On the existence of an almost generalized weakly-symmetric Sasakian manifold admitting quarter symmetric metric connection

Abstract

The object of the present work is to study an almost generalized weakly symmetric Sasakian manifold admitting quarter symmetric metric connection with a non-trivial example.

Keywords:

• Almost generalized weakly symmetric Sasakian manifolds
• quarter symmetric metric connection

Mathematics Subject Classification 2010:

• 53C15
• 53C25

1. Introduction

The notion of a weakly symmetric Riemannian manifold was initiated by Tamássy and Binh [1]. Thereafter, it becomes focus of interest for many geometers. For details, we refer to [2–13] and the references there in.  In analogy to [14], a weakly symmetric Riemannian manifold $(M^n,g)(n>2)$, is said to be an almost weakly pseudo symmetric manifold, if its curvature tensor $\overline{R}$ of type $(0,4)$ is not identically zero and satisfies the identity (1) $\begin{array}{rl}\left({\mathrm{\nabla }}_X\overline{R}\right)(Y,U,V,W)& =\left[A_1\right(X)+B_1(X\left)\right]\overline{R}(Y,U,V,W)\\ & +C_1\left(Y\right)\overline{R}(X,U,V,W)\\ & +C_1\left(U\right)\overline{R}(Y,X,V,W)\\ & +D_1\left(V\right)\overline{R}(Y,U,X,W)\\ & +D_1\left(W\right)\overline{R}(Y,U,V,X),\end{array}$(1) where $A_1,$ $B_1,$ $C_1$ & $D_1$ are non-zero 1-forms defined by $A_1\left(X\right)=g(X,{\sigma }_1)$, $B_1\left(X\right)=g(X,{\varrho }_1),$ $C_1\left(X\right)=g(X,{\pi }_1)$ and $D_1\left(X\right)=g(X,{\mathrm{\partial }}_1),$ for all X and $\overline{R}(Y,U,V,W)=g\left(R\right(Y,U)V,W),$ ∇ being the operator of the covariant differentiation with respect to the metric tensor g. An n-dimensional Riemannian manifold of this kind is denoted by $A(WPS{)}_n$-manifold.

Keeping in tune with Dubey [15], the author  in [16] has recently introduced the notion of an almost generalized weakly symmetric manifold (which is  abbreviated hereafter as $A(GWS{)}_n$-manifold) if it admits the equation (2) $\begin{array}{rl}\left({\mathrm{\nabla }}_X\overline{R}\right)(Y,U,V,W)& =\left[A_1\right(X)+B_1(X\left)\right]\overline{R}(Y,U,V,W)\\ & +C_1\left(Y\right)\overline{R}(X,U,V,W)\\ & +C_1\left(U\right)\overline{R}(Y,X,V,W)\\ & +D_1\left(V\right)\overline{R}(Y,U,X,W)\\ & +D_1\left(W\right)\overline{R}(Y,U,V,X)\\ & +\left[A_2\right(X)+B_2(X\left)\right]\overline{G}(Y,U,V,W)\\ & +C_2\left(Y\right)\overline{G}(X,U,V,W)\\ & +C_2\left(U\right)\overline{G}(Y,X,V,W)\\ & +D_2\left(V\right)\overline{G}(Y,U,X,W)\\ & +D_2\left(W\right)\overline{G}(Y,U,V,X)\end{array}$(2) where (3) $\begin{array}{r}\overline{G}(Y,U,V,W)=\left[g(U,V)g(Y,W)-g(Y,V)g(U,W)\right]\end{array}$(3) and $A_i,$ $B_i,$ $C_i$ & $D_i$ are non-zero 1-forms defined by $A_i\left(X\right)=g(X,{\sigma }_i)$, $B_i\left(X\right)=g(X,{\varrho }_i),$ $C_i\left(X\right)=g(X,{\pi }_i)$ and $D_i\left(X\right)=g(X,{\mathrm{\partial }}_i),$ for $i=1,2.$ The beauty of such $A(GWS{)}_n$-manifold is that it has the flavour of

1. locally symmetric space [17], (for $A_i=B_i=C_i=D_i=0$),

2. recurrent space [18], $R_n$ (for $A_1\ne 0$, $B_i=C_i=D_i=0$),

3. generalized recurrent space [15], $(GR{)}_n$ ($A_i\ne 0$ and $B_i=C_i=D_i=0$),

4. pseudo symmetric space [19], $(PS{)}_n$ (for $A_1=B_1=C_1=D_1\ne 0$ and $A_2=B_2=C_2=D_2=0$),

5. semi-pseudo symmetric space [20], $(SPS{)}_n$ (for $A_1=-B_1,C_1=D_1$ and $A_2=B_2=C_2=D_2=0$),

6. generalized semi-pseudo symmetric space [21], $(GSPS{)}_n$ (for $A_1=-B_1,C_1=D_1$ and $A_2=-B_2,C_2=D_2$),

7. generalized pseudo symmetric space [22], $(GPS{)}_n$ (for $A_i=B_i=C_i=D_i\ne 0$),

8. almost pseudo symmetric space [14], $A(PS{)}_n$ (for $B_1\ne 0$, $A_1=C_1=D_1\ne 0$ and $A_2=B_2=C_2=D_2=0$),

9. almost generalized pseudo symmetric space [16], $A(GPS{)}_n$ (for $B_i\ne 0$, $A_i=C_i=D_i\ne 0$) and

10. weakly symmetric space [1], $(WS{)}_n$ ( for $A_2=B_2=C_2=D_2=0$).

In [23], Golab defined and studied quarter-symmetric connection in a differentiable manifold with affine connection, which generalizes the thought of semi-symmetric connection. After Golab quarter symmetric connection has been studied by many geometers like as Mondal and De [24], Rastogi [25, 26], Mishra and Pandey [27], Yano and Imai [28] and others. A linear connection $\overline{\mathrm{\nabla }}$ on an n-dimensional Riemannian manifold (M, g) is called a quarter-symmetric connection [23] if its torsion tensor T of the connection $\overline{\mathrm{\nabla }}$ $T(X,Y)={\overline{\mathrm{\nabla }}}_{X}Y-{\overline{\mathrm{\nabla }}}_{Y}X-[X,Y]$ satisfies $T(X,Y)=\eta \left(Y\right)\phi X-\eta \left(X\right)\phi Y,$ where η is a 1-form and φ is a (1, 1) tensor field.

In particular, if $\phi X$ = X, then the quarter-symmetric connection reduces to the semi-symmetric connection [29, 30]. Thus the notion of quarter-symmetric connection generalizes that of the semi-symmetric connection.

