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∈TM, where C and D are two non-vanishing 1-forms such that C(X)=g(X,μ1), D(X)=g(X,μ2) 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∈TM. Here μ1 and μ2 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 μ1 and μ2 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 2n+1=2n−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 μ2 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 Sφ 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 Sφ, the φ-Ricci tensor on M and the symmetric tensor gφ is defined by gφ(X,Y)=g(φX,φY). 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((∇WR)(X,Y)Z,ξ)=−g((∇WR)(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 φZ 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)φW,U)=g(R(X,Y)W,φU) and g((∇WR)(X,Y)φZ,U)=g((∇WR)(X,Y)Z,φU), 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.