Tangent Bundles of P-Sasakian Manifolds Endowed with a Quarter-Symmetric Metric Connection

Abstract

The purpose of this study is to evaluate the curvature tensor and the Ricci tensor of a P-Sasakian manifold with respect to the quarter-symmetric metric connection on the tangent bundle $TM$. Certain results on a semisymmetric P-Sasakian manifold, generalized recurrent P-Sasakian manifolds, and pseudo-symmetric P-Sasakian manifolds on $TM$ are proved.

1. Introduction

Let M be a Riemannian manifold with a linear connection $\tilde{\nabla }$. If the torsion tensor T of $\tilde{\nabla }$

$$T(t_1,t_2)={\tilde{\nabla }}_{t_1}{t}_2-{\tilde{\nabla }}_{t_2}{t}_1-[t_1,t_2]$$

(1)

satisfies

$$T(t_1,t_2)=h(t_2)\varphi t_1-h(t_1)\varphi t_2,$$

(2)

where h is a 1-form and $\varphi$ is a (1, 1) tensor field, then the connection $\tilde{\nabla }$ is called a quarter-symmetric connection [1,2]. In addition, if $\tilde{\nabla }$ holds the relation

$$({\tilde{\nabla }}_{t_1}g)(t_2,t_3)=0,$$

(3)

$\forall t_1,t_2,t_3\in \Im (M)$, the set of all smooth vector fields on M, then $\tilde{\nabla }$ refers to the quarter-symmetric metric connection [3]. Many geometers such as [4,5,6,7,8,9,10,11,12,13,14,15,16] studied such connection on M and discussed some geometric properties of it. The quarter-symmetric connection generalizes the semi-symmetric connection that plays a key role in the geometry of Riemannian manifolds.

A Riemannian manifold M ($dimM=n\ge 3$) with respect to the Levi–Civita connection ∇ is said to be

• A generalized recurrent [17] if

  $$({\nabla }_{t_1}R)(t_2,t_3)t_4=\alpha (t_1)R(t_2,t_3)t_4+\beta (t_1)[g(t_3,t_4)t_2-g(t_2,t_4)t_3],$$

  (4)

  where $\alpha$ and $\beta$ are 1-forms of which $\beta \ne 0.$ If in Equation (4), $\alpha$ is non-zero and $\beta$ is zero, then the manifold is named a recurrent manifold [18].
• A pseudosymmetric [19] if

  $$\begin{array}{ccc}\hfill ({\nabla }_{t_1}R)(t_2,t_3)t_4& =& 2\alpha (t_1)R(t_2,t_3)t_4+\alpha (t_2)R(t_1,t_3)t_4\hfill \\ & +& \alpha (t_3)R(t_2,t_1)t_4+\alpha (t_4)R(t_2,t_3)t_1\hfill \\ & +& g(R(t_2,t_3)t_4,t_1)\rho ,\hfill \end{array}$$

  (5)

  for $\alpha \ne 0$. The 1-forms $\alpha$ and $\beta$ associated with the vector fields $\rho$ and $\sigma$ are defined as follows:

  $$\begin{array}{c}\hfill g(t_1,\rho )=\alpha (t_1),g(t_1,\sigma )=\beta (t_1).\end{array}$$

  (6)

On the other hand, Yano and Ishihara [20] proposed the notion of the lifting of tensor fields and connections to its tangent bundle and established the basic properties of curvature tensors. In [21], Manev studied tangent bundles with a complete lift of the base metric and almost hypercomplex Hermitian–Norden structure and characterized it. The metallic structures on the tangent bundle of a Riemannian manifold by using complete and horizontal lifts were studied by Azami [22]. Bilen [23] introduced the deformed Sasaki metric, which is a Berger type, studied the metric connection to the tangent bundle, established some curvature properties of this metric, and characterized the projective vector field. The geometric structures and the connections from a manifold to its tangent bundle have been studied by many authors such as [24,25,26,27] and many others.

Our main findings in the paper are as follows:

• Some results on the curvature tensor of a P-Sasakian manifold with respect to ${\tilde{\nabla }}^C$ on $TM$ are obtained.
• A theorem on a semisymmetric P-Sasakian manifold with respect to ${\tilde{\nabla }}^C$ on $TM$ is proved.
• A relationship between one and the forms ${\alpha }^C$ and ${\beta }^C$ on $TM$ of a generalized recurrent P-Sasakian manifold is established.
• An expression of a pseudosymmetric P-Sasakian manifold with respect to ${\tilde{\nabla }}^C$ on $TM$ is determined.

2. $\mathit{P}$-Sasakian Manifolds

Let M be a differentiable manifold ($dimM=n$) endowed with a tensor field $\varphi$ of type (1, 1), a characteristic vector field $\kappa$, and a 1-form h such that

$${\varphi }^2{t}_1=t_1-h(t_1)\kappa ,\varphi \kappa =0,h(\kappa )=1,h(\varphi t_1)=0$$

(7)

and let g be a Riemannian metric satisfying

$$g(\kappa ,t_1)=h(t_1),g(\varphi t_1,\varphi t_2)=g(t_1,t_2)-h(t_1)h(t_2);$$

(8)

then, the structure $(M,\varphi ,\kappa ,h,g)$ is said to be an almost para-contact metric manifold [28,29] If M holds:

$$\begin{array}{ccc}\hfill d\eta & =& 0,{\nabla }_{t_1}\kappa =\varphi t_1,\hfill \\ \hfill ({\nabla }_{t_1}\varphi )t_2& =& -g(t_1,t_2)\kappa -h(t_2)t_1+2\eta (t_1)h(t_2)\kappa ,\hfill \end{array}$$

