Lifts of a Quarter-Symmetric Metric Connection from a Sasakian Manifold to Its Tangent Bundle

Abstract

The objective of this paper is to explore the complete lifts of a quarter-symmetric metric connection from a Sasakian manifold to its tangent bundle. A relationship between the Riemannian connection and the quarter-symmetric metric connection from a Sasakian manifold to its tangent bundle was established. Some theorems on the curvature tensor and the projective curvature tensor of a Sasakian manifold with respect to the quarter-symmetric metric connection to its tangent bundle were proved. Finally, locally $\varphi$-symmetric Sasakian manifolds with respect to the quarter-symmetric metric connection to its tangent bundle were studied.

1. Introduction

The study of the tangent bundle is a powerful method in geometry that allows us to retrieve effective results while studying various connections and geometric structures, such as a quarter-symmetric metric connection, a semi-symmetric connection, an almost complex structure and a contact structure on the manifold M admitting lifts to its tangent bundle $TM$. Peyghan et al. [1] studied the members of a golden structure on $TM$ with a Riemannian metric and established the integrability condition of such a structure on $TM$. The complete lifts of connections such as quarter-symmetric metric connection and quarter-symmetric non-metric connection from the manifold M to $TM$ have been studied by Akpinar [2], Altunbas et al. ([3,4]), Kazan and Karadag [5], Khan [6]. For the recent studies on lifts of connections and geometric structures, we refer to ([7,8,9,10,11]) and many more.

The definition and discussion of a quarter-symmetric connection on a Riemannian manifold, on the other hand, were provided by Golab [12].

A linear connection $\tilde{\nabla }$ on a Riemannian manifold M (dim = $n$) with a Reimannian metric g is called a quarter-symmetric connection if its torsion tensor T of the connection $\tilde{\nabla }$

$$T({\zeta }_1,{\zeta }_2)={\tilde{\nabla }}_{{\zeta }_1}{\zeta }_2-{\tilde{\nabla }}_{{\zeta }_2}{\zeta }_1-[{\zeta }_1,{\zeta }_2]$$

(1)

satisfies

$$T({\zeta }_1,{\zeta }_2)=\eta \left({\zeta }_2\right)\varphi {\zeta }_1-\eta \left({\zeta }_1\right)\varphi {\zeta }_2,$$

(2)

where $\eta$ is a 1-form and $\varphi$ is a tensor field of type (1,1).

In addition, if $\tilde{\nabla }$ fulfills

$$\left({\tilde{\nabla }}_{{\zeta }_1}g\right)({\zeta }_2,{\zeta }_3)=0,$$

(3)

$\forall {\zeta }_1,{\zeta }_2,{\zeta }_3\in {\Im }_0^1\left(M\right)$, then $\tilde{\nabla }$ is called a quarter-symmetric metric connection; otherwise, it is called a quarter-symmetric non-metric connection ([13,14,15]). The quarter-symmetric metric connections on different manifolds such as Riemannian, Hermitian, Kaehlerian, Kenmotsu and Sasakian manifolds have been studied by Mondol and De [16], Mishra and Pandey [17], Mukhopadhyay et al. [18], Bahadir [19], Sular et al. [20] and many more.

We established certain curvature properties on $TM$ and explored the lifts of a quarter-symmetric metric connection from a Sasakian manifold to $TM$. The results of this paper are given as:

• We established a relationship between the Riemannian connection and the quarter-symmetric metric connection from a Sasakian manifold to $TM$.
• We derived the expression of the curvature tensor of a Sasakian manifold equipped with a quarter-symmetric metric connection to $TM$.
• We studied a $\xi$-projectively flat Sasakian manifold endowed with a quarter-symmetric metric connection to $TM$.
• We locally characterized a $\varphi$-symmetric Sasakian manifold admitting a quarter-symmetric metric connection to $TM$.

2. Preliminaries

Let us consider $TM$ to be the tangent bundle of a manifold M. The set of all tensor fields of type $(r,s)$ that are of contravariant degree r and covariant degree s in M and $TM$ are denoted by ${\Im }_s^r\left(M\right)$ and ${\Im }_s^r\left(TM\right)$, respectively. Let the function, a 1-form, a vector field and a tensor field of type (1,1) be symbolized as $f,\eta ,{\zeta }_1$ and $\varphi$, respectively. The complete and vertical lifts of $f,\eta ,{\zeta }_1,\varphi$ are symbolized as $f^C,{\eta }^C,{\zeta }_1^C,{\varphi }^C$ and $f^V,{\eta }^V,{\zeta }_1^V,{\varphi }^V$, respectively. The following operations on $f,\eta ,{\zeta }_1$ and $\varphi$ are defined by [21,22]

$$\begin{array}{c}\hfill {\left(f{\zeta }_1\right)}^V=f^V{\zeta }_1^V,{\left(f{\zeta }_1\right)}^C=f^C{\zeta }_1^V+f^V{\zeta }_1^C,\end{array}$$

(4)

$$\begin{array}{c}\hfill {\zeta }_1^V{f}^V=0,{\zeta }_1^V{f}^C={\zeta }_1^C{f}^V={\left({\zeta }_{1}f\right)}^V,{\zeta }_1^C{f}^C={\left({\zeta }_{1}f\right)}^C,\end{array}$$

(5)

$$\begin{array}{c}\hfill {\eta }^V\left(f^V\right)=0,{\eta }^V\left({\zeta }_1^C\right)={\eta }^C\left({\zeta }_1^V\right)=\eta {\left({\zeta }_1\right)}^V,{\eta }^C\left({\zeta }_1^C\right)=\eta {\left({\zeta }_1\right)}^C,\end{array}$$

(6)

$$\begin{array}{c}\hfill {\varphi }^V{\zeta }_1^C={\left(\varphi {\zeta }_1\right)}^V,{\varphi }^C{\zeta }_1^C={\left(\varphi {\zeta }_1\right)}^C,\end{array}$$

(7)

$$\begin{array}{c}\hfill {[{\zeta }_1,{\zeta }_2]}^V=[{\zeta }_1^C,{\zeta }_2^V]=[{\zeta }_1^V,{\zeta }_2^C],{[{\zeta }_1,{\zeta }_2]}^C=[{\zeta }_1^C,{\zeta }_2^C].\end{array}$$

(8)

$${\nabla }_{{\zeta }_1^C}^C{\zeta }_2^C={\left({\nabla }_{{\zeta }_1}{\zeta }_2\right)}^C,{\nabla }_{{\zeta }_1^C}^C{\zeta }_2^V={\left({\nabla }_{{\zeta }_1}{\zeta }_2\right)}^V,$$

(9)

where ∇ is the Levi–Civita connection.

Let M be a contact metric manifold of dimension n with a contact metric structure $(\varphi ,\xi ,\eta ,g)$ fulfilling the conditions [23]

$$\begin{array}{ccc}\hfill \eta \left({\zeta }_1\right)& =& g({\zeta }_1,\xi ),{\varphi }^2=-{\zeta }_1+\eta \left({\zeta }_1\right)\xi ,\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill \varphi \xi & =& 0,\eta \left(\xi \right)=1,\eta ·\varphi =0,\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill g(\varphi {\zeta }_1,\varphi {\zeta }_2)& =& g({\zeta }_1,{\zeta }_2)-\eta \left({\zeta }_1\right)\eta \left({\zeta }_2\right),\hfill \end{array}$$

(12)

where $\varphi$ is a (1,1) tensor, $\xi$ is a vector field, called the characteristic vector field, and $\eta$ is a 1-form. If M satisfies

$$\left({\nabla }_{{\zeta }_1}\varphi \right){\zeta }_2=g({\zeta }_1,{\zeta }_2)\xi -\eta \left({\zeta }_2\right){\zeta }_1,$$

(13)

then M is named a Sasakian manifold. In addition, the following properties hold on a Sasakian manifold M:

$$\begin{array}{ccc}\hfill {\nabla }_{{\zeta }_1}\xi & =& -\varphi {\zeta }_1,\hfill \end{array}$$

(14)

$$\begin{array}{ccc}\hfill \left({\nabla }_{{\zeta }_1}\eta \right){\zeta }_2& =& g({\zeta }_1,\varphi {\zeta }_2),\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill R({\zeta }_1,{\zeta }_2)\xi & =& \eta \left({\zeta }_2\right){\zeta }_1-\eta \left({\zeta }_1\right){\zeta }_2,\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill R(\xi ,{\zeta }_1){\zeta }_2& =& \left({\nabla }_{{\zeta }_1}\varphi \right){\zeta }_2),\hfill \end{array}$$

(17)

$$\begin{array}{ccc}\hfill S({\zeta }_1,\xi )& =& (n-1)\eta \left({\zeta }_1\right),\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill S(\varphi {\zeta }_1,\varphi {\zeta }_2)& =& S({\zeta }_1,{\zeta }_2)-(n-1)\eta \left({\zeta }_1\right)\eta \left({\zeta }_2\right),\hfill \end{array}$$

(19)

where $\forall {\zeta }_1,{\zeta }_2\in {\Im }_0^1\left(M\right)$, R and S indicate the curvature tensor and the Ricci tensor, respectively.

