Tangent Bundles Endowed with Quarter-Symmetric Non-Metric Connection (QSNMC) in a Lorentzian Para-Sasakian Manifold

Abstract

The purpose of the present paper is to study the complete lifts of a QSNMC from an LP-Sasakian manifold to its tangent bundle. The lifts of the curvature tensor, Ricci tensor, projective Ricci tensor, and lifts of Einstein manifold endowed with QSNMC in an LP-Sasakian manifold to its tangent bundle are investigated. Necessary and sufficient conditions for the lifts of the Ricci tensor to be symmetric and skew-symmetric and the lifts of the projective Ricci tensor to be skew-symmetric in the tangent bundle are given. An example of complete lifts of four-dimensional LP-Sasakian manifolds in the tangent bundle is shown.

1. Introduction

Tangent bundle geometry has long been a source of interest in differential geometry. Tangent bundle investigation introduces several novel challenges to the study of modern differential geometry. Using the lift function, it is convenient to generalize differentiable structures on any manifold M to its tangent bundle. The theory of vertical, complete, and horizontal lifts of geometrical structures and connections from a manifold to its tangent bundle was developed by Yano and Ishihara [1]. Numerous researchers have examined various connections and geometric structures on the tangent bundle like Yano and Kobayashi [2], Tani [3], Pandey and Chaturvedi [4], and Khan [5,6]. Different lifts of metallic structures to tangent bundles have been studied in [7,8,9]. Tangent bundles immersed with quarter-symmetric non-metric connections, semi-symmetric P-connections, and semi-symmetric non-metric connections on almost Hermitian manifolds, Kähler manifolds, Kenmotsu manifolds, Sasakian manifolds, para-Sasakian manifolds, Riemannian manifolds and their submanifolds, and statistical manifolds and their submanifolds have been studied in [5,10,11,12,13,14,15,16,17,18]. Recently, Khan et al. [19] studied the tangent bundle of P-Sasakian manifolds endowed with a quarter-symmetric metric connection (QSMC).

On the other hand, the notion of quarter-symmetric connection in a Riemannian manifold with affine connection was introduced by Golab in 1975 [20]. This was further developed by many geometers like Yano and Imai [21], Rastogi [22,23], Mishra and Pandey [24], Mukhopadhyay et al. [25], Biswas and De [26], Sengupta and Biswas [27], Singh and Pandey [28], and others.

Let ∇ be a linear connection on an n-dimensional differentiable manifold $M^n$ of class $C^{\infty }$. If the torsion tensor T of ∇ defined by

$$T(X_0,Y_0)={\mathrm{\nabla }}_{X_0}{Y}_0-{\mathrm{\nabla }}_{Y_0}{X}_0-[X_0,Y_0],$$

(1)

satisfies

$$T(X_0,Y_0)={\lambda }_0\left(Y_0\right){\varphi }_0{X}_0-{\lambda }_0\left(X_0\right){\varphi }_0{Y}_0,$$

(2)

where ${\lambda }_0$ is a 1-form and ${\varphi }_0$ is a $(1,1)$ tensor field, then the connection ∇ is called a quarter-symmetric connection [21,29,30]. Also, if ∇ satisfies

$$\left({\mathrm{\nabla }}_{X_0}g\right)(Y_0,Z_0)\ne 0,$$

(3)

for all $X_0,Y_0,Z_0\in \mathfrak{X}\left(M^n\right)$, the set of all vector fields on $M^n$, then ∇ is called a quarter-symmetric non-metric connection (QSNMC).

We start this paper with Section 1. Section 2 is devoted to preliminaries. In Section 3, a QSNMC in an LP-Sasakian manifold is studied. The complete lifts of LP-Sasakian manifolds and QSNMC in an LP-Sasakian manifold to its tangent bundle are investigated in Section 4 and Section 5. In Section 6 and Section 7, the complete lifts of the curvature tensor and symmetric and skew-symmetric condition of the Ricci tensor in an LP-Sasakian manifold endowed with QSNMC to its tangent bundle are investigated. The skew-symmetric properties of the projective Ricci tensor and Einstein manifold endowed with QSNMC in an LP-Sasakian manifold to its tangent bundle are studied in Section 8 and Section 9. Lastly, an example of the lift of four-dimensional LP-Sasakian manifolds to its tangent bundle is shown in Section 9, followed by a conclusion section.

2. Preliminaries

Let $M^n$ be a differentiable manifold and $T_0{M}^n={\bigcup }_{p\in M^n}{T}_{0p}{M}^n$ be the tangent bundle, where $T_{0p}{M}^n$ is the tangent space at a point $p\in M^n$ and $\pi :T_0{M}^n\to M^n$ is the natural bundle structure of $T_0{M}^n$ over $M^n$. For any co-ordinate system $(Q,x^h)$ in $M^n$, where $\left(x^h\right)$ is a local co-ordinate system in the neighborhood Q, then $({\pi }^{-1}\left(Q\right),x^h,y^h)$ is a co-ordinate system in $T_0{M}^n$, where $(x^h,y^h)$ is an induced co-ordinate system in ${\pi }^{-1}\left(Q\right)$ from $\left(x^h\right)$ [1].

2.1. Vertical and Complete Lifts

Let us define a vector field $X_0$, a tensor field $F_0$ of type $(1,1)$, a function $f_0$, a 1-form ${\omega }_0$, and an affine connection ∇ in $M^n$; their vertical and complete lifts are denoted by $f_0^v,X_0^v,{\omega }_0^v,F_0^v,{\mathrm{\nabla }}^v$, and $f_0^c,X_0^c,{\omega }_0^c,F_0^c,{\mathrm{\nabla }}^c$, respectively. The following formulas of complete and vertical lifts are defined by [1,5]

$$\begin{array}{c}\hfill {\left(f_0{X}_0\right)}^v=f_0^v{X}_0^v,{\left(f_0{X}_0\right)}^c=f_0^c{X}_0^v+f_0^v{X}_0^c,\end{array}$$

(4)

$$\begin{array}{c}\hfill X_0^v{f}_0^v=0,X_0^v{f}_0^c=X_0^c{f}_0^v={\left(X_0{f}_0\right)}^v,X_0^c{f}_0^c={\left(X_0{f}_0\right)}^c,\end{array}$$

(5)

$$\begin{array}{c}\hfill {\omega }_0\left(f_0^v\right)=0,{\omega }_0^v\left(X_0^c\right)={\omega }_0^c\left(X_0^v\right)={\omega }_0{\left(X_0\right)}^v,{\omega }_0^c\left(X_0^c\right)={\omega }_0{\left(X_0\right)}^c,\end{array}$$

(6)

$$\begin{array}{c}\hfill F_0^v{X}_0^c={\left(F_0{X}_0\right)}^v,F_0^c{X}_0^c={\left(F_0{X}_0\right)}^c,\end{array}$$

(7)

$$\begin{array}{c}\hfill [X_0,Y_0{]}^v=[X_0^c,Y_0^v]=[X_0^v,Y_0^c],{[X_0,Y_0]}^c=[X_0^c,Y_0^c],\end{array}$$

(8)

$$\begin{array}{c}\hfill {{\mathrm{\nabla }}^c}_{X_0^c}{Y}_0^c={\left({\mathrm{\nabla }}_{X_0}{Y}_0\right)}^c,{{\mathrm{\nabla }}^c}_{X_0^c}{Y}_0^v={\left({\mathrm{\nabla }}_{X_0}{Y}_0\right)}^v.\end{array}$$

(9)

Suppose $T_{0}M$ is the tangent bundle and let $X_0=X_0^i\frac{\partial }{\partial x^i}$ be a local vector field on M, then its vertical and complete lifts in the term of partial differential equations are

$$X_0^v=X_0^i\frac{\partial }{\partial y^i}\mathrm{and}{X}_0^c=X_0^i\frac{\partial }{\partial x^i}+\frac{\partial X_0^i}{\partial x^j}{y}^j\frac{\partial }{\partial y^i}.$$

2.2. LP-Sasakian Manifolds

An n-dimensional differentiable manifold $M^n$ is called a Lorentzian para-Sasakian (briefly LP-Sasakian) [31] of dimension n if it admits a $(1,1)$- tensor field ${\varphi }_0$, a contravariant vector field ${\xi }_0$, a 1-form ${\eta }_0$, and a Lorentzian metric g which satisfy

$$\begin{array}{c}\hfill {{\varphi }_0}^2\left(X_0\right)=X_0+{\eta }_0\left(X_0\right){\xi }_0,\end{array}$$

(10)

$$\begin{array}{c}\hfill {\eta }_0\left({\xi }_0\right)=-1,\end{array}$$

(11)

$$\begin{array}{c}\hfill g({\varphi }_0{X}_0,{\varphi }_0{Y}_0)=g(X_0,Y_0)+{\eta }_0\left(X_0\right){\eta }_0\left(Y_0\right),\end{array}$$

(12)

$$\begin{array}{c}\hfill g(X_0,{\xi }_0)={\eta }_0\left(X_0\right),\end{array}$$

(13)

$$\begin{array}{c}\hfill \left({\mathrm{\nabla }}_{X_0}{\varphi }_0\right)\left(Y_0\right)=g(X_0,Y_0){\xi }_0+{\eta }_0\left(Y_0\right)X_0+2{\eta }_0\left(X_0\right){\eta }_0\left(Y_0\right){\xi }_0,\end{array}$$

(14)

$$\begin{array}{c}\hfill {\mathrm{\nabla }}_{X_0}{\xi }_0={\varphi }_0{X}_0.\end{array}$$

(15)

In an LP-Sasakian manifold, the following relations also hold:

$$\begin{array}{c}\hfill {\varphi }_0{\xi }_0=0,{\eta }_0\circ {\varphi }_0=0,\end{array}$$