(9)

then M is called a para-Sasakian manifold or, briefly, a P-Sasakian manifold [30,31,32]. Moreover, if M satisfies

$$({\nabla }_{t_1}h)(t_2)=-g(t_1,t_2)+h(t_1)h(t_2),$$

(10)

then M is a called special para-Sasakian manifold or an $SP$-Sasakian manifold [33]. In a P-Sasakian manifold, we have [32]:

$$\begin{array}{ccc}\hfill S(t_1,\kappa )& =& -(n-1)h(t_1)\iff Q\kappa =-(n-1)\kappa ,\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill h(R(t_1,t_2)t_3)& =& g(t_1,t_3)h(t_2)-g(t_2,t_3)h(t_1),\hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill R(t_1,\kappa )t_2& =& g(t_1,t_2)\kappa -h(t_2)t_1,\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill R(t_1,t_2)\kappa & =& h(t_1)t_2-h(t_2)t_1,\hfill \end{array}$$

(14)

$$\begin{array}{ccc}\hfill S(\varphi t_1,\varphi t_2)& =& S(t_1,t_2)+(n-1)h(t_1)h(t_2),\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill h(R(t_1,t_2)\kappa )& =& 0,\hfill \end{array}$$

(16)

$\forall t_1,t_2,t_3\in \Im (M)$, where the curvature and the Ricci tensors are symbolized as R and S, respectively.

For further studies on P-Sasakian manifolds, we recommend the papers [31,32,34,35,36,37] and many others. An almost paracontact Riemannian manifold M is said to be an h-Einstein manifold if its Ricci tensor $S(\ne 0)$ satisfies

$$S(t_1,t_2)=ag(t_1,t_2)+bh(t_1)h(t_2),$$

where a and b are smooth functions on the manifold M. In particular, if $b=0$, then M is named as an Einstein manifold.

Definition 1. 

In an n-dimensional differentiable manifold M, $T_p(M)$ is the tangent space at a point p of M, i.e., the set of all tangent vectors of M at p. Then, the set $TM={\bigcup }_{p\in M}{T}_p(M)$ is the tangent bundle over M.

Definition 2. 

Let us consider $(x^i),i=1,\cdots ,n$ as a local co-ordinate system on M and let $(x^i,y^i),i=1,\cdots ,n$ be an induced local co-ordinate system on $TM$. If $t_1=X^i\frac{\partial }{\partial x^i}$ is a local vector field on M, then its vertical, complete, and horizontal lifts in terms of partial differential equations are provided by

$$\begin{array}{ccc}\hfill t_1^V& =& X^i\frac{\partial }{\partial y^i},\hfill \end{array}$$

(17)

$$\begin{array}{ccc}\hfill t_1^C& =& X^i\frac{\partial }{\partial x^i}+\frac{\partial X^i}{\partial x^j}{y}^j\frac{\partial }{\partial y^i}.\hfill \end{array}$$

(18)

Let $f,h,t_1$ and $\varphi$ represent a function, the 1-form, the vector field, and the tensor field type (1,1), respectively, on M. The complete and vertical lifts of such quantities are $f^C,f^V,h^C,h^V,t_1^C,t_1^V,{\varphi }^C,{\varphi }^V$ on the tangent bundle $TM$.

Let the mathematical operators ∇ and ${\nabla }^C$ be the Levi–Civita connections on M and $TM$. Then, we have [38,39,40]:

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

(19)

$$\begin{array}{c}\hfill t_1^V{f}^V=0,t_1^V{f}^C=t_1^C{f}^V={(t_{1}f)}^V,t_1^C{f}^C={(t_{1}f)}^C,\end{array}$$

(20)

$$\begin{array}{c}\hfill h^V(f^V)=0,h^V(t_1^C)=h^C(t_1^V)=h{(t_1)}^V,h^C(t_1^C)=h{(t_1)}^C,\end{array}$$

(21)

$$\begin{array}{c}\hfill {\varphi }^V{t}_1^C={(\varphi t_1)}^V,{\varphi }^C{t}_1^C={(\varphi t_1)}^C,\end{array}$$

(22)

$$\begin{array}{c}\hfill {[t_1,t_2]}^V=[t_1^C,t_2^V]=[t_1^V,t_2^C],{[t_1,t_2]}^C=[t_1^C,t_2^C],\end{array}$$

(23)

$${\nabla }_{t_1^C}^C{t}_2^C={({\nabla }_{t_1}{t}_2)}^C,{\nabla }_{t_1^C}^C{t}_2^V={({\nabla }_{t_1}{t}_2)}^V.$$

(24)

Employing the complete lift on (1)–(16), we acquire

$$\begin{array}{ccc}\hfill {({\varphi }^2{t}_1)}^C& =& t_1^C-h^C(t_1^C){\kappa }^V-h^V(t_1^C){\kappa }^C,\hfill \end{array}$$

(25)

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

(26)

$$\begin{array}{ccc}\hfill h^C({\kappa }^C)& =& h^V({\kappa }^V)=0,h^C({\kappa }^V)=h^V({\kappa }^C)=1,\hfill \end{array}$$

(27)

$$\begin{array}{ccc}\hfill h^C{(\varphi t_1)}^C& =& h^V{(\varphi t_1)}^V=h^C{(\varphi t_1)}^V=h^V{(\varphi t_1)}^C=0.\hfill \end{array}$$

(28)

Let $g^C$ on $TM$ be the complete lift of g on M, then

$$\begin{array}{ccc}\hfill g^C({\kappa }^C,t_1^C)& =& h^C(t_1^C),\hfill \end{array}$$