3. Complete Lifts from a Sasakian Manifold to Its Tangent Bundle

Let us consider $TM$ to be the tangent bundle of a Sasakian manifold M. Taking complete lifts on both sides of Equations (1), (2) and (10)–(32), we infer that

$$\begin{array}{ccc}\hfill T^C({\zeta }_1^C,{\zeta }_2^C)& =& {\tilde{\nabla }}_{{\zeta }_1^C}^C{\zeta }_2^C-{\tilde{\nabla }}_{{\zeta }_2^C}^C{\zeta }_1^C-[{\zeta }_1^C,{\zeta }_2^C],\hfill \\ \hfill T^C({\zeta }_1^C,{\zeta }_2^C)& =& {\pi }^C\left({\zeta }_2^C\right){\left(\varphi {\zeta }_1\right)}^V+{\pi }^V\left({\zeta }_2^C\right){\left(\varphi {\zeta }_1\right)}^C\hfill \end{array}$$

(20)

$$\begin{array}{ccc}& -& {\pi }^C\left({\zeta }_1^C\right){\left(\varphi {\zeta }_2\right)}^V-{\pi }^V\left({\zeta }_1^C\right){\left(\varphi {\zeta }_2\right)}^C,\hfill \end{array}$$

(21)

$$\begin{array}{ccc}\hfill {\pi }^C\left({\zeta }_1^C\right)& =& g^C({\xi }^C,{\zeta }_1^C),\hfill \\ \hfill (d\eta {({\zeta }_1,{\zeta }_2)}^C& =& g^C({\left(\varphi {\zeta }_1\right)}^C,{\zeta }_2^C),{\eta }^C\left({\zeta }_1^C\right)=g^C({\zeta }_1^C,{\xi }^C)\hfill \end{array}$$

(22)

$$\begin{array}{ccc}\hfill {\left({\varphi }^2\right)}^C{\zeta }_1& =& -{\zeta }_1^C+{\eta }^C\left({\zeta }_1^C\right){\xi }^V+{\eta }^V\left({\zeta }_1^C\right){\xi }^C,\hfill \\ \hfill {\varphi }^C{\xi }^V& =& {\varphi }^V{\xi }^C={\varphi }^V{\xi }^V={\varphi }^C{\xi }^C=0,\hfill \\ \hfill {\eta }^C{\xi }^C& =& {\eta }^V{\xi }^V=0,{\eta }^C{\xi }^V={\eta }^V{\xi }^C=1\hfill \end{array}$$

(23)

$$\begin{array}{ccc}\hfill {\eta }^V\circ {\varphi }^C& =& {\eta }^C\circ {\varphi }^V={\eta }^C\circ {\varphi }^C={\eta }^V\circ {\varphi }^V=0,\hfill \end{array}$$

(24)

$$\begin{array}{ccc}\hfill g({\left(\varphi {\zeta }_1\right)}^C,{\left(\varphi {\zeta }_2\right)}^C)& =& g^C({\zeta }_1^C,{\zeta }_2^C)-{\eta }^C\left({\zeta }_1^C\right){\eta }^V\left({\zeta }_2^C\right)\hfill \end{array}$$

$$\begin{array}{ccc}& -& {\eta }^V\left({\zeta }_1^C\right){\eta }^C\left({\zeta }_2^C\right),\hfill \\ \hfill \left({\nabla }_{{\zeta }_1^C}^C{\varphi }^C\right){\zeta }_2^C& =& g^C({\zeta }_1^C,{\zeta }_2^C){\xi }^V+g^C({\zeta }_1^V,{\zeta }_2^C){\xi }^C\hfill \end{array}$$

(25)

$$\begin{array}{ccc}& -& {\eta }^C\left({\zeta }_2^C\right){\zeta }_1^V-{\eta }^V\left({\zeta }_2^C\right){\zeta }_1^C,\hfill \end{array}$$

(26)

$$\begin{array}{ccc}\hfill {\nabla }_{{\zeta }_1^C}^C{\xi }^C& =& -{\left(\varphi {\zeta }_1\right)}^C,\hfill \end{array}$$

(27)

$$\begin{array}{ccc}\hfill \left({\nabla }_{{\zeta }_1^C}^C{\eta }^C\right){\zeta }_2^C& =& g^C({\zeta }_1^C,{\left(\varphi {\zeta }_2\right)}^C),\hfill \\ \hfill R^C({\zeta }_1^C,{\zeta }_2^C){\xi }^C& =& {\eta }^C\left({\zeta }_2^C\right){\zeta }_1^V+{\eta }^V\left({\zeta }_2^C\right){\zeta }_1^C-{\eta }^C\left({\zeta }_1^C\right){\zeta }_2^V\hfill \end{array}$$

(28)

$$\begin{array}{ccc}& -& {\eta }^V\left({\zeta }_1^C\right){\zeta }_2^C,\hfill \end{array}$$

(29)

$$\begin{array}{ccc}\hfill R^C({\xi }^C,{\zeta }_1^C){\zeta }_2^C& =& \left({\nabla }_{{\zeta }_1^C}^C{\varphi }^C\right){\zeta }_2^C,\hfill \end{array}$$

(30)

$$\begin{array}{ccc}\hfill S^C({\zeta }_1^C,{\xi }^C)& =& (n-1){\eta }^C\left({\zeta }_1^C\right),\hfill \\ \hfill S^C({\left(\varphi {\zeta }_1\right)}^C,{\left(\varphi {\zeta }_2\right)}^C)& =& S^C({\zeta }_1^C,{\zeta }_2^C)-(n-1)\{{\eta }^C\left({\zeta }_1^C\right){\eta }^V\left({\zeta }_2^C\right)\hfill \end{array}$$

(31)

$$\begin{array}{ccc}& +& {\eta }^V\left({\zeta }_1^C\right){\eta }^C\left({\zeta }_2^C\right)\},\hfill \end{array}$$

(32)

where $\forall {\zeta }_1^C,{\zeta }_2^C,{\xi }^C\in {\Im }_0^1\left(TM\right),{\varphi }^C\in {\Im }_1^1\left(TM\right)$.

4. Relation between the Riemannian Connection and the Quarter-Symmetric Metric Connection from a Sasakian Manifold to Its Tangent Bundle

Assuming that M is an almost contact metric manifold, let $\tilde{\nabla }$ be a linear connection and ∇ be a Riemannian connection. Then,

$${\tilde{\nabla }}_{X}Y={\nabla }_{X}Y+U({\zeta }_1,{\zeta }_2),$$

(33)

where $\forall {\zeta }_1,{\zeta }_2\in {\Im }_0^1\left(M\right),U\in {\Im }_2^1\left(M\right)$. Let $\tilde{\nabla }$ be a quarter-symmetric metric connection in M. Then [12],

$$U({\zeta }_1,{\zeta }_2)=\frac{1}{2}[T({\zeta }_1,{\zeta }_2)+T^{\prime }({\zeta }_1,{\zeta }_2)+T^{\prime }({\zeta }_2,{\zeta }_1),$$

(34)

where $T^{\prime }$ is a (1,2) tensor; that is, $T^{\prime }\in {\Im }_2^1\left(M\right)$ such that

$$g(T^{\prime }({\zeta }_1,{\zeta }_2),{\zeta }_3)=g(T^{\prime }({\zeta }_3,{\zeta }_1),{\zeta }_2).$$

