Generalized B-Curvature Tensor in Lorentzian Para-Kenmotsu Manifold with Semi-Symmetric Metric Connection

Abstract

The main object of this work is to study the generalized B-curvature tensor in an n-dimensional Lorentzian para-Kenmotsu (briefly, ${\left(LPK\right)}_n$) manifold along a semi-symmetric metric connection $\overline{\nabla }$. First, in an ${\left(LPK\right)}_n$-manifold, we explore certain flatness conditions, namely, $\overline{B}(Y,Z)X=0$, $\overline{B}(Y,Z)\zeta =0$, $g(\overline{B}(\phi Y,\phi Z)\phi X,\phi W)=0$, and $\overline{B}(Y,Z)·\phi =0$ conditions, which all result in an $\eta$-Einstein manifold. Furthermore, in an ${\left(LPK\right)}_n$-manifold, we study the curvature conditions $\overline{B}.Q=0$ and $\overline{B}.\overline{Q}$ = 0, which provide the scalar curvature. The generalized B-curvature tensor blends the features of different curvature tensors, allowing researchers to study conditions like semi-symmetry, pseudo-symmetry in a unified framework. Conditions like B-semi-symmetry correspond to conservation laws or stability properties in physical systems.

1. Introduction

In [1], Shaikh and Kundu introduced the equivalency of various geometric structures obtained by same restriction imposed on different curvature tensors. For this aim, they defined a (0, 4)-type tensor, which is linear combination of the Riemann-Christoffel curvature tensor, Ricci tensor, metric tensor, and scalar curvature, and describes various curvature tensors as its particular cases. The set of all B-tensors are denoted by $\mathcal{B}$.

The generalized B-curvature tensor is a refinement of classical curvature tensors that allows for deeper exploration of geometric structures. Standard tensors (like Riemann, Ricci, and Weyl) often fail to distinguish subtle geometric properties in ${\left(LPK\right)}_n$-manifold. The generalized B-tensor introduces additional flexibility to study vanishing, semi-symmetric, and more. Existing studies on ${\left(LPK\right)}_n$-manifold emphasize Ricci solitons or Weyl curvature conditions.

The generalized B-curvature tensor introduces new geometric invariants and conditions that were not previously studied, making it a novel tool for both classification and physical interpretation. Think of the manifold like a fabric stretched evenly in all directions (Einstein case). In the $\eta$-Einstein case, one thread (the $\zeta$-direction) is woven tighter or looser, giving the fabric a special anisotropy. That “thread” controls how the geometry bends differently along that axis. The stated results in Section 4.1, Section 4.2 and Section 4.3 and Section 5 that “$\eta$-Einstein manifold” means the manifold belongs to a special family where curvature is uniform, except in one distinguished direction. That makes the geometry easier to study and gives it potential physical interpretations.

The idea of almost para-contact manifolds was introduced by Sato [2]. According to Kaneyuki and his collaborator [3], the main variation among an almost para-contact manifold is the signature of metric. In 1989, Matsumoto [4] used a structure vector field ‘$\zeta$’ instead of ‘$\zeta$’ in an almost para contact manifold and associated a Lorentzian metric with this resulting structure, and called it a Lorentzian almost para contact manifold. The “para-Kenmotsu” condition encodes a specific interaction between the metric and a para-contact structure, giving a controlled way to study curvature and hypersurfaces in a pseudo-Riemannian setting. Afterwards, para-Kenmotsu manifolds were a great focus of geometers and brought forward the significant characteristics of such manifolds [5,6,7,8,9].

In 1924, Friedmann and Schouten proposed the concept of a semi-symmetric linear connection on differentiable manifold. A semi-symmetric metric connection preserves the metric but allows torsion, specifically tied to a 1-form. In [10], the authors studied a semi-symmetric metric connection on submanifolds of a Riemannian manifold. Certain properties of a semi-symmetric metric connection on a Riemannian manifold have been studied by De and De [11]. Recently, the authors Haseeb and Prasad investigated Kenmotsu and Lorentzian para-Sasakian manifolds with a semi-symmetric metric connection by satisfying certain curvature conditions [12,13]. The semi-symmetric metric connection on indefinite Kenmotsu manifold have been studied by the authors Kumar et al. [14]. In 2009, the authors Shukla et. al. [15] studied $\phi$-Ricci symmetric Kenmotsu and illustrated some of the results through an example.

This paper is organized as follows: Section 1 covers the introduction, corresponding concepts, and brief histories. Section 2 contains preliminaries, where some fundamental results are given, which are used in subsequent sections. In Section 3, the generalized B-curvature tensor of ${\left(LPK\right)}_n$-manifold with connection $\overline{\nabla }$ is described. In Section 4, we explore certain curvature conditions on ${\left(LPK\right)}_n$-manifolds, namely, $\overline{B}(Y,Z)X=0$, $\overline{B}(Y,Z)\zeta =0$, and $g(\overline{B}(\phi Y,\phi Z)\phi X,\phi W)=0$. In Section 5, we study $\phi$-$\overline{B}$ semi-symmetric ${\left(LPK\right)}_n$. In a ${\left(LPK\right)}_n$, the curvature conditions $\overline{B}.Q=0$ and $\overline{B}.\overline{Q}$ = 0 have been studied in Section 6 and Section 7, respectively.