(29)

$$\begin{array}{ccc}\hfill g^C({(\varphi t_1)}^C,{(\varphi t_2)}^C)& =& g^C(t_1^C,t_2^C)-h^C(t_1^C)h^V(t_2^C)\hfill \\ & -& h^V(t_1^C)h^C(t_2^C).\hfill \end{array}$$

(30)

If $(TM,g^C)$ satisfies

$$\begin{array}{ccc}\hfill {(d\eta )}^C& =& 0,{\nabla }_{t_1^C}^C{\kappa }^C={(\varphi t_1)}^C,\hfill \\ \hfill ({\nabla }_{t_1^C}^C{\varphi }^C)t_2^C& =& -g^C(t_1^C,t_2^C){\kappa }^V-g^C(t_1^V,t_2^C){\kappa }^C\hfill \\ & -& h^C(t_2^C)t_1^V-h^V(t_2^C)t_1^C+2\{h^C(t_1^C)h^C(t_2^C){\kappa }^V\hfill \\ & +& h^C(t_1^C)h^V(t_2^C){\kappa }^C+h^V(t_1^C)h^C(t_2^C){\kappa }^C\},\hfill \end{array}$$

(31)

$$({\nabla }_{t_1^C}^C{h}^C)(t_2^C)=-g^C(t_1^C,t_2^C)+h^C(t_1^C)h^V(t_2^C)+h^V(t_1^C)h^C(t_2^C),$$

(32)

then the $(TM,g^C)$ is called an $SP$-Sasakian manifold. Furthermore, we have

$$\begin{array}{ccc}\hfill S^C(t_1^C,{\kappa }^C)& =& -(n-1)h^C(t_1^C),{(Q\xi )}^C=-(n-1){\kappa }^C,\hfill \end{array}$$

(33)

$$\begin{array}{ccc}\hfill h^C(R^C(t_1^C,t_2^C)t_3^C)& =& g^C(t_1^C,t_3^C)h^V(t_2^C)+g^C(t_1^V,t_3^C)h^C(t_2^C)\hfill \\ & -& g^C(t_2^C,t_3^C)h^V(t_1^C)-g^C(t_2^V,t_3^C)h^C(t_1^C),\hfill \end{array}$$

(34)

$$\begin{array}{ccc}\hfill R^C(t_1^C,{\kappa }^C)t_2^C& =& g^C(t_1^C,t_2^C){\kappa }^V+g^C(t_1^V,t_2^C){\kappa }^C\hfill \\ & -& h^C(t_2^C)t_1^V-h^V(t_2^C)t_1^C,\hfill \end{array}$$

(35)

$$\begin{array}{ccc}\hfill R^C(t_1^C,t_2^C){\kappa }^C& =& h^C(t_1^C)t_2^V+h^V(t_1^C)t_2^C\hfill \\ & -& h^C(t_2^C)t_1^V+h^V(t_2^C)t_1^C,\hfill \end{array}$$

(36)

$$\begin{array}{ccc}\hfill S^C({(\varphi t_1)}^C,{(\varphi t_2)}^C)& =& S^C(t_1^C,t_2^C)+(n-1)\{h^C(t_1^C)h^V(t_2^C)\hfill \\ & +& h^V(t_1^C)h^C(t_2^C)\},\hfill \end{array}$$

(37)

$$\begin{array}{ccc}\hfill h^C(R^C(t_1^C,t_2^C){\kappa }^C)& =& 0,\hfill \end{array}$$

(38)

such that

$$g^C({(Q{t}_1)}^C,t_2^C)=S^C(t_1^C,t_2^C),$$

$$S^C(t_1^C,t_2^C)=a{g}^C(t_1^C,t_2^C)+b\{h^C(t_1^C)h^V(t_2^C)+h^V(t_1^C)h^C(t_2^C)\},$$

$\forall t_1^C,t_2^C,t_3^C\in \Im (TM)$.

3. Expression of the Curvature Tensor of a $\mathit{P}$-Sasakian Manifold with Respect to ${\tilde{\nabla }}^{\mathit{C}}$ on $\mathit{TM}$

Let $\tilde{\nabla }$ be a linear connection and ∇ be the Levi–Civita connection of a P-Sasakian manifold M such that

$${\tilde{\nabla }}_{t_1}{t}_2={\nabla }_{t_1}{t}_2+\mathcal{H}(t_1,t_2),$$

(39)

where $\mathcal{H}$ is a (1, 1)-type tensor and is provided by [1]

$$\begin{array}{c}\hfill \mathcal{H}(t_1,t_2)=\frac{1}{2}[T(t_1,t_2)+T^{\prime }(t_1,t_2)+T^{\prime }(t_2,t_1)],\end{array}$$

(40)

such that

$$\begin{array}{c}\hfill g(T^{\prime }(t_1,t_2),t_3)=g(T(t_3,t_1),t_2).\end{array}$$

(41)

Applying the complete lift on (1), (2), (6), and using (39)–(41), we infer

$$T^C(t_1^C,t_2^C)={\tilde{\nabla }}_{t_1^C}^C{t}_2^C-{\tilde{\nabla }}_{t_2^C}^V{t}_1^C-[t_1^C,t_2^C],$$