(16)

$$\begin{array}{c}\hfill \mathit{rank}{\varphi }_0=n-1.\end{array}$$

(17)

If we take a tensor field ${\mathrm{\Phi }}_0(X_0,Y_0)$ as

$${\mathrm{\Phi }}_0(X_0,Y_0)=g(X_0,{\varphi }_0{Y}_0),$$

(18)

for any vector fields $X_0$ and $Y_0$, then the tensor field ${\mathrm{\Phi }}_0(X_0,Y_0)$ is a symmetric $(0,2)$ tensor field [31]. Since the 1-form ${\eta }_0$ is closed in an LP-Sasakian manifold, we have [31,32]

$$\begin{array}{c}\hfill \left({\mathrm{\nabla }}_{X_0}{\eta }_0\right)\left(Y_0\right)={\mathrm{\Phi }}_0(X_0,Y_0),{\mathrm{\Phi }}_0(X_0,{\xi }_0)=0,\end{array}$$

(19)

for all $X_0,Y_0\in M^n$. In an LP-Sasakian manifold, the following relations hold [32,33]:

$$\begin{array}{c}\hfill g(R_0(X_0,Y_0)Z_0,{\xi }_0)=g(Y_0,Z_0){\eta }_0\left(X_0\right)-g(X_0,Z_0){\eta }_0\left(Y_0\right),\end{array}$$

(20)

$$\begin{array}{c}\hfill R_0({\xi }_0,X_0)Y_0=g(X_0,Y_0){\xi }_0-{\eta }_0\left(Y_0\right)X_0,\end{array}$$

(21)

$$\begin{array}{c}\hfill R_0(X_0,Y_0){\xi }_0={\eta }_0\left(Y_0\right)X_0-{\eta }_0\left(X_0\right)Y_0,\end{array}$$

(22)

$$\begin{array}{c}\hfill R_0({\xi }_0,X_0){\xi }_0=X_0+{\eta }_0\left(X_0\right){\xi }_0,\end{array}$$

(23)

$$\begin{array}{c}\hfill S_0(X_0,{\xi }_0)=(n-1){\eta }_0\left(X_0\right),\end{array}$$

(24)

$$\begin{array}{c}\hfill S_0({\varphi }_0{X}_0,{\varphi }_0{Y}_0)=S_0(X_0,Y_0)+(n-1){\eta }_0\left(X_0\right){\eta }_0\left(Y_0\right),\end{array}$$

(25)

where $R_0$ is the Riemannian curvature tensor and $S_0$ is the Ricci tensor of the manifold.

3. QSNMC

In an LP-Sasakian manifold $(M^n,g)$, the linear connection $\ddot{\mathrm{\nabla }}$ on $M^n$ is given by [29]

$${\ddot{\mathrm{\nabla }}}_{X_0}{Y}_0={\mathrm{\nabla }}_{X_0}{Y}_0+{\eta }_0\left(Y_0\right){\varphi }_0{X}_0+a_0\left(X_0\right){\varphi }_0{Y}_0,$$

(26)

where ${\eta }_0$ and $a_0$ are 1-form associated with vector field ${\xi }_0$ and $A_0$ on $M^n$ is given by

$$\begin{array}{c}\hfill {\eta }_0\left(X_0\right)=g(X_0,{\xi }_0),\end{array}$$

(27)

$$\begin{array}{c}\hfill a_0\left(X_0\right)=g(X_0,A_0),\end{array}$$

(28)

for all vector fields $X_0\in {\mathfrak{X}}_0\left(M^n\right)$, where ${\mathfrak{X}}_0\left(M^n\right)$ is the set of all differentiable vector fields on $M^n$ and the torsion tensor is given by

$$\ddot{T}(X_0,Y_0)={\eta }_0\left(Y_0\right){\varphi }_0{X}_0-{\eta }_0\left(X_0\right){\varphi }_0{Y}_0+a_0\left(X_0\right){\varphi }_0{Y}_0-a_0\left(Y_0\right){\varphi }_0{X}_0.$$

(29)

A linear connection satisfying (29) is called a quarter-symmetric connection. Further, by using (26), we have

$$\left({\ddot{\mathrm{\nabla }}}_{X_0}g\right)(Y_0,Z_0)=-{\eta }_0\left(Y_0\right)g({\varphi }_0{X}_0,Z_0)-{\eta }_0\left(Z_0\right)g({\varphi }_0{X}_0,Y_0)-2a_0\left(X_0\right)g({\varphi }_0{Y}_0,Z_0).$$

(30)

A linear connection $\ddot{\mathrm{\nabla }}$ defined by (26) which satisfies (29) and (30) is called QSNMC.

4. Complete Lifts from an LP-Sasakian Manifold to Its Tangent Bundle

Let the tangent bundle be denoted by $T_0{M}^n$ in an LP-Sasakian manifold $(M^n,g)$. Taking complete lifts by mathematical operators on (10)–(16) and (18)–(25), we obtain

$$\begin{array}{c}\hfill {\left({\varphi }_0^2\left(X_0\right)\right)}^c=X_0^c+{\eta }_0^c\left(X_0^c\right){\xi }_0^v+{\eta }_0^v\left(X_0^c\right){\xi }_0^c,\end{array}$$

(31)

$$\begin{array}{c}\hfill {\eta }_0^c\left({\xi }_0^c\right)={\eta }_0^v\left({\xi }_0^v\right)=0,{\eta }_0^c\left({\xi }_0^v\right)={\eta }_0^v\left({\xi }_0^c\right)=-1,\end{array}$$

(32)

$$\begin{array}{c}\hfill g^c\left({\left({\varphi }_0{X}_0\right)}^c,{\left({\varphi }_0{Y}_0\right)}^c\right)=g^c(X_0^c,Y_0^c)+{\eta }_0^c\left(X_0^c\right){\eta }_0^v\left(Y_0^c\right)+{\eta }_0^v\left(X^c\right){\eta }_0^c\left(Y_0^c\right),\end{array}$$

(33)

$$\begin{array}{c}\hfill g^c(X_0^c,{\xi }_0^c)={\eta }_0^c\left(X_0^c\right),\end{array}$$

(34)

$$\begin{array}{cc}\hfill \left({\mathrm{\nabla }}_{X_0^c}^c{\varphi }_0^c\right)Y_0^c& =g^c(X_0^c,Y_0^c){\xi }_0^v+g^c(X_0^v,Y_0^c){\xi }_0^c+{\eta }_0^c\left(Y_0^c\right)X_0^v+{\eta }_0^v\left(Y_0^c\right)X_0^c\hfill \\ \hfill & +2\left\{{\eta }_0^c\left(X_0^c\right){\eta }_0^c\left(Y_0^c\right){\xi }_0^v+{\eta }_0^c\left(X_0^c\right){\eta }_0^v\left(Y_0^c\right){\xi }_0^c+{\eta }_0^v\left(X_0^c\right){\eta }_0^c\left(Y_0^c\right){\xi }_0^c\right\},\hfill \end{array}$$

(35)

$$\begin{array}{c}\hfill {\mathrm{\nabla }}_{X_0^c}^c{\xi }_0^c={\left({\varphi }_0{X}_0\right)}^c,\end{array}$$

(36)

$$\begin{array}{c}\hfill {\varphi }_0^c{\xi }_0^c={\varphi }_0^v{\xi }_0^v={\varphi }_0^c{\xi }_0^v={\varphi }_0^v{\xi }_0^c=0,\end{array}$$

(37)

$$\begin{array}{c}\hfill {\eta }_0^c\circ {\varphi }_0^c={\eta }_0^v\circ {\varphi }_0^v={\eta }_0^c\circ {\varphi }_0^v={\eta }_0^v\circ {\varphi }_0^c=0,\end{array}$$

(38)

$$\begin{array}{c}\hfill {\mathrm{\Phi }}_0^c(X_0^c,Y_0^c)=g^c(X_0^c,{\varphi }_0^c{Y}_0^c),\end{array}$$

(39)

$$\begin{array}{c}\hfill \left({\mathrm{\nabla }}_{X_0^c}^c{\eta }_0^c\right)Y_0^c={\mathrm{\Phi }}_0^c(X_0^c,Y_0^c),\end{array}$$

(40)

$$\begin{array}{c}\hfill {\mathrm{\Phi }}_0^c(X_0^c,{\xi }_0^c)=0,\end{array}$$

(41)

$$\begin{array}{cc}\hfill g^c(R^c(X_0^c,Y_0^c)Z_0^c,{\xi }_0^c)& =g^c(Y_0^c,Z_0^c){\eta }_0^v\left(X_0^c\right)+g^c(Y_0^v,Z_0^c){\eta }_0^c\left(X_0^c\right)\hfill \\ \hfill & -g^c(X_0^c,Z_0^c){\eta }_0^v\left(Y_0^c\right)-g^c(X_0^v,Z_0^c){\eta }_0^c\left(Y_0^c\right),\hfill \end{array}$$

(42)

$$R^c({\xi }^c,X_0^c)Y_0^c=g^c(X_0^c,Y_0^c){\xi }_0^v+g^c(X_0^v,Y_0^c){\xi }_0^c-{\eta }_0^c\left(Y_0^c\right)X_0^v-{\eta }_0^v\left(Y_0^c\right)X_0^c,$$

(43)

$$R^c(X_0^c,Y_0^c){\xi }_0^c={\eta }_0^c\left(Y_0^c\right)X_0^v+{\eta }_0^v\left(Y_0^c\right)X_0^c-{\eta }_0^c\left(X_0^c\right)Y_0^v-{\eta }_0^v\left(X_0^c\right)Y_0^c,$$

(44)

$$R^c({\xi }_0^c,X_0^c){\xi }_0^c=X_0^c+{\eta }_0^c\left(X_0^c\right){\xi }_0^v+{\eta }_0^v\left(X_0^c\right){\xi }_0^c,$$