Furthermore, if a quarter-symmetric connection $\overline{\mathrm{\nabla }}$ admits the condition $\left({\overline{\mathrm{\nabla }}}_{X}g\right)(Y,Z)=0$ then $\overline{\mathrm{\nabla }}$ is said to be a quarter-symmetric metric connection, otherwise it is said to be a quarter-symmetric non-metric connection [31].

Our work is structured as follows. Section 2 is concerned with Sasakian manifolds and some known results. In Section 3, we have investigated almost generalized weakly symmetric Sasakian manifolds admitting a quarter-symmetric metric connection $\overline{\mathrm{\nabla }}$,  which will be abbreviated by $[A(GPS{)}_n,\overline{\mathrm{\nabla }}].$ It is observed that in a Sasakian manifold a necessary condition (i) for each of $[R_n,\overline{\mathrm{\nabla }}],$ $\left[\right(GR{)}_n,\overline{\mathrm{\nabla }}],$ $\left[\right(PS{)}_n,\overline{\mathrm{\nabla }}]$ and $\left[\right(GPS{)}_n,\overline{\mathrm{\nabla }}]$ to be, respectively, $[R_n,\mathrm{\nabla }],$ $\left[\right(GR{)}_n,\mathrm{\nabla }],$ $\left[\right(PS{)}_n,\mathrm{\nabla }]$ and $\left[\right(GPS{)}_n,\mathrm{\nabla }]$ is $A_1\left(\xi \right)=0$; (ii) for each of $\left[\right(SPS{)}_n,\overline{\mathrm{\nabla }}]$ and $\left[\right(GSPS{)}_n,\overline{\mathrm{\nabla }}]$ to be , respectively, $\left[\right(SPS{)}_n,\mathrm{\nabla }]$ and $\left[\right(GSPS{)}_n,\mathrm{\nabla }]$ is $C_1\left(\xi \right)=0$ & (iii) for $\left[\right(WS{)}_n,\overline{\mathrm{\nabla }}]$ to be $\left[\right(WS{)}_n,\mathrm{\nabla }]$ is $A_1\left(\xi \right)+C_1\left(\xi \right)+D_1\left(\xi \right)=0.$ Finally, we have constructed a non-trivial example of an $\left[A\right(GPS{)}_n,\overline{\mathrm{\nabla }}].$

2. Sasakian manifold and some known results

Let M be an $n=(2m+1)$-dimensional almost contact metric manifold equipped with an almost contact metric structure (φ, ξ, η, g) consisting of a (1, 1) tensor field φ, a vector field ξ, a 1-form η and a Riemannian metric g. Then (4) $\begin{array}{rl}{\phi }^2\left(X\right)& =-X+\eta \left(X\right)\xi ,\eta \left(\xi \right)=1,\\ \phi \xi & =0,\eta \left(\phi X\right)=0,\end{array}$(4) (5) $\begin{array}{rl}g(\phi X,\phi Y)& =g(X,Y)-\eta \left(X\right)\eta \left(Y\right),\mathrm{\forall }X,Y\in TM.\end{array}$(5) From (4(4) $\begin{array}{rl}{\phi }^2\left(X\right)& =-X+\eta \left(X\right)\xi ,\eta \left(\xi \right)=1,\\ \phi \xi & =0,\eta \left(\phi X\right)=0,\end{array}$(4) ) and (5(5) $\begin{array}{rl}g(\phi X,\phi Y)& =g(X,Y)-\eta \left(X\right)\eta \left(Y\right),\mathrm{\forall }X,Y\in TM.\end{array}$(5) ), it can be easily seen that (6) $g(X,\phi Y)=-g(\phi X,Y),g(X,\xi )=\eta \left(X\right),\mathrm{\forall }X,Y\in TM.$(6) An almost contact metric manifold M is said to be (a) a contact metric manifold if (7) $g(X,\phi Y)=d\eta (X,Y),\mathrm{\forall }X,Y\in TM;$(7) (b) a K-contact manifold if the vector field ξ is Killing equivalently (8) ${\mathrm{\nabla }}_X\xi =-\phi X,$(8) where ∇ is Riemannian connection and

(c) a Sasakian manifold if (9) $\left({\mathrm{\nabla }}_X\phi \right)Y=g(X,Y)\xi -\eta \left(Y\right)X,\mathrm{\forall }X,Y\in TM.$(9) A K-contact manifold is a contact metric manifold, while the converse is true if the Lie derivative of φ in the characteristic direction ξ vanishes identically. A Sasakian manifold is always a K-contact manifold. A 3-dimensional K-contact manifold is a Sasakian manifold.

It is well known that a contact metric manifold is Sasakian if and only if (10) $R(X,Y)\xi =\eta \left(Y\right)X-\eta \left(X\right)Y,\mathrm{\forall }X,Y\in TM.$(10) In a Sasakian manifold equipped with the structure (φ, ξ, η, g), the following relations also hold [32–34]: (11) $\begin{array}{rl}\left({\mathrm{\nabla }}_X\eta \right)Y& =g(X,\phi Y),\end{array}$(11) (12) $\begin{array}{rl}R(\xi ,X)Y& =g(X,Y)\xi -\eta \left(Y\right)X,\end{array}$(12) (13) $\begin{array}{rl}S(X,\xi )& =(n-1)\eta \left(X\right),\end{array}$(13) (14) $\begin{array}{rl}R(X,\xi )Y& =\eta \left(Y\right)X-g(X,Y)\xi ,\end{array}$(14) for all $X,Y,Z\in TM$, where S is the Ricci tensor.

The relation between the quarter-symmetric metric connection $\overline{\mathrm{\nabla }}$ and the Levi-Civita connection ∇ of ($M^n,g$) has been obtained by Yano and Imai [28], which is given by (15) ${\tilde{\mathrm{\nabla }}}_{X}Y={\mathrm{\nabla }}_{X}Y-\eta \left(Y\right)\phi X;$(15) given by.

A relation between the curvature tensor $R^{\mathrm{\nabla }}$of M with respect to the quarter-symmetric metric connection $\overline{\mathrm{\nabla }}$ and R of M with respect to the the Riemannian connection ∇, is given by (16) $\begin{array}{rl}{R}^{\mathrm{\nabla }}(X,Y)Z& =R(X,Y)Z-2g(X,\phi Y)\phi Z+\eta \left(X\right)g(Y,Z)\xi \\ & -\eta \left(Y\right)g(X,Z)\xi +\eta \left(Z\right)\left\{\eta \right(Y)X-\eta (X\left)Y\right\}.\end{array}$(16) which yields (17) $S^{\mathrm{\nabla }}(Y,Z)=S(Y,Z)-g(Y,Z)+n\eta \left(Y\right)\eta \left(Z\right),$(17) where $S^{\mathrm{\nabla }}$ and S are the Ricci tensors of the connections $\overline{\mathrm{\nabla }}$ and ∇ , respectively.

