ABSTRACT

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

Subject Areas:

Geometry

Keywords:

ò-Para Sasakian Manifolds, ϕ-Pseudo Symmetric and ϕ-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.

Cite this paper

Somashekhara, P. and Venkatesha (2019) ϕ-Pseudo Symmetric ò-Para Sasakian Manifolds. Open Access Library Journal, 6: e5273. https://doi.org/10.4236/oalib.1105273

References