2. Preliminaries

Let M be a n-dimensional Lorentzian metric manifold. If it is endowed with a structure ($\phi ,\zeta ,\eta ,g)$, where $\phi$ is a (1,1) tensor field, $\zeta$ is a vector field, $\eta$ is a 1-form on M, and g is a Lorentz metric, fulfilling [16]

$${\phi }^{2}Y=Y+\eta \left(Y\right)\zeta ,g(\phi Y,\phi Z)=g(Y,Z)+\eta \left(Y\right)\eta \left(Z\right),$$

(1)

$$\eta \left(\zeta \right)=-1,g(Y,\zeta )=\eta \left(Y\right),$$

(2)

for any vector fields Y, Z on M, then it is called a Lorentzian almost para-contact manifold. In a Lorentzian almost para-contact manifold, the coming relations are valid:

$$\phi \zeta =0,\eta \left(\phi Y\right)=0,$$

(3)

$$\mathrm{\Phi }(Y,Z)=\mathrm{\Phi }(Z,Y)$$

(4)

here $\mathrm{\Phi }(Y,Z)=g(Y,\phi Z)$.

Definition 1.

A Lorentzian almost para-contact manifold M is called a Lorentzian para-Kenmotsu (briefly, ${\left(LPK\right)}_n$) manifold if [17,18]

$$\left({\nabla }_Y\phi \right)Z=-g(\phi Y,Z)\zeta -\eta \left(Z\right)\phi Y,$$

(5)

for any vector fields Y, Z on ${\left(LPK\right)}_n$-manifold.

• In an ${\left(LPK\right)}_n$-manifold, we have

  $${\nabla }_Y\zeta =-Y-\eta \left(Y\right)\zeta ,$$

  (6)

  $$\left({\nabla }_Y\eta \right)Z=-g(Y,Z)-\eta \left(Y\right)\eta \left(Z\right),$$

  (7)

  where ∇ indicates the operator of covariant differentiation respecting to the Lorentzian metric g.

• Further, in an ${\left(LPK\right)}_n$-manifold, the following relations are valid [17,18,19]:

  $$g\left(R\right(Y,Z)X,\zeta )=\eta \left(R\right(Y,Z\left)X\right)=g(Z,X)\eta \left(Y\right)-g(Y,X)\eta \left(Z\right),$$

  (8)

  $$R(\zeta ,Y)Z=g(Y,Z)\zeta -\eta \left(Z\right)Y,$$

  (9)

  $$R(Y,Z)\zeta =\eta \left(Z\right)Y-\eta \left(Y\right)Z,$$

  (10)

  $$R(\zeta ,Y)\zeta =Y+\eta \left(Y\right)\zeta ,$$

  (11)

  $$S(Y,\zeta )=(n-1)\eta \left(Y\right),S(\zeta ,\zeta )=-(n-1),$$

  (12)

  $$r=n(n-1),Q\zeta =(n-1)\zeta ,$$

  (13)

  $$S(\phi Y,\phi Z)=S(Y,Z)+(n-1)\eta \left(Y\right)\eta \left(Z\right),$$

  (14)

  for any vector fields $Y,Z$ and X on M, where $\mathit{S},\mathit{R}$ and $\mathit{Q}$ denotes the Ricci tensor, the curvature tensor, and the Ricci operator on ${\left(LPK\right)}_n$.

Definition 2.

The generalized B-curvature tensor on a Riemannian (or semi-Riemannian) manifold is given by [1]

$$\begin{array}{ccc}\hfill B(Y,Z)X& =& a_{0}R(Y,Z)X+a_1[S(Z,X)Y-S(Y,X)Z\hfill \\ & +& g(Z,X)QY-g(Y,X)QZ]+2a_{2}r[g(Z,X)Y-g(Y,X)Z],\hfill \end{array}$$

(15)

where $a_0,a_1,a_2$ are scalars. Also, see [20,21,22].

From (15), we have

$$\begin{array}{ccc}\hfill B(\zeta ,Y)Z& =& (a_0+a_1(n-1)+2a_{2}r)[g(Y,X)\zeta -\eta \left(X\right)Y]\hfill \\ & +& a_1(S(Y,X)\zeta -\eta \left(X\right)QY)\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill B(Y,\zeta )X& =& (a_0+a_1(n-1)+2a_{2}r)[\eta \left(X\right)Y-g(Y,X)\zeta ]\hfill \\ & +& a_1(\eta \left(X\right)QY-S(Y,X)\zeta ),\hfill \end{array}$$

(17)

$$\begin{array}{ccc}\hfill B(Y,Z)\zeta & =& (a_0+a_1(n-1)+2a_{2}r)[\eta \left(Z\right)Y-\eta \left(Y\right)Z]\hfill \\ & +& a_1(\eta \left(Z\right)QY-\eta \left(Y\right)QZ).\hfill \end{array}$$

(18)

These equations will be applied in subsequent sections.