(45)

$$S^c(X_0^c,{\xi }_0^c)=(n-1){\eta }_0^c\left(X_0^c\right),$$

(46)

$$\begin{array}{c}\hfill S^c({\varphi }_0^c{X}_0^c,{\varphi }_0^c{Y}_0^c)=S^c(X_0^c,Y_0^c)+(n-1)\left\{{\eta }_0^c\left(X_0^c\right){\eta }_0^v\left(Y_0^c\right)+{\eta }_0^v\left(X_0^c\right){\eta }_0^c\left(Y_0^c\right)\right\}.\end{array}$$

(47)

5. Complete Lifts of QSNMC of an LP-Sasakian Manifold in the Tangent Bundle

In an LP-Sasakian manifold $(M^n,g)$ and its tangent bundle $T_0{M}^n$, let us take complete lifts by mathematical operators on Equations (26)–(30), and we have

$$\begin{array}{cc}\hfill {\ddot{\mathrm{\nabla }}}_{X_0^c}^c{Y}_0^c& ={\mathrm{\nabla }}_{X_0^c}^c{Y}_0^c+{\eta }_0^c\left(Y_0^c\right){\left({\varphi }_0{X}_0\right)}^v+{\eta }_0^v\left(Y_0^c\right){\left({\varphi }_0{X}_0\right)}^c+a^c\left(X_0^c\right){\left({\varphi }_0{Y}_0\right)}^v\hfill \\ \hfill & +a^v\left(X_0^c\right){\left({\varphi }_0{Y}_0\right)}^c,\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill {\ddot{T}}^c(X_0^c,Y_0^c)& ={\eta }_0^c\left(Y_0^c\right){\left({\varphi }_0{X}_0\right)}^v+{\eta }_0^v\left(Y_0^c\right){\left({\varphi }_0{X}_0\right)}^c-{\eta }_0^c\left(X_0^c\right){\left({\varphi }_0{Y}_0\right)}^v\hfill \\ \hfill & -{\eta }_0^v\left(X_0^c\right){\left({\varphi }_0{Y}_0\right)}^c+a_0^c\left(X_0^c\right){\left({\varphi }_0{Y}_0\right)}^v+a_0^v\left(X_0^c\right){\left({\varphi }_0{Y}_0\right)}^c\hfill \\ \hfill & -a_0^c\left(Y_0^c\right){\left({\varphi }_0{X}_0\right)}^v-a_0^v\left(Y_0^c\right){\left({\varphi }_0{X}_0\right)}^c,\hfill \end{array}$$

(49)

$$\begin{array}{c}\hfill {\eta }_0^c\left(X_0^c\right)=g^c(X_0^c,{\xi }_0^c),\end{array}$$

(50)

$$\begin{array}{c}\hfill a^c\left(X_0^c\right)=g^c(X_0^c,A_0^c),\end{array}$$

(51)

$$\begin{array}{cc}\hfill \left({\ddot{\mathrm{\nabla }}}_{X_0^c}^c{g}^c\right)(Y_0^c,Z_0^c)& =-{\eta }_0^c\left(Y_0^c\right)g^c\left({\left({\varphi }_0{X}_0\right)}^v,Z_0^c\right)-{\eta }_0^v\left(Y_0^c\right)g^c\left({\left({\varphi }_0{X}_0\right)}^c,Z_0^c\right)\hfill \\ \hfill & -{\eta }_0^c\left(Z_0^c\right)g^c\left({\left({\varphi }_0{X}_0\right)}^v,Y_0^c\right)-{\eta }_0^v\left(Z_0^c\right)g^c\left({\left({\varphi }_0{X}_0\right)}^c,Y_0^c\right)\hfill \\ \hfill & -2a^c\left(X_0^c\right)g^c\left({\left({\varphi }_0{Y}_0\right)}^v,Z_0^c\right)-2a^v\left(X_0^c\right)g^c\left({\left({\varphi }_0{Y}_0\right)}^c,Z_0^c\right).\hfill \end{array}$$

(52)

The connection given by Equation (48) is said to be a QSNMC on an LP-Sasakian manifold in its tangent bundle if the torsion tensor ${\ddot{T}}^c$ of $T_0{M}^n$ endowed with ${\ddot{\mathrm{\nabla }}}^c$ satisfies Equation (49) and the complete lifts of Lorentzian metric $g^c$ fulfill the relation (52).

Theorem 1.

If an LP-Sasakian manifold $(M^n,g)$ with an almost Lorentzian para-contact metric structure $({\varphi }_0,{\xi }_0,{\eta }_0,g)$ admitting a QSNMC $\ddot{\mathrm{\nabla }}$ which satisfies (49) and (52), then the QSNMC in the tangent bundle is given by

$${\ddot{\mathrm{\nabla }}}_{X_0^c}^c{Y}_0^c={\mathrm{\nabla }}_{X_0^c}^c{Y}_0+{\eta }_0^c\left(Y_0^c\right){\left({\varphi }_0{X}_0\right)}^v+{\eta }_0^v\left(Y_0^c\right){\left({\varphi }_0{X}_0\right)}^c+a^c\left(X_0^c\right){\left({\varphi }_0{Y}_0\right)}^v+a^v\left(X_0^c\right){\left({\varphi }_0{Y}_0\right)}^c.$$

Proof. 

Let ${\ddot{\mathrm{\nabla }}}^c$ be the complete lifts of a linear connection in $M^n$ given by

$${\ddot{\mathrm{\nabla }}}_{X_0^c}^c{Y}_0^c={\mathrm{\nabla }}_{X_0^c}^c{Y}_0^c+H_0^c(X_0^c,Y_0^c).$$

(53)

Now, we shall determine the complete lifts of the tensor field $H_0^c$ such that ${\ddot{\mathrm{\nabla }}}^c$ satisfies (49) and (52). From (53), we have

$${\ddot{T}}^c(X_0^c,Y_0^c)=H_0^c(X_0^c,Y_0^c)-H_0^c(Y_0^c,X_0^c).$$

(54)

We denote

$$G_0^c(X_0^c,Y_0^c,Z_0^c)=\left({\ddot{\mathrm{\nabla }}}_{X_0^c}^c{g}^c\right)(Y_0^c,Z_0^c).$$

(55)

From (53) and (55), we have

$$g^c\left(H_0^c(X_0^c,Y_0^c),Z_0^c\right)+g^c\left(H_0^c(X_0^c,Z_0^c),Y_0^c\right)=-G_0^c(X_0^c,Y_0^c,Z_0^c).$$

(56)

Using (52), (53), (55), and (56) we have

$$\begin{array}{cc}\hfill & g^c\left({\ddot{T}}^c(X_0^c,Y_0^c),Z_0^c\right)+g^c\left({\ddot{T}}^c(Z_0^c,X_0^c),Y_0^c\right)+g^c\left({\ddot{T}}^c(Z_0^c,Y_0^c),X_0^c\right)\hfill \\ \hfill & =g^c\left(H_0^c(X_0^c,Y_0^c),Z_0^c\right)-g^c\left(H_0^c(Y_0^c,X_0^c),Z_0^c\right)+g^c\left(H_0^c(Z_0^c,X_0^c),Y_0^c\right)\hfill \\ \hfill & -g^c\left(H_0^c(X_0^c,Z_0^c),Y_0^c\right)+g^c\left(H_0^c(Z_0^c,Y_0^c),X_0^c\right)-g^c\left(H_0^c(Y_0^c,Z_0^c),X_0^c\right)\hfill \\ \hfill & =g^c\left(H_0^c(X_0^c,Y_0^c),Z_0^c\right)-g^c\left(H_0^c(X_0^c,Z_0^c),Y_0^c\right)-G_0^c(Z_0^c,X_0^c,Y_0^c)+G_0^c(Y_0^c,X_0^c,Z_0^c)\hfill \\ \hfill & =2g^c\left(H_0^c(X_0^c,Y_0^c),Z_0^c\right)+G_0^c(X_0^c,Y_0^c,Z_0^c)+G_0^c(Y_0^c,X_0^c,Z_0^c)-G_0^c(Z_0^c,X_0^c,Y_0^c)\hfill \\ \hfill & =2g^c\left(H_0^c(X_0^c,Y_0^c),Z_0^c\right)-2\left\{{\eta }_0^c\left(Z_0^c\right)g^c\left({\left({\varphi }_0{X}_0\right)}^v,Y_0^c\right)+{\eta }_0^v\left(Z_0^c\right)g^c\left({\left({\varphi }_0{X}_0\right)}^c,Y_0^c\right)\right\}\hfill \\ \hfill & -2\left\{a_0^c\left(X_0^c\right)g^c\left({\left({\varphi }_0{Y}_0\right)}^v,Z_0^c\right)+a_0^v\left(X_0^c\right)g^c\left({\left({\varphi }_0{Y}_0\right)}^c,Z_0^c\right)\right\}-2\{a_0^c\left(Y_0^c\right)g^c\left({\left({\varphi }_0{X}_0\right)}^v,Z_0^c\right)\hfill \\ \hfill & +a_0^v\left(Y_0^c\right)g^c\left({\left({\varphi }_0{X}_0\right)}^c,Z_0^c\right)\}+2\left\{a_0^c\left(Z_0^c\right)g^c\left({\left({\varphi }_0{X}_0\right)}^v,Y_0^c\right)+a_0^v\left(Z_0^c\right)g^c\left({\left({\varphi }_0{X}_0\right)}^c,Y_0^c\right)\right\},\hfill \end{array}$$

