1. Introduction
Majority of present approaches to mathematical general relativity launch with the concept of a manifold. The standpoint of physics and relativity is to the investigation of manifolds with indefinite metrics. Several authors have studied manifold with indefinite matrices. Bejancu and Duggal [1] originated the concept of ò-Sasakian manifolds in 1993. De and Sarkar [2] pioneered (ò)-Kenmotsu manifolds and investigated some curvature conditions on it. Pandey and Tiwari [3] constructed the relation between semi-symmetric metric connection and Riemannian connection of (ò)-Kenmotsu manifolds and have studied several curvature conditions. The notion of (ò)-Para Sasakian Manifolds was pioneered by Tripathi et al. [4] in 2009.
The Riemannian symmetric spaces were introduced by French mathematician Carton during the nineteenth century and play a main tool in differential geometry. A Riemannian manifold is locally symmetric [5] , if
, where R 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 recurrent manifold by Walker [6] , semisymmetric manifold by Szabó [7] , pseudosymmetric manifold in the sense of Deszcz [8] , a non-flat Riemannian manifold
is said to be pseudosymmetric in the sense of Chaki [9] if it satisfies the relation
(1.1)
i.e.,
(1.2)
for any
and where R is the Riemannian curvature tensor of the manifold, A is a non-zero 1-form such that
for every vector field X. Every recurrent manifold is pseudosymmetric in the sense of Chaki [9] but not conversely. The pseudosymmetry in the sense of Chaki is not equivalent to that in the sense of Deszcz [8] . However, the pseudosymmetry by Chaki will be the pseudosymmetry by Deszcz if and only if the non-zero 1-form associated with n-dimensional pseudosymmetry is closed. Pseudosymmetric manifolds also have been studied by Chaki and De [10] , Özen and Altay [11] , Tarafder [12] , De, Murathan and Özgür [13] , Tarafder and De [14] and others. Many authors have been weakened by Ricci symmetry that has been differently extended such as a Ricci recurrent, Ricci symmetric and pseudo Ricci symmetric for past two decades.
A non-flat Riemannian manifold
is said to be pseudo-Ricci symmetric [15] if its Ricci tensor S of type
is not identically zero and satisfies the condition
(1.3)
for any
where A is a nowhere vanishing 1-form and
refers the operator of covariant differentiation with respect to the metric tensor g. Such a n-dimensional manifold is denoted by
. The pseudo-Ricci symmetric manifolds have also been studied by Arslan et al. [16] , De and Mazumder [17] and many others. The notion of locally ϕ-symmetric Sasakian manifold was introduced by Takahashi [18] due to a weaker version of locally symmetry. Generating the notion of locally ϕ-symmetric Sasakian manifolds, De et al. [19] , introduce the notion of ϕ-recurrent Sasakian manifolds also Shukla et al. [20] studied ϕ-symmetric and ϕ Ricci symmetric para Sasakian manifolds.
Inspired by above studies this paper makes an attempt to study of ϕ-pseudo symmetric and ϕ-pseudo Ricci symmetric ò-para Sasakian manifolds. It is organized as follows. Section 2 is related with ò-para Sasakian manifolds. Section 3 is dealt with the study of ϕ pseudo symmetric ò-para Sasakian manifolds. In Section 4, we study of ϕ-pseudo Concircularly symmetric ò-para Sasakian manifold. In Section 5, we study ϕ-pseudo Ricci symmetric ò-para Sasakian manifold. The relation (1.3) can be written as
(1.4)
where
is the vector field associated to the 1-form A such that
and Q is the Ricci operator, i.e.,
.
2. Preliminaries
Let
be an almost paracontact manifold is equipped with an almost paracontact structure
consisting of a tensor field ϕ of type
, a vector field
and a 1-form
satisfying
(2.1)
(2.2)
(2.3)
where
, in this case
is called an (ò)-almost paracontact metric manifold equipped with an (ò)-almost paracontact structure
[21] . In particular,
, then (ò)-almost paracontact metric manifold will be called a Lorentzian almost paracontact metric manifold. In view of equation [22] [23] , we have
(2.4)
(2.5)
for any
, the structure of a vector field
is a never light like. An (ò)-almost paracontact metric manifold (respectively a Lorentzian almost paracontact manifold
is said to be space-like (ò)-almost paracontact metric manifold (respectively a space-like Lorentzian almost paracontact manifold), if
and
is said to be a time-like (ò)-almost paracontact metric manifold (respectively a Lorentzian almost paracontact manifold), if
. An (ò)-almost paracontact metric structure is called an (ò)-Para Sasakian structure if
(2.6)
where
is the Levi-Civita connection. A manifold
endowed with an (ò)-para Sasakian structure is called an (ò)-para Sasakian manifold. For
and g is a Riemannian,
is the usual para Sasakian manifold [24] . For
, g Lorentzian and
replaced by
,
becomes a Lorentzian para Sasakian manifold [23] . In an (ò)-para Sasakian manifold, we have
(2.7)
(2.8)
(2.9)
for any
, where
is the fundamental 2-form. In an (ò)-almost para Sasakian manifold
, the following relations are hold.
