Articles
Kyungpook Mathematical Journal -0001; 56(3): 951-964
Published online November 30, -0001
Copyright © Kyungpook Mathematical Journal.
On Concircular Curvature Tensor with respect to the Semi-symmetric Non-metric Connection in a Kenmotsu Manifold
Abdul Haseeb
Department of Mathematics, Faculty of Science, Jazan University, Jazan, Kingdom of Saudi Arabia
Received: May 1, 2015; Accepted: July 25, 2016
Abstract
- Abstract
- 1. Introduction
- 2. Kenmotsu Manifolds
- 3. Curvature Tensor on a Kenmotsu Manifold with respect to the Semi-symmetric Non-metric Connection
- 4. Concircular Curvature Tensor on a Kenmotsu Manifold with respect to the Semi-symmetric Non-metric Connection
- 5. Kenmotsu Manifolds with respect to the Semi-symmetric Non-metric Connection Satisfying the Curvature Condition
- 6. Concircularly flat and
ξ -concircularly Flat Kenmotsu Manifolds with respect to the Semi-symmetric Non-metric Connection - 7. Concircularly symmetric Kenmotsu manifolds with respect to the semi-symmetric non-metric connection
- Acknowledgement
- References
The objective of the present paper is to study some new results on concircular curvature tensor in a Kenmotsu manifold with respect to the semi-symmetric non-metric connection.
Keywords: Kenmotsu manifold, semi-symmetric non-metric connection, $eta$-Einstein manifold, concircular curvature tensor, concircularly flat manifold, concircularly symmetric manifold.
1. Introduction
- Abstract
- 1. Introduction
- 2. Kenmotsu Manifolds
- 3. Curvature Tensor on a Kenmotsu Manifold with respect to the Semi-symmetric Non-metric Connection
- 4. Concircular Curvature Tensor on a Kenmotsu Manifold with respect to the Semi-symmetric Non-metric Connection
- 5. Kenmotsu Manifolds with respect to the Semi-symmetric Non-metric Connection Satisfying the Curvature Condition
- 6. Concircularly flat and
ξ -concircularly Flat Kenmotsu Manifolds with respect to the Semi-symmetric Non-metric Connection - 7. Concircularly symmetric Kenmotsu manifolds with respect to the semi-symmetric non-metric connection
- Acknowledgement
- References
In 1969, S. Tanno classified connected almost contact metric manifolds whose automorphism groups possess the maximum dimension [14]. For such a manifold, the sectional curvature of plane sections containing
In 1924, the idea of semi-symmetric linear connection on a differentiable manifold was introduced by A. Friedmann and J. A. Schouten [8]. In 1930, E. Bartolotti [2] gave a geometrical meaning of such a connection. In 1932, H. A. Hayden [10] introduced semi-symmetric metric connection in Riemannian manifolds and this was studied systematically by K. Yano [15]. The semi-symmetric non-metric connection in a Riemannian manifold have been studied by N. S. Agashe and M. R. Chafle [1], U. C. De and S. C. Biswas [7], L. S. Das, M. Ahmad and A. Haseeb [6], S. K. Chaubey and R. H. Ojha [5] and others. Recently, A. Haseeb, M. A. Khan and M. D. Siddiqi studied an
Let ∇ be a linear connection in an
The connection ∇ is said to be
A linear connection ∇ is said to be
where
Semi-symmetric connections play an important role in the study of Riemannian manifolds. There are various physical problems involving the semi-symmetric metric connection. For example, if a man is moving on the surface of the earth always facing one definite point, say Jaruselam or Mekka or the North pole, then this displacement is semi-symmetric and metric [8].