(35)

Taking complete lifts on both sides of Equations (34)–(36), we infer that

$$\begin{array}{ccc}\hfill {\tilde{\nabla }}_{{\zeta }_1^C}^C{\zeta }_2^C& =& {\nabla }_{{\zeta }_1^C}^C{\zeta }_2^C+U^C({\zeta }_1^C,{\zeta }_2^C),\hfill \\ \hfill U^C({\zeta }_1^C,{\zeta }_2^C)& =& \frac{1}{2}[T^C({\zeta }_1^C,{\zeta }_2^C)+{T^{\prime }}^C({\zeta }_1^C,{\zeta }_2^C)\hfill \end{array}$$

(36)

$$\begin{array}{ccc}& +& {T^{\prime }}^C({\zeta }_2^C,{\zeta }_1^C)],\hfill \end{array}$$

(37)

$$\begin{array}{ccc}\hfill g^C({T^{\prime }}^C({\zeta }_1^C,{\zeta }_2^C),{\zeta }_3^C)& =& g^C({T^{\prime }}^C({\zeta }_3^C,{\zeta }_1^C),{\zeta }_2^C),\hfill \end{array}$$

(38)

where $U^C,{\nabla }^C,T^C$ and ${T^{\prime }}^C$ are complete lifts of $U,\nabla ,T$ and $T^{\prime }$, respectively.

From (21) and (38), we infer that

$$\begin{array}{ccc}\hfill {T^{\prime }}^C({\zeta }_1^C,{\zeta }_2^C)& =& g^C({\left(\varphi {\zeta }_2\right)}^C,{\zeta }_1^C){\xi }^V+g^C({\left(\varphi {\zeta }_2\right)}^V,{\zeta }_1^C){\xi }^C\hfill \\ & -& {\eta }^C\left({\zeta }_1^C\right){\left(\varphi {\zeta }_2\right)}^V-{\eta }^V\left({\zeta }_1^C\right){\left(\varphi {\zeta }_2\right)}^C.\hfill \end{array}$$

(39)

Using (21) and (39) in (37), we provide

$$U^C({\zeta }_1^C,{\zeta }_2^C)=-{\eta }^C\left({\zeta }_1^C\right){\left(\varphi {\zeta }_2\right)}^V-{\eta }^V\left({\zeta }_1^C\right){\left(\varphi {\zeta }_2\right)}^C.$$

Hence, a quarter-symmetric metric connection ${\tilde{\nabla }}^C$ on a Sasakian manifold on $TM$ is defined by

$${\tilde{\nabla }}_{{\zeta }_1^C}^C{\zeta }_2^C={\nabla }_{{\zeta }_1^C}^C{\zeta }_2^C-{\eta }^C\left({\zeta }_1^C\right){\left(\varphi {\zeta }_2\right)}^V-{\eta }^V\left({\zeta }_1^C\right){\left(\varphi {\zeta }_2\right)}^C.$$

(40)

In contrast, we demonstrate that a linear connection $\tilde{\nabla }$ on a Sasakian manifold defined by

$${\tilde{\nabla }}_{{\zeta }_1^C}^C{\zeta }_2^C={\nabla }_{{\zeta }_1^C}^C{\zeta }_2^C-{\eta }^C\left({\zeta }_1^C\right){\left(\varphi {\zeta }_2\right)}^V-{\eta }^V\left({\zeta }_1^C\right){\left(\varphi {\zeta }_2\right)}^C.$$

(41)

denotes a quarter-symmetric metric connection on $TM$.

In view of (41), the torsion tensor of the connection ${\tilde{\nabla }}^C$ on $TM$ is defined by

$$\begin{array}{ccc}\hfill T^C({\zeta }_1^C,{\zeta }_2^C)& =& {\eta }^C\left({\zeta }_2^C\right){\left(\varphi {\zeta }_1\right)}^V+{\eta }^V\left({\zeta }_2^C\right){\left(\varphi {\zeta }_1\right)}^C\hfill \\ & -& {\eta }^C\left({\zeta }_1^C\right){\left(\varphi {\zeta }_2\right)}^V-{\eta }^V\left({\zeta }_1^C\right){\left(\varphi {\zeta }_2\right)}^C.\hfill \end{array}$$

(42)

The Equation (42) implies that ${\tilde{\nabla }}^C$ is a quarter-symmetric connection on $TM$. Further, we infer that

$$\begin{array}{ccc}\hfill ({\tilde{\nabla }}_{{\zeta }_1^C}^C{g}^C)({\zeta }_2^C,{\zeta }_3^C)& =& {\zeta }_1^C{g}^C({\zeta }_2^V,{\zeta }_3^C)+{\zeta }_1^V{g}^C({\zeta }_2^C,{\zeta }_3^C)\hfill \\ & -& g^C({\tilde{\nabla }}_{{\zeta }_1^C}^C{\zeta }_2^C,{\zeta }_3^C).\hfill \end{array}$$

(43)

In view of (42) and (43), ${\tilde{\nabla }}^C$ is a quarter-symmetric metric connection on $TM$. The relationship between the Riemannian connection and the quarter-symmetric metric connection on a Sasakian manifold on $TM$ is given by (41).

5. Expression of the Curvature Tensor of a Sasakian Manifold to Its Tangent Bundle

The two curvature tensors R and $\tilde{R}$ corresponding to the connections $\tilde{\nabla }$ and ∇, respectively, are related by the formula [24]

$$\begin{array}{ccc}\hfill \tilde{R}({\zeta }_1,{\zeta }_2){\zeta }_3& =& R({\zeta }_1,{\zeta }_2){\zeta }_3-2d\eta ({\zeta }_1,{\zeta }_2)\varphi {\zeta }_3+\eta \left({\zeta }_1\right)g({\zeta }_2,{\zeta }_3)\xi \hfill \\ & -& \eta \left({\zeta }_2\right)g({\zeta }_1,{\zeta }_3)\xi +\{\eta \left({\zeta }_2\right){\zeta }_1-\eta \left({\zeta }_1\right){\zeta }_2\}\eta \left({\zeta }_3\right),\hfill \end{array}$$

(44)

where $R({\zeta }_1,{\zeta }_2){\zeta }_3$ indicates the Riemannian curvature of M.

Taking complete lifts on both sides of (44), we infer that

$$\begin{array}{ccc}\hfill {\tilde{R}}^C({\zeta }_1^C,{\zeta }_2^C){\zeta }_3^C& =& R^C({\zeta }_1^C,{\zeta }_2^C){\zeta }_3^C-2d{\eta }^C({\zeta }_1^C,{\zeta }_2^C){\left(\varphi {\zeta }_3\right)}^V\hfill \\ & -& 2d{\eta }^V({\zeta }_1^C,{\zeta }_2^C){\left(\varphi {\zeta }_3\right)}^C+{\eta }^C\left({\zeta }_1^C\right)g^C({\zeta }_2^C,{\zeta }_3^C){\xi }^V\hfill \\ & +& {\eta }^C\left({\zeta }_1^C\right)g^C({\zeta }_2^V,{\zeta }_3^C){\xi }^C+{\eta }^V\left({\zeta }_1^C\right)g^C({\zeta }_2^C,{\zeta }_3^C){\xi }^C\hfill \\ & -& {\eta }^C\left({\zeta }_2^C\right)g^C({\zeta }_1^C,{\zeta }_3^C){\xi }^V-{\eta }^C\left({\zeta }_2^C\right)g^C({\zeta }_1^V,{\zeta }_3^C){\xi }^C\hfill \\ & -& {\eta }^V\left({\zeta }_2^C\right)g^C({\zeta }_1^C,{\zeta }_3^C){\xi }^C+{\eta }^C\left({\zeta }_2^C\right){\eta }^C\left({\zeta }_3^C\right){\zeta }_1^V\hfill \\ & +& {\eta }^C\left({\zeta }_2^C\right){\eta }^V\left({\zeta }_3^C\right){\zeta }_1^C+{\eta }^V\left({\zeta }_2^C\right){\eta }^C\left({\zeta }_3^C\right){\zeta }_1^C\hfill \\ & -& \{{\eta }^C\left({\zeta }_1^C\right){\eta }^C\left({\zeta }_3^C\right){\zeta }_2^V+{\eta }^C\left({\zeta }_1^C\right){\eta }^V\left({\zeta }_3^C\right){\zeta }_2^C\hfill \\ & +& {\eta }^V\left({\zeta }_1^C\right){\eta }^C\left({\zeta }_3^C\right){\zeta }_2^C\},\hfill \end{array}$$

(45)

where $R^C({\zeta }_1^C,{\zeta }_2^C){\zeta }_3^C$ is the complete lift of $R({\zeta }_1,{\zeta }_2){\zeta }_3$.