Definition 3.

A linear connection $\overline{\nabla }$ on M is called a semi-symmetric metric connection if its torsion tensor $\overline{T}$ satisfies

$$\overline{T}(Y,Z)={\overline{\nabla }}_{Y}Z-{\overline{\nabla }}_{Z}Y-[Y,Z],$$

satisfies

$$\overline{T}(Y,Z)=\eta \left(Z\right)Y-\eta \left(Y\right)Z,$$

for all $Y,Z$$\in \chi \left(M\right)$, the torsion tesnor is skew-symmetric and expressible as the tensor product of 1-form and a vector-field. Here, $\chi \left(M\right)$ is the set of differentiable vector fields on M.

The connection $\overline{\nabla }$ is called a semi-symmetric metric connection [23], if ${\overline{\nabla }}_{Y}g=0$.

A relation between the connections $\overline{\nabla }$ and ∇ is given by

$${\overline{\nabla }}_{Y}Z={\nabla }_{Y}Z+\eta \left(Z\right)Y-g(Y,Z)\zeta ,$$

(19)

here, ∇ represents the Levi-Civita connection.

In an ${\left(LPK\right)}_n$-manifold, we have

$${\overline{\nabla }}_X\xi =-2X-2\eta \left(X\right)\xi ,\left({\overline{\nabla }}_X\eta \right)Y=-2g(X,Y)-2\eta \left(X\right)\eta \left(Y\right).$$

(20)

The Riemannian Christoffel curvature tensor with a connection $\overline{\nabla }$ is given by

$$\overline{R}(Y,Z)X={\overline{\nabla }}_Y{\overline{\nabla }}_{Z}X-{\overline{\nabla }}_Z{\overline{\nabla }}_{Y}X-{\overline{\nabla }}_{[Y,Z]}X.$$

(21)

Using relation (19) in (21), we have

$$\begin{array}{ccc}\hfill \overline{R}(Y,Z)X& =& R(Y,Z)X+3g(Z,X)Y-3g(Y,X)Z+2\eta \left(Z\right)\eta \left(X\right)Y\hfill \\ & -& 2\eta \left(Y\right)\eta \left(X\right)Z+2g(Z,X)\eta \left(Y\right)\zeta -2g(Y,X)\eta \left(Z\right)\zeta ,\hfill \end{array}$$

(22)

$$\begin{array}{c}\hfill \overline{R}(\zeta ,Z)X=2g(Z,X)\zeta -2\eta \left(X\right)Z,\end{array}$$

(23)

$$\begin{array}{c}\hfill \overline{R}(Y,\zeta )X=2\eta \left(X\right)Y-2g(Y,X)\zeta ,\end{array}$$

(24)

$$\begin{array}{c}\hfill \overline{R}(Y,Z)\zeta =2\eta \left(Z\right)Y-2\eta \left(Y\right)Z.\end{array}$$

(25)

Let $\left\{e_i\right\},i=1,2,...n$ be an orthonormal basis of the tangent space at any point of the manifold. Then, putting $Y=W=e_i$ and taking summation over i, we have

$$\begin{array}{ccc}\hfill \sum _{i=1}^{n}g(\overline{R}(e_i,Z)X,e_i)& =& \sum _{i=1}^n[g(R(e_i,Z)X,e_i)+3g(Z,X)g(e_i,e_i)-3g(e_i,X)g(Z,e_i)\hfill \\ & +& 2\eta \left(Z\right)\eta \left(X\right)g(e_i,e_i)-2g(e_i,\zeta )\eta \left(X\right)g(Z,e_i)\hfill \\ & +& 2g(Z,X)g(e_i,\zeta )g(\zeta ,e_{i,})-2g(e_i,X)\eta \left(Z\right)g(\zeta ,e_i)].\hfill \end{array}$$

On simplification, the above relation gives

$$\overline{S}(Z,X)=S(Z,X)+(3n-5)g(Z,X)+2(n-2)\eta \left(Z\right)\eta \left(X\right).$$

(26)

By contracting (26) over X and Z, we have

$$\sum _{i=1}^n\overline{S}(e_i,e_i)=\sum _{i=1}^n[S(e_i,e_i)+(3n-5)g(e_i,e_i)+2(n-2)g(e_i,\zeta )g(e_i,\zeta )],$$

which gives

$$\overline{r}=r+3n^2-7n+4.$$

(27)

Putting X= $\zeta$ in (26) and using (12), we have

$$\overline{S}(Z,\zeta )=2(n-1)\eta \left(Z\right).$$

(28)

Also, from (26) it follows that

$$\overline{Q}Z=QZ+(3n-5)Z+2(n-2)\eta \left(Z\right)\zeta .$$

(29)

Here, $\overline{S}$, $\overline{R}$, $\overline{r}$, and $\overline{Q}$ denote the Ricci tensor, the curvature tensor, the scalar curvature, and the Ricci operator with respect to the connection $\overline{\nabla }$ on the ${\left(LPK\right)}_n$-manifold.

