In the paper [19], the authors studied quasi generalized recurrent manifolds and obtain some interesting results. Motivated by this work, we extend the notion called quasi generalized *φ*-recurrent manifolds. In this section, we study quasi generalized *φ*-recurrent Sasakian manifolds.

###
**Definition 3**

A Sasakian manifold *M* is said to be quasi generalized *φ*-recurrent manifold if its curvature tensor *R* satisfies the condition

$$\begin{array}{*{20}l} \varphi^{2}\left((\nabla_{W} R)(X,Y)Z\right)=C(W)R(X,Y)Z+D(W)F(X,Y)Z, \end{array} $$

(46)

for all *X*,*Y*,*Z*∈*T**M*, where *C* and *D* are two non-vanishing 1-forms such that *C*(*X*)=*g*(*X*,*μ*_{1}), *D*(*X*)=*g*(*X*,*μ*_{2}) and the tensor *F* is defined by

$$\begin{array}{*{20}l} F(X,Y)Z=&g(Y,Z)X-g(X,Z)Y+\eta(Y)\eta(Z)X-\eta(X)\eta(Z)Y \\ &+g(Y,Z)\eta(X)\xi-g(X,Z)\eta(Y)\xi, \end{array} $$

(47)

for all *X*,*Y*,*Z*∈*T**M*. Here *μ*_{1} and *μ*_{2} are vector fields associated with 1-forms *C* and *D* respectively. Especially, if the 1-form *D* vanishes, then (46) turns into the notion of *φ*-recurrent manifold.

**Note:** In view of (46) and (47), we say that locally quasi generalized *φ*-recurrent Sasakian manifold is a locally generalized *φ*-recurrent manifold.

We begin this section with the following:

###
**Theorem 5**

A quasi generalized *φ*-recurrent Sasakian manifold *M* is an Einstein manifold and moreover the associated vector fields *μ*_{1} and *μ*_{2} of the 1-forms *C* and *D* respectively are co-directional.

###
*Proof*

Using the same steps as in the proof of Theorem 1, we get the relation

$$\begin{array}{*{20}l} -(\nabla_{W} S)(Y,Z)&=C(W)S(Y,Z)+D(W)(2n+1)g(Y,Z)\\ &\quad+D(W)(2n-1)\eta(Y)\eta(Z). \end{array} $$

(48)

Again using the same steps as in the Theorem 2, we get the equations

$$\begin{array}{*{20}l} &S(Y,W)=2ng(Y,W),\qquad \text{and} \end{array} $$

(49)

$$\begin{array}{*{20}l} &C(W)=\frac{1-4n^{2}}{2n}D(W), \end{array} $$

(50)

for all *Y*,*W*. This completes the proof of the theorem. □

Equation (48) leads to the following:

###
**Theorem 6**

A quasi generalized *φ*-recurrent Sasakian manifold is a super generalized Ricci-recurrent manifold.

From (48), it follows that

$$\begin{array}{*{20}l} -dr(W)=rC(W)+2n(2n+3)D(W). \end{array} $$

(51)

This leads to the following:

###
**Theorem 7**

In a quasi generalized *φ*-recurrent Sasakian manifold, the 1-forms *C* and *D* are related by the Eq. (51).

###
**Corollary 7.1**

In a quasi generalized *φ*-recurrent Sasakian manifold with non-zero constant scalar curvature, the associated 1-forms *C* and *D* are related by

$$\begin{array}{*{20}l} rC(W)+2n(2n+3)D(W)=0. \end{array} $$

Now suppose that quasi generalized *φ*-recurrent Sasakian manifold is quasi generalized Ricci-recurrent [19]. Then from (48) we have 2*n*+1=2*n*−1, which is not possible. Therefore we can state the following;

###
**Theorem 8**

A quasi generalized *φ*-recurrent Sasakian manifold can not be a quasi generalized Ricci-recurrent manifold.

In view of (46) and (6), we obtain

$$\begin{array}{*{20}l} (\nabla_{W} R)(X,Y)Z=\eta((\nabla_{W} R)(X,Y)Z)\xi-C(W)R(X,Y)Z-D(W)F(X,Y)Z. \end{array} $$

(52)

From (52) and second Bianchi identity we get

$$\begin{array}{*{20}l} &C(W)R(X,Y,Z,U)+D(W)F(X,Y,Z,U)+C(X)R(Y,W,Z,U) \\ &+D(X)F(Y,W,Z,U)+C(Y)R(W,X,Z,U)+D(Y)F(W,X,Z,U)=0. \end{array} $$

(53)

Contracting the above relation over *Y* and *Z* and using (47), we get

$$\begin{array}{*{20}l} C(W)S(X,U)&+D(W)\left\{(2n+1)g(X,U)+(2n-1)\eta(X)\eta(U)\right\}\\ &-C(X)S(W,U)-D(X)\left\{(2n+1)g(W,U)+(2n-1)\eta(W)\eta(U)\right\} \\ &-C(R(W,X)U)+D(X)\left\{g(W,U)+\eta(W)\eta(U)\right\}-D(W)\{g(X,U) \\ &+\eta(X)\eta(U)\}+D(\xi)\left\{\eta(X)g(W,U)-\eta(W)g(X,U)\right\}=0. \end{array} $$

(54)

Again contracting (54) over *X* and *U* and using (50), we get

$$ S(W,\mu_{2})=\beta g(W,\mu_{2})+\gamma \eta(W)\eta(\mu_{2}),\\ $$

(55)

where \(\beta =\frac {r}{2}+\frac {2n\left (2n^{2}-1\right)}{1-4n^{2}}\) and \(\gamma =\frac {2n(1-4n)}{1-4n^{2}}\). Hence we can state the following;

###
**Theorem 9**

In a quasi generalized *φ*-recurrent Sasakian manifold, the Ricci tensor *S* and vector field *μ*_{2} are related by the Eq. (55).

