ø-Pseudo Symmetric ε-Para Sasakian Manifolds

P. Somashekhara, Venkatesha Venkatesha

Department of Mathematics, Kuvempu University, Shimoga, India.

DOI: 10.4236/oalib.1105273                

Abstract

The present paper focuses on the study of ø-pseudo symmetric, ø-pseudo concircularly symmetric and ø-pseudo Ricci symmetric on ε-Para Sasakian Manifolds. Also interesting results are obtained.

Keywords

ε-Para Sasakian Manifolds, ø-Pseudo Symmetric and ø-Pseudo Ricci Symmetric

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 $\nabla R=0$ , where R is the Riemannian curvature tensor of $\left(M\mathrm{,}g\right)$ . 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 $\left(M^n,g\right)\left(n>2\right)$ is said to be pseudosymmetric in the sense of Chaki [9] if it satisfies the relation

$\begin{array}{l}\left({\nabla }_{W}R\right)\left(X,Y,Z,U\right)\\ =2A\left(W\right)R\left(X,Y,Z,U\right)+A\left(X\right)R\left(W,Y,Z,U\right)+A\left(Y\right)R\left(X,W,Z,U\right)\\ +A\left(Z\right)R\left(X,Y,W,U\right)+A\left(U\right)R\left(X,Y,Z,W\right),\end{array}$ (1.1)

i.e.,

$\begin{array}{l}\left({\nabla }_{W}R\right)\left(X\mathrm{,}Y\right)Z\\ =2A\left(W\right)R\left(X\mathrm{,}Y\right)Z+A\left(X\right)R\left(W\mathrm{,}Y\right)Z+A\left(Y\right)R\left(X\mathrm{,}W\right)Z\\ +A\left(Z\right)R\left(X\mathrm{,}Y\right)W+g\left(R\left(X\mathrm{,}Y\right)Z\mathrm{,}W\right)\rho \mathrm{,}\end{array}$ (1.2)

for any $X\mathrm{,}Y\mathrm{,}Z\mathrm{,}U\mathrm{,}W\in T_P\left(M\right)$ and where R is the Riemannian curvature tensor of the manifold, A is a non-zero 1-form such that $g\left(X,\rho \right)=A\left(X\right)$ 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 $\left(M^n\mathrm{,}g\right)$ is said to be pseudo-Ricci symmetric [15] if its Ricci tensor S of type $\left(\mathrm{0,2}\right)$ is not identically zero and satisfies the condition

$\left({\nabla }_{X}S\right)\left(Y\mathrm{,}Z\right)=2A\left(X\right)S\left(Y\mathrm{,}Z\right)+A\left(Y\right)S\left(X\mathrm{,}Z\right)+A\left(Z\right)S\left(Y\mathrm{,}X\right)\mathrm{,}$ (1.3)

for any $X\mathrm{,}Y\mathrm{,}Z\in T_P\left(M\right)$ where A is a nowhere vanishing 1-form and $\nabla$ refers the operator of covariant differentiation with respect to the metric tensor g. Such a n-dimensional manifold is denoted by ${\left(PRS\right)}_n$ . 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

$\left({\nabla }_{X}Q\right)\left(Y\right)=2A\left(X\right)Q\left(Y\right)+A\left(Y\right)Q\left(X\right)+S\left(Y\mathrm{,}X\right)\rho \mathrm{,}$ (1.4)

where $\rho$ is the vector field associated to the 1-form A such that $A\left(X\right)=g\left(X,\rho \right)$ and Q is the Ricci operator, i.e., $g\left(QX,Y\right)=S\left(X,Y\right)$ .

2. Preliminaries

Let $\left(M^n\mathrm{,}g\right)$ be an almost paracontact manifold is equipped with an almost paracontact structure $\left(\varphi \mathrm{,}\xi \mathrm{,}\eta \right)$ consisting of a tensor field ϕ of type $\left(\mathrm{1,1}\right)$ , a vector field $\xi$ and a 1-form $\eta$ satisfying

${\varphi }^{2}X=X-\eta \left(X\right)\xi ,$ (2.1)

$\eta \left(\xi \right)=-1,\varphi \xi =0,\eta \circ \varphi =0,$ (2.2)