or,

$$\begin{array}{cc}\hfill H_0^c(X_0^c,Y_0^c)& =\frac{1}{2}\left\{{\ddot{T}}^c(X_0^c,Y_0^c)+{}^{\prime }{\ddot{T}}^c(X_0^c,Y_0^c)+{}^{\prime }{\ddot{T}}^c(Y_0^c,X_0^c)\right\}+a_0^c\left(X_0^c\right){\left({\varphi }_0{Y}_0\right)}^v\hfill \\ \hfill & +a_0^v\left(X_0^c\right){\left({\varphi }_0{Y}_0\right)}^c+a_0^c\left(Y_0^c\right){\left({\varphi }_0{X}_0\right)}^v+a_0^v\left(Y_0^c\right){\left({\varphi }_0{X}_0\right)}^c\hfill \\ \hfill & +g^c\left({\varphi }_0{X}_0{)}^c,Y_0^c\right){\xi }_0^v+g^c\left({\varphi }_0{X}_0{)}^v,Y_0^c\right){\xi }_0^c\hfill \\ \hfill & -g^c\left({\varphi }_0{X}_0{)}^c,Y_0^c\right)A_0^v-g^c\left({\varphi }_0{X}_0{)}^v,Y_0^c\right)A_0^c,\hfill \end{array}$$

where ${}^{\prime }{\ddot{T}}^c$ is a tensor field of type $(1,2)$ defined by

$$g^c\left({}^{\prime }{\ddot{T}}^c(X_0^c,Y_0^c),Z_0^c\right)=g^c\left({\ddot{T}}^c(Z_0^c,X_0^c),Y_0^c\right),$$

or,

$$H_0^c(X_0^c,Y_0^c)={\eta }_0^c\left(Y_0^c\right){\left({\varphi }_0{X}_0\right)}^v+{\eta }_0^v\left(Y_0^c\right){\left({\varphi }_0{X}_0\right)}^c+a^c\left(X_0^c\right){\left({\varphi }_0{Y}_0\right)}^v+a^v\left(X_0^c\right){\left({\varphi }_0{Y}_0\right)}^c,$$

which gives

$${\ddot{\mathrm{\nabla }}}_{X_0^c}^c{Y}_0^c={\mathrm{\nabla }}_{X_0^c}^c{Y}_0+{\eta }_0^c\left(Y_0^c\right){\left({\varphi }_0{X}_0\right)}^v+{\eta }_0^v\left(Y_0^c\right){\left({\varphi }_0{X}_0\right)}^c+a^c\left(X_0^c\right){\left({\varphi }_0{Y}_0\right)}^v+a^v\left(X_0^c\right){\left({\varphi }_0{Y}_0\right)}^c.$$

□

6. Curvature Tensor of LP-Sasakian Manifolds Endowed with QSNMC to Tangent Bundle

Let ${\ddot{R}}_0^c$ and $R_0^c$ be the curvature tensors of the connections ${\ddot{\mathrm{\nabla }}}^c$ and ${\mathrm{\nabla }}^c$ to tangent bundle $T_0{M}^n$, respectively.

$${\ddot{R}}_0^c(X_0^c,Y_0^c)Z_0^c={\ddot{\mathrm{\nabla }}}_{X_0^c}^c{\ddot{\mathrm{\nabla }}}_{Y_0^c}^c{Z}_0^c-{\ddot{\mathrm{\nabla }}}_{Y_0^c}^c{\ddot{\mathrm{\nabla }}}_{X_0^c}^c{Z}_0^c-{\ddot{\mathrm{\nabla }}}_{[X_0^c,Y_0^c]}^c{Z}_0^c.$$

(57)

Using (48) in (57), we have

$$\begin{array}{cc}\hfill {\ddot{R}}_0^c(X_0^c,Y_0^c)Z_0^c& =R_0^c(X_0^c,Y_0^c)Z_0^c+g^c\left({\left({\varphi }_0{X}_0\right)}^c,Z_0^c\right){\left({\varphi }_0{Y}_0\right)}^v\hfill \\ \hfill & +g^c\left({\left({\varphi }_0{X}_0\right)}^v,Z_0^c\right){\left({\varphi }_0{Y}_0\right)}^c-g^c\left({\left({\varphi }_0{Y}_0\right)}^c,Z_0^c\right){\left({\varphi }_0{X}_0\right)}^v\hfill \\ \hfill & -g^c\left({\left({\varphi }_0{Y}_0\right)}^v,Z_0^c\right){\left({\varphi }_0{X}_0\right)}^c+{\eta }_0^c\left(Y_0^c\right){\eta }_0^c\left(Z_0^c\right)X_0^v\hfill \\ \hfill & +{\eta }_0^c\left(Y_0^c\right){\eta }_0^v\left(Z_0^c\right)X_0^c+{\eta }_0^v\left(Y_0^c\right){\eta }_0^c\left(Z_0^c\right)X_0^c\hfill \\ \hfill & -{\eta }_0^c\left(X_0^c\right){\eta }_0^c\left(Z_0^c\right)Y_0^v-{\eta }_0^c\left(X_0^c\right){\eta }_0^v\left(Z_0^c\right)Y_0^c\hfill \\ \hfill & -{\eta }_0^v\left(X_0^c\right){\eta }_0^c\left(Z_0^c\right)Y_0^c+a_0^c\left(Y_0^c\right)g^c(X_0^c,Z_0^c){\xi }_0^v\hfill \\ \hfill & +a_0^c\left(Y_0^c\right)g^c(X_0^v,Z_0^c){\xi }_0^c+a_0^v\left(Y_0^c\right)g^c(X_0^c,Z_0^c){\xi }_0^c\hfill \\ \hfill & -a_0^c\left(X_0^c\right)g^c(Y_0^c,Z_0^c){\xi }_0^v-a_0^c\left(X_0^c\right)g^c(Y_0^v,Z_0^c){\xi }_0^c\hfill \\ \hfill & -a_0^v\left(X_0^c\right)g^c(Y_0^c,Z_0^c){\xi }_0^c+a_0^c\left(Y_0^c\right){\eta }_0^c\left(X_0^c\right){\eta }_0^c\left(Z_0^c\right){\xi }_0^v\hfill \\ \hfill & +a_0^c\left(Y_0^c\right){\eta }_0^c\left(X_0^c\right){\eta }_0^v\left(Z_0^c\right){\xi }_0^c+a_0^c\left(Y_0^c\right){\eta }_0^v\left(X_0^c\right){\eta }_0^c\left(Z_0^c\right){\xi }_0^c\hfill \\ \hfill & +a_0^v\left(Y_0^c\right){\eta }_0^c\left(X_0^c\right){\eta }_0^c\left(Z_0^c\right){\xi }_0^c-a_0^c\left(X_0^c\right){\eta }_0^c\left(Y_0^c\right){\eta }_0^c\left(Z_0^c\right){\xi }_0^v\hfill \\ \hfill & -a_0^c\left(X_0^c\right){\eta }_0^c\left(Y_0^c\right){\eta }_0^v\left(Z_0^c\right){\xi }_0^c-a_0^c\left(X_0^c\right){\eta }_0^v\left(Y_0^c\right){\eta }_0^c\left(Z_0^c\right){\xi }_0^c\hfill \\ \hfill & -a_0^v\left(X_0^c\right){\eta }_0^c\left(Y_0^c\right){\eta }_0^c\left(Z_0^c\right){\xi }_0^c+{\mathit{da}}_0^c(X_0^c,Y_0^c){\left({\varphi }_0{Z}_0\right)}^v\hfill \\ \hfill & +{\mathit{da}}_0^v(X_0^c,Y_0^c){\left({\varphi }_0{Z}_0\right)}^c,\hfill \end{array}$$

(58)

where

$$R_0^c(X_0^c,Y_0^c)Z_0^c={\mathrm{\nabla }}_{X_0^c}^c{\mathrm{\nabla }}_{Y_0^c}^c{Z}_0-{\mathrm{\nabla }}_{Y_0^c}^c{\mathrm{\nabla }}_{X_0^c}^c{Z}_0-{\mathrm{\nabla }}_{[X_0^c,Y_0^c]}^c{Z}_0^c,$$

(59)

is the curvature tensor of ${\mathrm{\nabla }}^c$ with respect to the Riemannian connection. Contracting (58), we obtain

$$\begin{array}{cc}\hfill {\ddot{S}}_0^c(Y_0^c,Z_0^c)& =S_0^c(Y_0^c,Z_0^c)-{\gamma }^c{g}^c\left({\left({\varphi }_0{Y}_0\right)}^c,Z_0^c\right)+\left[1-a_0^c\left({\xi }_0^c\right)\right]g^c(Y_0^c,Z_0^c)\hfill \\ \hfill & +\left[{\eta }_0^c-a_0^c\left({\xi }_0^c\right)\right]\left[{\eta }_0^c\left(Y_0^c\right){\eta }_0^v\left(Z_0^c\right)+{\eta }_0^v\left(Y_0^c\right){\eta }_0^c\left(Z_0^c\right)\right]\hfill \\ \hfill & +{\mathit{da}}_0^c\left({\left({\varphi }_0{Z}_0\right)}^c,Y_0^c\right),\hfill \end{array}$$

(60)

and

$${\ddot{r_0}}^c=r_0^c-(n-1)a_0^c\left({\xi }_0^c\right)+{\lambda }_0^c-{\gamma }^{c^2},$$

(61)

where ${\ddot{S}}_0^c$ and ${\ddot{r_0}}^c$ are the Ricci tensor and scalar curvature with respect to ${\ddot{\mathrm{\nabla }}}^c$.

$${\lambda }_0^c=\mathit{trace}{\mathit{da}}_0^c\left({\left({\varphi }_0{Z}_0\right)}^c,Y_0^c\right)\mathrm{and}{\gamma }^c=\mathit{trace}{\varphi }_0^c.$$

(62)

Theorem 2.