On contracting (45), we infer that

$$\begin{array}{ccc}\hfill {\tilde{S}}^C({\zeta }_2^C,{\zeta }_3^C)& =& S^C({\zeta }_2^C,{\zeta }_3^C)-2d{\eta }^C({\left(\varphi {\zeta }_3\right)}^C,{\zeta }_2^C)+g^C({\zeta }_2^C,{\zeta }_3^C)\hfill \\ & +& (n-2)\{{\eta }^C\left({\zeta }_2^C\right){\eta }^V\left({\zeta }_3^C\right)+{\eta }^V\left({\zeta }_2^C\right){\eta }^C\left({\zeta }_3^C\right)\},\hfill \end{array}$$

(46)

where ${\tilde{S}}^C$ and $S^C$ are the complete lifts of the Ricci tensors $\tilde{S}$ and S of the connections $\tilde{\nabla }$ and ∇, respectively. From (46), we infer that the Ricci tensor with regard to ${\tilde{\nabla }}^C$ on a Sasakian manifold on $TM$ is symmetric.

Again contracting (46), we infer that

$${\tilde{r}}^C=r^C+2(n-1),$$

where ${\tilde{r}}^C$ and $r^C$ on $TM$ are the complete lifts of the scalar curvatures $\tilde{r}$ and r of the connections $\tilde{\nabla }$ and ∇, respectively.

6. Expression of the Projective Curvature Tensor of a Sasakian Manifold to Its Tangent Bundle

The projective curvature tensor of a Sasakian manifold with regard to $\tilde{\nabla }$ is given by [17]

$$\begin{array}{ccc}\hfill \tilde{P}({\zeta }_1,{\zeta }_2){\zeta }_3& =& \tilde{R}({\zeta }_1,{\zeta }_2){\zeta }_3+\frac{1}{n+1}[\tilde{S}({\zeta }_1,{\zeta }_2){\zeta }_3-\tilde{S}({\zeta }_2,{\zeta }_1){\zeta }_3]\hfill \\ & +& \frac{1}{n^2-1}[\{n\tilde{S}({\zeta }_1,{\zeta }_3)+\tilde{S}({\zeta }_3,{\zeta }_1)\}{\zeta }_2\hfill \\ & -& \{n\tilde{S}({\zeta }_2,{\zeta }_3)+\tilde{S}({\zeta }_3,{\zeta }_2)\}{\zeta }_1].\hfill \end{array}$$

(47)

Due to the symmetric property of the Ricci tensor $\tilde{S}$ of M with regard to $\tilde{\nabla }$, the projective curvature tensor $\tilde{P}$ becomes

$$\begin{array}{ccc}\hfill \tilde{P}({\zeta }_1,{\zeta }_2){\zeta }_3& =& \tilde{R}({\zeta }_1,{\zeta }_2){\zeta }_3+\frac{1}{n+1}[\tilde{S}({\zeta }_1,{\zeta }_2){\zeta }_3-\tilde{S}({\zeta }_2,{\zeta }_1){\zeta }_3].\hfill \end{array}$$

(48)

Taking complete lifts on both sides of (48), we acquire

$$\begin{array}{ccc}\hfill {\tilde{P}}^C({\zeta }_1^C,{\zeta }_2^C){\zeta }_3^C& =& {\tilde{R}}^C({\zeta }_1^C,{\zeta }_2^C){\zeta }_3^C\hfill \\ & +& \frac{1}{n+1}[{\tilde{S}}^C({\zeta }_1^C,{\zeta }_2^C){\zeta }_3^V+{\tilde{S}}^V({\zeta }_1^C,{\zeta }_2^C){\zeta }_3^C\hfill \\ & -& {\tilde{S}}^C({\zeta }_2^C,{\zeta }_1^C){\zeta }_3^V-{\tilde{S}}^V({\zeta }_2^C,{\zeta }_1^C){\zeta }_3^C].\hfill \end{array}$$

(49)

Using (45) and (46), (49) reduces to

$$\begin{array}{ccc}\hfill {\tilde{P}}^C({\zeta }_1^C,{\zeta }_2^C){\zeta }_3^C& =& P^C({\zeta }_1^C,{\zeta }_2^C){\zeta }_3^C-2d{\eta }^C({\zeta }_1^C,{\zeta }_2^C){\left(\varphi {\zeta }_3\right)}^V\hfill \\ & -& 2d{\eta }^V({\zeta }_1^C,{\zeta }_2^C){\left(\varphi {\zeta }_3\right)}^C+{\eta }^C\left({\zeta }_1^C\right)g^C({\zeta }_2^C,{\zeta }_3^C){\xi }^V\hfill \\ & +& {\eta }^C\left({\zeta }_1^C\right)g^C({\zeta }_2^V,{\zeta }_3^C){\xi }^C+{\eta }^V\left({\zeta }_1^C\right)g^C({\zeta }_2^C,{\zeta }_3^C){\xi }^C\hfill \\ & -& {\eta }^C\left({\zeta }_2^C\right)g^C({\zeta }_1^C,{\zeta }_3^C){\xi }^V-{\eta }^C\left({\zeta }_2^C\right)g^C({\zeta }_1^V,{\zeta }_3^C){\xi }^C\hfill \\ & -& {\eta }^V\left({\zeta }_2^C\right)g^C({\zeta }_1^C,{\zeta }_3^C){\xi }^C+\frac{2}{n-1}[d{\eta }^C({\left(\varphi {\zeta }_3\right)}^C,{\zeta }_2^C){\zeta }_1^V\hfill \\ & +& d{\eta }^V({\left(\varphi {\zeta }_3\right)}^C,{\zeta }_2^C){\zeta }_1^C]-d{\eta }^C({\left(\varphi {\zeta }_3\right)}^C,{\zeta }_1^C){\zeta }_2^V\hfill \\ & +& d{\eta }^V({\left(\varphi {\zeta }_3\right)}^C,{\zeta }_1^C){\zeta }_2^C+\frac{1}{n-1}\{{\eta }^C\left({\zeta }_2^C\right){\eta }^C\left({\zeta }_3^C\right){\zeta }_1^V\hfill \\ & +& {\eta }^C\left({\zeta }_2^C\right){\eta }^V\left({\zeta }_3^C\right){\zeta }_1^C+{\eta }^V\left({\zeta }_2^C\right){\eta }^C\left({\zeta }_3^C\right){\zeta }_1^C\hfill \\ & -& {\eta }^C\left({\zeta }_1^C\right){\eta }^C\left({\zeta }_3^C\right){\zeta }_2^V-{\eta }^C\left({\zeta }_1^C\right){\eta }^V\left({\zeta }_3^C\right){\zeta }_2^C\hfill \\ & -& {\eta }^V\left({\zeta }_1^C\right){\eta }^C\left({\zeta }_3^C\right){\zeta }_2^C-g^C({\zeta }_2^C,{\zeta }_3^C){\zeta }_1^V\hfill \\ & -& g^C({\zeta }_2^V,{\zeta }_3^C){\zeta }_1^C+g^C({\zeta }_1^C,{\zeta }_3^C){\zeta }_2^V\hfill \\ & +& g^C({\zeta }_1^V,{\zeta }_3^C){\zeta }_2^C\},\hfill \end{array}$$

(50)

where $P^C$ is the complete lift of the projective curvature tensor P defined by

$$P({\zeta }_1,{\zeta }_2){\zeta }_3=R({\zeta }_1,{\zeta }_2){\zeta }_3-\frac{1}{n-1}\{S({\zeta }_2,{\zeta }_3){\zeta }_1-S({\zeta }_1,{\zeta }_3){\zeta }_2\}.$$

(51)

Mondol and De [16] defined that “A Sasakian manifold M is called $\xi$-projectively flat if the condition $P({\zeta }_1,{\zeta }_2)\xi =0$ holds on M”.