$g\left(\varphi X\mathrm{,}\varphi Y\right)=g\left(X\mathrm{,}Y\right)-ϵ\eta \left(X\right)\eta \left(Y\right)\mathrm{,}$ (2.3)

where $ϵ=\pm 1$ , in this case $\left(M^n\mathrm{,}g\right)$ is called an (ò)-almost paracontact metric manifold equipped with an (ò)-almost paracontact structure $\left(\varphi \mathrm{,}\xi \mathrm{,}\eta \mathrm{,}g\right)$ [21] . In particular, $\text{index}\left(g\right)=1$ , then (ò)-almost paracontact metric manifold will be called a Lorentzian almost paracontact metric manifold. In view of equation [22] [23] , we have

$g\left(\varphi X\mathrm{,}Y\right)=g\left(X\mathrm{,}\varphi Y\right)\mathrm{,}$ (2.4)

$g\left(x\mathrm{,}\xi \right)=ϵ\eta \left(X\right)\mathrm{,}$ (2.5)

for any $X\mathrm{,}Y\in T_{p}M$ , the structure of a vector field $\xi$ is a never light like. An (ò)-almost paracontact metric manifold (respectively a Lorentzian almost paracontact manifold $\left(M^n\mathrm{,}\varphi \mathrm{,}\xi \mathrm{,}g\mathrm{,}ϵ\right)$ is said to be space-like (ò)-almost paracontact metric manifold (respectively a space-like Lorentzian almost paracontact manifold), if $ϵ=1$ and $\left(M^n\mathrm{,}g\right)$ is said to be a time-like (ò)-almost paracontact metric manifold (respectively a Lorentzian almost paracontact manifold), if $ϵ=-1$ . An (ò)-almost paracontact metric structure is called an (ò)-Para Sasakian structure if

$\left({\nabla }_X\varphi \right)\left(Y\right)=-g\left(X\mathrm{,}\varphi Y\right)\xi -ϵ\eta \left(Y\right){\varphi }^{2}X\mathrm{,}$ (2.6)

where $\nabla$ is the Levi-Civita connection. A manifold $\left(M^n\mathrm{,}g\right)$ endowed with an (ò)-para Sasakian structure is called an (ò)-para Sasakian manifold. For $ϵ=1$ and g is a Riemannian, $\left(M^n\mathrm{,}g\right)$ is the usual para Sasakian manifold [24] . For $ϵ=-1$ , g Lorentzian and $\xi$ replaced by $-\xi$ , $\left(M^n\mathrm{,}g\right)$ becomes a Lorentzian para Sasakian manifold [23] . In an (ò)-para Sasakian manifold, we have

${\nabla }_X\xi =ϵ\varphi X\mathrm{,}$ (2.7)

$g\left(\xi \mathrm{,}\xi \right)=\pm 1=ϵ\mathrm{,}$ (2.8)

$\left({\nabla }_X\eta \right)\left(Y\right)=ϵg\left(\varphi X,Y\right)=\Omega \left(X,Y\right),$ (2.9)

for any $X\mathrm{,}Y\in T_{p}M$ , where $\Omega$ is the fundamental 2-form. In an (ò)-almost para Sasakian manifold $\left(M^n\mathrm{,}g\right)$ , the following relations are hold.

$\eta \left(R\left(X\mathrm{,}Y\right)Z\right)=ϵ\left[g\left(Y\mathrm{,}Z\right)\eta \left(X\right)-g\left(X\mathrm{,}Z\right)\eta \left(Y\right)\right]\mathrm{,}$ (2.10)

$R\left(\xi \mathrm{,}X\right)Y=-ϵg\left(X\mathrm{,}Y\right)\xi -ϵ\eta \left(Y\right)X\mathrm{,}$ (2.11)

$R\left(X\mathrm{,}Y\right)\xi =-ϵ\eta \left(Y\right)X+ϵ\eta \left(X\right)Y\mathrm{,}$ (2.12)

$\left({\nabla }_{X}R\right)\left(Y\mathrm{,}Z\right)\xi ={ϵ}^2\left[g\left(\varphi X\mathrm{,}Y\right)Z-g\left(\varphi X\mathrm{,}Z\right)Y\right]\mathrm{.}$ (2.13)

In an n-dimensional (ò)-para Sasakian manifold $\left(M^n\mathrm{,}g\right)$ , the Ricci tensor satisfies

