Article
Kyungpook Mathematical Journal 2018; 58(2): 347-359
Published online June 23, 2018
Copyright © Kyungpook Mathematical Journal.
On Generalized φ -recurrent Kenmotsu Manifolds with respect to Quarter-symmetric Metric Connection
Shyamal Kumar Hui and Richard Santiago Lemence*
Department of Mathematics, The University of Burdwan, Burdwan –713104, West Bengal, India, e-mail: shyamal_hui@yahoo.co.in, Institute of Mathematics, College of Science, University of the Philippines, Diliman, Quezon City 1101, Philippines, e-mail: rslemence@math.upd.edu.ph
Received: March 25, 2015; Accepted: March 21, 2018
Abstract
A Kenmotsu manifold
for all
for any vector field
Keywords: generalized φ-recurrent, generalized φ Ricci-recurrent, Kenmotsu manifold, η-Einstein manifold, quarter-symmetric metric connection
1. Introduction
Tanno, in [43], classified connected almost contact metric manifolds whose automorphism groups possess the maximum dimension. For such a manifold, the sectional curvature of plane sections containing
As a generalization of both Sasakian and Kenmotsu manifolds, In [29], Oubiña introduced the notion of trans-Sasakian manifolds, which are closely related to the locally conformal Kähler manifolds. A trans-Sasakian manifold of type (0, 0), (
The study of Riemann symmetric manifolds began with the work of Cartan (see [8]). A Riemannian manifold (
During the last five decades, the notion of locally symmetric manifolds has been weakened by many authors in several ways to a different extent such as recurrent manifold by Walker [46], semisymmetric manifold by Szabó [41], pseudosymmetric manifold in the sense of Deszcz [14], pseudosymmetric manifold in the sense of Chaki [9], generalized recurrent manifold by Dubey [16].
A Riemannian manifold (
where
for all
A Riemannian manifold is said to be Ricci symmetric if its Ricci tensor
Again, the notion of generalized Ricci-recurrent manifolds has been introduced and studied by De, Guha and Kamilya [11]. A Riemannian manifold (
where
As a weaker version of local symmetry, the notion of locally
In [30], Özgür studied generalized recurrent Kenmotsu manifolds. Generalizing the notion of Özgür [30], and De, Yildiz and Yaliniz [13], Basari and Murathan [5] introduced the notion of generalized
Definition 1
A Kenmotsu manifold
for all
In particular, if
Definition 2
A Kenmotsu manifold
for any vector field
In particular if
Friedmann and Schouten, in [17], introduced the notion of semisymmetric linear connection on a differentiable manifold. Then in 1932 Hayden [19] introduced the idea of metric connection with torsion on a Riemannian manifold. A systematic study of the semisymmetric metric connection on a Riemannian manifold has been given by Yano in 1970 [47]. In 1975, Golab introduced the idea of a quarter symmetric linear connection in differentiable manifolds.
A linear connection ∇̄ in an
where
for all
Motivated by the above studies the present paper deals with the study of generalized
2. Preliminaries
A smooth manifold (
for all vector fields
An almost contact metric manifold
where ∇ denotes the Riemannian connection of
In a Kenmotsu manifold, the following relations hold [24]:
for any vector field
Let
where
and
From (
Using (
Hence a quarter symmetric metric connection ∇̄ in a Kenmotsu manifold is given by
If
From (
where
Also from (
where
From (
Again from (
Definition 3
A Kenmotsu manifold
where
3. Generalized φ -recurrent Kenmotsu Manifolds with respect to Quarter Symmetric Metric Connection
Definition 4
A Kenmotsu manifold
for any vector field
In particular if
We now consider a Kenmotsu manifold
from which it follows that
Taking an orthonormal frame field and then contracting (
Using (
By virtue of (
This leads to the following:
Theorem 1
Setting
In view of (
We know that
Using (
By virtue of (
Contracting (
This leads to the following:
Theorem 2
If, in particular,
which implies that the manifold under consideration is
Corollary 1
([22])
In view of (
Using (
for arbitrary vector fields
This leads to the following:
Theorem 3
We now take a generalized
it follows from (
From (
Theorem 4
4. Generalized φ -Ricci Recurrent Kenmotsu Manifolds with respect to Quarter Symmetric Metric Connection
Definition 5
A Kenmotsu manifold
for any vector field
In particular, if
let us take a Kenmotsu manifold
from which it follows that
Putting
This leads to the following:
Theorem 5
In particular if
which implies that the manifold under consideration is
Corollary 2
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