In an LP-Sasakian manifold $(M^n,g)$ with tangent bundle $T_0{M}^n$ admitting QSNMC, we have the following:

• The complete lifts of curvature tensor ${\ddot{R}}_0^c$ are given by Equation (58).
• The complete lifts of Ricci tensor ${\ddot{S}}_0^c$ are given by Equation (60).
• The complete lifts of scalar curvature $\ddot{r_0}$ are given by Equation (61).

Let us consider that ${\ddot{R}}_0^c(X_0^c,Y_0^c)=0$ in (58), and by contracting it we also obtain

$$\begin{array}{cc}\hfill S_0^c(Y_0^c,Z_0^c)& ={\gamma }^c{g}^c\left({\left({\varphi }_0{Y}_0\right)}^c,Z_0^c\right)-\left[1-a_0^c\left({\xi }_0^c\right)\right]g^c(Y_0^c,Z_0^c)\hfill \\ \hfill & -\left[{\eta }_0^c-a_0^c\left({\xi }_0^c\right)\right]\left[{\eta }_0^c\left(Y_0^c\right){\eta }_0^v\left(Z_0^c\right)+{\eta }_0^v\left(Y_0^c\right){\eta }_0^c\left(Z_0^c\right)\right]\hfill \\ \hfill & -{\mathit{da}}_0^c\left({\left({\varphi }_0{Z}_0\right)}^c,Y_0^c\right),\hfill \end{array}$$

(63)

which gives

$$r_0^c=(n-1)a_0^c\left({\xi }_0^c\right)-{\lambda }_0^c+{\gamma }^{c^2}.$$

(64)

Theorem 3.

In an LP-Sasakian manifold, $(M^n,g)$, with tangent bundle $T_0{M}^n$ endowed with QSNMC whose curvature tensor vanishes, then the complete lift of $r_0^c$ is given by (64).

From (58), it follows that

$${}^{\prime }{\ddot{R}}_0^c(X_0^c,Y_0^c,Z_0^c,W_0^c)+{}^{\prime }{\ddot{R}}_0^c(Y_0^c,X_0^c,Z_0^c,W_0^c)=0,$$

(65)

$$\begin{array}{cc}\hfill & {}^{\prime }{\ddot{R}}_0^c(X_0^c,Y_0^c,Z_0^c,W_0^c)+{}^{\prime }{\ddot{R}}_0^c(X_0^c,Y_0^c,W_0^c,Z_0^c)\hfill \\ \hfill & ={\eta }_0^c\left(Y_0^c\right){\eta }_0^v\left(Z_0^c\right)g^c(X_0^c,W_0^c)+{\eta }_0^v\left(Y_0^c\right){\eta }_0^c\left(Z_0^c\right)g^c(X_0^c,W_0^c)\hfill \\ \hfill & -{\eta }_0^c\left(X_0^c\right){\eta }_0^v\left(Z_0^c\right)g^c(Y_0^c,W_0^c)-{\eta }_0^v\left(X_0^c\right){\eta }_0^c\left(Z_0^c\right)g^c(Y_0^c,W_0^c)\hfill \\ \hfill & +{\eta }_0^c\left(Y_0^c\right){\eta }_0^v\left(W_0^c\right)g^c(X_0^c,Z_0^c)+{\eta }_0^v\left(Y_0^c\right){\eta }_0^c\left(W_0^c\right)g^c(X_0^c,Z_0^c)\hfill \\ \hfill & -{\eta }_0^c\left(X_0^c\right){\eta }_0^v\left(W_0^c\right)g^c(Y_0^c,Z_0^c)-{\eta }_0^v\left(X_0^c\right){\eta }_0^c\left(W_0^c\right)g^c(Y_0^c,Z_0^c)\hfill \\ \hfill & +a_0^c\left(Y_0^c\right){\eta }_0^v\left(W_0^c\right)g^c(X_0^c,Z_0^c)+a_0^v\left(Y_0^c\right){\eta }_0^c\left(W_0^c\right)g^c(X_0^c,Z_0^c)\hfill \\ \hfill & -a_0^c\left(X_0^c\right){\eta }_0^v\left(W_0^c\right)g^c(Y_0^c,Z_0^c)-a_0^v\left(X_0^c\right){\eta }_0^c\left(W_0^c\right)g^c(Y_0^c,Z_0^c)\hfill \\ \hfill & +a_0^c\left(Y_0^c\right){\eta }_0^v\left(Z_0^c\right)g^c(X_0^c,W_0^c)+a_0^v\left(Y_0^c\right){\eta }_0^c\left(Z_0^c\right)g^c(X_0^c,W_0^c)\hfill \\ \hfill & -a_0^c\left(X_0^c\right){\eta }_0^v\left(Z_0^c\right)g^c(Y_0^c,W_0^c)-a_0^v\left(X_0^c\right){\eta }_0^c\left(Z_0^c\right)g^c(Y_0^c,W_0^c)\hfill \\ \hfill & +2[a_0^c\left(Y_0^c\right){\eta }_0^c\left(X_0^c\right){\eta }_0^c\left(Z_0^c\right){\eta }_0^v\left(W_0^c\right)+a_0^c\left(Y_0^c\right){\eta }_0^c\left(X_0^c\right){\eta }_0^v\left(Z_0^c\right){\eta }_0^c\left(W_0^c\right)\hfill \\ \hfill & +a_0^c\left(Y_0^c\right){\eta }_0^v\left(X_0^c\right){\eta }_0^c\left(Z_0^c\right){\eta }_0^c\left(W_0^c\right)+a_0^v\left(Y_0^c\right){\eta }_0^c\left(X_0^c\right){\eta }_0^c\left(Z_0^c\right){\eta }_0^c\left(W_0^c\right)]\hfill \\ \hfill & -2[a_0^c\left(X_0^c\right){\eta }_0^c\left(Y_0^c\right){\eta }_0^c\left(Z_0^c\right){\eta }_0^v\left(W_0^c\right)+a_0^c\left(X_0^c\right){\eta }_0^c\left(Y_0^c\right){\eta }_0^v\left(Z_0^c\right){\eta }_0^c\left(W_0^c\right)\hfill \\ \hfill & +a_0^c\left(X_0^c\right){\eta }_0^v\left(Y_0^c\right){\eta }_0^c\left(Z_0^c\right){\eta }_0^c\left(W_0^c\right)+a_0^v\left(X_0^c\right){\eta }_0^c\left(Y_0^c\right){\eta }_0^c\left(Z_0^c\right){\eta }_0^c\left(W_0^c\right)]\hfill \\ \hfill & +2{\mathit{da}}_0^c(X_0^c,Y_0^c)g^c\left({\left({\varphi }_0{Z}_0\right)}^c,W_0^c\right).\hfill \end{array}$$

(66)

$$\begin{array}{cc}\hfill & {}^{\prime }{\ddot{R}}_0^c(X_0^c,Y_0^c,Z_0^c,W_0^c)-{}^{\prime }{\ddot{R}}_0^c(Z_0^c,W_0^c,X_0^c,Y_0^c)\hfill \\ \hfill & ={\eta }_0^c\left(Y_0^c\right){\eta }_0^v\left(Z_0^c\right)g^c(X_0^c,W_0^c)+{\eta }_0^v\left(Y_0^c\right){\eta }_0^c\left(Z_0^c\right)g^c(X_0^c,W_0^c)\hfill \\ \hfill & -{\eta }_0^c\left(X_0^c\right){\eta }_0^v\left(W_0^c\right)g^c(Y_0^c,Z_0^c)-{\eta }_0^v\left(X_0^c\right){\eta }_0^c\left(W_0^c\right)g^c(Y_0^c,Z_0^c)\hfill \\ \hfill & +a_0^c\left(Y_0^c\right){\eta }_0^v\left(W_0^c\right)g^c(X_0^c,Z_0^c)+a_0^v\left(Y_0^c\right){\eta }_0^c\left(W_0^c\right)g^c(X_0^c,Z_0^c)\hfill \\ \hfill & -a_0^c\left(X_0^c\right){\eta }_0^v\left(W_0^c\right)g^c(Y_0^c,Z_0^c)-a_0^v\left(X_0^c\right){\eta }_0^c\left(W_0^c\right)g^c(Y_0^c,Z_0^c)\hfill \\ \hfill & +a_0^c\left(Z_0^c\right){\eta }_0^v\left(Y_0^c\right)g^c(X_0^c,W_0^c)+a_0^v\left(Z_0^c\right){\eta }_0^c\left(Y_0^c\right)g^c(X_0^c,W_0^c)\hfill \\ \hfill & -a_0^c\left(W_0^c\right){\eta }_0^v\left(Y_0^c\right)g^c(X_0^c,Z_0^c)-a_0^v\left(W_0^c\right){\eta }_0^c\left(Y_0^c\right)g^c(X_0^c,Z_0^c)\hfill \\ \hfill & +a_0^c\left(Y_0^c\right){\eta }_0^c\left(X_0^c\right){\eta }_0^c\left(Z_0^c\right){\eta }_0^v\left(W_0^c\right)+a_0^c\left(Y_0^c\right){\eta }_0^c\left(X_0^c\right){\eta }_0^v\left(Z_0^c\right){\eta }_0^c\left(W_0^c\right)\hfill \\ \hfill & +a_0^c\left(Y_0^c\right){\eta }_0^v\left(X_0^c\right){\eta }_0^c\left(Z_0^c\right){\eta }_0^c\left(W_0^c\right)+a_0^v\left(Y_0^c\right){\eta }_0^c\left(X_0^c\right){\eta }_0^c\left(Z_0^c\right){\eta }_0^c\left(W_0^c\right)\hfill \\ \hfill & -a_0^c\left(X_0^c\right){\eta }_0^c\left(Y_0^c\right){\eta }_0^c\left(Z_0^c\right){\eta }_0^v\left(W_0^c\right)-a_0^c\left(X_0^c\right){\eta }_0^c\left(Y_0^c\right){\eta }_0^v\left(Z_0^c\right){\eta }_0^c\left(W_0^c\right)\hfill \\ \hfill & -a_0^c\left(X_0^c\right){\eta }_0^v\left(Y_0^c\right){\eta }_0^c\left(Z_0^c\right){\eta }_0^c\left(W_0^c\right)-a_0^v\left(X_0^c\right){\eta }_0^c\left(Y_0^c\right){\eta }_0^c\left(Z_0^c\right){\eta }_0^c\left(W_0^c\right)\hfill \\ \hfill & +a_0^c\left(Z_0^c\right){\eta }_0^c\left(X_0^c\right){\eta }_0^c\left(Y_0^c\right){\eta }_0^v\left(W_0^c\right)+a_0^c\left(Z_0^c\right){\eta }_0^c\left(X_0^c\right){\eta }_0^v\left(Y_0^c\right){\eta }_0^c\left(W_0^c\right)\hfill \\ \hfill & +a_0^c\left(Z_0^c\right){\eta }_0^v\left(X_0^c\right){\eta }_0^c\left(Y_0^c\right){\eta }_0^c\left(W_0^c\right)+a_0^v\left(Z_0^c\right){\eta }_0^c\left(X_0^c\right){\eta }_0^c\left(Y_0^c\right){\eta }_0^c\left(W_0^c\right)\hfill \\ \hfill & -a_0^c\left(W_0^c\right){\eta }_0^c\left(X_0^c\right){\eta }_0^c\left(Y_0^c\right){\eta }_0^v\left(Z_0^c\right)-a_0^c\left(W_0^c\right){\eta }_0^c\left(X_0^c\right){\eta }_0^v\left(Y_0^c\right){\eta }_0^c\left(Z_0^c\right)\hfill \\ \hfill & -a_0^c\left(W_0^c\right){\eta }_0^v\left(X_0^c\right){\eta }_0^c\left(Y_0^c\right){\eta }_0^c\left(Z_0^c\right)-a_0^v\left(W_0^c\right){\eta }_0^c\left(X_0^c\right){\eta }_0^c\left(Y_0^c\right){\eta }_0^c\left(Z_0^c\right)\hfill \\ \hfill & +{\mathit{da}}_0^c(X_0^c,Y_0^c)g^c\left({\left({\varphi }_0{Z}_0\right)}^c,W_0^c\right)-{\mathit{da}}_0^c(Z_0^c,W_0^c)g^c\left({\left({\varphi }_0{X}_0\right)}^c,Y_0^c\right),\hfill \end{array}$$