(42)

which satisfies

$$T^C(t_1^C,t_2^C)=h^C(t_2^C){(\varphi t_1)}^C-h^C(t_1^C){(\varphi t_2)}^C,$$

(43)

$$({\tilde{\nabla }}_{t_1^C}^C{g}^C)(t_2^C,t_3^C)=0,$$

(44)

$$\begin{array}{c}\hfill g^C(t_1^C,{\rho }^C)={\alpha }^C(t_1^C),\end{array}$$

(45)

$${\tilde{\nabla }}_{t_1^C}^C{t}_2^C={\nabla }_{t_1^C}^C{t}_2^C+t_5^C(t_1^C,t_2^C),$$

(46)

where

$$\begin{array}{c}\hfill {\mathcal{H}}^C(t_1^C,t_2^C)=\frac{1}{2}[T^C(t_1^C,t_2^C)+{T^{\prime }}^C(t_1^C,t_2^C)+{T^{\prime }}^C(t_2^C,t_1^C)],\end{array}$$

(47)

and

$$\begin{array}{c}\hfill g^C({T^{\prime }}^C(t_1^C,t_2^C),t_3^C)=g^C(T^C(t_3^C,t_1^C),t_2^C).\end{array}$$

(48)

From (43) and (48), we lead to

$$\begin{array}{ccc}\hfill {T^{\prime }}^C(t_1^C,t_2^C)& =& h^C(t_1^C){(\varphi t_2)}^C+h^V(t_1^C){(\varphi t_2)}^C\hfill \\ & -& g^C({(\varphi t_1)}^C,t_2^C){\kappa }^V-g^C({(\varphi t_1)}^V,t_2^C){\kappa }^C.\hfill \end{array}$$

(49)

Using (43) and (49) in (47), we have

$$\begin{array}{ccc}\hfill {\mathcal{H}}^C(t_1^C,t_2^C)& =& h^C(t_2^C){(\varphi t_1)}^V+h^V(t_2^C){(\varphi t_1)}^C\hfill \\ & -& g^C({(\varphi t_1)}^C,t_2^C){\kappa }^V+g^C({(\varphi t_1)}^V,t_2^C){\kappa }^C.\hfill \end{array}$$

(50)

Therefore, a quarter-symmetric metric connection ${\tilde{\nabla }}^C$ on $TM$ is provided by

$$\begin{array}{ccc}\hfill {\tilde{\nabla }}_{t_1^C}^C{t}_2^C& =& {\nabla }_{t_1^C}^C{t}_2^C+h^C(t_2^C){(\varphi t_1)}^V+h^V(t_2^C){(\varphi t_1)}^C\hfill \\ & -& g^C({(\varphi t_1)}^C,t_2^C){\kappa }^V+g^C({(\varphi t_1)}^V,t_2^C){\kappa }^C.\hfill \end{array}$$

(51)

Let ${\tilde{R}}^C$ and $R^C$ be the curvature tensors in respect of the connections ${\tilde{\nabla }}^C$ and ${\nabla }^C$ on $TM$, respectively. Then, from (51), we have

$$\begin{array}{ccc}\hfill {\tilde{R}}^C(t_1^C,t_2^C)t_5^C& =& R^C(t_1^C,t_2^C)t_5^C\hfill \\ & +& 3\{g^C({(\varphi t_1)}^C,t_5^C){(\varphi t_2)}^V+g^C({(\varphi t_1)}^V,t_5^C){(\varphi t_2)}^C\}\hfill \\ & -& 3\{g^C({(\varphi t_2)}^C,t_5^C){(\varphi t_1)}^V-g^C({(\varphi t_2)}^V,t_5^C){(\varphi t_1)}^C\}\hfill \\ & +& h^C(t_5^C)h^C(t_1^C)t_2^V+h^C(t_5^C)h^V(t_1^C)t_2^C\hfill \\ & +& h^V(t_5^C)h^C(t_1^C)t_2^C-h^C(t_5^C)h^C(t_2^C)t_1^V\hfill \\ & -& h^C(t_5^C)h^V(t_2^C)t_1^C-h^V(t_5^C)h^C(t_2^C)t_1^C\hfill \\ & -& h^C(t_1^C)g^C(t_2^C,t_5^C){\kappa }^V-h^C(t_1^C)g^C(t_2^V,t_5^C){\kappa }^C\hfill \\ & -& h^V(t_1^C)g^C(t_2^C,t_5^C){\kappa }^C+h^C(t_2^C)g^C(t_1^C,t_5^C){\kappa }^V\hfill \\ & +& h^C(t_2^C)g^C(t_1^V,t_5^C){\kappa }^C\hfill \\ & +& h^V(t_2^C)g^C(t_1^C,t_5^C){\kappa }^C,\hfill \end{array}$$

(52)

where ${\tilde{R}}^C(t_1^C,t_2^C)t_5^C={\tilde{\nabla }}_{t_1^C}^C{\tilde{\nabla }}_{t_2^C}^C{t}_5^C-{\tilde{\nabla }}_{t_2^C}^C{\tilde{\nabla }}_{t_1^C}^C{t}_5^C-{\tilde{\nabla }}_{[t_1^C,t_2^C]}^C{t}_5^C$, and $t_1^C,t_2^C,t_3^C\in \Im (TM)$. By using an appropriate contraction, from (52), we obtain that

$$\begin{array}{ccc}\hfill {\tilde{S}}^C(t_2^C,t_5^C)& =& S^C(t_2^C,t_5^C)+2g^C(t_2^C,t_5^C)\hfill \\ & -& (n+1)\{h^C(t_2^C)h^V(t_5^C)+h^V(t_2^C)h^C(t_5^C)\}\hfill \\ & -& 3trace{\varphi }^C{g}^C({(\varphi t_2)}^C,t_5^C),\hfill \end{array}$$

(53)

where ${\tilde{S}}^C$ and $S^C$ are the Ricci tensors of ${\tilde{\nabla }}^C$ and ${\nabla }^C$ on $TM$, respectively. This leads to the following theorem:

Theorem 1. 