(2.10)
(2.11)
(2.12)
(2.13)
In an n-dimensional (ò)-para Sasakian manifold
, the Ricci tensor satisfies
(2.14)
(2.15)
3. ϕ-Pseudo Symmetric on ò-Para Sasakian Manifold
Definition 3.1. A ò-Para Sasakian manifold
is said to be a ϕ-pseudo symmetric if the curvature tensor R satisfies
(3.1)
for any
. If
the manifold is said to be ϕ-symmetric.
By virtue of (2.1), it follows that
(3.2)
from which it follows that
(3.3)
Taking an orthonormal frame field and contracting (3.3) over X and U, then by using (2.2) and (2.5), we get
(3.4)
Using (2.11) and (2.13), we have
(3.5)
by virtue of (3.5), it follows from (3.4) that
(3.6)
This leads to the following:
Theorem 3.1. A ϕ-pseudo symmetric on a ò-para Sasakian manifold is Pseudo-Ricci symmetric if and only if
.
Putting
in (3.2), by using (2.10), (2.12) and (2.13), we have
(3.7)
This leads to the following:
Theorem 3.2. A ϕ-pseudo symmetric on a ò-para Sasakian manifold, the curvature tensor satisfies the relation (3.7).
From (3.7) follows that
(3.8)
replacing Y by
and W by
and using (2.3), (2.14), we have
(3.9)
Hence we can state the following:
Theorem 3.3. A ϕ-pseudo symmetric on a ò-para Sasakian manifold, the curvature tensor satisfies the relation (3.9), provided
.
4. ϕ-Pseudo Concircularly Symmetric ò-Para Sasakian Manifold
Definition 4.2. A n-dimensional ò-para Sasakian manifold is said to be ϕ-pseudo Concircularly symmetric, if its Concircular curvature tensor
is given by [25]
(4.1)
Satisfies the relation
(4.2)
for any
, where A is a non-zero 1-forms, such that
.
by virtue of (2.1), it follows from (4.2)
(4.3)
which follows that
(4.4)
Taking an orthonormal frame field and contracting (4.4) over X and U, by using (2.1) and (4.1), we get
(4.5)
by virtue of (3.5) and from (4.1), yields
(4.6)
In view of (4.6) from (4.5), we have
(4.7)
This leads to the following:
Theorem 4.4. A ϕ-pseudo Concircularly symmetric ò-para Sasakian manifold is pseudo-Ricci symmetric if and only if
(4.8)
Putting
in (4.3) and using (2.10), (2.12), (2.15) and (4.1), we obtain
(4.9)
Hence we can state the following:
Theorem 4.5. In a ϕ-pseudo Concircularly symmetric ò-para Sasakian manifold, the curvature tensor satisfies the relation (4.9).
Next, we take inner product of (4.9) with U and taking an orthonormal frame field and contracting (4.9) over X and U, yields
(4.10)
Replacing
by
and
by
, we obtain
(4.11)
This leads to the following:
Theorem 4.6. A ϕ-pseudo Concircularly symmetric ò-para Sasakian manifold, the curvature tensor satisfies the relation (4.11).
5. ϕ-Pseudo Ricci Symmetric ò-Para Sasakian Manifold
Definition 5.3. A n-dimensional ò-para Sasakian manifold is said to be ϕ-pseudo Ricci symmetric, if the Ricci operator Q satisfies
(5.1)
for any
, where A is a non zero 1-form.
In particular if
, then (5.1) turns into ϕ-Ricci symmetric ò-para Sasakian manifold.
In view of (2.1), then (5.1) becomes
(5.2)
which follows
(5.3)
putting
in (5.3), using (2.7) and (2.15), we get
(5.4)
Replacing
by
,
by
in (5.4) and using (2.14), we get
(5.5)
This leads to the following:
Theorem 5.7. A ϕ-pseudo Ricci symmetric ò-para Sasakian manifold, the curvature tensor satisfies the relation (5.5).