According to the above definition, from (50), we acquire $\tilde{P}({\zeta }_1,{\zeta }_2)\xi =P({\zeta }_1,{\zeta }_2)\xi$.

Hence, we conclude the following:

Theorem 1. 

Let $TM$ be the tangent bundle of a Sasakian manifold M with the Riemannian connection ∇. The Riemannian connection ${\nabla }^C$ on $TM$ is ${\xi }^C$-projectively flat if and only if ${\tilde{\nabla }}^C$ is so.

Özgür [25] defined that “a Sasakian manifold fulfilling

$${\varphi }^{2}P(\varphi {\zeta }_1,\varphi {\zeta }_2)\varphi {\zeta }_3=0$$

(52)

is called $\varphi$-projectively flat”.

In the case of the quarter-symmetric metric connection $\tilde{\nabla }$, we see that ${\varphi }^2\tilde{P}(\varphi {\zeta }_1,\varphi {\zeta }_2)\varphi {\zeta }_3=0$ remain invariant if and only if

$$g(\tilde{P}(\varphi {\zeta }_1,\varphi {\zeta }_2)\varphi {\zeta }_3,\varphi {\zeta }_4)=0,$$

(53)

for ${\zeta }_1,{\zeta }_2,{\zeta }_3,{\zeta }_4\in {\Im }_0^1\left(M\right)$.

In view of (49) and (53), $\varphi$-projectively flat means

$$\begin{array}{ccc}\hfill & & g^C({\tilde{R}}^C({(\varphi {\zeta }_1)}^C,{(\varphi {\zeta }_2)}^C){(\varphi {\zeta }_3)}^C,{(\varphi {\zeta }_4)}^C)\hfill \\ & =& \frac{1}{n-1}\{{\tilde{S}}^C({(\varphi {\zeta }_2)}^C,{(\varphi {\zeta }_3)}^C)g^V({(\varphi {\zeta }_1)}^C,{(\varphi {\zeta }_4)}^C)\hfill \\ & +& {\tilde{S}}^V({(\varphi {\zeta }_2)}^C,{(\varphi {\zeta }_3)}^C)g^C({(\varphi {\zeta }_1)}^C,{(\varphi {\zeta }_4)}^C)\hfill \\ & -& -{\tilde{S}}^C({(\varphi {\zeta }_1)}^C,{(\varphi {\zeta }_3)}^C)g^V({(\varphi {\zeta }_2)}^C,{(\varphi {\zeta }_4)}^C)\hfill \\ & -& {\tilde{S}}^V({(\varphi {\zeta }_1)}^C,{(\varphi {\zeta }_3)}^C)g^C({(\varphi {\zeta }_2)}^C,{(\varphi {\zeta }_4)}^C)\}.\hfill \end{array}$$

(54)

If $(e_1^C,e_2^C,\dots \dots ,e_{n-1}^C,{\xi }^C)\in TM$, then

$({(\varphi e_1)}^C,{(\varphi e_2)}^C,\dots \dots ,{(\varphi e_{n-1})}^C,{\xi }^C)\in TM$.

Substituting ${\zeta }_1={\zeta }_4=e_i$ into (54) and summing up with regard to $i=1,2,\dots \dots ,n-1$, we acquire

$$\begin{array}{ccc}\hfill & & g^C({\tilde{R}}^C({(\varphi e_i)}^C,{(\varphi {\zeta }_2)}^C){(\varphi {\zeta }_3)}^C,{(\varphi e_i)}^C)\hfill \\ & =& \frac{1}{n-1}\{{\tilde{S}}^C({(\varphi {\zeta }_2)}^C,{(\varphi {\zeta }_3)}^C)g^V({(\varphi e_i)}^C,{(\varphi e_i)}^C)\hfill \\ & +& {\tilde{S}}^V({(\varphi {\zeta }_2)}^C,{(\varphi {\zeta }_3)}^C)g^C({(\varphi e_i)}^C,{(\varphi e_i)}^C)\hfill \\ & -& {\tilde{S}}^C({(\varphi e_i)}^C,{(\varphi {\zeta }_3)}^C)g^V({(\varphi {\zeta }_2)}^C,{(\varphi e_i)}^C)\hfill \\ & -& {\tilde{S}}^V({(\varphi e_i)}^C,{(\varphi {\zeta }_3)}^C)g^C({(\varphi {\zeta }_2)}^C,{(\varphi e_i)}^C)\}.\hfill \end{array}$$

(55)

Using (23), (24), (28) and (46), the following equations are obtained:

$$\begin{array}{ccc}\hfill & & g^C({\tilde{R}}^C({(\varphi e_i)}^C,{(\varphi {\zeta }_2)}^C){(\varphi {\zeta }_3)}^C,{(\varphi e_i)}^C)\hfill \\ & =& g^C({\tilde{R}}^C({(\varphi e_i)}^C,{(\varphi {\zeta }_2)}^C){(\varphi {\zeta }_3)}^C,{(\varphi e_i)}^C)\hfill \\ & -& 2g^C({(\varphi {\zeta }_2)}^C,{(\varphi {\zeta }_3)}^C)\hfill \\ & =& S^C({\zeta }_2^C,{\zeta }_3^C)-R^C({\xi }^C,{\zeta }_2^C,{\zeta }_3^C,{\xi }^C)\hfill \\ & -& (n-1)\{{\eta }^C({\zeta }_2^C){\eta }^V({\zeta }_3^C)+{\eta }^V({\zeta }_2^C){\eta }^C({\zeta }_3^C)\}\hfill \\ & -& 2g^C({(\varphi {\zeta }_2)}^C,{(\varphi {\zeta }_3)}^C)\hfill \\ & =& {\tilde{S}}^C({\zeta }_2^C,{\zeta }_3^C)-6g^C({\zeta }_2^C,{\zeta }_3^C)\hfill \end{array}$$

(56)

$$\begin{array}{ccc}& -& 2(n-4)\{{\eta }^C({\zeta }_2^C){\eta }^V({\zeta }_3^C)+{\eta }^V({\zeta }_2^C){\eta }^C({\zeta }_3^C)\},\hfill \end{array}$$

(57)

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{i=1}^{n-1}}{g}^C({(\varphi e_i)}^C,{(\varphi e_i)}^C)& =& n-1,\hfill \end{array}$$

(58)

$$\begin{array}{ccc}\hfill {\displaystyle \sum _{i=1}^{n-1}}{(\tilde{S}(\varphi e_i,\varphi {\zeta }_3)g(\varphi {\zeta }_2,\varphi e_i))}^C& =& {\tilde{S}}^C({(\varphi {\zeta }_2)}^C,{(\varphi {\zeta }_3)}^C).\hfill \end{array}$$

(59)

In view of (56), (58) and (59), Equation (55) becomes

$$\begin{array}{ccc}\hfill {\tilde{S}}^C({\zeta }_2^C,{\zeta }_3^C)-6g^C({\zeta }_2^C,{\zeta }_3^C)& -& 2(n-4)\{{\eta }^C({\zeta }_2^C){\eta }^V({\zeta }_3^C)+{\eta }^V({\zeta }_2^C){\eta }^C({\zeta }_3^C)\}\hfill \\ & =& \frac{n-2}{n-1}{\tilde{S}}^C({(\varphi {\zeta }_2)}^C,{(\varphi {\zeta }_3)}^C).\hfill \end{array}$$

(60)

In view of (31) and (46), (60) becomes

$${\tilde{S}}^C({\zeta }_2^C,{\zeta }_3^C)=6g^C({\zeta }_2^C,{\zeta }_3^C)-4(n-1)\{{\eta }^C({\zeta }_2^C){\eta }^V({\zeta }_3^C)+{\eta }^V({\zeta }_2^C){\eta }^C({\zeta }_3^C)\}.$$

(61)

Hence, we conclude the following:

Theorem 2. 

Let $TM$ be the tangent bundle of a Sasakian manifold M with regard to $\tilde{\nabla }$. If a Sasakian manifold M on $TM$ is ${\varphi }^C$-projectively flat with regard to ${\tilde{\nabla }}^C$, then the manifold is an ${\eta }^C$-Einstein manifold with regard to ${\tilde{\nabla }}^C$ on $TM$.

