We inaugurate distinct results associated to almost pseudo symmetric K ̈hler manifolds gratifying the condition of having a special type of semi-symmetric nonmetric connection. 010 Mathematics Subject Classification: 53B15, 53C15, 53C21, 53C25, 53C55.

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International Journal of Mechanical Engineering and Technology (IJMET)
Volume 10, Issue 02, February 2019, pp. 248–254, Article ID: IJMET_10_02_027
Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=10&IType=02
ISSN Print: 0976-6340 and ISSN Online: 0976-6359
© IAEME Publication Scopus Indexed
A STUDY ON ALMOST PSEUDO SYMMETRIC
K ̈ HLER MANIFOLD
M. M. Praveena and Harshitha Urs A S
Department of Mathematics,
Sri Venkateshwara College of Engineering,
Banagaluru, Karnataka-562157.
C. S. Bagewadi
Department of Mathematics,
Kuvempu University,
Shankaraghatta-577 451,
ABSTRACT
We inaugurate distinct results associated to almost pseudo symmetric K ̈hler
manifolds gratifying the condition of having a special type of semi-symmetric non-
metric connection. 010 Mathematics Subject Classification: 53B15, 53C15, 53C21,
53C25, 53C55.
Key Words: Semi-symmetric non-metric connection, K ̈hler Manifold, Almost pseudo
symmetric manifold, Projective flat manifold, Ricci soliton.
Cite this Article: M. M. Praveena, Harshitha Urs A S and C. S. Bagewadi, A Study o
st s u tr r , International Journal of Mechanical
Engineering and Technology, 10(2), 2019, pp. 248–254
http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=10&IType=02
1. INTRODUCTION
The Riemannian symmetric spaces were introduced by French mathematician Cartan during
Nineteenth century and play a main tool in differential geometry. A Riemannian manifold is
Called locally symmetric [3] if , where is the Riemannian curvature tensor of
During the last five decades the notion of locally symmetric manifolds has been studied by
Many authors in several ways to a different extent such as semi-symmetric manifolds, weakly
symmetric manifolds, pseudo symmetric manifolds, almost pseudo symmetric manifold and
Almost pseudo Ricci-symmetric manifolds.
A Riemannian manifold is said to be almost pseudo symmetric manifold [8] if its
curvature tensor satisfies the condition

2. M. M. Praveena, Harshitha Urs A S and C. S. Bagewadi
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[ ]
(1.1)
Where A, B are two non-zero 1-forms defined by
(1.2)
For all vector fields denotes the operator of covariant differentiation with respect to the
Metric
The notions of almost pseudo symmetric and almost pseudo Ricci-symmetric manifolds
were introduced by De and Gazi [8] and Chaki and Kawaguchi [5] respectively. These are
extended class of pseudo symmetric and pseudo Ricci-symmetric manifolds introduced by
Chaki [4] and Chaki and Kawaguchi [5] respectively. Here we note that the notion of pseudo
symmetry in the sense of Chaki is different from that of Deszcz [11]. However, pseudo
symmetry defined by Chaki will be pseudo symmetry defined by Deszcz if and only if the non-
zero 1-form associated with pseudo symmetric is closed. It may be mentioned that the almost
pseudo symmetric manifold is not a particular case of a weakly symmetric manifold introduced
by Tamassy and Bink [15]. Tamassy et. al., [16] found interesting results on weakly symmetric
and weakly Ricci-symmetric K ̈hler manifolds in 2000. Also Shaikh et. al., [14] discussed on
quasi-conformally flat almost pseudo Ricci-symmetric manifolds in 2010. Chathurvadi and
Pandey [6] studied semi-symmetric non-metric connections in K ̈hler manifolds. Then in 2015,
Chathurvadi and Pandey [7] studied special type of semi-symmetric metric connection in a
weakly symmetric K ̈hler manifold. Based on the above work in this paper, we have made an
attempt to study admitting special type of semi-symmetric non-metric connection on almost
pseudo symmetric K ̈hler manifolds, projective flat almost pseudo symmetric K ̈hler
manifolds, almost pseudo symmetric K ̈hler manifolds with parallel projective curvature tensor
and almost pseudo projective symmetric K ̈hler manifolds.
2. PRELIMINARIES
A K ̈hler manifold is an n (even)-dimensional manifold, with a complex structure and a
positive-definite metric which satisfies the following conditions;
(2.1)
Where means covariant derivative according to the Levi-Civita connection.
The formulas [2]
(2.4.)
Are well known for a K ̈hler manifold.
If is the Levi-Civita connection of the manifold then a semi-symmetric non-metric
connection is given by [1]
̃ (2.5.)
For every vector field . It is called a special type of semi symmetric non-metric
connection
If the torsion tensor and the curvature tensor ̃ of the connection ̃ satisfy the following
conditions;

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(̃ ) (2.7)
And ̃
N. S. Agashe [1] proved in 1992 that the curvature tensor ̃ of the semi-symmetric non-
metric
Connection ̃ defined by is given by
̃
Where is a tensor field of type defined by?
(2.10)
In 2009 A. De [10] proved that almost pseudo symmetric manifold admitting a special type
of semi-symmetric non-metric connection is given by
(2.11)
[ ] (2.12)
In a paper [5] Chaki and T. Kawaguchi introduced a type of non-flat Riemannian manifold
whose Ricci tensor of type satisfies the condition , where and have
the meaning already stated. Such a manifold was called an almost pseudo Ricci-symmetric
manifold.
Putting and taking sum over we have
Putting (for all unit vector fields) in (2.13) we get
Putting and taking sum over we have
(2.15)
3. ALMOST PSEUDO SYMMETRIC K ̈ HLER MANIFOLDS
In this section we suppose that is an almost pseudo symmetric and K ̈hler manifold.
Then we can be written as
Taking the covariant derivative of , we get
Using in , we get
(3.3)
Putting after simplification we get
(3.4.)
And by replacing in (3.4) and summing over , we infer
(3.5.)
Using in we get,
This implies either , or . Now if we take then the connection ̃
defined by will change. Hence never vanishes. Hence we can state the following
result;