Let $TM$ be the tangent bundle of the P-Sasakian manifold with ${\tilde{\nabla }}^C$. Then, we have

(1)

(52)provides $R^C$;

(2)

${\tilde{S}}^C$is symmetric;

(3)

${\tilde{R}}^C(t_1^C,t_2^C,t_3^C,t_4^C)+{\tilde{R}}^C(t_1^C,t_2^C,t_4^C,t_3^C)=0;$

(4)

${\tilde{R}}^C(t_1^C,t_2^C,t_3^C,t_4^C)+{\tilde{R}}^C(t_2^C,t_1^C,t_3^C,t_4^C)=0;$

(5)

${\tilde{R}}^C(t_1^C,t_2^C,t_3^C,t_4^C)={\tilde{R}}^C(t_3^C,t_4^C,t_1^C,t_2^C);$

(6)

${\tilde{S}}^C(t_2^C,{\kappa }^C)=-2(n-1)h^C(t_2^C);$

for all $t_1^C,t_2^C,t_3^C\in \Im (TM)$.

With the help of (25)–(28), (35) and (36) from (52) we obtain

$$\begin{array}{ccc}\hfill {\tilde{R}}^C({\kappa }^C,t_2^C)t_5^C& =& 2[h^C(t_5^C)t_2^V+h^V(t_5^C)t_2^C\hfill \\ & -& g^C(t_5^C,t_2^C){\kappa }^V-g^C(t_5^V,t_2^C){\kappa }^C],\hfill \end{array}$$

(54)

and

$$\begin{array}{ccc}\hfill {\tilde{R}}^C(t_1^C,t_2^C){\kappa }^C& =& 2[h^C(t_1^C)t_2^V+h^V(t_1^C)t_2^C\hfill \\ & -& h^C(t_2^C)t_1^V-h^V(t_2^C)t_1^C],\hfill \end{array}$$

(55)

where $t_1^C,t_2^C\in \Im (TM)$.

4. Expression of Semi-Symmetric $\mathit{P}$-Sasakian Manifolds with Respect to ${\tilde{\nabla }}^{\mathit{C}}$ on $\mathit{TM}$

In 2015, Mandal and De [41] characterized semisymmetric P-Sasakian manifolds with respect to the quarter-symmetric metric connection, that is, the curvature tensor satisfies the condition:

$$\tilde{R}(\kappa ,t_2)·\tilde{R}(t_5,t_6)t_4=0.$$

This implies

$$\begin{array}{ccc}\hfill \tilde{R}(\kappa ,t_2)\tilde{R}(t_5,t_6)t_4& -& \tilde{R}(\tilde{R}(\kappa ,t_2)t_5,t_6)t_4-\tilde{R}(t_5,\tilde{R}(\kappa ,t_2)t_6)t_4\hfill \\ & -& \tilde{R}(t_5,t_6)\tilde{R}(\kappa ,t_2)t_4=0.\hfill \end{array}$$

(56)

Applying the complete lift on (56), we infer

$$\begin{array}{ccc}\hfill {(\tilde{R}(\kappa ,t_2)\tilde{R}(t_5,t_6)t_4)}^C& -& {(\tilde{R}(\tilde{R}(\kappa ,t_2)t_5,t_6)t_4)}^C-{(\tilde{R}(t_5,\tilde{R}(\kappa ,t_2)t_6)t_4)}^C\hfill \\ & -& {(\tilde{R}(t_5,t_6)\tilde{R}(\kappa ,t_2)t_4)}^C=0.\hfill \end{array}$$

(57)

Using (54) and (57) yields

$$\begin{array}{ccc}\hfill h^C{(\tilde{R}(t_5,t_6)t_4)}^C{t}_2^C& -& 2\{g^C(t_2^C,{(\tilde{R}(t_5,t_6)t_4)}^C){\kappa }^V+g^C(t_2^V,{(\tilde{R}(t_5,t_6)t_4)}^C){\kappa }^C\}\hfill \\ & -& 2\{h^C(t_5^C){(\tilde{R}(t_2,t_6)t_4)}^V+h^V(t_5^C){(\tilde{R}(t_2,t_6)t_4)}^C\}\hfill \\ & +& 2\{g^C(t_2^C,t_5^C){(\tilde{R}(\kappa ,t_6)t_4)}^V+g^C(t_2^V,t_5^C){(\tilde{R}(\kappa ,t_6)t_4)}^C\}\hfill \\ & -& 2\{h^C(t_6^C){(\tilde{R}(t_5,t_2)t_4)}^V+h^V(t_6^C){(\tilde{R}(t_5,t_2)t_4)}^C\}\hfill \\ & +& 2\{g^C(V^C,t_2^C){(\tilde{R}(t_5,\kappa )t_4)}^V+g^C(t_6^V,t_2^C){(\tilde{R}(t_5,\kappa )t_4)}^C\}\hfill \\ & -& 2\{h^C(t_4^C){(\tilde{R}(t_5,t_6)t_2)}^V+h^V(t_4^C){(\tilde{R}(t_5,t_6)t_2)}^C\}\hfill \\ & +& 2\{g^C(t_2^C,t_4^C){(\tilde{R}(t_5,t_6)\kappa )}^V\hfill \\ & +& g^C(t_2^V,t_4^C){(\tilde{R}(t_5,t_6)\kappa )}^C\}=0.\hfill \end{array}$$

(58)