Definition 4.

An ${\left(LPK\right)}_n$-manifold is called an η-Einstein manifold, if its $S(\ne 0)$ is of the form

$$S(Y,Z)=ag(Y,Z)+b\eta \left(Y\right)\eta \left(Z\right),$$

here, a and b are scalar functions on $\mathit{M}$. If $b=0$, then the manifold becomes an Einstein manifold [24].

3. Generalized B-Curvature Tensor in (LPK)_n-Manifold with Semi-Symmetric Metric Connection

In this section, we deal with the generalized B-curvature tensor with semi-symmetric metric connection in the ${\left(LPK\right)}_n$-manifold.

• For this purpose, we derive the generalized B-curvature tensor [25,26] with the connection $\overline{\nabla }$ as

  $$\begin{array}{ccc}\hfill \overline{B}(Y,Z)X& =& a_0\overline{R}(Y,Z)X+a_1[\overline{S}(Z,X)Y-\overline{S}(Y,X)Z+g(Z,X)\overline{Q}Y\hfill \\ & & -g(Y,X)\overline{Q}Z]+2a_2\overline{r}[g(Z,X)Y-g(Y,X)Z],\hfill \end{array}$$

Now, using (22), (26), (27), and (29) in the above relation, we obtain

$$\begin{array}{ccc}\hfill \overline{B}(Y,Z)X& =& a_{0}R(Y,Z)X+a_1[S(Z,X)Y-S(Y,X)Z+g(Z,X)QY\hfill \\ & -& g(Y,X)QZ]+(3a_0+2(3n-5)a_1+2a_2(r+3n^2-7n+4))\hfill \\ & & \times [g(Z,X)Y-g(Y,X)Z]+2(a_0+(n-2)a_1)[\eta \left(Z\right)\eta \left(X\right)Y\hfill \\ & -& \eta \left(Y\right)\eta \left(X\right)Z+g(Z,X)\eta \left(Y\right)\zeta -g(Y,X)\eta \left(Z\right)\zeta ].\hfill \end{array}$$

(30)

From (30), we can easily find

$$\begin{array}{ccc}\hfill \overline{B}(\zeta ,Z)X& =& (2a_0+(5n-7)a_1+2a_2(r+3n^2-7n+4))[g(Z,X)\zeta -\eta \left(X\right)Z]\hfill \\ & +& a_1[S(Z,X)\zeta -\eta \left(X\right)QZ],\hfill \end{array}$$

(31)

$$\begin{array}{ccc}\hfill \overline{B}(Y,\zeta )X& =& (2a_0+(5n-7)a_1+2a_2(r+3n^2-7n+4))[\eta \left(X\right)Y-g(Y,X)\zeta ]\hfill \\ & +& a_1[\eta \left(X\right)QY-S(Y,X)\zeta ],\hfill \end{array}$$

(32)

$$\begin{array}{ccc}\hfill \overline{B}(Y,Z)\zeta & =& (2a_0+(5n-7)a_1+2a_2(r+3n^2-7n+4))[\eta \left(Z\right)Y-\eta \left(Y\right)Z]\hfill \\ & +& a_1[\eta \left(Z\right)QY-\eta \left(Y\right)QZ].\hfill \end{array}$$

(33)

4. Certain Flatness Conditions on (LPK)_n-Manifold

This section deals with the study of certain flatness conditions in the framework of the ${\left(LPK\right)}_n$-manifold.

4.1. Generalized $\overline{B}$-Flat (LPK)_n-Manifold

Definition 5.

An ${\left(LPK\right)}_n$-manifold is said to be generalized $\overline{B}$-flat if

$$\overline{B}(Y,Z)X=0,$$

(34)

for any $Y,Z,X$ on ${\left(LPK\right)}_n$.

Let an ${\left(LPK\right)}_n$-manifold be generalized $\overline{B}$-flat, i.e., $\overline{B}(Y,Z)X=0$. Then, (30) takes the form 0

$$\begin{array}{ccc}\hfill a_{0}R(Y,Z)X& +a_1[S(Z,X)Y\hfill & -S(Y,X)Z+g(Z,X)QY\hfill \\ & -g(Y,X)QZ]\hfill & +(3a_0+2(3n-5)a_1+2a_2(r+3n^2-7n+4))\hfill \\ & \hfill \left[g\right(Z,X)Y& -g\left(Y,X\right)Z]+2(a_0+(n-2)a_1)[\eta \left(Z\right)\eta \left(X\right)Y \hfill \\ & & \hfill -\eta \left(Y\right)\eta \left(X\right)Z+g(Z,X)\eta \left(Y\right)\zeta -g(Y,X)\eta \left(Z\right)\zeta ]=0.\end{array}$$

(35)