3. Sasakian manifold with $\left[A\right(GPS{)}_n,\overline{\mathrm{\nabla }}]$

For an $[A(GPS{)}_n,\overline{\mathrm{\nabla }}],$ we have (18) $\begin{array}{rl}& \left({\mathrm{\nabla }}_X{\overline{R}}^{\mathrm{\nabla }}\right)(Y,U,V,W)=\left[A_1\right(X)+B_1(X\left)\right]{\overline{R}}^{\mathrm{\nabla }}(Y,U,V,W)\\ & +C_1\left(Y\right){\overline{R}}^{\mathrm{\nabla }}(X,U,V,W)+C_1\left(U\right){\overline{R}}^{\mathrm{\nabla }}(Y,X,V,W)\\ & +D_1\left(V\right){\overline{R}}^{\mathrm{\nabla }}(Y,U,X,W)+D_1\left(W\right){\overline{R}}^{\mathrm{\nabla }}(Y,U,V,X)\\ & \left[A_2\right(X)+B_2(X\left)\right]G(Y,U,V,W)\\ & +C_2\left(Y\right)G(X,U,V,W)+C_2\left(U\right)G(Y,X,V,W)\\ & +D_2\left(V\right)G(Y,U,X,W)+D_2\left(W\right)G(Y,U,V,X)\end{array}$(18) for all X, Y, Z, U. Making use of (15(15) ${\tilde{\mathrm{\nabla }}}_{X}Y={\mathrm{\nabla }}_{X}Y-\eta \left(Y\right)\phi X;$(15) ), we can find (19) $\begin{array}{rl}\left({\tilde{\mathrm{\nabla }}}_X{\tilde{R}}^{\mathrm{\nabla }}\right)(Y,U,V,W)& =\left({\mathrm{\nabla }}_X{\tilde{R}}^{\mathrm{\nabla }}\right)(Y,U,V,W)\\ & -\eta \left({\tilde{R}}^{\mathrm{\nabla }}\right(Y,U\left)V\right)g(\phi X,W)\\ & +\eta \left(Y\right){\tilde{R}}^{\mathrm{\nabla }}(\phi X,U,V,W)\\ & +\eta \left(U\right){\tilde{R}}^{\mathrm{\nabla }}(Y,\phi X,V,W)\\ & +\eta \left(V\right){\tilde{R}}^{\mathrm{\nabla }}(Y,U,\phi X,W)\\ & +\eta \left(W\right){\tilde{R}}^{\mathrm{\nabla }}(Y,U,V,\phi X).\end{array}$(19) Now, using (16(16) $\begin{array}{rl}{R}^{\mathrm{\nabla }}(X,Y)Z& =R(X,Y)Z-2g(X,\phi Y)\phi Z+\eta \left(X\right)g(Y,Z)\xi \\ & -\eta \left(Y\right)g(X,Z)\xi +\eta \left(Z\right)\left\{\eta \right(Y)X-\eta (X\left)Y\right\}.\end{array}$(16) ) in the foregoing equation, we have (20) $\begin{array}{rl}\left({\tilde{\mathrm{\nabla }}}_X{\tilde{R}}^{\mathrm{\nabla }}\right)(Y,U,V,W)& =\left({\mathrm{\nabla }}_X\overline{R}\right)(Y,U,V,W)\\ & -2\left({\mathrm{\nabla }}_{X}g\right)(Y,\phi U)g(\phi V,W)\\ & -2g(Y,\phi U)\left({\mathrm{\nabla }}_{X}g\right)(\phi V,W)\\ & +\left({\mathrm{\nabla }}_X\eta \right)\left(Y\right)g(U,V)\eta \left(W\right)\\ & +\eta \left(Y\right)g(U,V)\left({\mathrm{\nabla }}_X\eta \right)\left(W\right)\\ & -\left({\mathrm{\nabla }}_X\eta \right)\left(U\right)g(Y,V)\eta \left(W\right)\\ & -\eta \left(U\right)g(Y,V)\left({\mathrm{\nabla }}_X\eta \right)\left(W\right)\\ & +\left({\mathrm{\nabla }}_X\eta \right)\left(V\right)\left\{g\right(Y,W\left)\eta \right(U)\\ & -g(U,W)\eta \left(Y\right)\}\\ & +\eta \left(V\right)\left\{g\right(Y,W\left)\right({\mathrm{\nabla }}_X\eta \left)\right(U)\\ & -g(U,W)\left({\mathrm{\nabla }}_X\eta \right)\left(Y\right)\}\\ & -\left[\eta \right(R(Y,U)V)+g(U,V\left)\eta \right(Y)\\ & -g(Y,V)\eta \left(U\right)\left]g\right(\phi X,W)\\ & +\eta \left(Y\right)\left[R\right(\phi X,U,V,W)\\ & -2g(\phi X,\phi U)g(\phi V,W)\\ & -g(\phi X,V)\eta \left(U\right)\eta \left(W\right)\\ & +g(\phi X,W)\eta \left(U\right)\eta \left(V\right)]\\ & +\eta \left(U\right)\left[R\right(Y,\phi X,V,W)\\ & +2g(\phi Y,\phi X)g(\phi V,W)\\ & +g(\phi X,V)\eta \left(Y\right)\eta \left(W\right)\\ & -g(\phi X,W)\eta \left(Y\right)\eta \left(V\right)]\\ & +\eta \left(V\right)\left[R\right(Y,U,\phi X,W)\\ & +2g(Y,\phi U)g(\phi X,\phi W)\\ & +g(\phi X,U)\eta \left(Y\right)\eta \left(W\right)\\ & -g(\phi X,Y)\eta \left(U\right)\eta \left(W\right)]\\ & +\eta \left(W\right)\left[R\right(Y,U,V,\phi X)\\ & -2g(Y,\phi U)g(\phi V,\phi X)\\ & +g(Y,\phi X)\eta \left(U\right)\eta \left(V\right)\\ & -g(\phi X,U)\eta \left(Y\right)\eta \left(V\right)].\end{array}$(20)

Theorem 3.1

An $[A(GPS{)}_n,\overline{\mathrm{\nabla }}]$  is an $\left[A\right(GPS{)}_n,\mathrm{\nabla }]$, if the 1-forms are related by the following relation (21) $A_1\left(\xi \right)+B_1\left(\xi \right)+C_1\left(\xi \right)+D_1\left(\xi \right)=0.$(21)

Proof.

