Article
Kyungpook Mathematical Journal 2018; 58(2): 389-398
Published online June 23, 2018
Copyright © Kyungpook Mathematical Journal.
From the Eisenhart Problem to Ricci Solitons in Quaternion Space Forms
Mundalamane Manjappa Praveena and Channabasappa Shanthappa Bagewadi*
Department of Mathematics, Sri Venkateshwara College of Engineering, Bengaluru-562157, Karnataka, India, e-mail: mmpraveenamaths@gmail.com.in, Department of Mathematics, Kuvempu University, Shankaraghatta-577451, Shimoga, Karnataka, India, e-mail: prof_bagewadi@yahoo.co.in
Received: June 19, 2016; Accepted: April 25, 2018
Abstract
In this paper we obtain the condition for the existence of Ricci solitons in non-flat quaternion space form by using Eisenhart problem. Also it is proved that if (
Keywords: quaternion Kä,hler manifolds, quaternion space form, parallel second order covariant tensor field, Einstein space, Ricci soliton
1. Introduction
During 1982, Hamilton [7] made the fundamental observation that Ricci flow is an excellent tool for simplifying the structure of the manifold. It is defined for Riemannain manifolds of any dimension. It is a process which deforms the metric of a Riemannian manifold analogous to the diffusion of heat there by smoothing out the regularity in the metric. It is given by
where
Let
If
A Ricci soliton is a generalization of an Einstein metric and is defined on a Riemannian manifold (
where
In 1923, Eisenhart [6] proved that if a positive definite Riemannian manifold (
Motivated by these ideas, in this paper, we study Ricci solitons of quaternion space form which is a generalization of complex space form using L.P. Eisenhart problem. The paper is organized as follows: The section two consists of definitions, notions and basic results of quaternion space form. The third section deals with parallel symmetric second order covariant tensor and Ricci soliton in a non-flat quaternion space form. In section 4 we give semi-symmetry on quaternion space forms.
2. Preliminaries
Let
where
Further the condition
Let
A quaternion Kaehlerian manifold is called a
The formulae [13]
are well known for a quaternion Kaehler manifold where
Symmetry of the manifold is one of the most important property among all its geometrical properties. Symmetry of the manifold basically depends on curvature tensor and the Ricci tensor of the manifold.
Definition 2.1
([5, 12]) An
Definition 2.2
We shall give the concepts of symmetrization and anti-symmetri zation as below:
If
3. Parallel Symmetric Second Order Covariant Tensor and Ricci Soliton in a Non-flat Quaternion Space Forms
Let
we obtain the relation[11]:
Plugging the value of
where
where
Replacing
Again changing
Multiply the
Substituting
Likewise; changing
Again replacing
Substituting (
Substituting
By summing (
where
The above equation implies
where Ω1 =
Now as
Theorem 3.1
Corollary 3.1
Corollary 3.2
If
Remark 3.1
The following statements for non-flat quaternion space form are equivalent:
(1) Einstein.
(2) locally Ricci symmetric.
(3) Ricci semi-symmetric that is
R ·S = 0.
The statements (1) → (2) → (3) are trivial. Now, we prove the statement (3) → (1). Here
where
Using
If
Therefore we conclude the following.
Lemma 3.1
Corollary 3.3
From Theorem 3.1 and Corollary 3.2, we have
Simplifying the above we have
By virtue of (
Suppose
Therefore
Lemma 3.2
Using
Putting
The above equation implies
If
Thus
4. Semi-symmetric Quaternion Space Forms
Using Definition 2.1 we have
This implies
Taking inner product with tangent vector
Use
This implies,
where
That is quaternion space form is an Einstein manifold.
Hence we can state following result;
Theorem 4.1
Using
Putting
If
Thus we can state the following from (
Lemma 4.1
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