Using the inner product of (58) with $\kappa$ and then using (52), (54) and (55), we obtain from (58) that

$$\begin{array}{ccc}\hfill g^C({(R(t_5,t_6)t_4)}^C,t_2^C)& +& 3\{g^C({(\varphi t_5)}^C,t_4^C)g^C({(\varphi t_6)}^V,t_2^C)\hfill \\ & +& g^C({(\varphi t_6)}^V,t_4^C)g^C({(\varphi t_6)}^C,t_2^C)\}\hfill \\ & -& 3\{g^C({(\varphi t_6)}^V,t_4^C)g^C({(\varphi t_6)}^C,t_2^C)\hfill \\ & +& g^C({(\varphi t_6)}^C,t_4^C)g^C({(\varphi t_5)}^V,t_2^C)\}\hfill \\ & +& g^C(t_6^C,t_2^C)h^C(t_5^C)h^V(t_4^C)+g^C(t_6^C,t_2^C)h^V(t_5^C)h^C(t_4^C)\hfill \\ & +& g^C(t_6^V,t_2^C)h^C(t_5^C)h^C(t_4^C)-g^C(t_5^C,t_2^C)h^C(t_6^C)h^V(t_4^C)\hfill \\ & -& g^C(t_5^C,t_2^C)h^V(t_6^C)h^C(t_4^C)-g^C(t_5^V,t_2^C)h^C(t_6^C)h^C(t_4^C)\hfill \\ & -& g^C(t_6^C,t_4^C)h^C(t_5^C)h^V(t_2^C)-g^C(t_6^C,t_4^C)h^V(t_5^C)h^C(t_2^C)\hfill \\ & -& g^C(t_6^V,t_4^C)h^C(t_5^C)h^C(t_2^C)+g^C(t_5^C,t_4^C)h^C(t_6^C)h^V(t_2^C)\hfill \\ & +& g^C(t_5^C,t_4^C)h^V(t_6^C)h^C(t_2^C)\hfill \\ & +& g^C(t_5^V,t_4^C)h^C(t_6^C)h^C(t_2^C)+2\{g^C(t_2^C,t_5^C)g^C(t_6^V,t_4^C)\hfill \\ & +& g^C(t_2^V,t_5^C)g^C(t_6^C,t_4^C)-g^C(t_5^C,t_4^C)g^C(t_6^V,t_2^C)\hfill \\ & -& g^C(t_5^V,t_4^C)g^C(t_6^C,t_2^C)=0.\hfill \end{array}$$

(59)

By contracting the above equation over $t_4$ and $t_6$, we infer

$$\begin{array}{ccc}\hfill S^C(t_5^C,t_2^C)& =& -2n{g}^C(t_5^C,t_2^C)+(n+1)\{h^V(t_5^C)h^C(t_2^C)\hfill \\ & +& h^C(t_5^C)h^V(t_2^C)\}+3trace{\varphi }^C{g}^C({(\varphi t_5)}^C,t_2^C).\hfill \end{array}$$

(60)

In view of (53) and (60), we obtain

$${\tilde{S}}^C(t_5^C,t_2^C)=-2(n-1)g^C(t_5^C,t_2^C).$$

(61)

By contracting (61), we obtain

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

(62)

This leads to the following theorem:

Theorem 2. 

The tangent bundle $TM$ of a quarter-symmetric P-Sasakian manifold M is an Eienstein manifold with0 respect to ${\tilde{\nabla }}^C$ and ${\tilde{r}}^C=-2n(n-1).$

5. Expression of Generalized Recurrent $\mathit{P}$-Sasakian Manifolds in Respect of ${\tilde{\nabla }}^{\mathit{C}}$ on $\mathit{TM}$

In this section, we consider generalized recurrent P-Sasakian manifolds with respect to the quarter-symmetric metric connection $\tilde{\nabla }$. Equation (4) with respect to $\tilde{\nabla }$ can be expressed as

$$({\tilde{\nabla }}_{t_1}\tilde{R})(t_2,t_3)t_4=\alpha (t_1)\tilde{R}(t_2,t_3)t_4+\beta (t_1)[g(t_3,t_4)t_2-g(t_2,t_4)t_3].$$

(63)

Applying the complete lift on (63), we infer

$$\begin{array}{ccc}\hfill {(({\tilde{\nabla }}_{t_1}\tilde{R})(t_2,t_3)t_4)}^C& =& (\alpha (t_1){(\tilde{R}(t_2,t_3)t_4)}^C+{\beta }^C(t_1^C)g^C(t_3^C,t_4^C)t_2^V\hfill \\ & +& {\beta }^C(t_1^C)g^C(t_3^V,t_4^C)t_2^C+{\beta }^V(t_1^C)g^C(t_3^C,t_4^C)t_2^C\hfill \\ & -& {\beta }^C(t_1^C)g^C(t_2^C,t_4^C)t_3^V-{\beta }^C(t_1^C)g^C(t_2^V,t_4^C)t_3^C\hfill \\ & -& {\beta }^V(t_1^C)g^C(t_2^C,t_4^C)t_3^C\hfill \end{array}$$

(64)

for $t_1,t_2,t_3,t_4\in \Im (M)$. Substituting $t_2=t_4=\kappa$ in (64),

$$\begin{array}{ccc}\hfill {(({\tilde{\nabla }}_{t_1}\tilde{R})(\kappa ,t_3)\kappa )}^C& =& {\alpha }^C(t_1^C){(\tilde{R}(\kappa ,t_3)\kappa )}^V+{\alpha }^V(t_1^C){(\tilde{R}(\kappa ,t_3)\kappa )}^C\hfill \\ & +& {\beta }^C(t_1^C)h^C(t_3^C){\kappa }^V+{\beta }^C(t_1^C)h^V(t_3^C){\kappa }^C\hfill \\ & +& {\beta }^V(t_1^C)h^C(t_3^C){\kappa }^C-{\beta }^C(t_1^C)t_3^V-{\beta }^V(t_1^C)t_3^C.\hfill \end{array}$$