$S\left(\varphi X\mathrm{,}\varphi Y\right)=S\left(X\mathrm{,}Y\right)+\left(n-1\right)\eta \left(X\right)\eta \left(Y\right)\mathrm{,}$ (2.14)

$S\left(X\mathrm{,}\xi \right)=S\left(\xi \mathrm{,}X\right)=-\left(n-1\right)\eta \left(X\right)\mathrm{.}$ (2.15)

3. ϕ-Pseudo Symmetric on ò-Para Sasakian Manifold

Definition 3.1. A ò-Para Sasakian manifold $\left(M^n\right)\left(\varphi \mathrm{,}\xi \mathrm{,}\eta \mathrm{,}g\right)$ is said to be a ϕ-pseudo symmetric if the curvature tensor R satisfies

$\begin{array}{l}{\varphi }^2\left(\left({\nabla }_{W}R\right)\left(X\mathrm{,}Y\right)Z\right)\\ =2A\left(W\right)R\left(X\mathrm{,}Y\right)Z+A\left(X\right)R\left(W\mathrm{,}Y\right)Z+A\left(Y\right)R\left(X\mathrm{,}W\right)Z\\ +A\left(Z\right)R\left(X\mathrm{,}Y\right)W+g\left(R\left(X\mathrm{,}Y\right)Z\mathrm{,}W\right)\rho \mathrm{,}\end{array}$ (3.1)

for any $X\mathrm{,}Y\mathrm{,}Z\mathrm{,}W\in T_{P}M$ . If $A=0$ the manifold is said to be ϕ-symmetric.

By virtue of (2.1), it follows that

$\begin{array}{l}\left({\nabla }_{W}R\right)\left(X,Y\right)Z-\eta \left(\left({\nabla }_{W}R\right)\left(X,Y\right)Z\right)\xi \\ =2A\left(W\right)R\left(X,Y\right)Z+A\left(X\right)R\left(W,Y\right)Z+A\left(Y\right)R\left(X,W\right)Z\\ +A\left(Z\right)R\left(X,Y\right)W+g\left(R\left(X,Y\right)Z,W\right)\rho ,\end{array}$ (3.2)

from which it follows that

$\begin{array}{l}g\left(\left({\nabla }_{W}R\right)\left(X,Y\right)Z,U\right)-\eta \left(\left({\nabla }_{W}R\right)\left(X,Y\right)Z\right)\eta \left(U\right)\\ =2A\left(W\right)g\left(R\left(X,Y\right)Z,U\right)+A\left(X\right)g\left(R\left(W,Y\right)Z,U\right)\\ +A\left(Y\right)g\left(R\left(X,W\right)Z,U\right)+A\left(Z\right)g\left(R\left(X,Y\right)W,U\right)\\ +g\left(R\left(X,Y\right)Z,W\right)A\left(U\right).\end{array}$ (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

$\begin{array}{l}\left({\nabla }_{W}S\right)\left(Y,Z\right)-g\left(\left({\nabla }_{W}R\right)\left(\xi ,Y\right)Z,\xi \right)\\ =2A\left(W\right)S\left(Y,Z\right)+A\left(Y\right)S\left(W,Z\right)+A\left(Z\right)S\left(Y,W\right)\\ +A\left(R\left(W,Y\right)Z\right)+A\left(R\left(W,Z\right)Y\right).\end{array}$ (3.4)

Using (2.11) and (2.13), we have

$g\left(\left({\nabla }_{W}R\right)\left(\xi ,Y\right)Z,\xi \right)=0,$ (3.5)

by virtue of (3.5), it follows from (3.4) that

$\begin{array}{c}\left({\nabla }_{W}S\right)\left(Y\mathrm{,}Z\right)=2A\left(W\right)S\left(Y\mathrm{,}Z\right)+A\left(Y\right)S\left(W\mathrm{,}Z\right)+A\left(Z\right)S\left(Y\mathrm{,}W\right)\\ +A\left(R\left(W\mathrm{,}Y\right)Z\right)+A\left(R\left(W\mathrm{,}Z\right)Y\right)\mathrm{,}\end{array}$ (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 $A\left(R\left(W,Y\right)Z\right)+A\left(R\left(W,Z\right)Y\right)=0$ .