As a direct consequence of (16(16) $\begin{array}{rl}{R}^{\mathrm{\nabla }}(X,Y)Z& =R(X,Y)Z-2g(X,\phi Y)\phi Z+\eta \left(X\right)g(Y,Z)\xi \\ & -\eta \left(Y\right)g(X,Z)\xi +\eta \left(Z\right)\left\{\eta \right(Y)X-\eta (X\left)Y\right\}.\end{array}$(16) ), (18(18) $\begin{array}{rl}& \left({\mathrm{\nabla }}_X{\overline{R}}^{\mathrm{\nabla }}\right)(Y,U,V,W)=\left[A_1\right(X)+B_1(X\left)\right]{\overline{R}}^{\mathrm{\nabla }}(Y,U,V,W)\\ & +C_1\left(Y\right){\overline{R}}^{\mathrm{\nabla }}(X,U,V,W)+C_1\left(U\right){\overline{R}}^{\mathrm{\nabla }}(Y,X,V,W)\\ & +D_1\left(V\right){\overline{R}}^{\mathrm{\nabla }}(Y,U,X,W)+D_1\left(W\right){\overline{R}}^{\mathrm{\nabla }}(Y,U,V,X)\\ & \left[A_2\right(X)+B_2(X\left)\right]G(Y,U,V,W)\\ & +C_2\left(Y\right)G(X,U,V,W)+C_2\left(U\right)G(Y,X,V,W)\\ & +D_2\left(V\right)G(Y,U,X,W)+D_2\left(W\right)G(Y,U,V,X)\end{array}$(18) ) and (20(20) $\begin{array}{rl}\left({\tilde{\mathrm{\nabla }}}_X{\tilde{R}}^{\mathrm{\nabla }}\right)(Y,U,V,W)& =\left({\mathrm{\nabla }}_X\overline{R}\right)(Y,U,V,W)\\ & -2\left({\mathrm{\nabla }}_{X}g\right)(Y,\phi U)g(\phi V,W)\\ & -2g(Y,\phi U)\left({\mathrm{\nabla }}_{X}g\right)(\phi V,W)\\ & +\left({\mathrm{\nabla }}_X\eta \right)\left(Y\right)g(U,V)\eta \left(W\right)\\ & +\eta \left(Y\right)g(U,V)\left({\mathrm{\nabla }}_X\eta \right)\left(W\right)\\ & -\left({\mathrm{\nabla }}_X\eta \right)\left(U\right)g(Y,V)\eta \left(W\right)\\ & -\eta \left(U\right)g(Y,V)\left({\mathrm{\nabla }}_X\eta \right)\left(W\right)\\ & +\left({\mathrm{\nabla }}_X\eta \right)\left(V\right)\left\{g\right(Y,W\left)\eta \right(U)\\ & -g(U,W)\eta \left(Y\right)\}\\ & +\eta \left(V\right)\left\{g\right(Y,W\left)\right({\mathrm{\nabla }}_X\eta \left)\right(U)\\ & -g(U,W)\left({\mathrm{\nabla }}_X\eta \right)\left(Y\right)\}\\ & -\left[\eta \right(R(Y,U)V)+g(U,V\left)\eta \right(Y)\\ & -g(Y,V)\eta \left(U\right)\left]g\right(\phi X,W)\\ & +\eta \left(Y\right)\left[R\right(\phi X,U,V,W)\\ & -2g(\phi X,\phi U)g(\phi V,W)\\ & -g(\phi X,V)\eta \left(U\right)\eta \left(W\right)\\ & +g(\phi X,W)\eta \left(U\right)\eta \left(V\right)]\\ & +\eta \left(U\right)\left[R\right(Y,\phi X,V,W)\\ & +2g(\phi Y,\phi X)g(\phi V,W)\\ & +g(\phi X,V)\eta \left(Y\right)\eta \left(W\right)\\ & -g(\phi X,W)\eta \left(Y\right)\eta \left(V\right)]\\ & +\eta \left(V\right)\left[R\right(Y,U,\phi X,W)\\ & +2g(Y,\phi U)g(\phi X,\phi W)\\ & +g(\phi X,U)\eta \left(Y\right)\eta \left(W\right)\\ & -g(\phi X,Y)\eta \left(U\right)\eta \left(W\right)]\\ & +\eta \left(W\right)\left[R\right(Y,U,V,\phi X)\\ & -2g(Y,\phi U)g(\phi V,\phi X)\\ & +g(Y,\phi X)\eta \left(U\right)\eta \left(V\right)\\ & -g(\phi X,U)\eta \left(Y\right)\eta \left(V\right)].\end{array}$(20) ) one can say that an almost generalized weakly symmetric Sasakian manifold admitting quarter symmetric connection $\overline{\mathrm{\nabla }}$ reduces to an almost generalized weakly symmetric Sasakian manifold admitting Riemannian metric connection ∇, if the following relation holds (22) $\begin{array}{rl}& -2\left({\mathrm{\nabla }}_{X}g\right)(Y,\phi U)g(\phi V,W)-2g(Y,\phi U)\left({\mathrm{\nabla }}_{X}g\right)(\phi V,W)\\ & +\left({\mathrm{\nabla }}_X\eta \right)\left(Y\right)g(U,V)\eta \left(W\right)+\eta \left(Y\right)g(U,V)\left({\mathrm{\nabla }}_X\eta \right)\left(W\right)\\ & -\left({\mathrm{\nabla }}_X\eta \right)\left(U\right)g(Y,V)\eta \left(W\right)-\eta \left(U\right)g(Y,V)\left({\mathrm{\nabla }}_X\eta \right)\left(W\right)\\ & +\left({\mathrm{\nabla }}_X\eta \right)\left(V\right)\left\{g\right(Y,W\left)\eta \right(U)-g(U,W\left)\eta \right(Y\left)\right\}\\ & +\eta \left(V\right)\left\{g\right(Y,W\left)\right({\mathrm{\nabla }}_X\eta \left)\right(U)-g(U,W\left)\right({\mathrm{\nabla }}_X\eta \left)\right(Y\left)\right\}\\ & -\left[\eta \right(R(Y,U)V)+g(U,V\left)\eta \right(Y)\\ & -g(Y,V)\eta \left(U\right)\left]g\right(\phi X,W)+\eta (Y\left)\right[R(\phi X,U,V,W)\\ & -2g(\phi X,\phi U)g(\phi V,W)-g(\phi X,V)\eta \left(U\right)\eta \left(W\right)\\ & +g(\phi X,W)\eta \left(U\right)\eta \left(V\right)]+\eta (U\left)\right[R(Y,\phi X,V,W)\\ & +2g(\phi Y,\phi X)g(\phi V,W)+g(\phi X,V)\eta \left(Y\right)\eta \left(W\right)\\ & -g(\phi X,W)\eta \left(Y\right)\eta \left(V\right)]+\eta (V\left)\right[R(Y,U,\phi X,W)\\ & +2g(Y,\phi U)g(\phi X,\phi W)+g(\phi X,U)\eta \left(Y\right)\eta \left(W\right)\\ & -g(\phi X,Y)\eta \left(U\right)\eta \left(W\right)]+\eta (W\left)\right[R(Y,U,V,\phi X)\\ & -2g(Y,\phi U)g(\phi V,\phi X)+g(Y,\phi X)\eta \left(U\right)\eta \left(V\right)\\ & -g(\phi X,U)\eta \left(Y\right)\eta \left(V\right)]+[A_1\left(X\right)+B_1\left(X\right)]\\ & \times \left[2g\right(Y,\phi U\left)g\right(\phi V,W)\\ & -g(U,V)\eta \left(Y\right)\eta \left(W\right)+g(Y,V)\eta \left(U\right)\eta \left(W\right)\\ & -\left\{g\right(Y,W\left)\eta \right(U)-g(U,W\left)\eta \right(Y\left)\right\}\eta \left(V\right)]\\ & +C_1\left(Y\right)\left[2g\right(X,\phi U\left)g\right(\phi V,W)-g(U,V\left)\eta \right(X\left)\eta \right(W)\\ & +g(X,V)\eta \left(U\right)\eta \left(W\right)-\left\{g\right(X,W\left)\eta \right(U)\\ & -g(U,W)\eta \left(X\right)\left\}\eta \right(V\left)\right]+C_1\left(U\right)\left[2g\right(Y,\phi X\left)g\right(\phi V,W)\\ & -g(X,V)\eta \left(Y\right)\eta \left(W\right)+g(Y,V)\eta \left(X\right)\eta \left(W\right)\\ & -\left\{g\right(Y,W\left)\eta \right(X)-g(X,W\left)\eta \right(Y\left)\right\}\eta \left(V\right)]\\ & +D_1\left(V\right)\left[2g\right(Y,\phi U\left)g\right(\phi X,W)-g(U,X\left)\eta \right(Y\left)\eta \right(W)\\ & +g(Y,X)\eta \left(U\right)\eta \left(W\right)-\left\{g\right(Y,W\left)\eta \right(U)\\ & -g(U,W)\eta \left(Y\right)\left\}\eta \right(X\left)\right]+D_1\left(W\right)\left[2g\right(Y,\phi U\left)g\right(\phi V,X)\\ & -g(U,V)\eta \left(Y\right)\eta \left(X\right)+g(Y,V)\eta \left(U\right)\eta \left(X\right)\\ & -\left\{g\right(Y,X\left)\eta \right(U)-g(U,X\left)\eta \right(Y\left)\right\}\eta \left(V\right)]\\ & =0\end{array}$(22) which yields $A_1\left(\xi \right)+B_1\left(\xi \right)+C_1\left(\xi \right)+D_1\left(\xi \right)=0$ for $X=U=V=\xi$.