Motivated by the above studies, in this paper we obtain some new results on concircular curvature tensor in a Kenmotsu manifold with respect to the semi-symmetric non-metric connection. The paper is organized as follows : In Section 2, we give a brief introduction of a Kenmotsu manifold and define semi-symmetric non-metric connection. In Section 3, we find the curvature tensor, Ricci tensor and scalar curvature in a Kenmotsu manifold with respect to the semi-symmetric non-metric connection. Section 4 deals with the study of concircular curvature tensor in a Kenmotsu manifold with respect to the semi-symmetric non-metric connection and we establish the relation between concircular curvature tensors with respect to the connections ∇ and ∇̄. In Section 5, we show that the Kenmotsu manifold satisfying the condition
2. Kenmotsu Manifolds
- Abstract
- 1. Introduction
- 2. Kenmotsu Manifolds
- 3. Curvature Tensor on a Kenmotsu Manifold with respect to the Semi-symmetric Non-metric Connection
- 4. Concircular Curvature Tensor on a Kenmotsu Manifold with respect to the Semi-symmetric Non-metric Connection
- 5. Kenmotsu Manifolds with respect to the Semi-symmetric Non-metric Connection Satisfying the Curvature Condition
- 6. Concircularly flat and
ξ -concircularly Flat Kenmotsu Manifolds with respect to the Semi-symmetric Non-metric Connection - 7. Concircularly symmetric Kenmotsu manifolds with respect to the semi-symmetric non-metric connection
- Acknowledgement
- References
An
for arbitrary vector fields
An almost contact metric manifold
where ∇ denotes the Riemannian connection of
Also the following relations hold in a Kenmotsu manifold [12]:
where
Example
We consider the three dimensional manifold
which are linearly independent at each point of
Let
for any vector fields
Let ∇ be the Levi-Civita connection with respect to the metric
The Riemannian connection ∇ with respect to the metric
From above equation which is known as Koszul’s formula, we have
Using the above relations, for any vector field
for
Definition 2.1([16])
A Kenmotsu manifold
where
Definition 2.2([4])
The
where
Definition 2.3
A Kenmotsu manifold
for all vector fields
Definition 2.4
A Kenmotsu manifold
for all vector fields
A linear connection ∇̄ in
satisfies
Further, a semi-symmetric connection is called
Let
3. Curvature Tensor on a Kenmotsu Manifold with respect to the Semi-symmetric Non-metric Connection
- Abstract
- 1. Introduction
- 2. Kenmotsu Manifolds
- 3. Curvature Tensor on a Kenmotsu Manifold with respect to the Semi-symmetric Non-metric Connection
- 4. Concircular Curvature Tensor on a Kenmotsu Manifold with respect to the Semi-symmetric Non-metric Connection
- 5. Kenmotsu Manifolds with respect to the Semi-symmetric Non-metric Connection Satisfying the Curvature Condition
- 6. Concircularly flat and
ξ -concircularly Flat Kenmotsu Manifolds with respect to the Semi-symmetric Non-metric Connection - 7. Concircularly symmetric Kenmotsu manifolds with respect to the semi-symmetric non-metric connection
- Acknowledgement
- References
The curvature tensor
Using Equations (
where
is the Riemannian curvature tensor of the connection ∇.
Contracting
where
This gives
Contracting again
where
Lemma 3.1
By taking
Thus we are in a position to prove the following theorem.
Theorem 3.2
In order to prove the theorem (
Taking
Using (
By making use of (
This completes the proof.
4. Concircular Curvature Tensor on a Kenmotsu Manifold with respect to the Semi-symmetric Non-metric Connection
- Abstract
- 1. Introduction
- 2. Kenmotsu Manifolds
- 3. Curvature Tensor on a Kenmotsu Manifold with respect to the Semi-symmetric Non-metric Connection
- 4. Concircular Curvature Tensor on a Kenmotsu Manifold with respect to the Semi-symmetric Non-metric Connection
- 5. Kenmotsu Manifolds with respect to the Semi-symmetric Non-metric Connection Satisfying the Curvature Condition
- 6. Concircularly flat and
ξ -concircularly Flat Kenmotsu Manifolds with respect to the Semi-symmetric Non-metric Connection - 7. Concircularly symmetric Kenmotsu manifolds with respect to the semi-symmetric non-metric connection
- Acknowledgement
- References
Analogous to the Definition 2.2, the concircular curvature tensor
where
By making use of (
where
Interchanging
On adding (
From (
Equation (
Next, taking inner product of (
Using (
5. Kenmotsu Manifolds with respect to the Semi-symmetric Non-metric Connection Satisfying the Curvature Condition
- Abstract
- 1. Introduction
- 2. Kenmotsu Manifolds
- 3. Curvature Tensor on a Kenmotsu Manifold with respect to the Semi-symmetric Non-metric Connection
- 4. Concircular Curvature Tensor on a Kenmotsu Manifold with respect to the Semi-symmetric Non-metric Connection
- 5. Kenmotsu Manifolds with respect to the Semi-symmetric Non-metric Connection Satisfying the Curvature Condition
- 6. Concircularly flat and
ξ -concircularly Flat Kenmotsu Manifolds with respect to the Semi-symmetric Non-metric Connection - 7. Concircularly symmetric Kenmotsu manifolds with respect to the semi-symmetric non-metric connection
- Acknowledgement
- References
Now we consider a Kenmotsu manifold with respect to the semi-symmetric non-metric connection ∇̄ satisfying the condition
Then we have
By virtue of (
From (
which on using (
By using (
This implies that either the scalar curvature of
Thus we can state the following theorem.