Putting $Z=\xi$ in (3.2), by using (2.10), (2.12) and (2.13), we have

$\begin{array}{l}A\left(\xi \right)R\left(X,Y\right)W\\ ={ϵ}^2\left[g\left(\varphi W,Y\right)X-g\left(\varphi W,X\right)Y\right]+ϵ\{2A\left(W\right)\left[\eta \left(Y\right)X-\eta \left(X\right)Y\right]\\ +A\left(X\right)\left[\eta \left(Y\right)W-\eta \left(W\right)Y\right]+A\left(Y\right)\left[\eta \left(X\right)W-\eta \left(W\right)X\right]\\ +\left[\eta \left(Y\right)g\left(X,W\right)-\eta \left(X\right)g\left(Y,W\right)\right]\rho \}.\end{array}$ (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

$\begin{array}{c}A\left(\xi \right)S\left(Y,W\right)={ϵ}^2\left(n-1\right)\Omega \left(W,Y\right)+ϵ\{\left(n-1\right)\eta \left(Y\right)A\left(W\right)\\ +\left(n-1\right)\eta \left(W\right)A\left(Y\right)+\eta \left(Y\right)\eta \left(W\right)+g\left(Y,W\right)\},\end{array}$ (3.8)

replacing Y by $\varphi Y$ and W by $\varphi W$ and using (2.3), (2.14), we have

$S\left(Y,W\right)=\frac{1}{A\left(\xi \right)}\left\{ϵg\left(Y,W\right)-\left[{ϵ}^2+\left(n-1\right)A\left(\xi \right)\right]\eta \left(Y\right)\eta \left(W\right)+{ϵ}^2\left(n-1\right)\Omega \left(Y,W\right)\right\}.$

(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 $A\left(\xi \right)\ne 0$ .

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 $\tilde{C}$ is given by [25]

$\tilde{C}\left(X,Y\right)Z=R\left(X,Y\right)Z-\frac{r}{n\left(n-1\right)}\left[g\left(Y,Z\right)X-g\left(X,Z\right)Y\right].$ (4.1)

Satisfies the relation

$\begin{array}{l}{\varphi }^2\left(\left({\nabla }_W\tilde{C}\right)\left(X\mathrm{,}Y\right)Z\right)\\ =2A\left(W\right)\tilde{C}\left(X\mathrm{,}Y\right)Z+A\left(X\right)\tilde{C}\left(W\mathrm{,}Y\right)Z+A\left(Y\right)\tilde{C}\left(X\mathrm{,}W\right)Z\\ +A\left(Z\right)\tilde{C}\left(X\mathrm{,}Y\right)W+g\left(\tilde{C}\left(X\mathrm{,}Y\right)Z\mathrm{,}W\right)\rho \mathrm{,}\end{array}$ (4.2)

for any $X\mathrm{,}Y\mathrm{,}Z\mathrm{,}W\in T_{P}M$ , where A is a non-zero 1-forms, such that $A\left(X\right)=g\left(X,\rho \right)$ .

by virtue of (2.1), it follows from (4.2)

$\begin{array}{l}\left({\nabla }_W\tilde{C}\right)\left(X,Y\right)Z-\eta \left(\left({\nabla }_W\tilde{C}\right)\left(X,Y\right)Z\right)\xi \\ =2A\left(W\right)\tilde{C}\left(X,Y\right)Z+A\left(X\right)\tilde{C}\left(W,Y\right)Z+A\left(Y\right)\tilde{C}\left(X,W\right)Z\\ +A\left(Z\right)\tilde{C}\left(X,Y\right)W+g\left(\tilde{C}\left(X,Y\right)Z,W\right)\rho ,\end{array}$ (4.3)

which follows that

$\begin{array}{l}g\left(\left({\nabla }_W\tilde{C}\right)\left(X,Y\right)Z,U\right)-\eta \left(\left({\nabla }_W\tilde{C}\right)\left(X,Y\right)Z\right)\eta \left(U\right)\\ =2A\left(W\right)g\left(\tilde{C}\left(X,Y\right)Z,U\right)+A\left(X\right)g\left(\tilde{C}\left(W,Y\right)Z,U\right)\\ +A\left(Y\right)g\left(\tilde{C}\left(X,W\right)Z,U\right)+A\left(Z\right)g\left(\tilde{C}\left(X,Y\right)W,U\right)\\ +g\left(\tilde{C}\left(X,Y\right)Z,W\right)A\left(U\right).\end{array}$ (4.4)