From, the above one can state the followings

Claim 3.2

In a Sasakian manifold, a necessary condition for each of $[R_n,\overline{\mathrm{\nabla }}],$ $\left[\right(GR{)}_n,\overline{\mathrm{\nabla }}],$ $\left[\right(PS{)}_n,\overline{\mathrm{\nabla }}]$ and $\left[\right(GPS{)}_n,\overline{\mathrm{\nabla }}]$ to be $[R_n,\mathrm{\nabla }],$ $\left[\right(GR{)}_n,\mathrm{\nabla }],$ $\left[\right(PS{)}_n,\mathrm{\nabla }]$ and $\left[\right(GPS{)}_n,\mathrm{\nabla }]$ is $A_1\left(\xi \right)=0.$

Claim 3.3

In a Sasakian manifold, a necessary condition for each of $\left[\right(SPS{)}_n,\overline{\mathrm{\nabla }}]$ and $\left[\right(GSPS{)}_n,\overline{\mathrm{\nabla }}],$ to be $\left[\right(SPS{)}_n,\mathrm{\nabla }]$ and $\left[\right(GSPS{)}_n,\mathrm{\nabla }]$ is $C_1\left(\xi \right)=0.$

Claim 3.4

In a Sasakian manifold, a necessary condition for $\left[\right(WS{)}_n,\overline{\mathrm{\nabla }}]$ to be $\left[\right(WS{)}_n,\mathrm{\nabla }]$ is $A_1\left(\xi \right)+C_1\left(\xi \right)+D_1\left(\xi \right)=0.$