7. Locally $\varphi$-Symmetric Sasakian Manifold with regard to the Quarter-Symmetric Metric Connection to its Tangent Bundle

Takahashi [26] defined that “A Sasakian manifold is said to be locally $\varphi$-symmetric if

$${\varphi }^2({\nabla }_{{\zeta }_4}R)({\zeta }_1,{\zeta }_2){\zeta }_3=0,$$

(62)

for all vector fields ${\zeta }_4,{\zeta }_1,{\zeta }_2,{\zeta }_3$ orthogonal to $\xi$, where $\xi$ is the characteristic vector field of the Sasakian manifold M.” Further, Mondal and De [16] defined locally $\varphi$-symmetric Sasakian manifold with regard to $\tilde{\nabla }$ as

$${\varphi }^2({\tilde{\nabla }}_{{\zeta }_4}\tilde{R})({\zeta }_1,{\zeta }_2){\zeta }_3=0,$$

(63)

where ${\zeta }_4,{\zeta }_1,{\zeta }_2,{\zeta }_3$ are orthogonal to $\xi$. In view of (40), we infer that

$$\begin{array}{ccc}\hfill {(({\tilde{\nabla }}_{{\zeta }_4}\tilde{R})({\zeta }_1,{\zeta }_2){\zeta }_3)}^C& =& {(({\nabla }_{{\zeta }_4}\tilde{R})({\zeta }_1,{\zeta }_2){\zeta }_3)}^C-{\eta }^C({\zeta }_4^C){(\varphi \tilde{R}({\zeta }_1,{\zeta }_2){\zeta }_3)}^V\hfill \\ & -& {\eta }^V({\zeta }_4^C){(\varphi \tilde{R}({\zeta }_1,{\zeta }_2){\zeta }_3)}^C.\hfill \end{array}$$

(64)

Now, differentiating (44) with regard to ${\zeta }_4$, we infer that

$$\begin{array}{ccc}\hfill {(({\tilde{\nabla }}_{{\zeta }_4}\tilde{R})({\zeta }_1,{\zeta }_2){\zeta }_3)}^C& =& {(({\nabla }_{{\zeta }_4}\tilde{R})({\zeta }_1,{\zeta }_2){\zeta }_3)}^C-2d{\eta }^C({\zeta }_1^C,{\zeta }_2^C){(({\nabla }_{{\zeta }_4}\varphi ){\zeta }_3)}^V\hfill \\ & -& 2d{\eta }^V({\zeta }_1^C,{\zeta }_2^C){(({\nabla }_{{\zeta }_4}\varphi ){\zeta }_3)}^C+{({\nabla }_{{\zeta }_4}\eta )}^C({\zeta }_1^C)g^C({\zeta }_2^C,{\zeta }_3^C){\xi }^V\hfill \\ & +& {({\nabla }_{{\zeta }_4}\eta )}^C({\zeta }_1^C)g^C({\zeta }_2^V,{\zeta }_3^C){\xi }^C+{({\nabla }_{{\zeta }_4}\eta )}^V({\zeta }_1^C)g^C({\zeta }_2^C,{\zeta }_3^C){\xi }^C\hfill \\ & -& {({\nabla }_{{\zeta }_4}\eta )}^C({\zeta }_2^C)g^C({\zeta }_1^C,{\zeta }_3^C){\xi }^V-{({\nabla }_{{\zeta }_4}\eta )}^C({\zeta }_2^C)g^C({\zeta }_1^V,{\zeta }_3^C){\xi }^C\hfill \\ & -& {({\nabla }_{{\zeta }_4}\eta )}^V({\zeta }_2^C)g^C({\zeta }_1^C,{\zeta }_3^C){\xi }^C-{\eta }^C({\zeta }_2^C)g^C({\zeta }_1^C,{\zeta }_3^C){({\nabla }_{{\zeta }_4}\xi )}^V\hfill \\ & -& {\eta }^C({\zeta }_2^C)g^C({\zeta }_1^V,{\zeta }_3^C){({\nabla }_{{\zeta }_4}\xi )}^C-{\eta }^V({\zeta }_2^C)g^C({\zeta }_1^C,{\zeta }_3^C){({\nabla }_{{\zeta }_4}\xi )}^C\hfill \\ & -& {\eta }^C({\zeta }_1^C)g^C({\zeta }_2^C,{\zeta }_3^C){({\nabla }_{{\zeta }_4}\xi )}^V-{\eta }^C({\zeta }_1^C)g^C({\zeta }_2^V,{\zeta }_3^C){({\nabla }_{{\zeta }_4}\xi )}^C\hfill \\ & -& {\eta }^V({\zeta }_1^C)g^C({\zeta }_2^C,{\zeta }_3^C){({\nabla }_{{\zeta }_4}\xi )}^C+{({\nabla }_{{\zeta }_4}\eta )}^C({\zeta }_2^C){\eta }^C({\zeta }_3^C){\zeta }_1^V\hfill \\ & +& {({\nabla }_{{\zeta }_4}\eta )}^C({\zeta }_2^C){\eta }^V({\zeta }_3^C){\zeta }_1^V+{({\nabla }_{{\zeta }_4}\eta )}^V({\zeta }_2^C){\eta }^C({\zeta }_3^C){\zeta }_1^C\hfill \\ & +& {({\nabla }_{{\zeta }_4}\eta )}^C({\zeta }_3^C){\eta }^C({\zeta }_2^C){\zeta }_1^V+{({\nabla }_{{\zeta }_4}\eta )}^C({\zeta }_3^C){\eta }^V({\zeta }_2^C){\zeta }_1^V\hfill \\ & +& {({\nabla }_{{\zeta }_4}\eta )}^V({\zeta }_3^C){\eta }^C({\zeta }_2^C){\zeta }_1^C-{({\nabla }_{{\zeta }_4}\eta )}^C({\zeta }_1^C){\eta }^C({\zeta }_3^C){\zeta }_2^V\hfill \\ & -& {({\nabla }_{{\zeta }_4}\eta )}^C({\zeta }_1^C){\eta }^V({\zeta }_3^C){\zeta }_2^V-{({\nabla }_{{\zeta }_4}\eta )}^V({\zeta }_1^C){\eta }^C({\zeta }_3^C){\zeta }_2^C\hfill \\ & -& {({\nabla }_{{\zeta }_4}\eta )}^C({\zeta }_3^C){\eta }^C({\zeta }_1^C){\zeta }_2^V+{({\nabla }_{{\zeta }_4}\eta )}^C({\zeta }_3^C){\eta }^V({\zeta }_1^C){\zeta }_2^V\hfill \\ & +& {({\nabla }_{{\zeta }_4}\eta )}^V({\zeta }_3^C){\eta }^C({\zeta }_1^C){\zeta }_2^C.\hfill \end{array}$$

(65)

Using (25), (26) and (27), we infer that

$$\begin{array}{ccc}\hfill {(({\tilde{\nabla }}_{{\zeta }_4^C}^C\tilde{R})({\zeta }_1,{\zeta }_2){\zeta }_3)}^C& =& {(({\nabla }_{{\zeta }_4}\tilde{R})({\zeta }_1,{\zeta }_2){\zeta }_3)}^C-2d{\eta }^C({\zeta }_1^C,{\zeta }_2^C)g^C({\zeta }_3^C,{\zeta }_4^C){\xi }^V\hfill \\ & -& 2d{\eta }^C({\zeta }_1^C,{\zeta }_2^C)g^C({\zeta }_3^V,{\zeta }_4^C){\xi }^C\hfill \\ & -& 2d{\eta }^C({\zeta }_1^C,{\zeta }_2^C)g^C({\zeta }_3^C,{\zeta }_4^C){\xi }^C\hfill \\ & +& 4d{\eta }^C({\zeta }_1^C,{\zeta }_2^C)g^C({(\varphi {\zeta }_1)}^C,{\zeta }_2^C){\eta }^C({\zeta }_3^C){\zeta }_4^V\hfill \\ & +& 4d{\eta }^C({\zeta }_1^C,{\zeta }_2^C)g^C({(\varphi {\zeta }_1)}^C,{\zeta }_2^C){\eta }^V({\zeta }_3^C){\zeta }_4^C\hfill \\ & +& 4d{\eta }^C({\zeta }_1^C,{\zeta }_2^C)g^C({(\varphi {\zeta }_1)}^V,{\zeta }_2^C){\eta }^C({\zeta }_3^C){\zeta }_4^C\hfill \\ & +& 4d{\eta }^V({\zeta }_1^C,{\zeta }_2^C)g^C({(\varphi {\zeta }_1)}^C,{\zeta }_2^C){\eta }^C({\zeta }_3^C){\zeta }_4^C\hfill \\ & +& g^C({(\varphi {\zeta }_4)}^C,{\zeta }_2^C)g^C({\zeta }_1^C,{\zeta }_3^C){\xi }^V\hfill \end{array}$$