Taking an orthonormal frame field and contracting (4.4) over X and U, by using (2.1) and (4.1), we get

$\begin{array}{l}\left({\nabla }_{X}S\right)\left(Y\mathrm{,}Z\right)-\frac{dr\left(W\right)}{n}g\left(Y\mathrm{,}Z\right)+g\left(\left({\nabla }_W\tilde{C}\right)\left(\xi \mathrm{,}Y\right)Z\mathrm{,}\xi \right)\\ =2A\left(W\right)S\left(Y\mathrm{,}Z\right)+A\left(Y\right)S\left(W\mathrm{,}Z\right)+A\left(Z\right)S\left(Y\mathrm{,}W\right)-\frac{r}{n}[2A\left(W\right)g\left(Y\mathrm{,}Z\right)\\ +A\left(Y\right)g\left(W\mathrm{,}Z\right)+A\left(Z\right)g\left(Y\mathrm{,}W\right)]+A\left(\tilde{C}\left(W\mathrm{,}Y\right)Z\right)+A\left(\tilde{C}\left(W\mathrm{,}Z\right)Y\right)\mathrm{,}\end{array}$ (4.5)

by virtue of (3.5) and from (4.1), yields

$g\left(\left({\nabla }_W\tilde{C}\right)\left(\xi \mathrm{,}Y\right)Z\mathrm{,}\xi \right)=-\frac{dr\left(W\right)}{n\left(n-1\right)}\left[g\left(Y\mathrm{,}Z\right)-\eta \left(Y\right)\eta \left(Z\right)\right]\mathrm{.}$ (4.6)

In view of (4.6) from (4.5), we have

$\begin{array}{l}\left({\nabla }_{X}S\right)\left(Y\mathrm{,}Z\right)-\frac{dr\left(W\right)}{n}g\left(Y\mathrm{,}Z\right)-\frac{dr\left(W\right)}{n\left(n-1\right)}\left[g\left(Y\mathrm{,}Z\right)-\eta \left(Y\right)\eta \left(Z\right)\right]\\ =2A\left(W\right)S\left(Y\mathrm{,}Z\right)+A\left(Y\right)S\left(W\mathrm{,}Z\right)+A\left(Z\right)S\left(Y\mathrm{,}W\right)-\frac{r}{n}[2A\left(W\right)g\left(Y\mathrm{,}Z\right)\\ +A\left(Y\right)g\left(W\mathrm{,}Z\right)+A\left(Z\right)g\left(Y\mathrm{,}W\right)]+A\left(\tilde{C}\left(W\mathrm{,}Y\right)Z\right)+A\left(\tilde{C}\left(W\mathrm{,}Z\right)Y\right)\mathrm{.}\end{array}$ (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

$\begin{array}{l}\frac{dr\left(W\right)}{n}g\left(Y,Z\right)+\frac{dr\left(W\right)}{n\left(n-1\right)}\left[g\left(Y,Z\right)-\eta \left(Y\right)\eta \left(Z\right)\right]\\ -\frac{r}{n}\left[2A\left(W\right)g\left(Y,Z\right)+A\left(Y\right)g\left(W,Z\right)+A\left(Z\right)g\left(Y,W\right)\right]\\ +A\left(\tilde{C}\left(W,Y\right)Z\right)+A\left(\tilde{C}\left(W,Z\right)Y\right)=0.\end{array}$ (4.8)

Putting $Z=\xi$ in (4.3) and using (2.10), (2.12), (2.15) and (4.1), we obtain

$\begin{array}{l}{ϵ}^2\left[\Omega \left(W,X\right)Y-\Omega \left(W,Y\right)X\right]-ϵ\frac{dr\left(W\right)}{n\left(n-1\right)}\left[\eta \left(Y\right)X-\eta \left(X\right)Y\right]\\ +ϵ\left[1-\frac{r}{n\left(n-1\right)}\right]\left[\eta \left(Y\right)g\left(X,W\right)-\eta \left(X\right)g\left(Y,W\right)\right]\rho \\ \frac{r}{n\left(n-1\right)}A\left(\xi \right)\left[g\left(Y,W\right)X-g\left(X,W\right)Y\right]\\ +ϵ\left[1-\frac{r}{n\left(n-1\right)}\right]\{2A\left(W\right)\left[\eta \left(Y\right)X-\eta \left(X\right)Y\right]\\ +A\left(X\right)\left[\eta \left(Y\right)W-\eta \left(W\right)Y\right]+A\left(Y\right)\left[\eta \left(W\right)X-\eta \left(X\right)W\right]\}\\ =A\left(\xi \right)R\left(X,Y\right)W.\end{array}$ (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