Again, contracting Equation (18(18) $\begin{array}{rl}& \left({\mathrm{\nabla }}_X{\overline{R}}^{\mathrm{\nabla }}\right)(Y,U,V,W)=\left[A_1\right(X)+B_1(X\left)\right]{\overline{R}}^{\mathrm{\nabla }}(Y,U,V,W)\\ & +C_1\left(Y\right){\overline{R}}^{\mathrm{\nabla }}(X,U,V,W)+C_1\left(U\right){\overline{R}}^{\mathrm{\nabla }}(Y,X,V,W)\\ & +D_1\left(V\right){\overline{R}}^{\mathrm{\nabla }}(Y,U,X,W)+D_1\left(W\right){\overline{R}}^{\mathrm{\nabla }}(Y,U,V,X)\\ & \left[A_2\right(X)+B_2(X\left)\right]G(Y,U,V,W)\\ & +C_2\left(Y\right)G(X,U,V,W)+C_2\left(U\right)G(Y,X,V,W)\\ & +D_2\left(V\right)G(Y,U,X,W)+D_2\left(W\right)G(Y,U,V,X)\end{array}$(18) ), we have (23) $\begin{array}{rl}\left({\overline{\mathrm{\nabla }}}_X{S}^{\mathrm{\nabla }}\right)(U,V)& =\left[A_1\right(X)+B_1(X\left)\right]S^{\mathrm{\nabla }}(U,V)\\ & +C_1\left(R^{\mathrm{\nabla }}\right(X,U\left)V\right)+C_1\left(U\right)S^{\mathrm{\nabla }}(X,V)\\ & +D_1\left(V\right)S^{\mathrm{\nabla }}(U,X)-D_1\left(R^{\mathrm{\nabla }}\right(V,X\left)U\right)\\ & +\left[A_2\right(X)+B_2(X\left)\right](n-1)g(U,V)\\ & +\left[C_2\right(X\left)g\right(U,V)-C_2(U\left)g\right(V,X\left)\right]\\ & +(n-1)C_2\left(U\right)g(X,V)\\ & +(n-1)D_2\left(V\right)g(U,X)\\ & +\left[D_2\right(X\left)g\right(U,V)-D_2(V\left)g\right(U,X\left)\right].\end{array}$(23) Replacing V by ξ in the above equation and then using the relations (16(16) $\begin{array}{rl}{R}^{\mathrm{\nabla }}(X,Y)Z& =R(X,Y)Z-2g(X,\phi Y)\phi Z+\eta \left(X\right)g(Y,Z)\xi \\ & -\eta \left(Y\right)g(X,Z)\xi +\eta \left(Z\right)\left\{\eta \right(Y)X-\eta (X\left)Y\right\}.\end{array}$(16) ), (17(17) $S^{\mathrm{\nabla }}(Y,Z)=S(Y,Z)-g(Y,Z)+n\eta \left(Y\right)\eta \left(Z\right),$(17) ), (10(10) $R(X,Y)\xi =\eta \left(Y\right)X-\eta \left(X\right)Y,\mathrm{\forall }X,Y\in TM.$(10) ), (13(13) $\begin{array}{rl}S(X,\xi )& =(n-1)\eta \left(X\right),\end{array}$(13) ) and (14(14) $\begin{array}{rl}R(X,\xi )Y& =\eta \left(Y\right)X-g(X,Y)\xi ,\end{array}$(14) ),  we get (24) $\begin{array}{rl}0& =2(n-1)\left[A_1\right(X)+B_1(X\left)\right]+2\left[C_1\right(X)+D_1(X\left)\right]\\ & +2(n-2)\left[C_1\right(\xi )+D_1(\xi \left)\right]\eta \left(X\right)\\ & +(n-1)\left[A_2\right(X)+B_2(X\left)\right]+C_2\left(X\right)+D_2\left(X\right)\\ & +(n-2)\left[C_2\right(\xi )+D_2(\xi \left)\right]\eta \left(X\right).\end{array}$(24)

Thus we can state the followings

Theorem 3.5

In a $\left[\right(AGPS{)}_n,\overline{\mathrm{\nabla }}],$  the 1-forms are related by (24(24) $\begin{array}{rl}0& =2(n-1)\left[A_1\right(X)+B_1(X\left)\right]+2\left[C_1\right(X)+D_1(X\left)\right]\\ & +2(n-2)\left[C_1\right(\xi )+D_1(\xi \left)\right]\eta \left(X\right)\\ & +(n-1)\left[A_2\right(X)+B_2(X\left)\right]+C_2\left(X\right)+D_2\left(X\right)\\ & +(n-2)\left[C_2\right(\xi )+D_2(\xi \left)\right]\eta \left(X\right).\end{array}$(24) ).

4. Example of a Sasakian manifold with $[A(GPS{)}_n,\overline{\mathrm{\nabla }}]$

Chose a 3-dimensional manifold spanned by a set of vector fields {$e_1,e_2,e_3$} defined by $\begin{array}{rl}{e}_1& =x_1\left(\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_1}+\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_2}\right)-2x_2\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_3},e_2=\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^2},\\ e_3& =\xi =\frac{\mathrm{\partial }}{\mathrm{\partial }{x}^3},\end{array}$ where {$x^1;x^2;x^3$} is a standard coordinates in $R^3.$ Define 1-form η, characteristic vector field ξ, Riemannian metric g and (1-1) tensor φ by $\eta \left(Z\right)=g(Z,e_3),$ $\xi =\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_3},$ $g(e_i,e_j)={\delta }_{ij}$ and $\phi e_1=-e_2$, $\phi e_2=e_1$and $\phi e_3=0$. Let ∇be the Levi-Civita connection with respect to the Riemannian metric g. Then we have $[e_1,e_2]=2e_3$, $[e_1,e_3]=0,$ $[e_2,e_3]=0.$ Thus, M(φ, ξ, η, g) defines a Sasakian manifold.

The Levi-Civita connection ∇ of the metric tensor g can be obtained by using Koszul's formula which are as follows: $\begin{array}{rl}{\mathrm{\nabla }}_{e_1}{e}_3& =-e_2,{\mathrm{\nabla }}_{e_1}{e}_2=e_3,{\mathrm{\nabla }}_{e_1}{e}_1=0,\\ {\mathrm{\nabla }}_{e_2}{e}_3& =e_1,{\mathrm{\nabla }}_{e_2}{e}_2=0,{\mathrm{\nabla }}_{e_2}{e}_1=-e_3,\\ {\mathrm{\nabla }}_{e_3}{e}_3& =0,{\mathrm{\nabla }}_{e_3}{e}_2=e_1,{\mathrm{\nabla }}_{e_3}{e}_1=-e_2.\end{array}$