###
**Definition 4**

[25] Let *M* be an almost contact metric manifold with Ricci tensor *S*. The ∗-Ricci tensor and ∗-scalar curvature of *M* are defined repectively by

$$\begin{array}{*{20}l} S^{*}(X,Y)=\sum_{i=1}^{2n+1}R(X,e_{i},\varphi e_{i}, \varphi Y),\quad \text{and}\quad r^{*}=\sum_{i=1}^{2n+1}S^{*}(e_{i},e_{i}). \end{array} $$

(56)

###
**Definition 5**

[26] An almost contact metric manifold *M* is said to be weakly *φ*-Einstein if

$$\begin{array}{*{20}l} S^{\varphi}(X,Y)=\beta g^{\varphi}(X,Y),\quad X,Y\in TM, \end{array} $$

for some function *β*. Here *S*^{φ} denotes the symmetric part of *S*^{∗}, that is,

$$\begin{array}{*{20}l} S^{\varphi}(X,Y)=\frac{1}{2}\left\{S^{*}(X,Y)+S^{*}(Y,X)\right\}, \quad X,Y\in TM, \end{array} $$

we call *S*^{φ}, the *φ*-Ricci tensor on *M* and the symmetric tensor *g*^{φ} is defined by *g*^{φ}(*X*,*Y*)=*g*(*φ**X*,*φ**Y*). When *β* is constant, *M* is said to be *φ*-Einstein.

In a Sasakian manifold we know the following relation

$$\begin{array}{*{20}l} (\nabla_{W} R)(X,Y)\xi=g(W,\varphi Y)X-g(W,\varphi X)Y+R(X,Y)\varphi W. \end{array} $$

(57)

Using (57) and the relation *g*((∇_{W}*R*)(*X*,*Y*)*Z*,*ξ*)=−*g*((∇_{W}*R*)(*X*,*Y*)*ξ*,*Z*) in (52), we have

$$\begin{array}{*{20}l} (\nabla_{W} R)(X,Y)Z=&g(W,\varphi X)g(Y,Z)\xi-g(W,\varphi Y)g(X,Z)\xi-g(R(X,Y)\varphi W,Z)\xi \\ &-C(W)R(X,Y)Z-D(W)F(X,Y)Z, \end{array} $$

(58)

from which it follows that

$$\begin{array}{*{20}l} g((\nabla_{W} R)(X,Y)Z, U)&=g(W,\varphi X)g(Y,Z)\eta(U)-g(W,\varphi Y)g(X,Z)\eta(U)\\ &\quad+g(R(X,Y)Z, \varphi W)\eta(U)-C(W)g(R(X,Y)Z,U)\\ &\quad-D(W)g(F(X,Y)Z, U). \end{array} $$

(59)

Replacing *Z* by *φ**Z* in the foregoing equation, we obtain

$$\begin{array}{*{20}l} g((\nabla_{W} R)(X,Y)\varphi Z, U)&=g(W,\varphi X)g(Y,\varphi Z)\eta(U)-g(W,\varphi Y)g(X,\varphi Z)\eta(U)\\ &\quad+g(R(X,Y)\varphi Z, \varphi W)\eta(U) -C(W)g(R(X,Y)\varphi Z,U)\\ &\quad-D(W)g(F(X,Y)\varphi Z, U). \end{array} $$

(60)

Since *g*(*R*(*X*,*Y*)*φ**W*,*U*)=*g*(*R*(*X*,*Y*)*W*,*φ**U*) and *g*((∇_{W}*R*)(*X*,*Y*)*φ**Z*,*U*)=*g*((∇_{W}*R*)(*X*,*Y*)*Z*,*φ**U*), using these equation in (60), we get

$$\begin{array}{*{20}l} g((\nabla_{W} R)(X,Y)Z,\varphi U)&=g(W,\varphi X)g(Y,\varphi Z)\eta(U)-g(W,\varphi Y)g(X,\varphi Z)\eta(U)\\ &\quad+g(R(X,Y)\varphi Z, \varphi W)\eta(U) -C(W)g(R(X,Y) Z,\varphi U)\\ &\quad-D(W)g(F(X,Y)\varphi Z, U). \end{array} $$

(61)

Contracting (61) over *Y* and *Z* and using (47), we get

$$\begin{array}{*{20}l} (\nabla_{W} S)(X,\varphi U)=&-g(\varphi X,\varphi W)\eta(U)+S^{*}(X,W)\eta(U)\\ &-C(W)S(X,\varphi U)+D(W)g(X,\varphi U). \end{array} $$

(62)

In view of (48), we have

$$\begin{array}{*{20}l} S^{*}(X,W)=g(\varphi X,\varphi W)-\frac{(2n+2)D(W)}{\eta(U)}g(X,\varphi U). \end{array} $$

(63)

Substituting *U*=*ξ* in (63), we get

$$\begin{array}{*{20}l} S^{*}(X,W)=g(\varphi X,\varphi W). \end{array} $$

(64)

From (64) and Definition 5, we conclude that it is *φ*-Einstein. Hence we can state the following;

###
**Theorem 10**

A quasi generalized *φ*-recurrent Sasakian manifold is an *φ*-Einstein manifold.

In view of (7) and (64), we have the following;

###
**Theorem 11**

A quasi generalized *φ*-recurrent Sasakian manifold is an ∗- *η*-Einstein manifold.