Applying the inner product on (35) with W, we have

$$\begin{array}{c}{a}_{0}g(R(Y,Z)X,W)+a_1[S(Z,X)g(Y,W)-S(Y,X)g(Z,W)+g(Z,X)S(Y,W)\hfill \\ -g(Y,X)S(Z,W)]+(3a_0+2(3n-5)a_1+2a_2(r+3n^2-7n+4))[g(Z,X)g(Y,W)\\ -g(Y,X)g(Z,W)]+2(a_0+(n-2)a_1)[\eta \left(Z\right)\eta \left(X\right)g(Y,W)-\eta \left(Y\right)\eta \left(X\right)g(Z,W)\\ \hfill +\eta \left(Y\right)\eta \left(W\right)g(Z,X)-\eta \left(Z\right)\eta \left(W\right)g(Y,X)]=0.\end{array}$$

Let $e_i,i=1,2,...n$ be an orthonormal basis of the tangent space at any point of the manifold. Then, putting $Y=W=e_i$ and taking summation over i, we have

$$\begin{array}{c}{a}_0\sum _{i=1}^{n}g(R(e_i,Z)X,e_i)+a_1\sum _{i=1}^n[S(Z,X)g(e_i,e_i)-S(e_i,X)g(Z,e_i)+g(Z,X)S(e_i,e_i)\hfill \\ -g(e_i,X)S(Z,e_i)]+(3a_0+2(3n-5)a_1+2a_2(r+3n^2-7n+4))\sum _{i=1}^n[g(Z,X)g(e_i,e_i)\\ -g(e_i,X)g(Z,e_i)]+2(a_0+(n-2)a_1)\sum _{i=1}^n[\eta \left(Z\right)\eta \left(X\right)g(e_i,e_i)-g(e_i,\zeta )\eta \left(X\right)g(Z,e_i)\\ \hfill +g(e_i,\zeta )g(e_i,\zeta )g(Z,X)-\eta \left(Z\right)g(e_i,\zeta )g(e_i,X)]=0,\end{array}$$

which after some steps calculations gives

$$\begin{array}{c}\hfill S(Z,X)={\lambda }_{1}g(Z,X)+{\lambda }_2\eta \left(Z\right)\eta \left(X\right),\end{array}$$

(36)

where ${\lambda }_1=-{\displaystyle \frac{[(3n-5)a_0+2(3n^2-9n+7)a_1+(a_1+2(n-1)a_2)(r+3n^2-7n+4)]}{a_0+a_1(n-2)}}$ and ${\lambda }_2=-2(n-2)$. Hence, we state the following theorem:

Theorem 1.

An ${\left(LPK\right)}_n$-manifold along with semi-symmetric metric connection satisfying condition generalized $\overline{B}$-flat, then the manifold is an η-Einstein manifold of the form (36), provided $a_0+a_1(n-2)\ne 0$ (i.e., $\overline{B}$ defines a non-degenerate curvature structure).

The $\eta$-Einstein condition essentially decomposes the Ricci tensor into two geometrically meaningful components: one proportional to the metric (isotropic curvature) and another proportional to the tensor product of the contact 1-form $\eta$ (anisotropic curvature) with itself.

4.2. Generalized $\zeta$-$\overline{B}$-Flat ${\left(LPK\right)}_n$-Manifold

In this subsection, we study generalized $\zeta$-$\overline{B}$-flat ${\left(LPK\right)}_n$-manifold, i.e., $\overline{B}(Y,Z)\zeta =0$. Thus, it follows from (30) that

$$\begin{array}{ccc}& & a_{0}R(Y,Z)\zeta +a_1[S(Z,\zeta )Y-S(Y,\zeta )Z+g(Z,\zeta )QY-g(Y,\zeta )QZ]\hfill \\ & & +(3a_0+2(3n-5)a_1+2a_2(r+3n^2-7n+4))[g(Z,\zeta )Y-g(Y,\zeta )Z]\hfill \\ & & +2(a_0+(n-2)a_1)[\eta \left(Z\right)\eta \left(\zeta \right)Y-\eta \left(Y\right)\eta \left(\zeta \right)Z]=0.\hfill \end{array}$$

(37)

Taking the inner product of (37) with W and using (2), (10) and (12), we have

$$\begin{array}{ccc}& & (2a_0+(5n-7)a_1+2a_2(r+3n^2-7n+4))[\eta \left(Z\right)g(Y,W)-\eta \left(Y\right)g(Z,W)]\hfill \\ & & +a_1[\eta \left(Z\right)S(Y,W)-\eta \left(Y\right)S(Z,W)]=0.\hfill \end{array}$$

(38)

Substitute $Z=\zeta$ in (38), we lead to

$$\begin{array}{c}S(Y,W)=-{\displaystyle \frac{[2a_0+(5n-7)a_1+2a_2(r+3n^2-7n+4)]}{a_1}}g(Y,W)\hfill \\ \hfill -{\displaystyle \frac{2[a_0+(3n-4)a_1+a_2(r+3n^2-7n+4)]}{a_1}}\eta \left(Y\right)\eta \left(W\right),a_1\ne 0.\end{array}$$

(39)

Hence, we state the following theorem:

Theorem 2.