In view of the above, one can easily obtain the following: $\begin{array}{rl}{\overline{\mathrm{\nabla }}}_{e_1}{e}_3& =0,{\overline{\mathrm{\nabla }}}_{e_1}{e}_2=e_3,{\overline{\mathrm{\nabla }}}_{e_1}{e}_1=0,\\ {\overline{\mathrm{\nabla }}}_{e_2}{e}_3& =0,{\overline{\mathrm{\nabla }}}_{e_2}{e}_2=0,{\overline{\mathrm{\nabla }}}_{e_2}{e}_1=-e_3,\\ {\overline{\mathrm{\nabla }}}_{e_3}{e}_3& =0,{\overline{\mathrm{\nabla }}}_{e_3}{e}_2=e_1,{\overline{\mathrm{\nabla }}}_{e_3}{e}_1=-e_2.\end{array}$ By virtue of the above results, we can easily obtain the non-vanishing components of the curvature tensors as follows: $R(e_1,e_2)e_2=-3e_1,R(e_1,e_3)e_3=e_1,R(e_2,e_3)e_3=e_2;$ and $\begin{array}{rl}& \tilde{R}(e_1,e_2)e_2=-5e_1,\tilde{R}(e_1,e_3)e_3=2e_1,\\ & \tilde{R}(e_2,e_3)e_3=2e_2.\end{array}$ Since {$e_1,e_2,e_3$} forms a basis of the Sasakian manifold, any vector field Y, U, V, $W\in$ χ(M) can be written as $\begin{array}{rl}Y& =a_1{e}_1+b_1{e}_2+c_1{e}_3;U=a_2{e}_1+b_2{e}_2+c_2{e}_3;\\ V& =a_3{e}_1+b_3{e}_2+c_3{e}_{3}W=a_4{e}_1+b_4{e}_2+c_4{e}_3;\end{array}$ where $a_i,b_i,c_i\in R^{+}$ (the set of all positive real numbers). Then $\begin{array}{rl}\overline{G}(Y,U,V,W)& =(a_2{a}_3+b_2{b}_3+c_2{c}_3)\\ & \times (a_1{a}_4+b_1{b}_4+c_1{c}_4)\\ & -(a_1{a}_3+b_1{b}_3+c_1{c}_3)\\ & \times (a_2{a}_4+b_2{b}_4+c_2{c}_4)=\theta \left(\mathrm{s}\mathrm{a}\mathrm{y}\right)\\ \overline{G}(e_1,U,V,W)& =a_4(a_2{a}_3+b_2{b}_3+c_2{c}_3)\\ & -a_3(a_2{a}_4+b_2{b}_4+c_2{c}_4)={\theta }_1\left(\mathrm{s}\mathrm{a}\mathrm{y}\right)\\ \overline{G}(e_2,U,V,W)& =b_4(a_2{a}_3+b_2{b}_3+c_2{c}_3)\\ & -b_3(a_2{a}_4+b_2{b}_4+c_2{c}_4)={\theta }_2\left(\mathrm{s}\mathrm{a}\mathrm{y}\right)\\ \overline{G}(e_3,U,V,W)& =c_4(a_2{a}_3+b_2{b}_3+c_2{c}_3)\\ & -c_3(a_2{a}_4+b_2{b}_4+c_2{c}_4)={\theta }_3\left(\mathrm{s}\mathrm{a}\mathrm{y}\right)\\ \overline{G}(Y,e_1,V,W)& =a_3(a_1{a}_4+b_1{b}_4+c_1{c}_4)\\ & -a_4(a_1{a}_3+b_1{b}_3+c_1{c}_3)={\theta }_4\left(\mathrm{s}\mathrm{a}\mathrm{y}\right)\\ \overline{G}(Y,e_2,V,W)& =b_3(a_1{a}_4+b_1{b}_4+c_1{c}_4)\\ & -b_4(a_1{a}_3+b_1{b}_3+c_1{c}_3)={\theta }_5\left(\mathrm{s}\mathrm{a}\mathrm{y}\right)\\ \overline{G}(Y,e_3,V,W)& =c_3(a_1{a}_4+b_1{b}_4+c_1{c}_4)\\ & -c_4(a_1{a}_3+b_1{b}_3+c_1{c}_3)={\theta }_6\left(\mathrm{s}\mathrm{a}\mathrm{y}\right)\\ \overline{G}(Y,U,e_1,W)& =a_2(a_1{a}_4+b_1{b}_4+c_1{c}_4)\\ & -a_1(a_2{a}_4+b_2{b}_4+c_2{c}_4)={\theta }_7\left(\mathrm{s}\mathrm{a}\mathrm{y}\right)\\ \overline{G}(Y,U,e_2,W)& =b_2(a_1{a}_4+b_1{b}_4+c_1{c}_4)\\ & -b_1(a_2{a}_4+b_2{b}_4+c_2{c}_4)={\theta }_8\left(\mathrm{s}\mathrm{a}\mathrm{y}\right)\\ \overline{G}(Y,U,e_3,W)& =c_2(a_1{a}_4+b_1{b}_4+c_1{c}_4)\\ & -c_1(a_2{a}_4+b_2{b}_4+c_2{c}_4)={\theta }_9\left(\mathrm{s}\mathrm{a}\mathrm{y}\right)\\ \overline{G}(Y,U,V,e_1)& =a_1(a_2{a}_3+b_2{b}_3+c_2{c}_3)\\ & -a_1(a_1{a}_3+b_1{b}_3+c_1{c}_3)={\theta }_{10}\left(\mathrm{s}\mathrm{a}\mathrm{y}\right)\\ \overline{G}(Y,U,V,e_2)& =b_1(a_2{a}_3+b_2{b}_3+c_2{c}_3)\\ & -b_1(a_1{a}_3+b_1{b}_3+c_1{c}_3)={\theta }_{11}\left(\mathrm{s}\mathrm{a}\mathrm{y}\right)\\ \overline{G}(Y,U,V,e_3)& =c_1(a_2{a}_3+b_2{b}_3+c_2{c}_3)\\ & -c_1(a_1{a}_3+b_1{b}_3+c_1{c}_3)={\theta }_{12}\left(\mathrm{s}\mathrm{a}\mathrm{y}\right)\end{array}$ $\begin{array}{rl}{R}^{\mathrm{\nabla }}(Y,U,V,W)& =-5(a_1{b}_2-a_2{b}_1)(a_3{b}_4-a_4{b}_3)\\ & +2(a_1{c}_2-a_2{c}_1)(a_3{c}_4-a_4{c}_3)\\ & +2(b_1{c}_2-b_2{c}_1)(b_3{c}_4-b_4{c}_3)\\ & =\lambda \left(\mathrm{s}\mathrm{a}\mathrm{y}\right)\\ R^{\mathrm{\nabla }}(e_1,U,V,W)& =-5b_2(a_3{b}_4-a_4{b}_3)+2c_2(a_3{c}_4\\ & -a_4{c}_3)={\lambda }_1\left(\mathrm{s}\mathrm{a}\mathrm{y}\right)\\ R^{\mathrm{\nabla }}(e_2,U,V,W)& =-5a_2(a_3{b}_4-a_4{b}_3)\\ & +2c_2(b_3{c}_4-b_4{c}_3)={\lambda }_2\left(\mathrm{s}\mathrm{a}\mathrm{y}\right)\\ R^{\mathrm{\nabla }}(e_3,U,V,W)& =2c_2(a_3{c}_4-a_4{c}_3)-2b_2(b_3{c}_4-b_4{c}_3)\\ & ={\lambda }_3\left(\mathrm{s}\mathrm{a}\mathrm{y}\right)\\ R^{\mathrm{\nabla }}(Y,e_1,V,W)& =5b_1(a_3{b}_4-a_4{b}_3)-2c_1(a_3{c}_4-a_4{c}_3)\\ & ={\lambda }_4\left(\mathrm{s}\mathrm{a}\mathrm{y}\right)\\ R^{\mathrm{\nabla }}(Y,e_2,V,W)& =-5a_1(a_3{b}_4-a_4{b}_3)\\ & +2c_1(b_3{c}_4-b_4{c}_3)={\lambda }_5\left(\mathrm{s}\mathrm{a}\mathrm{y}\right)\\ R^{\mathrm{\nabla }}(Y,e_3,V,W)& =2c_1(a_3{c}_4-a_4{c}_3)-2b_1(b_3{c}_4-b_4{c}_3)\\ & ={\lambda }_6\left(\mathrm{s}\mathrm{a}\mathrm{y}\right)\\ R^{\mathrm{\nabla }}(Y,U,e_1,W)& =-5b_4(a_1{b}_2-a_2{b}_1)\\ & +2c_4(a_1{c}_2-a_2{c}_1)={\lambda }_7\left(\mathrm{s}\mathrm{a}\mathrm{y}\right)\\ R^{\mathrm{\nabla }}(Y,U,e_2,W)& =-5a_4(a_1{b}_2-a_2{b}_1)\\ & +2c_4(b_1{c}_2-b_2{c}_1)={\lambda }_8\left(\mathrm{s}\mathrm{a}\mathrm{y}\right)\\ R^{\mathrm{\nabla }}(Y,U,e_3,W)& =2c_4(a_1{c}_2-a_2{c}_1)-2b_4(b_1{c}_2-b_2{c}_1)\\ & ={\lambda }_9\left(\mathrm{s}\mathrm{a}\mathrm{y}\right)\\ R^{\mathrm{\nabla }}(Y,U,V,e_1)& =-5b_3(a_1{b}_2-a_2{b}_1)\\ & +2c_3(a_1{c}_2-a_2{c}_1)={\lambda }_{10}\left(\mathrm{s}\mathrm{a}\mathrm{y}\right)\\ R^{\mathrm{\nabla }}(Y,U,V,e_2)& =-5a_3(a_1{b}_2-a_2{b}_1)\\ & +2c_3(b_1{c}_2-b_2{c}_1)={\lambda }_{11}\left(\mathrm{s}\mathrm{a}\mathrm{y}\right)\\ R^{\mathrm{\nabla }}(Y,U,V,e_3)& =2c_3(a_1{c}_2-a_2{c}_1)-2b_3(b_1{c}_2-b_2{c}_1)\\ & ={\lambda }_{12}\left(\mathrm{s}\mathrm{a}\mathrm{y}\right).\end{array}$ Making use of the above results we obtain the covariant derivative as follows: $\begin{array}{rl}\left({\overline{\mathrm{\nabla }}}_{e_1}{R}^{\mathrm{\nabla }}\right)(Y,U,V,W)& =-[b_1{\lambda }_3+b_2{\lambda }_6+b_3{\lambda }_9+b_4{\lambda }_{12}]\\ \left({\overline{\mathrm{\nabla }}}_{e_2}{R}^{\mathrm{\nabla }}\right)(Y,U,V,W)& =[a_1{\lambda }_3+a_2{\lambda }_6+a_3{\lambda }_9+a_4{\lambda }_{12}]\\ \left({\overline{\mathrm{\nabla }}}_{e_2}{R}^{\mathrm{\nabla }}\right)(Y,U,V,W)& =[a_1{\lambda }_2+a_2{\lambda }_5+a_3{\lambda }_8+a_4{\lambda }_{11}]\\ & -[b_1{\lambda }_1+b_2{\lambda }_4+b_3{\lambda }_7+b_4{\lambda }_{10}]\end{array}$ where $R^{\mathrm{\nabla }}(Y,U,V,W)=g\left(\overline{R}\right(Y,U)V,W).$ For the following choice of the one forms $\begin{array}{rl}{A}_1\left(e_1\right)& =-\frac{b_1{\lambda }_3}{T_1},B_1\left(e_1\right)=-\frac{b_2{\lambda }_6}{T_1},\\ A_2\left(e_1\right)& =-\frac{b_3{\lambda }_9}{T_2},B_2\left(e_1\right)=-\frac{b_4{\lambda }_{12}}{T_2},\\ A_1\left(e_2\right)& =\frac{a_1{\lambda }_3}{T_1},B_1\left(e_2\right)=\frac{a_2{\lambda }_6}{T_1},\\ A_2\left(e_2\right)& =\frac{a_3{\lambda }_9}{T_2},B_2\left(e_2\right)=\frac{a_4{\lambda }_{12}}{T_2},\\ A_1\left(e_3\right)& =\frac{a_3{\lambda }_8+a_4{\lambda }_{11}}{T_1},B_1\left(e_3\right)=\frac{a_1{\lambda }_2+a_2{\lambda }_5}{T_1},\\ C_1\left(e_3\right)& =\frac{1}{a_3{\lambda }_3+b_3{\lambda }_6},C_2\left(e_3\right)=\frac{1}{a_3{\theta }_3+b_3{\theta }_6},\\ D_1\left(e_3\right)& =-\frac{1}{c_3{\lambda }_9+d_3{\lambda }_{12}},D_2\left(e_3\right)=-\frac{1}{c_3{\theta }_9+d_3{\theta }_{12}},\\ A_2\left(e_3\right)& =-\frac{b_1{\lambda }_1+b_2{\lambda }_4}{T_2},B_2\left(e_3\right)=-\frac{b_3{\lambda }_7+b_4{\lambda }_{10}}{T_2},\end{array}$ one can easily verify the relations $\begin{array}{rl}\left({\overline{\mathrm{\nabla }}}_{e_i}{R}^{\mathrm{\nabla }})\right(X,Y,U,V)& =\left[A_1\right(e_i)+B_1(e_i\left)\right]R^{\mathrm{\nabla }}(X,Y,U,V)\\ & +C_1\left(X)R^{\mathrm{\nabla }}\right(e_i,Y,U,V)\\ & +C_1\left(Y)R^{\mathrm{\nabla }}\right(X,e_i,U,V)\\ & +D_1\left(U)R^{\mathrm{\nabla }}\right(X,Y,e_i,V)\\ & +D_1\left(V)R^{\mathrm{\nabla }}\right(X,Y,U,e_i)\\ & +\left[A_2\right(e_i)+B_2(e_i\left)\right]\overline{G}(X,Y,U,V)\\ & +C_2\left(X)\overline{G}\right(e_i,Y,U,V)\\ & +C_2\left(Y)\overline{G}\right(X,e_i,U,V)\\ & +D_2\left(U)\overline{G}\right(X,Y,e_i,V)\\ & +D_2\left(V)\overline{G}\right(X,Y,U,e_i)\end{array}$ for $1,2,3.$ From the above, we can state that

Theorem 4.1

There exists a Sasakian manifold ($M^3,g$) which is an $\left[A\right(GPS{)}_n,\overline{\mathrm{\nabla }}].$

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Kanak Kanti Baishya http://orcid.org/0000-0001-9331-7960

Fusun Nurcan http://orcid.org/0000-0003-0146-992X

Sanjib Kumar Jana http://orcid.org/0000-0001-8114-6156