Theorem 5.1
6. Concircularly flat and ξ -concircularly Flat Kenmotsu Manifolds with respect to the Semi-symmetric Non-metric Connection
- Abstract
- 1. Introduction
- 2. Kenmotsu Manifolds
- 3. Curvature Tensor on a Kenmotsu Manifold with respect to the Semi-symmetric Non-metric Connection
- 4. Concircular Curvature Tensor on a Kenmotsu Manifold with respect to the Semi-symmetric Non-metric Connection
- 5. Kenmotsu Manifolds with respect to the Semi-symmetric Non-metric Connection Satisfying the Curvature Condition
- 6. Concircularly flat and
ξ -concircularly Flat Kenmotsu Manifolds with respect to the Semi-symmetric Non-metric Connection - 7. Concircularly symmetric Kenmotsu manifolds with respect to the semi-symmetric non-metric connection
- Acknowledgement
- References
Firstly, we assume that the manifold
Taking inner product of the above equation with
In view of (
This implies that either the scalar curvature of
Putting
Taking
Hence we can state the following theorem.
Theorem 6.1
Secondly, we assume that the manifold
In view of (
Taking
Taking inner product of (
Now taking
By making use (
This implies that either the scalar curvature of
Hence we can state the following theorem.
Theorem 6.2
Next, taking
Making use of (
Since
Theorem 6.3
7. Concircularly symmetric Kenmotsu manifolds with respect to the semi-symmetric non-metric connection
- Abstract
- 1. Introduction
- 2. Kenmotsu Manifolds
- 3. Curvature Tensor on a Kenmotsu Manifold with respect to the Semi-symmetric Non-metric Connection
- 4. Concircular Curvature Tensor on a Kenmotsu Manifold with respect to the Semi-symmetric Non-metric Connection
- 5. Kenmotsu Manifolds with respect to the Semi-symmetric Non-metric Connection Satisfying the Curvature Condition
- 6. Concircularly flat and
ξ -concircularly Flat Kenmotsu Manifolds with respect to the Semi-symmetric Non-metric Connection - 7. Concircularly symmetric Kenmotsu manifolds with respect to the semi-symmetric non-metric connection
- Acknowledgement
- References
Using (
Now differentiating (
Making use of (
From (
In view of (
Now differentiating (
From (
from which we have
where
If the scalar curvature
Hence we can state the following theorem.
Theorem 7.1
Acknowledgement
- Abstract
- 1. Introduction
- 2. Kenmotsu Manifolds
- 3. Curvature Tensor on a Kenmotsu Manifold with respect to the Semi-symmetric Non-metric Connection
- 4. Concircular Curvature Tensor on a Kenmotsu Manifold with respect to the Semi-symmetric Non-metric Connection
- 5. Kenmotsu Manifolds with respect to the Semi-symmetric Non-metric Connection Satisfying the Curvature Condition
- 6. Concircularly flat and
ξ -concircularly Flat Kenmotsu Manifolds with respect to the Semi-symmetric Non-metric Connection - 7. Concircularly symmetric Kenmotsu manifolds with respect to the semi-symmetric non-metric connection
- Acknowledgement
- References
The author wishes to express his sincere thanks and gratitude to the referee for valuable suggestions towards the improvement of the paper.
References
- Abstract
- 1. Introduction
- 2. Kenmotsu Manifolds
- 3. Curvature Tensor on a Kenmotsu Manifold with respect to the Semi-symmetric Non-metric Connection
- 4. Concircular Curvature Tensor on a Kenmotsu Manifold with respect to the Semi-symmetric Non-metric Connection
- 5. Kenmotsu Manifolds with respect to the Semi-symmetric Non-metric Connection Satisfying the Curvature Condition
- 6. Concircularly flat and
ξ -concircularly Flat Kenmotsu Manifolds with respect to the Semi-symmetric Non-metric Connection - 7. Concircularly symmetric Kenmotsu manifolds with respect to the semi-symmetric non-metric connection
- Acknowledgement
- References
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