(67)

and

$$\begin{array}{cc}\hfill {}^{\prime }{\ddot{R}}_0^c& (X_0^c,Y_0^c,Z_0^c,W_0^c)+{}^{\prime }{\ddot{R}}_0^c(Y_0^c,Z_0^c,X_0^c,W_0^c)+{}^{\prime }{\ddot{R}}_0^c(Z_0^c,X_0^c,Y_0^c,W_0^c)\hfill \\ \hfill & ={\mathit{da}}_0^c(X_0^c,Y_0^c)g^c\left({\left({\varphi }_0{Z}_0\right)}^c,W_0^c\right)+{\mathit{da}}_0^c(Y_0^c,Z_0^c)g^c\left({\left({\varphi }_0{X}_0\right)}^c,W_0^c\right)\hfill \\ \hfill & +{\mathit{da}}_0^c(Z_0^c,X_0^c)g^c\left({\left({\varphi }_0{Y}_0\right)}^c,W_0^c\right).\hfill \end{array}$$

(68)

If the 1-form $a_0^c$ is closed, then from (68) we have

$$\begin{array}{cc}\hfill {}^{\prime }{\ddot{R}}_0^c& (X_0^c,Y_0^c,Z_0^c,W_0^c)+{}^{\prime }{\ddot{R}}_0^c(Y_0^c,Z_0^c,X_0^c,W_0^c)\hfill \\ \hfill & +{}^{\prime }{\ddot{R}}_0^c(Z_0^c,X_0^c,Y_0^c,W_0^c)=0,\hfill \end{array}$$

(69)

where

$$\begin{array}{cc}\hfill & {}^{\prime }{\ddot{R}}_0^c(X_0^c,Y_0^c,Z_0^c,W_0^c)=g^c\left({\ddot{R}}_0^c(X_0^c,Y_0^c)Z_0^c,W_0^c\right)\hfill \\ \hfill & \mathrm{and}{}^{\prime }{R}_0^c(X_0^c,Y_0^c,Z_0^c,W_0^c)=g^c\left(R_0^c(X_0^c,Y_0^c)Z_0^c,W_0^c\right).\hfill \end{array}$$

Theorem 4.

In an LP-Sasakian manifold, $(M^n,g)$ with tangent bundle $T_0{M}^n$ endowed with a QSNMC, the curvature tensor satisfies relations (65)–(68). In particular, if the complete lift of 1-form $a_0^c$ is closed, then

$${}^{\prime }{\ddot{R}}_0^c(X_0^c,Y_0^c,Z_0^c,W_0^c)+{}^{\prime }{\ddot{R}}_0^c(Y_0^c,Z_0^c,X_0^c,W_0^c)+{}^{\prime }{\ddot{R}}_0^c(Z_0^c,X_0^c,Y_0^c,W_0^c)=0.$$

7. Symmetric and Skew-Symmetric Condition of the Ricci Tensor of ${\ddot{\mathrm{\nabla }}}^{\mathit{c}}$ in an LP-Sasakian Manifold Endowed with a QSNMC to Tangent Bundle

From Equation (60), we have

$$\begin{array}{cc}\hfill {\ddot{S}}_0^c(Z_0^c,Y_0^c)& =S_0^c(Z_0^c,Y_0^c)-{\gamma }^c{g}^c\left({\left({\varphi }_0{Z}_0\right)}^c,Y_0^c\right)\hfill \\ \hfill & +\left[1-a_0^c\left({\xi }_0^c\right)\right]g^c(Y_0^c,Z_0^c)+\left[{\eta }_0^c-a_0^c\left({\xi }_0^c\right)\right][{\eta }_0^c\left(Y_0^c\right){\eta }_0^v\left(Z_0^c\right)\hfill \\ \hfill & +{\eta }_0^v\left(Y_0^c\right){\eta }_0^c\left(Z_0^c\right)]+{\mathit{da}}_0^c\left({\left({\varphi }_0{Y}_0\right)}^c,Z_0^c\right).\hfill \end{array}$$

(70)

From (60) and (70), we have

$${\ddot{S}}_0^c(Y_0^c,Z_0^c)-{\ddot{S}}_0^c(Z_0^c,Y_0^c)={\mathit{da}}_0^c\left({\left({\varphi }_0{Z}_0\right)}^c,Y_0^c\right)-{\mathit{da}}_0^c\left({\left({\varphi }_0{Y}_0\right)}^c,Z_0^c\right).$$

(71)

If ${\ddot{S}}_0^c(Y_0^c,Z_0^c)$ is symmetric, then the left-hand side of (71) vanishes, and then

$${\mathit{da}}_0^c\left({\left({\varphi }_0{Z}_0\right)}^c,Y_0^c\right)={\mathit{da}}_0^c\left({\left({\varphi }_0{Y}_0\right)}^c,Z_0^c\right).$$

(72)

Moreover, if Equation (72) holds, then from (71), ${\ddot{S}}_0^c(Y_0^c,Z_0^c)$ is symmetric.

Theorem 5.

In an LP-Sasakian manifold $(M^n,g)$ with tangent bundle $T_0{M}^n$ endowed with QSNMC ${\ddot{\mathrm{\nabla }}}^c$, the Ricci tensor ${\ddot{S}}_0^c(Y_0^c,Z_0^c)$ with respect to QSNMC is symmetric if and only if relation (72) holds.

From (60) and (70), we have

$$\begin{array}{cc}\hfill {\ddot{S}}_0^c(Y_0^c,Z_0^c)+{\ddot{S}}_0^c(Z_0^c,Y_0^c)& =2S_0^c(Y_0^c,Z_0^c)-2{\gamma }^c{g}^c\left({\left({\varphi }_0{Y}_0\right)}^c,Z_0^c\right)\hfill \\ \hfill & +2\left[1-a_0^c\left({\xi }_0^c\right)\right]g^c(Y_0^c,Z_0^c)\hfill \\ \hfill & +2\left[n-a_0^c\left({\xi }_0^c\right)\right][{\eta }_0^c\left(Y_0^c\right){\eta }_0^v\left(Z_0^c\right)\hfill \\ \hfill & +{\eta }_0^v\left(Y_0^c\right){\eta }_0^c\left(Z_0^c\right)]+{\mathit{da}}_0^c\left({\left({\varphi }_0{Y}_0\right)}^c,Z_0^c\right)\hfill \\ \hfill & +{\mathit{da}}_0^c\left({\left({\varphi }_0{Z}_0\right)}^c,Y_0^c\right).\hfill \end{array}$$

(73)

By taking the skew-symmetry of ${\ddot{S}}_0^c(Y_0^c,Z_0^c)$, the left-hand side of (73) will vanish and we have

$$\begin{array}{cc}\hfill S_0^c(Y_0^c,Z_0^c)& ={\gamma }^c{g}^c\left({\left({\varphi }_0{Y}_0\right)}^c,Z_0^c\right)\hfill \\ \hfill & -\left[1-a_0^c\left({\xi }_0^c\right)\right]g^c(Y_0^c,Z_0^c)-\left[n-a_0^c\left({\xi }_0^c\right)\right]\hfill \\ \hfill & \left[{\eta }_0^c\left(Y_0^c\right){\eta }_0^v\left(Z_0^c\right)-{\eta }_0^v\left(Y_0^c\right){\eta }_0^c\left(Z_0^c\right)\right]\hfill \\ \hfill & -\frac{1}{2}\left[{\mathit{da}}_0^c\left({\left({\varphi }_0{Y}_0\right)}^c,Z_0^c\right)+{\mathit{da}}_0^c\left({\left({\varphi }_0{Z}_0\right)}^c,Y_0^c\right)\right].\hfill \end{array}$$

(74)

Moreover if $S_0^c(Y_0^c,Z_0^c)$ is given by (74), then from (73), we have

$$S_0^c(Y_0^c,Z_0^c)+S_0^c(Z_0^c,Y_0^c)=0.$$

Theorem 6.

