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International Journal of Statistics and Applied Mathematics 2021; 6(5): 59-61
ISSN: 2456-1452
Maths 2021; 6(5): 59-61
© 2021 Stats & Maths
www.mathsjournal.com
Received: 25-07-2021
Accepted: 27-08-2021
E Nafula
School of Mathematics,
University of Nairobi, Kenya
Stephen K Moindi
School of Mathematics,
University of Nairobi, Kenya
Peter W Njori
School of Pure and Applied
Schiences, Kirinyaga University,
Kenya
Corresponding Author:
E Nafula
School of Mathematics,
University of Nairobi, Kenya
A study of
curvature tensor in para Kenmotsu
manifolds
E Nafula, Stephen K Moindi and Peter W Njori
Abstract
The object of the present paper is to study certain curvature conditions in Para Kenmotsu manifolds
admitting a
7
-curvature tensor.
Keywords: para Kenmotsu manifold,
7
-curvature tensor, symmetric, semi-symmetric, and
7
-flat,
recurrent.
1. Introduction
A manifold
is called an almost para-contact metric manifolds if there exists in
a (1,1)
tensor field ϕ, a vector field ξ, and a 1-form η such that
(1.1) () = 1,
2
() = − ()
(1.2) (, ) = (), (, ) = (, ) − ()()
(1.3) ξ = 0, η(ϕX) = 0, =−1
Where g is the Riemmannian metric.
If in addition the manifold
satisfies
(1.4) (∇
)() − (∇
) = 0
(1.5) (∇
∇
) = [−(, ) + ()()]() + [−(, ) + ()()]()
(1.6) ∇
=
2
= − ()
Then it is called a Para-Kenmotsu manifold or briefly P-Kenmostu manifold.
It is well known that is a P-Kenmotsu manifold,
(1.7) (, ) = −( − 1)()
(1.8) (
)(, ) = 2(, ) + 2( − 1)()()
Where is the Ricci tensor of type (0,2) and denotes the Lie derivative.
2.
Curvature Tensor in Para Kenmotsu Manifold
Mishra and Pokhariyal
[2]
gave the definition of
7
curvature tensor as
(2.1)
7
= (, ) +
1
−1
[(, ) − (, )]
Or
(2.2)
7
′
(, , , ) =
′
(, , , ) +
1
−1
[(, )(, ) − (, )(, )
Definition 2.1 A Para Kenmotsu manifold is said to be flat if the Riemannian curvature tensor
vanishes identically i.e. (, ) = 0
Definition 2.2 A Para Kenmotsu manifold is said to be
7
flat if
7
curvature tensor vanishes
identically i.e.
7
(, ) = 0
Theorem 2.1 A
7
flat Para Kenmotsu manifold is a flat manifold