An ${\left(LPK\right)}_n$-manifold along with semi-symmetric connection satisfying condtion generelized ζ-$\overline{B}$-flat, then manifold is an η-Einstein manifold of the form (39), provided $a_1\ne 0$.

Next, from (30), we have

$$\begin{array}{ccc}\hfill \overline{B}(Y,Z)\zeta & =& a_{0}R(Y,Z)\zeta +a_1[S(Z,\zeta )Y-S(Y,\zeta )Z+g(Z,\zeta )QY-g(Y,\zeta )QZ]\hfill \\ & & +[3a_0+2(3n-5)a_1+2a_2(r+3n^2-7n+4)-2(a_0+(n-2)a_1)]\hfill \\ & & \times \left[\eta \right(Z)Y-\eta (Y\left)Z\right],\hfill \end{array}$$

which can be written as

$$\begin{array}{c}\hfill \overline{B}(Y,Z)\zeta =B(Y,Z)\zeta +[a_0+(4n-6)a_1+2(3n^2-7n+4)a_2]\eta \left(Y\right)\eta \left(Z\right),\end{array}$$

(40)

where

$$\begin{array}{ccc}\hfill B(Y,Z)\zeta & =& a_{0}R(Y,Z)\zeta +a_1[S(Z,\zeta )Y-S(Y,\zeta )Z+g(Z,\zeta )QY-g(Y,\zeta )QZ]\hfill \\ & +& 2a_{2}r(\eta \left(Z\right)Y-\eta \left(Y\right)Z).\hfill \end{array}$$

If the scalars $a_0,a_1,a_2$ are related by $a_0+2(2n-3)a_1+2(n-1)(3n-4)a_2=0$, then (40) reduces to $\overline{B}(Y,Z)\zeta =B(Y,Z)\zeta$.

Thus, we have the following corollary:

Corollary 1.

An ${\left(LPK\right)}_n$-manifold is generalized ζ-B-flat with respect to the semi-symmetric connection if and only if the manifold is also generelized ζ-B-flat with respect to the Levi-Civita connection, provided $a_0+2(2n-3)a_1+2(n-1)(3n-4)a_2=0$.

4.3. $\phi$-Generalized $\overline{B}$-Flat ${\left(LPK\right)}_n$-Manifold

In this subsection, we study $\phi$-generalized $\overline{B}$-flat ${\left(LPK\right)}_n$-manifold, i.e., $g(\overline{B}(\phi Y,\phi Z)\phi X,\phi W)=0$. Thus, in account of (30), we have

$$\begin{array}{ccc}& & a_{0}g(R(\phi Y,\phi Z)\phi X,\phi W)+a_1[S(\phi Z,\phi X)g(\phi Y,\phi W)-S(\phi Y,\phi X)g(\phi Z,\phi W)\hfill \\ & & +g(\phi Z,\phi X)S(\phi Y,\phi W)-g(\phi Y,\phi X)S(\phi Z,\phi W)]\hfill \\ & & +(3a_0+2(3n-5)a_1+2a_2(r+3n^2-7n+4))[g(\phi Z,\phi X)g(\phi Y,\phi W)\hfill \\ & & -g(\phi Y,\phi X)g(\phi Z,\phi W)]\hfill \\ & & +2(a_0+(n-2)a_1)[\eta \left(\phi Z\right)\eta \left(\phi X\right)g(\phi Y,\phi W)-\eta \left(\phi Y\right)\eta \left(\phi X\right)g(\phi Z,\phi W)\hfill \\ & & +\eta \left(\phi Y\right)\eta \left(\phi W\right)g(\phi Z,\phi X)-\eta \left(\phi Z\right)\eta \left(\phi W\right)g(\phi Y,\phi X)]=0.\hfill \end{array}$$

In view of (1) and (14), the above expression takes the form

$$\begin{array}{ccc}& & a_{0}g(R(\phi Y,\phi Z)\phi X,\phi W)+a_1[S(Z,X)g(Y,W)-S(Y,X)g(Z,W)\hfill \\ & & +g(Z,X)S(Y,W)-g(Y,X)S(Z,W)+S(Z,X)\eta \left(Y\right)\eta \left(W\right)-S(Y,X)\eta \left(Z\right)\eta \left(W\right)\hfill \\ & & +S(Y,W)\eta \left(Z\right)\eta \left(X\right)-S(Z,W)\eta \left(Y\right)\eta \left(X\right)+(n-1)g(Y,W)\eta \left(Z\right)\eta \left(X\right)\hfill \\ & & -(n-1)g(Z,W)\eta \left(Y\right)\eta \left(X\right)+(n-1)g(Z,X)\eta \left(Y\right)\eta \left(W\right)-(n-1)g(Y,X)\eta \left(Z\right)\eta \left(W\right)]\hfill \\ & & +(3a_0+2(3n-5)a_1+2a_2(r+3n^2-7n+4))[g(Z,X)g(Y,W)-g(Y,X)g(Z,W)\hfill \\ & & +g(Y,W)\eta \left(Z\right)\eta \left(X\right)+g(Z,X)\eta \left(Y\right)\eta \left(W\right)-g(Z,W)\eta \left(Y\right)\eta \left(X\right)-g(Y,X)\eta \left(Z\right)\eta \left(W\right)]=0.\hfill \end{array}$$