(65)

Using (55) in (65), we obtain

$${(({\tilde{\nabla }}_{t_1}\tilde{R})(t_2,t_3)\kappa )}^C=2[(({\tilde{\nabla }}_{t_1^C}^C{h}^C)t_2^C)t_3^C-({\tilde{\nabla }}_{t_1^C}^C{h}^C)t_3^C)t_2^C].$$

(66)

On the other hand, using (9), (44) and (51) we obtain

$$({\tilde{\nabla }}_{t_1^C}^C{h}^C)t_2^C=2g^C(t_2^C,{(\varphi t_1)}^C).$$

(67)

Thus, from the differential Equations (66) and (67), we have

$$\begin{array}{ccc}\hfill {(({\tilde{\nabla }}_{t_1}\tilde{R})(t_2,t_3)\kappa )}^C& =& 4[g^C(t_2^C,{(\varphi t_1)}^C{t}_3^V+g^C(t_2^V,{(\varphi t_1)}^C{t}_3^C\hfill \\ & -& g^C(t_3^C,{(\varphi t_1)}^C{t}_2^V-g^C(t_3^V,{(\varphi t_1)}^C{t}_2^C],\hfill \end{array}$$

which, by putting $t_2=\kappa$, yields

$$\begin{array}{ccc}\hfill {(({\tilde{\nabla }}_{t_1}\tilde{R})(\kappa ,t_3)\kappa )}^C& =& -4g^C(t_3^C,{(\varphi t_1)}^C{\xi }^V-g^C(t_3^V,{(\varphi t_1)}^C{\kappa }^C.\hfill \end{array}$$

(68)

Again, from (55), we have

$${(\tilde{R}(\kappa ,t_3)\kappa )}^C=2[t_3^C-h^C(t_3^C){\kappa }^V-h^V(t_3^C){\kappa }^C].$$

(69)

Thus, from (65) and (69), we obtain

$$\begin{array}{ccc}\hfill {(({\tilde{\nabla }}_{t_1}\tilde{R})(\kappa ,t_3)\kappa )}^C& =& {\alpha }^C(t_1^C)[t_3^C-h^C(t_3^C){\kappa }^V-h^V(t_3^C){\kappa }^C]\hfill \\ & +& {\beta }^C(t_1^C)[h^C(t_3^C){\kappa }^V+h^V(t_3^C){\kappa }^C-t_3^C].\hfill \end{array}$$

(70)

In view of (68) and (70), we obtain

$$\begin{array}{ccc}\hfill -4\{g^C(t_3^C,{(\varphi t_1)}^C){\kappa }^V& +& g^C(t_3^V,{(\varphi t_1)}^C){\kappa }^C\}\hfill \\ & =& 2{\alpha }^C(t_1^C)[t_3^C-h^C(t_3^C){\kappa }^V-h^V(t_3^C){\kappa }^C]\hfill \\ & -& {\beta }^C(t_1^C)[t_3^C-h^C(t_3^C){\kappa }^V-h^V(t_3^C){\kappa }^C].\hfill \end{array}$$

(71)

By applying $\varphi$ on (71) and using (25)–(28), we infer

$${\beta }^C(t_1^C)=2{\alpha }^C(t_1^C).$$

(72)

This leads to the following theorem:

Theorem 3. 

The 1-forms ${\alpha }^C$ and ${\beta }^C$ on $TM$ of a generalized recurrent P-Sasakian manifold are related by ${\beta }^C=2{\alpha }^C$.

Next, applying the complete lift on (4), we infer

$$\begin{array}{ccc}\hfill {(({\tilde{\nabla }}_{t_1}\tilde{R})(t_2,t_3)t_4)}^C& =& {\alpha }^C(t_1^C){(\tilde{R}(t_2,t_3)t_4)}^V+{\alpha }^V(t_1^C){(\tilde{R}(t_2,t_3)t_4)}^C\hfill \\ & +& {\beta }^C(t_1^C)g^C(t_3^C,t_4^C)t_2^V+{\beta }^C(t_1^C)g^C(t_3^V,t_4^C)t_2^C\hfill \\ & +& {\beta }^V(t_1^C)g^C(t_3^C,t_4^C)t_2^C-{\beta }^C(t_1^C)g^C(t_2^C,t_4^C)t_3^V\hfill \\ & -& {\beta }^C(t_1^C)g^C(t_2^V,t_4^C)t_3^C\hfill \\ & -& {\beta }^V(t_1^C)g^C(t_3^C,t_4^C)t_2^C,\hfill \end{array}$$

(73)

where ${\tilde{\nabla }}^C$ is the complete lift of $\tilde{\nabla }$. From the above equation, it follows that

$${(({\tilde{\nabla }}_{t_1}\tilde{R})(t_2,t_3)t_4)}^C={\alpha }^C(t_1^C){(\tilde{R}(t_2,t_3)t_4)}^V+{\alpha }^V(t_1^C){(\tilde{R}(t_2,t_3)t_4)}^C,$$

(74)

$$\begin{array}{c}\hfill \forall t_1^C,t_2^C,t_3^C,t_4^C\in \Im (TM).\end{array}$$

Thus, in view of Theorem 3, we obtain ${\alpha }^C(t_1^C)=0.$ Hence, we have the following corollary:

Corollary 1. 