$$\begin{array}{ccc}\hfill & +& g^C({(\varphi {\zeta }_4)}^C,{\zeta }_2^C)g^C({\zeta }_1^V,{\zeta }_3^C){\xi }^C\hfill \\ & +& g^C({(\varphi {\zeta }_4)}^V,{\zeta }_2^C)g^C({\zeta }_1^C,{\zeta }_3^C){\xi }^C\hfill \\ & -& g^C({(\varphi {\zeta }_4)}^C,{\zeta }_1^C)g^C({\zeta }_2^C,{\zeta }_3^C){\xi }^V\hfill \\ & -& g^C({(\varphi {\zeta }_4)}^C,{\zeta }_1^C)g^C({\zeta }_2^V,{\zeta }_3^C){\xi }^C\hfill \\ & -& g^C({(\varphi {\zeta }_4)}^V,{\zeta }_1^C)g^C({\zeta }_2^C,{\zeta }_3^C){\xi }^C\hfill \\ & +& {\eta }^C({\zeta }_2^C)g^C({\zeta }_1^C,{\zeta }_3^C){(\varphi {\zeta }_4)}^V\hfill \\ & +& {\eta }^C({\zeta }_2^C)g^C({\zeta }_1^V,{\zeta }_3^C){(\varphi {\zeta }_4)}^C+{\eta }^V({\zeta }_2^C)g^C({\zeta }_1^C,{\zeta }_3^C){(\varphi {\zeta }_4)}^C\hfill \\ & -& {\eta }^C({\zeta }_1^C)g^C({\zeta }_2^C,{\zeta }_3^C){(\varphi {\zeta }_4)}^V-{\eta }^C({\zeta }_1^C)g^C({\zeta }_2^V,{\zeta }_3^C){(\varphi {\zeta }_4)}^C\hfill \\ & -& {\eta }^V({\zeta }_1^C)g^C({\zeta }_2^C,{\zeta }_3^C){(\varphi {\zeta }_4)}^C-g^C({(\varphi {\zeta }_4)}^C,{\zeta }_2^C){\eta }^C({\zeta }_3^C){\zeta }_1^V\hfill \\ & -& g^C({(\varphi {\zeta }_4)}^C,{\zeta }_2^C){\eta }^V({\zeta }_3^C){\zeta }_1^C-g^C({(\varphi {\zeta }_4)}^V,{\zeta }_2^C){\eta }^C({\zeta }_3^C){\zeta }_1^C\hfill \\ & -& g^C({(\varphi {\zeta }_4)}^C,{\zeta }_3^C){\eta }^C({\zeta }_2^C){\zeta }_1^V-g^C({(\varphi {\zeta }_4)}^C,{\zeta }_3^C){\eta }^V({\zeta }_2^C){\zeta }_1^C\hfill \\ & -& g^C({(\varphi {\zeta }_4)}^V,{\zeta }_3^C){\eta }^C({\zeta }_1^C){\zeta }_1^C\hfill \\ & +& g^C({(\varphi {\zeta }_4)}^C,{\zeta }_1^C){\eta }^C({\zeta }_3^C){\zeta }_2^V+g^C({(\varphi {\zeta }_4)}^C,{\zeta }_1^C){\eta }^V({\zeta }_3^C){\zeta }_2^C\hfill \\ & +& g^C({(\varphi {\zeta }_4)}^V,{\zeta }_1^C){\eta }^C({\zeta }_3^C){\zeta }_2^C\hfill \\ & +& g^C({(\varphi {\zeta }_4)}^C,{\zeta }_3^C){\eta }^C({\zeta }_1^C){\zeta }_2^V+g^C({(\varphi {\zeta }_4)}^C,{\zeta }_3^C){\eta }^V({\zeta }_1^C){\zeta }_2^C\hfill \\ & +& g^C({(\varphi {\zeta }_4)}^V,{\zeta }_3^C){\eta }^C({\zeta }_1^C){\zeta }_2^C.\hfill \end{array}$$

(66)

Using (66) and (24) in (64), we infer that

$$\begin{array}{ccc}\hfill {({\varphi }^2({\tilde{\nabla }}_{{\zeta }_4}\tilde{R})({\zeta }_1,{\zeta }_2){\zeta }_3)}^C& =& {({\varphi }^2({\nabla }_{W}R)({\zeta }_1,{\zeta }_2){\zeta }_3)}^C-2d{\eta }^C({\zeta }_1^C,{\zeta }_2^C){\eta }^C({\zeta }_3^C){\zeta }_4^V\hfill \\ & -& 2d{\eta }^C({\zeta }_1^C,{\zeta }_2^C){\eta }^V({\zeta }_3^C){\zeta }_4-2d{\eta }^V({\zeta }_1^C,{\zeta }_2^C){\eta }^C({\zeta }_3^C){\zeta }_4^C\hfill \\ & +& 2d{\eta }^C({\zeta }_1^C,{\zeta }_2^C){\eta }^C({\zeta }_3^C){\eta }^C({\zeta }_4^C){\xi }^V\hfill \\ & +& 2d{\eta }^C({\zeta }_1^C,{\zeta }_2^C){\eta }^C({\zeta }_3^C){\eta }^V({\zeta }_4^C){\xi }^C\hfill \\ & +& 2d{\eta }^C({\zeta }_1^C,{\zeta }_2^C){\eta }^V({\zeta }_3^C){\eta }^C({\zeta }_4^C){\xi }^C\hfill \\ & +& 2d{\eta }^V({\zeta }_1^C,{\zeta }_2^C){\eta }^C({\zeta }_3^C){\eta }^C({\zeta }_4^C){\xi }^C\hfill \\ & -& {\eta }^C({\zeta }_2^C)g^C({\zeta }_1^C,{\zeta }_3^C){(\varphi {\zeta }_4)}^V\hfill \\ & -& {\eta }^C({\zeta }_2^C)g^C({\zeta }_1^V,{\zeta }_3^C){(\varphi {\zeta }_4)}^C-{\eta }^V({\zeta }_2^C)g^C({\zeta }_1^C,{\zeta }_3^C){(\varphi {\zeta }_4)}^C\hfill \\ & +& {\eta }^C({\zeta }_1^C)g^C({\zeta }_2^C,{\zeta }_3^C){(\varphi {\zeta }_4)}^V+{\eta }^C({\zeta }_1^C)g^C({\zeta }_2^V,{\zeta }_3^C){(\varphi {\zeta }_4)}^C\hfill \\ & +& {\eta }^V({\zeta }_1^C)g^C({\zeta }_2^C,{\zeta }_3^C){(\varphi {\zeta }_4)}^C+g^C({(\varphi {\zeta }_4)}^C,{\zeta }_2^C){\eta }^C({\zeta }_3^C){\zeta }_1^V\hfill \\ & +& g^C({(\varphi {\zeta }_4)}^C,{\zeta }_2^C){\eta }^V({\zeta }_3^C){\zeta }_1^C+g^C({(\varphi {\zeta }_4)}^V,{\zeta }_2^C){\eta }^C({\zeta }_3^C){\zeta }_1^C\hfill \\ & -& g^C({(\varphi {\zeta }_4)}^C,{\zeta }_2^C){\eta }^C({\zeta }_3^C){\eta }^C({\zeta }_1^C){\xi }^V\hfill \\ & -& g^C({(\varphi {\zeta }_4)}^C,{\zeta }_2^C){\eta }^C({\zeta }_3^C){\eta }^V({\zeta }_1^C){\xi }^C\hfill \\ & -& g^C({(\varphi {\zeta }_4)}^C,{\zeta }_2^C){\eta }^V({\zeta }_3^C){\eta }^C({\zeta }_1^C){\xi }^C\hfill \\ & -& g^C({(\varphi {\zeta }_4)}^V,{\zeta }_2^C){\eta }^C({\zeta }_3^C){\eta }^C({\zeta }_1^C){\xi }^C\hfill \\ & +& g^C({(\varphi {\zeta }_4)}^C,{\zeta }_3^C){\eta }^C({\zeta }_2^C){\zeta }_1^V+g^C({(\varphi {\zeta }_4)}^C,{\zeta }_3^C){\eta }^V({\zeta }_2^C){\zeta }_1^C\hfill \\ & +& g^C({(\varphi {\zeta }_4)}^V,{\zeta }_3^C){\eta }^C({\zeta }_2^C){\zeta }_1^C-g^C({(\varphi {\zeta }_4)}^C,{\zeta }_1^C){\eta }^C({\zeta }_3^C){\zeta }_2^V\hfill \\ & -& g^C({(\varphi {\zeta }_4)}^C,{\zeta }_1^C){\eta }^V({\zeta }_3^C){\zeta }_2^C-g^C({(\varphi {\zeta }_4)}^V,{\zeta }_1^C){\eta }^C({\zeta }_3^C){\zeta }_2^C\hfill \\ & +& g^C({(\varphi {\zeta }_4)}^C,{\zeta }_1^C){\eta }^C({\zeta }_3^C){\eta }^C({\zeta }_2^C){\xi }^V\hfill \\ & +& g^C({(\varphi {\zeta }_4)}^C,{\zeta }_1^C){\eta }^C({\zeta }_3^C){\eta }^V({\zeta }_2^C){\xi }^C\hfill \\ & +& g^C({(\varphi {\zeta }_4)}^C,{\zeta }_1^C){\eta }^V({\zeta }_3^C){\eta }^C({\zeta }_2^C){\xi }^C\hfill \\ & +& g^C({(\varphi {\zeta }_4)}^V,{\zeta }_1^C){\eta }^C({\zeta }_3^C){\eta }^C({\zeta }_2^C){\xi }^C\hfill \\ & -& g^C({(\varphi {\zeta }_4)}^C,{\zeta }_3^C){\eta }^C({\zeta }_1^C){\zeta }_2^V-g^C({(\varphi {\zeta }_4)}^C,{\zeta }_3^C){\eta }^V({\zeta }_1^C){\zeta }_2^C\hfill \\ & -& g^C({(\varphi {\zeta }_4)}^V,{\zeta }_3^C){\eta }^C({\zeta }_1^C){\zeta }_2^C-{\eta }^C({\zeta }_4^C){({\varphi }^2(\varphi \tilde{R})({\zeta }_1,{\zeta }_2){\zeta }_3)}^V\hfill \\ & -& {\eta }^V({\zeta }_4^C){({\varphi }^2(\varphi \tilde{R})({\zeta }_1,{\zeta }_2){\zeta }_3)}^C.\hfill \end{array}$$