Let $e_i,i=1,2,...n$ be an orthonormal basis of the tangent space at any point of the manifold. Then, putting $Y=W=e_i$ and taking summation over i, we have

$$\begin{array}{c}\hfill a_0\sum _{i=1}^{n}g(R(\phi e_i,\phi Z)\phi X,\phi e_i)+a_1\sum _{i=1}^n[S(Z,X)g(e_i,e_i)-S(e_i,X)g(Z,e_i)\\ \hfill +g(Z,X)S(e_i,e_i)-g(e_i,X)S(Z,e_i)+S(Z,X)g(e_i,\zeta )g(e_i,\zeta )\\ \hfill +S(e_i,e_i)\eta \left(Z\right)\eta \left(X\right)-S(Z,e_i)g(e_i,\zeta )\eta \left(X\right)+(n-1)g(e_i,e_i)\eta \left(Z\right)\eta \left(X\right)\\ \hfill -(n-1)g(Z,e_i)g(e_i,\zeta )\eta \left(X\right)+(n-1)g(Z,X)g(e_i,\zeta )g(e_i,\zeta )\\ \hfill -(n-1)g(e_i,X)\eta \left(Z\right)g(e_i,\zeta )]+(3a_0+2(3n-5)a_1+2a_2(r+3n^2-7n+4))\\ \hfill \sum _{i=1}^n[g(Z,X)g(e_i,e_i)-g(e_i,X)g(Z,e_i)+g(e_i,e_i)\eta \left(Z\right)\eta \left(X\right)\\ \hfill +g(Z,X)g(e_i,\zeta )g(e_i,\zeta )-g(Z,e_i)g(e_i,\zeta )\eta \left(X\right)-g(e_i,X)\eta \left(Z\right)g(e_i,\zeta )]=0.\end{array}$$

This gives

$$\begin{array}{ccc}\hfill (a_0+(n-3)a_1)S(Z,X)& =& -[(3n-7)(a_0+(2n-3)a_1)+2(n-2)(3n^2-7n+4)a_2\hfill \\ & +& (a_1+2(n-2)a_2)r]g(Z,X)-[4(n-2)a_0+(7n^2-27n+24)a_1\hfill \\ & +& 2(n-2)(3n^2-7n+4)a_2+(a_1+2(n-2)a_2)r]\eta \left(Z\right)\eta \left(X\right),\hfill \end{array}$$

which is of the form

$$S(Z,X)={\lambda }_{1}g(Z,X)+{\lambda }_2\eta \left(Z\right)\eta \left(X\right),$$

(41)

where

$$\begin{array}{c}\hfill {\lambda }_1=-{\displaystyle \frac{(3n-7)(a_0+(2n-3)a_1)+2(n-2)(3n^2-7n+4)a_2+(a_1+2(n-2)a_2)r}{a_0+(n-3)a_1}},\end{array}$$

and

$$\begin{array}{c}\hfill {\lambda }_2=-{\displaystyle \frac{4(n-2)a_0+(7n^2-27n+24)a_1+2(n-2)(3n^2-7n+4)a_2+(a_1+2(n-2)a_2)r}{a_0+(n-3)a_1}}.\end{array}$$

Thus, we state the following result:

Theorem 3.

An ${\left(LPK\right)}_n$-manifold along with a semi-symmetric metric connection is φ-generalized $\overline{B}$-flat, then the manifold is an η-Einstein manifold of the form (41), provided $a_0+a_1(n-3)\ne 0$.

Definition 6.

An (LP-K)_n-manifold is called φ-Ricci-symmetric with a semi-symmetric metric connection $\overline{\nabla }$ if

$${\phi }^2\left(\left({\overline{\nabla }}_{X}Q̧\right)\left(Y\right)\right)=0,$$

(42)

for any $X,Y$ on (LP-K)_n-manifold [27]. In case, $X,Y$ are orthogonal to ζ, then (LP-K)_n is named locally φ-Ricci-symmetric.

