformal almost cosymplectic manifold. They gave a geometric interpretation of
By an
Let
for any vector fields
where
For an
The condition
In a 3-dimensional Riemannian manifold, we always have
In a 3-dimensional
where
From (
and (
Example.([8])
We consider the three-dimensional manifold
are linearly independent at each point of
Let
Then using linearity of
for any
The Riemannian connection ∇ of the metric tensor
Using (
Hence
From the above it follows that the manifold satisfies ∇
3. Almost Ricci Soliton
In this section we consider almost Ricci solitons on 3-dimensional
Using (
Putting
Putting
Putting the value of
Applying
Taking wedge product of (
Since
Using (
In view of (
Theorem 3.1
The converse of the above theorem is not true, in general. However if we take
Let
Now using (
Therefore
From
Theorem 3.2
Now let
Now, in view of (
Putting
Assuming that
Theorem 3.3
4. Gradient Almost Ricci Soliton
This section is devoted to study 3-dimensional
where
Differentiating (
Similarly, we get
and
In view of (
We get from (
Differentiating (
In view of (
This implies
Also, we have from (
In view of (
Assuming that the scalar curvature
Applying
Using (
This implies
Theorem 4.1
Acknowledgements
The author is thankful to the referee for hisher valuable comments towards the improvement of this article.
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