$\begin{array}{c}A\left(\xi \right)S\left(Y\mathrm{,}W\right)={ϵ}^2\left(1-n\right)\Omega \left(Y\mathrm{,}W\right)-ϵ\frac{dr\left(W\right)}{n}\eta \left(Y\right)\\ +ϵ\left[1-\frac{r}{n\left(n-1\right)}\right]\left[\eta \left(Y\right)\eta \left(W\right)-g\left(Y\mathrm{,}W\right)\right]+\frac{r}{n}A\left(\xi \right)g\left(Y\mathrm{,}W\right)\\ +ϵ\left[1-\frac{r}{n\left(n-1\right)}\right]\left(n-1\right)\left[2A\left(W\right)\eta \left(Y\right)+A\left(Y\right)\eta \left(W\right)\right]\mathrm{.}\end{array}$ (4.10)

Replacing $Y$ by $\varphi Y$ and $W$ by $\varphi W$ , we obtain

$\begin{array}{c}S\left(Y,W\right)=\frac{{ϵ}^2\left(1-n\right)}{A\left(\xi \right)}\Omega \left(Y,W\right)+\left[\frac{r}{n}-\frac{ϵ}{A\left(\xi \right)}\left[1-\frac{r}{n\left(n-1\right)}\right]g\left(Y,W\right)\right]\\ +\left[\frac{{ϵ}^2}{A\left(\xi \right)}\left[1-\frac{r}{n\left(n-1\right)}\right]-\frac{ϵ}{A\left(\xi \right)}-\left(n-1\right)\right]\eta \left(Y\right)\eta \left(W\right).\end{array}$ (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

${\varphi }^2\left(\left({\nabla }_{W}Q\right)\left(Y\right)\right)=2A\left(X\right)QY+A\left(Y\right)QX+S\left(Y\mathrm{,}X\right)\rho \mathrm{,}$ (5.1)

for any $X\mathrm{,}Y\in T_{P}M$ , where A is a non zero 1-form.

In particular if $A=0$ , then (5.1) turns into ϕ-Ricci symmetric ò-para Sasakian manifold.

In view of (2.1), then (5.1) becomes

$\left({\nabla }_{W}Q\right)\left(Y\right)-\eta \left(\left({\nabla }_{W}Q\right)\left(Y\right)\right)\xi =2A\left(X\right)QY+A\left(Y\right)QX+S\left(Y,X\right)\rho ,$ (5.2)

which follows

$\begin{array}{l}g(({\nabla }_{W}Q)(Y),Z)-S({\nabla }_{W}Y,Z)-\eta (({\nabla }_{W}Q)(Y))\eta (Z)\\ =2A(X)S(Y,Z)+A(Y)S(X,Z)+S(Y,X)A(Z\eta ),\end{array}$ (5.3)

putting $Y=\xi$ in (5.3), using (2.7) and (2.15), we get

$\begin{array}{l}A\left(\xi \right)S\left(X,Z\right)+ϵS\left(\varphi X,Z\right)\\ =\left(n-1\right)\left[ϵg\left(\varphi X,Z\right)+2A\left(X\right)\eta \left(Z\right)+\eta \left(X\right)A\left(Z\right)\right].\end{array}$ (5.4)

Replacing $X$ by $\varphi X$ , $Z$ by $\varphi Z$ in (5.4) and using (2.14), we get

$S\left(X,Z\right)=\frac{ϵ}{A\left(\xi \right)}\left[\left(n-1\right)\Omega \left(X,Z\right)-S\left(\varphi X,Z\right)\right]-\left(n-1\right)\eta \left(X\right)\eta \left(Z\right).$ (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).

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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