(67)

If we take ${\zeta }_4,{\zeta }_1,{\zeta }_2,{\zeta }_3$ orthogonal to $\xi$, (67) reduces to

$${({\varphi }^2({\tilde{\nabla }}_{{\zeta }_4}\tilde{R})({\zeta }_1,{\zeta }_2){\zeta }_3)}^C={({\varphi }^2({\nabla }_{W}R)({\zeta }_1,{\zeta }_2){\zeta }_3)}^C.$$

Hence, the following theorem can be stated as:

Theorem 3. 

Let $TM$ be the tangent bundle of a Sasakian manifold M. Then, ${\nabla }^C$ is locally ${\varphi }^C$-symmetric on $TM$ if and only if ${\tilde{\nabla }}^C$ on $TM$ is so.

8. Example

Let us consider a three-dimensional differentiable manifold $M=\{(u,v,w):u,v,w\in {\Re }^3,$$z\ne 0\}$, where ℜ is a set of real numbers and $TM$ its tangent bundle. Let $e_1,e_2,e_3$ be linearly independent vector fields on M given by

$$e_1=-u\frac{\partial }{\partial u},e_2=u\frac{\partial }{\partial v},e_3=u\frac{\partial }{\partial w}.$$

Let g be the Riemannian metric and $\eta$ be a 1-form on M given by

$$g(e_1,e_2)=g(e_1,e_3)=g(e_2,e_3)=0,g(e_1,e_1)=g(e_2,e_2)=g(e_3,e_3)=1$$

and

$$\eta ({\zeta }_3)=g({\zeta }_3,e_1),{\zeta }_3\in {\Im }_0^1(M).$$

Let $\varphi$ be the (1,1) tensor field defined by $\varphi e_1=0,\varphi e_2=e_2,\varphi e_3=e_3$. Using the linearity of $\varphi$ and g, we acquire $\eta (e_1)=1,{\varphi }^2{\zeta }_3=-{\zeta }_3+\eta ({\zeta }_3)e_1$ and $g(\varphi {\zeta }_1,\varphi {\zeta }_2)=g({\zeta }_1,{\zeta }_2)-\eta ({\zeta }_1)\eta ({\zeta }_2)$.

Thus, for $e_1=\xi$, the $(\varphi ,\xi ,\eta ,g)$ is a contact metric structure on M and M is called a contact metric manifold. In addition, M satisfies

$$({\nabla }_{{\zeta }_1}\varphi ){\zeta }_2=g({\zeta }_1,{\zeta }_2)e_1-\eta ({\zeta }_2){\zeta }_1.$$

Hence, for $e_1=\xi$, M is a Sasakian manifold.

Let $e_1^C,e_2^C,e_3^C$ and $e_1^V,e_2^V,e_3^V$ be the complete and vertical lifts on $TM$ of $e_1,e_2,e_3$ on M. Let $g^C$ be the complete lift of a Riemannian metric g on $TM$ such that

$$\begin{array}{ccc}\hfill g^C({\zeta }_1^V,e_1^C)& =& {(g^C({\zeta }_1,e_1))}^V={(\eta ({\zeta }_1))}^V,\hfill \end{array}$$

(68)

$$\begin{array}{ccc}\hfill g^C({\zeta }_1^C,e_1^C)& =& {(g^C({\zeta }_1,e_1))}^C={(\eta ({\zeta }_1))}^C,\hfill \\ \hfill g^C(e_1^C,e_1^C)& =& 1,g^V({\zeta }_1^V,e_1^C)=0,g^V(e_1^V,e_1^V)=0\hfill \end{array}$$

(69)

and so on. Let ${\varphi }^C$ and ${\varphi }^V$ be the complete and vertical lifts of the (1,1) tensor field $\varphi$ defined by

$${\varphi }^V(e_1^V)={\varphi }^C(e_1^C)=0,$$

$${\varphi }^V(e_2^V)=e_2^V,{\varphi }^C(e_2^C)=e_2^C,$$

$${\varphi }^V(e_3^V)=e_3^V,{\varphi }^C(e_3^C)=e_3^C.$$

By using the linearity of $\varphi$ and g, we infer that

$${({\varphi }^2{\zeta }_1)}^C=-{\zeta }_1^C+{\eta }^V({\zeta }_1)e_1^C+{\eta }^C({\zeta }_1)e_1^V,$$

(70)

$$\begin{array}{ccc}\hfill g^C({(\varphi e_1)}^C,{(\varphi e_2)}^C)& =& g^C(e_1^C,e_2^C)-{(\eta (e_1))}^C{(\eta (e_2))}^V\hfill \\ & -& {(\eta (e_1))}^V{(\eta (e_2))}^C.\hfill \end{array}$$

Thus, for $e_1=\xi$ in (68)–(70), the structure $({\varphi }^C,{\xi }^C,{\eta }^C,g^C)$ is a contact metric structure on $TM$ and satisfies the relation

$$\begin{array}{ccc}\hfill ({\nabla }_{e_1^C}^C{\varphi }^C)e_2^C& =& g^C(e_1^C,e_2^C){\xi }^V+g^C(e_1^V,e_2^C){\xi }^C\hfill \\ & -& {\eta }^C(e_2^C)e_1^V-e^V(e_2^C)e_1^C,\hfill \end{array}$$

Then, $({\varphi }^C,{\xi }^C,{\eta }^C,g^C,TM)$ is a Sasakian manifold.