The necessary and sufficient condition for the Ricci tensor of ${\ddot{\mathrm{\nabla }}}^c$ in an LP-Sasakian manifold $(M^n,g)$ endowed with QSNMC ${\ddot{\mathrm{\nabla }}}^c$ in the tangent bundle $T_0{M}^n$ to be skew-symmetric is that the Ricci tensor of the Levi-Civita connection ${\mathrm{\nabla }}^c$ is given by (74).

8. Skew-Symmetric Properties of the Projective Ricci Tensor in an LP-Sasakian Manifold Endowed with QSNMC ${\ddot{\mathrm{\nabla }}}^{\mathit{c}}$ in the Tangent Bundle

Chaki and Saha defined the projective Ricci tensor in a Riemannian manifold as [34]

$$P_0(X_0,Y_0)=\frac{n}{n-1}\left[S_0(X_0,Y_0)-\frac{r_0}{n}g(X_0,Y_0)\right].$$

(75)

So, the projective Ricci tensor with respect to QSNMC $\ddot{\mathrm{\nabla }}$ is defined as

$${\ddot{P}}_0(X_0,Y_0)=\frac{n}{n-1}\left[{\ddot{S}}_0(X_0,Y_0)-\frac{{\ddot{r}}_0}{n}g(X_0,Y_0)\right].$$

(76)

Taking a complete lift by mathematical operators on (76), we have

$${\ddot{P}}_0^c(X_0^c,Y_0^c)=\frac{n}{n-1}\left[{\ddot{S}}_0^c(X_0^c,Y_0^c)-\frac{{\ddot{r}}_0^c}{n}{g}^c(X_0^c,Y_0^c)\right].$$

(77)

Using (60) and (61) in (77), we have

$$\begin{array}{cc}\hfill {\ddot{P}}_0^c(X_0^c,Y_0^c)& =\frac{n}{n-1}[S_0^c(X_0^c,Y_0^c)-{\gamma }^c{g}^c\left({\left({\varphi }_0{X}_0\right)}^c,Y_0^c\right)\hfill \\ \hfill & +\left(1-a_0^c\left({\xi }_0^c\right)\right)g^c(X_0^c,Y_0^c)+\left(n-a_0^c\left({\xi }_0^c\right)\right)({\eta }_0^c\left(X_0^c\right){\eta }_0^v\left(Y_0^c\right)\hfill \\ \hfill & +{\eta }_0^v\left(X_0^c\right){\eta }_0^c\left(Y_0^c\right))+{\mathit{da}}_0^c\left({\left({\varphi }_0{Y}_0\right)}^c,X_0^c\right)\hfill \\ \hfill & -\frac{1}{n}\left(r_0^c-(n-1)a_0^c\left({\xi }_0^c\right)+{\lambda }_0^c-{\gamma }^{c^2}\right)g^c(X_0^c,Y_0^c)].\hfill \end{array}$$

(78)

Similarly, we have

$$\begin{array}{cc}\hfill {\ddot{P}}_0^c(Y_0^c,X_0^c)& =\frac{n}{n-1}[S_0^c(Y_0^c,X_0^c)-{\gamma }^c{g}^c\left({\left({\varphi }_0{Y}_0\right)}^c,X_0^c\right)\hfill \\ \hfill & +\left(1-a_0^c\left({\xi }_0^c\right)\right)g^c(Y_0^c,X_0^c)+\left(n-a_0^c\left({\xi }_0^c\right)\right)({\eta }_0^c\left(X_0^c\right){\eta }_0^v\left(Y_0^c\right)\hfill \\ \hfill & +{\eta }_0^v\left(X_0^c\right){\eta }_0^c\left(Y_0^c\right))+{\mathit{da}}_0^c\left({\left({\varphi }_0{X}_0\right)}^c,Y_0^c\right)\hfill \\ \hfill & -\frac{1}{n}\left(r_0^c-(n-1)a_0^c\left({\xi }_0^c\right)+{\lambda }_0^c-{\gamma }^{c^2}\right)g^c(Y_0^c,X_0^c)].\hfill \end{array}$$

(79)

From (78) and (79), we have

$$\begin{array}{cc}\hfill & {\ddot{P}}_0^c(X_0^c,Y_0^c)+{\ddot{P}}_0^c(Y_0^c,X_0^c)\hfill \\ \hfill & =\frac{n}{n-1}[2S_0^c(X_0^c,Y_0^c)-2{\gamma }^c{g}^c\left({\left({\varphi }_0{X}_0\right)}^c,Y_0^c\right)\hfill \\ \hfill & +2\left(1-a_0^c\left({\xi }_0^c\right)\right)g^c(X_0^c,Y_0^c)+2\left(n-a_0^c\left({\xi }_0^c\right)\right)\hfill \\ \hfill & \left({\eta }_0^c\left(X_0^c\right){\eta }_0^v\left(Y_0^c\right)+{\eta }_0^v\left(X_0^c\right){\eta }_0^c\left(Y_0^c\right)\right)\hfill \\ \hfill & -\frac{2}{n}\left(r_0^c-(n-1)a_0^c\left({\xi }_0^c\right)+{\lambda }_0^c-{\gamma }^{c^2}\right)g^c(X_0^c,Y_0^c)\hfill \\ \hfill & +{\mathit{da}}_0^c\left({\left({\varphi }_0{X}_0\right)}^c,Y_0^c\right)+{\mathit{da}}_0^c\left({\left({\varphi }_0{Y}_0\right)}^c,X_0^c\right)].\hfill \end{array}$$

(80)

If ${\ddot{P}}_0^c(X_0^c,Y_0^c)$ is skew-symmetric, then the left-hand side of (80) vanishes and we have

$$\begin{array}{cc}\hfill S_0^c(X_0^c,Y_0^c)& =[{\gamma }^c{g}^c\left({\left({\varphi }_0{X}_0\right)}^c,Y_0^c\right)-\left(1-a_0^c\left({\xi }_0^c\right)\right)g^c(X_0^c,Y_0^c)\hfill \\ \hfill & -\left(n-a_0^c\left({\xi }_0^c\right)\right)\left({\eta }_0^c\left(X_0^c\right){\eta }_0^v\left(Y_0^c\right)+{\eta }_0^v\left(X_0^c\right){\eta }_0^c\left(Y_0^c\right)\right)\hfill \\ \hfill & +\frac{1}{n}\left(r_0^c-(n-1)a_0^c\left({\xi }_0^c\right)+{\lambda }_0^c-{\gamma }^{c^2}\right)g^c(X_0^c,Y_0^c)\hfill \\ \hfill & -\frac{1}{2}\left({\mathit{da}}_0^c\left({\left({\varphi }_0{X}_0\right)}^c,Y_0^c\right)+{\mathit{da}}_0^c\left({\left({\varphi }_0{Y}_0\right)}^c,X_0^c\right)\right)].\hfill \end{array}$$

(81)

Moreover, if $S_0^c(X_0^c,Y_0^c)$ is given by (81), then from (80) we obtain

$${\ddot{P}}_0^c(X_0^c,Y_0^c)+{\ddot{P}}_0^c(Y_0^c,X_0^c)=0\mathrm{s}.\mathrm{t}{\ddot{P}}_0^c(X_0^c,Y_0^c)=-{\ddot{P}}_0^c(Y_0^c,X_0^c).$$

(82)

which gives a skew-symmetric condition of the projective Ricci tensor of ${\ddot{\mathrm{\nabla }}}^c$.

Theorem 7.

The necessary and sufficient condition for the projective Ricci tensor of ${\ddot{\mathrm{\nabla }}}^c$ in an LP-Sasakian manifold $(M^n,g)$ endowed with QSNMC ${\ddot{\mathrm{\nabla }}}^c$ in the tangent bundle $T_0{M}^n$ to be skew-symmetric is that the Ricci tensor of the Levi-Civita connection ${\ddot{\mathrm{\nabla }}}^c$ is given by (81).