4. M. M. Praveena, Harshitha Urs A S and C. S. Bagewadi
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Theorem 3.1. Let be an almost pseudo symmetric K ̈hler manifold allowing a special
type of semi-symmetric non-metric connection ̃, then is a manifold of zero scalar curvature
with respect to Levi-Civita connection .
A. De proved in his paper [10], if is an almost pseudo symmetric manifold admitting a
special type of semi-symmetric non-metric connection ̃ then the manifold is of constant
curvature and almost pseudo Ricci-symmetric manifold with non-zero scalar curvature. In a
paper [13] we proved, let be an almost pseudo Ricci-symmetric K ̈hler manifold and almost
pseudo Bochner Ricci-symmetric K ̈hler manifold then it is Ricci flat and Einstein manifold
respectively. Also we proved, let be a Ricci soliton in an almost pseudo Bochner Ricci
symmetric K ̈hler manifold. Then is solenoidal if and only if it is shrinking, steady and
expanding depending upon the sign of scalar curvature. So we conclude the following
theorems.
Theorem 3.2. Let be an almost pseudo symmetric K ̈hler manifold initiating a special
type of semi-symmetric non-metric connection ̃, then it is Ricci flat.
Theorem 3.3. An almost pseudo Bochner symmetric K ̈hler manifold admitting a special
type of semi-symmetric non-metric connection ̃ is an Einstein manifold.
Corollary 3.1. Let be a Ricci soliton in an almost pseudo Bochner symmetric
K ̈hler manifold admitting a special type of semi-symmetric non-metric connection ̃. Then is
solenoidal if and only if it is shrinking, steady and expanding depending upon the sign of scalar
curvature.
4. PROJECTIVE FLAT ALMOST PSEUDO SYMMETRIC K ̈ HLER
MANIFOLDS
Now the projective curvature tensor ̃ of connection ̃ is given by
̃ [ ̃ ̃ ] (4.1)
In 2016 U. C. De [9] proved that, if a Riemannian manifold admits a type of the semi-
symmetric non-metric connection whose torsion tensor is pseudo symmetric, then the projective
curvature tensor with respect to the semi-symmetric non-metric connection is equal to the
projective curvature tensor with respect to the Levi-Civita connection i.e.,
(4.2.)
If the manifold is projective flat with respect to ̃ then the manifold will be projective flat
with respect to the connection . i.e.
̃
Now equation implies,
[ ] (4.4)
Using in , we have
[ ] (4.5)
Hence from the above equation, we conclude;
Theorem 4.4. If is a projective flat Riemannian manifold with respect to a special type of
semi-symmetric non-metric connection ̃, then it is the manifold of constant curvature. Equation
and are identical, and therefore Theorems can be stated as follows;
Theorem 4.5. Let be an almost pseudo symmetric projective fat K ̈hler manifold
allowing

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a special type of semi-symmetric non-metric connection ̃, then is a manifold of zero
scalar curvature with respect to Levi-Civita connection .
5. ALMOST PSEUDO SYMMETRIC K ̈ HLER MANIFOLDS WITH
PARALLEL PROJECTIVE CURVATURE TENSOR
Assume that the projective curvature of an almost pseudo symmetric K ̈hler manifold is
parallel i.e., .
Using the properties of K ̈hler manifolds and using the equation we can be
expressed as
[ ] (5.1.)
in , where ) is an orthonormal basis of the
tangent space at each point of the manifold and taking summation over , we get
Taking covariant differentiation of and our assumption yields
In view of the covariant derivative can be expressed in the following form
[ ]
Using in we obtain,
Theorem 5.6. An almost pseudo symmetric K ̈hler manifold initiate a special type of
semi-symmetric non-metric connection ̃, then is projectively symmetric if and only if it is
locally symmetric.
6. ALMOST PSEUDO PROJECTIVE SYMMETRIC K ̈ HLER
MANIFOLDS:
Definition 6.1. A Riemannian manifold is called almost pseudo projective symmetric
manifold if its Projective curvature tensor of type is satisfies the condition
[ ]
where A, B are two non-zero 1-forms and is given in .
Setting in and taking summation over , we get
[ ]
[ ][ ]
[ ] [ ]
[ ]
[ ]

6. M. M. Praveena, Harshitha Urs A S and C. S. Bagewadi
http://www.iaeme.com/IJMET/index.asp 253 editor@iaeme.com
Again putting in and taking summation over , we get
[ ] [ ] [ ] [ ]
Using equation in we get
[ ]
Theorem 6.7. The scalar curvature tensor r of an almost pseudo projective symmetric
K ̈hler
manifold allowing a special type of semi-symmetric non-metric connection ̃ satisfies the
following relation
[ ]
Let us consider an almost pseudo projective symmetric K ̈hler manifold of constant scalar
curvature. Thus from and not equal to , we get
Using equation in equation then we get
Theorem 6.8. If the two associated 1-forms are linearly independent in an almost pseudo
projective symmetric K ̈hler manifold admitting a special type of semi-symmetric non-metric
connection ̃, then its non-zero constant scalar curvature.
Theorem 6.9. If an almost pseudo projective symmetric K ̈hler manifold admitting a
special type of semi-symmetric non-metric connection ̃, then it is reduced to a special type of
projective symmetric Kahler manifold provided that non zero constant scalar curvature.
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