The 1-form ${\alpha }^C$ on $TM$ of a generalized recurrent P-Sasakian manifold vanishes.

6. Expression of Pseudosymmetric $\mathit{P}$-Sasakian Manifolds with Respect to ${\tilde{\nabla }}^{\mathit{C}}$ on $\mathit{TM}$

In this section, we prove the following theorem:

Theorem 4. 

There is no pseudosymmetric P-Sasakian manifold with respect to ${\tilde{\nabla }}^C$ on $TM$.

Proof. 

Let us suppose that $TM$ is the tangent bundle of a pseudosymmetric P-Sasakian manifold with respect to ${\tilde{\nabla }}^C$. Using the complete lift on (5), we obtain

$$\begin{array}{ccc}\hfill {(({\tilde{\nabla }}_{t_1}\tilde{R})(t_2,t_3)t_4)}^C& =& 2{(\alpha (t_1)\tilde{R}(t_2,t_3)t_4)}^C+{(\alpha (t_2)\tilde{R}(t_1,t_3)t_4)}^C\hfill \\ & +& {(\alpha (t_3)\tilde{R}(t_2,t_1)t_4)}^C+{(\alpha (t_4)\tilde{R}(t_2,t_3)t_1)}^C\hfill \\ & +& {(g(\tilde{R}(t_2,t_3)t_4,t_1)\rho )}^C.\hfill \end{array}$$

(75)

By contracting $t_2$ in (75) and substituting $t_4=\kappa$, we have

$$\begin{array}{ccc}\hfill {(({\tilde{\nabla }}_{t_1}\tilde{S})(t_3,\kappa ))}^C& =& 2{(\alpha (t_1)\tilde{S}(t_3,\kappa ))}^C+(\alpha {(\tilde{R}(t_1,t_3)\kappa )}^C\hfill \\ & +& {(\alpha (t_3)\tilde{S}(t_1,\kappa ))}^C+{(\alpha (\kappa )\tilde{S}(t_3,t_1))}^C\hfill \\ & +& {(g(\tilde{R}(\rho ,t_3)\kappa ,t_1))}^C.\hfill \end{array}$$

(76)

In view of Theorem 1, we acquire

$${\tilde{S}}^C(t_3^C,{\kappa }^C)=-2(n-1)h^C(t_3^C).$$

In consequence of (67), we infer

$${\tilde{\nabla }}_{t_1^C}^C{\tilde{S}}^C(t_3^C,{\kappa }^C)=-4(n-1)g^C(t_3^C,{(\varphi t_1)}^C).$$

(77)

Next, the consequences of (25)–(28) and Theorem 1, we infer

$$\begin{array}{ccc}\hfill {\tilde{\nabla }}_{t_1^C}^C{\tilde{S}}^C(t_3^C,{\kappa }^C)& =& -4n\{{\alpha }^C(t_1^C)h^V(t_3^C)+{\alpha }^C(t_1^C)h^C(t_3^V)\}\hfill \\ & +& 2\{h^C(t_1^C){\alpha }^V(t_3^C)+h^C(t_1^C){\alpha }^C(t_3^V)\}\hfill \\ & -& 2(n-1)\{{\alpha }^C(t_3^C)h^V(t_1^C)+{\alpha }^C(t_3^C)h^C(t_1^V)\}\hfill \\ & +& 2\{{\alpha }^C({\kappa }^C)g^C(t_1^V,t_3^C)+{\alpha }^V({\kappa }^C)g^C(t_1^C,t_3^C)\}\hfill \\ & +& {\alpha }^C({\kappa }^C){\tilde{S}}^C(t_1^V,t_3^C)+{\alpha }^V({\kappa }^C){\tilde{S}}^C(t_1^C,t_3^C).\hfill \end{array}$$

(78)

Equating the differential Equations (77) and (78) and then using $t_1=\kappa$, we obtain

$$\begin{array}{ccc}\hfill -4(n-1)g^C(t_3^C,{(\varphi \kappa )}^C)& =& -4n\{{\alpha }^C({\kappa }^C)h^V(t_3^C)+{\alpha }^C({\kappa }^C)h^C(t_3^V)\}\hfill \\ & +& 2\{h^C({\kappa }^C){\alpha }^V(t_3^C)+h^C({\kappa }^C){\alpha }^C(t_3^V)\}\hfill \\ & -& 2(n-1)\{{\alpha }^C(t_3^C)h^V({\kappa }^C)+{\alpha }^C(t_3^C)h^C({\kappa }^V)\}\hfill \\ & +& 2\{{\alpha }^C({\kappa }^C)h^V(t_3^C)+{\alpha }^V({\kappa }^C)h^C(t_3^C)\}\hfill \\ & +& {\alpha }^C({\kappa }^C){\tilde{S}}^C({\kappa }^V,t_3^C)+{\alpha }^V({\kappa }^C){\tilde{S}}^C({\kappa }^C,t_3^C).\hfill \end{array}$$

(79)

By using (25)–(30), (45), and Theorem 1 in (79), we lead to

$$(2-3n)\{{\alpha }^C({\kappa }^C)h^V(t_3^C)+{\alpha }^C({\kappa }^C)h^C(t_3^V)\}+(2-n){\alpha }^C(t_3^C)=0.$$

(80)

By replacing $t_3$ by $\kappa$ in (80), we obtain ${\alpha }^C{\kappa }^C=0,$ which, used in (80), provides

$${\alpha }^C{t}_3^C=0\Rightarrow {\alpha }^C=0.$$

This goes against what we assumed. This completes the proof. □