9. Lifts of Einstein Manifold Endowed with QSNMC ${\ddot{\mathrm{\nabla }}}^{\mathit{c}}$ in an LP-Sasakian Manifold to the Tangent Bundle

A Riemannian manifold $(M^n,g)$ is called an Einstein manifold with respect to Riemannian connection if

$$S_0^c(X_0^c,Y_0^c)=\frac{r_0^c}{n}{g}^c(X_0^c,Y_0^c).$$

(83)

Then, the Einstein manifold with respect to QSNMC ${\ddot{\mathrm{\nabla }}}^c$ is given by

$${\ddot{S}}_0^c(X_0^c,Y_0^c)=\frac{{\ddot{r}}_0^c}{n}{g}^c(X_0^c,Y_0^c).$$

(84)

Using (60) and (61) in (84), we have

$$\begin{array}{cc}\hfill & {\ddot{S}}_0^c(X_0^c,Y_0^c)-\frac{{\ddot{r}}_0^c}{n}{g}^c(X_0^c,Y_0^c)\hfill \\ \hfill & =S_0^c(X_0^c,Y_0^c)-\frac{r_0^c}{n}{g}^c(X_0^c,Y_0^c)-{\gamma }^c{g}^c\left({\left({\varphi }_0{X}_0\right)}^c,Y_0^c\right)\hfill \\ \hfill & +{\mathit{da}}_0^c\left({\left({\varphi }_0{Y}_0\right)}^c,X_0^c\right)+\frac{1}{n}\left[n+{\gamma }^{c^2}-{\lambda }_0^c-a_0^c\left({\xi }_0^c\right)\right]g^c(X_0^c,Y_0^c)\hfill \\ \hfill & +\left(n-a_0^c\left({\xi }_0^c\right)\right)\left[{\eta }_0^c\left(X_0^c\right){\eta }_0^v\left(Y_0^c\right)+{\eta }_0^v\left(X_0^c\right){\eta }_0^c\left(Y_0^c\right)\right].\hfill \end{array}$$

(85)

If

$$\begin{array}{cc}\hfill & {\gamma }^c{g}^c\left({\left({\varphi }_0{X}_0\right)}^c,Y_0^c\right)+{\mathit{da}}_0^c\left(X_0^c,{\left({\varphi }_0{Y}_0\right)}^c\right)\hfill \\ \hfill & =\frac{1}{n}\left[n+{\gamma }^{c^2}-{\lambda }_0^c-a_0^c\left({\xi }_0^c\right)\right]g^c(X_0^c,Y_0^c)\hfill \\ \hfill & +\left(n-a_0^c\left({\xi }_0^c\right)\right)\left[{\eta }_0^c\left(X_0^c\right){\eta }_0^v\left(Y_0^c\right)+{\eta }_0^v\left(X_0^c\right){\eta }_0^c\left(Y_0^c\right)\right],\hfill \end{array}$$

(86)

then from (85), we have

$${\ddot{S}}_0^c(X_0^c,Y_0^c)-\frac{{\ddot{r}}_0^c}{n}{g}^c(X_0^c,Y_0^c)=S_0^c(X_0^c,Y_0^c)-\frac{r_0^c}{n}{g}^c(X_0^c,Y_0^c).$$

(87)

Theorem 8.

In an LP-Sasakian manifold $(M^n,g)$ with tangent bundle $T_0{M}^n$ admitting QSNMC if Equation (86) holds, then the manifold reduces to an Einstein manifold for the Riemannian connection if and only if it is an Einstein manifold for the connection ${\ddot{\mathrm{\nabla }}}^c$.

10. Example

Let M be a four-dimensional manifold defined as

$$M=\left\{(x_1,x_2,x_3,x_4)\in {\mathbb{R}}^4;x_4\ne 0\right\},$$

(88)

where $\mathbb{R}$ is the set of real numbers. Let $x_1,x_2,x_3,x_4$ be given by

$$e_1=\frac{x_1}{x_4}\frac{\partial }{\partial x_1},e_2=\frac{x_2}{x_4}\frac{\partial }{\partial x_2},e_3=\frac{x_3}{x_4}\frac{\partial }{\partial x_3},e_4=x_4\frac{\partial }{\partial x_4},$$

where $\{e_1,e_2,e_3,e_4\}$ are a linearly independent global frame on M. Let the 1-form ${\eta }_0$ be given by

$${\eta }_0\left(X_0\right)=g(X_0,e_4).$$

The Lorentzian metric g is defined by

$$g(e_i,e_j)=\left\{\begin{array}{cc}-1,\hfill & i=j=4\hfill \\ 1,\hfill & i=j=1,2,3\hfill \\ 0,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

Let ${\varphi }_0$ be the tensor field defined by

$${\varphi }_0{e}_i=\left\{\begin{array}{cc}0,\hfill & i=4\hfill \\ e_i,\hfill & i=1,2,3.\hfill \end{array}\right.$$

Using the linearity of ${\varphi }_0$ and g, we acquire ${\eta }_0\left(e_4\right)=-1$, ${\varphi }_0^2{X}_0=-X_0+{\eta }_0\left(X_0\right)e_4$ and $g({\varphi }_0{X}_0,{\varphi }_0{Y}_0)=g(X_0,Y_0)+{\eta }_0\left(X_0\right){\eta }_0\left(Y_0\right).$ Thus, for $e_4={\xi }_0$, then the structure $(\varphi ,{\xi }_0,{\eta }_0,g)$ is an almost para-contact metric structure on M and M is called an almost para-contact metric manifold. In addition, M satisfies

$$\left({\mathrm{\nabla }}_{X_0}{\varphi }_0\right)Y_0=g(X_0,Y_0)e_4+{\eta }_0\left(Y_0\right)X_0+2{\eta }_0\left(X_0\right){\eta }_0\left(Y_0\right)e_4.$$

Here, for $e_4={\xi }_0$, M is an LP-Sasakian manifold. In tangent bundle $T_{0}M$, let the complete and vertical lifts of $e_1,e_2,e_3,e_4$ be $e_1^c,e_2^c,e_3^c,e_4^c$ and $e_1^v,e_2^v,e_3^v,e_4^v$ on M and let $g^c$ be the complete lift of the Lorentzian metric g on $T_{0}M$ such that

$$\begin{array}{c}\hfill g^c(X_0^v,e_4^c)={\left(g^c(X_0,e_4)\right)}^v={\left({\eta }_0\left(X_0\right)\right)}^v\end{array}$$

(89)

$$\begin{array}{c}\hfill g^c(X_0^c,e_4^c)={\left(g^c(X_0,e_4)\right)}^c={\left({\eta }_0\left(X_0\right)\right)}^c\end{array}$$

(90)

$$\begin{array}{c}\hfill g^c(e_4^c,e_4^c)=-1,g^v(X_0^v,e_4^c)=0,g^v(e_4^v,e_4^c)=0,\end{array}$$

(91)

and so on. Let ${\varphi }_0^c$ and ${\varphi }_0^v$ be the complete and vertical lifts of the $(1,1)$ tensor field ${\varphi }_0$ defined by

$$\begin{array}{c}\hfill {\varphi }_0^v\left(e_4^v\right)={\varphi }_0^c\left(e_4^c\right)=0,\end{array}$$

(92)

$$\begin{array}{c}\hfill {\varphi }_0^v\left(e_1^v\right)=e_1^v,{\varphi }_0^c\left(e_1^c\right)=e_1^c,\end{array}$$

(93)

$$\begin{array}{c}\hfill {\varphi }_0^v\left(e_2^v\right)=e_2^v,{\varphi }_0^c\left(e_2^c\right)=e_2^c,\end{array}$$

(94)

$$\begin{array}{c}\hfill {\varphi }_0^v\left(e_3^v\right)=e_3^v,{\varphi }_0^c\left(e_3^c\right)=e_3^c.\end{array}$$

(95)

Using the linearity of ${\varphi }_0$ and g, we infer that

$$\begin{array}{c}\hfill {\left({\varphi }_0^2{X}_0\right)}^c=X_0^c+{\eta }_0^c\left(X_0\right)e_4^v+{\eta }_0^v\left(X_0\right)e_4^c,\end{array}$$

(96)

$$\begin{array}{c}\hfill g^c\left({\left({\varphi }_0{e}_4\right)}^c,{\left({\varphi }_0{e}_3\right)}^c\right)=g^c(e_4^c,e_3^c)+{\eta }_0^c\left(e_4^c\right){\eta }_0^v\left(e_3^c\right)+{\eta }_0^v\left(e_4^c\right){\eta }_0^c\left(e_3^c\right).\end{array}$$

(97)

Thus, for $e_4={\xi }_0$ in (89)–(91) and (96), the structure $({\varphi }_0^c,{\xi }_0^c,{\eta }_0^c,g^c)$ is an almost para-contact metric structure on $T_{0}M$ and satisfies the relation

$$\begin{array}{cc}\hfill \left({\mathrm{\nabla }}_{e_4^c}^c{\varphi }_0^c\right)e_3^c& =g^c(e_4^c,e_3^c){\xi }_0^v+g^c(e_4^v,e_3^c){\xi }_0^c+{\eta }_0^c\left(e_3^c\right)e_4^v+{\eta }_0^v\left(e_3^c\right)e_4^c\hfill \\ \hfill & +2\left\{{\eta }_0^c\left(e_4^c\right){\eta }_0^c\left(e_3^c\right){\xi }_0^v+{\eta }_0^c\left(e_4^c\right){\eta }_0^v\left(e_3^c\right){\xi }_0^c+{\eta }_0^v\left(e_4^c\right){\eta }_0^c\left(e_3^c\right){\xi }_0^c\right\}.\hfill \end{array}$$

Thus, $({\varphi }_0^c,{\xi }_0^c,{\eta }_0^c,g^c,T_{0}M)$ is an LP-Sasakian manifold.

11. Conclusions

The current work investigates the lifts of a QSNMC and LP-Sasakian manifold to the tangent bundle. First, the LP-Sasakian manifold lifts to the tangent bundle are presented. The relationship between the Riemannian connection and the QSNMC from an LP-Sasakian manifold to the tangent bundle is established. An expression of the curvature tensor of the lifts of an LP-Sasakian manifold associated with QSNMC to its tangent bundle is given. The Ricci tensor and the scalar curvature lifts to the tangent bundle are provided. Some theorems regarding the properties of the lifts of the curvature tensor of an LP-Sasakian manifold endowed with QSNMC in an LP-Sasakian manifold to the tangent bundle are given.

Necessary and sufficient conditions for the symmetric and skew-symmetric properties of the lifts of the Ricci tensor are investigated. Sufficient conditions for the skew-symmetric property of the lifts of the projective Ricci tensor in the tangent bundle are provided. The lifts of the Einstein manifold associated with QSNMC on an LP-Sasakian manifold to the tangent bundle are also established. An example of the lifts of LP-Sasakian manifolds in the tangent bundle is constructed.