• From (41), it follows that

$$QZ={\lambda }_{1}Z+{\lambda }_2\eta \left(Z\right)\xi .$$

(43)

Differentiating covariantly (43) along Y w. r. t. $\overline{\nabla }$, we have

$$\begin{array}{ccc}\hfill \left({\overline{\nabla }}_{Y}Q\right)Z+Q\left({\overline{\nabla }}_{Y}Z\right)& =& {\lambda }_1{\overline{\nabla }}_{Y}Z-{\displaystyle \frac{(a_1+2(n-2)a_2)dr\left(Y\right)}{a_0+(n-1)a_1}}(Z+\eta \left(Z\right)\xi )\hfill \\ & +& {\lambda }_2[\left({\overline{\nabla }}_Y\eta \right)\left(Z\right)\xi +\eta \left({\overline{\nabla }}_{Y}Z\right)\xi +\eta \left(Z\right){\overline{\nabla }}_Y\xi ].\hfill \end{array}$$

(44)

By virtue of (43), (44) takes the form

$$\begin{array}{ccc}\hfill \left({\overline{\nabla }}_{Y}Q\right)Z& =& -{\displaystyle \frac{(a_1+2(n-2)a_2)dr\left(Y\right)}{a_0+(n-1)a_1}}(Z+\eta \left(Z\right)\xi )\hfill \\ & & +{\lambda }_2[\left({\overline{\nabla }}_Y\eta \right)\left(Z\right)\xi +\eta \left(Z\right){\overline{\nabla }}_Y\xi ].\hfill \end{array}$$

(45)

By using (20) in (45), we have

$$\begin{array}{ccc}\hfill \left({\overline{\nabla }}_{Y}Q\right)Z& =& -{\displaystyle \frac{(a_1+2(n-2)a_2)dr\left(Y\right)}{a_0+(n-1)a_1}}(Z+\eta \left(Z\right)\xi )\hfill \\ & & -2{\lambda }_2[g(Y,Z)\xi +\eta \left(Z\right)Y+2\eta \left(Y\right)\eta \left(Z\right)\xi ].\hfill \end{array}$$

(46)

By operating ${\phi }^2$ on both sides of (46) and using (1), (3), we arrive at

$${\phi }^2\left(\left({\overline{\nabla }}_{Y}Q\right)Z\right)=-{\displaystyle \frac{(a_1+2(n-2)a_2)dr\left(Y\right)}{a_0+(n-3)a_1}}[Z+\eta \left(Z\right)\xi ]-2{\lambda }_2\eta \left(Z\right)[Y+\eta \left(Y\right)\xi ].$$

(47)

If Z is orthogonal to $\xi$, then (47) provides

$${\phi }^2\left(\left({\overline{\nabla }}_{Y}Q\right)Z\right)=-{\displaystyle \frac{(a_1+2(n-2)a_2)dr\left(Y\right)}{a_0+(n-3)a_1}}Z.$$

(48)

If r is constant, then from (48), it follows that

$${\phi }^2\left(\left({\overline{\nabla }}_{Y}Q\right)Z\right)=0.$$

(49)

Thus, we have the following corollary:

Corollary 2.

A φ-generalized $\overline{B}$-flat ${\left(LPK\right)}_n$-manifold of a constant scalar curvature with a semi-symmetric metric connection $\overline{\nabla }$ is locally φ-Ricci-symmetric.

5. φ-Generalized $\overline{\mathit{B}}$-Semi-Symmetric (LPK)_n-Manifold

Definition 7.

An ${\left(LPK\right)}_n$-manifold is said to be φ-generalized $\overline{B}$-semi-symmetric if

$$\overline{B}(Y,Z)·\phi =0,$$

(50)

for any $Y,Z$ on M. See [28].

In this section, we study a $\phi$-generalized $\overline{B}$-semi-symmetric (LPK)_n-manifold, i.e., $\overline{B}(Y,Z)·\phi =0$. This implies that

$$(\overline{B}(Y,Z)·\phi )X=\overline{B}(Y,Z)\phi X-\phi \overline{B}(Y,Z)X=0.$$

(51)

Making use of (30) in (51), we have

$$\begin{array}{c}{a}_{0}R(Y,Z)\phi X+a_1[S(Z,\phi X)Y-S(Y,\phi X)Z+g(Z,\phi X)QY-g(Y,\phi X)QZ]\hfill \\ +(3a_0+2(3n-5)a_1+2a_2(r+3n^2-7n+4))[g(Z,\phi X)Y-g(Y,\phi X)Z]+2(a_0+(n-2)a_1)\\ \left[g\right(Z,\phi X)\eta \left(Y\right)\zeta -g(Y,\phi X)\eta \left(Z\right)\zeta ]-a_0\phi \left(R(Y,Z)X\right)-a_{1}S(Z,X)\phi Y+a_{1}S(Y,X)\phi Z\\ -a_{1}g(Z,X)\phi \left(QY\right)+a_{1}g(Y,X)\phi \left(QZ\right)-(3a_0+2(3n-5)a_1+2a_2(r+3n^2-7n+4))\\ \hfill \left[g\right(Z,\phi Y-g(Y,X)\phi Z]-2(a_0+(n-2)a_1)[\eta \left(Z\right)\eta \left(X\right)\phi Y-\eta \left(Y\right)\eta \left(X\right)\phi Z]=0.\end{array}$$

(52)

Putting $Y=\zeta$ in (52) and making use of (2), (3), (9), and (12), we have

$$\begin{array}{ccc}& & a_{1}S(Z,\phi X)\zeta +[2a_0+(5n-7)a_1+2a_2(r+3n^2-7n+4)]\times \hfill \\ & & [g(Z,\phi X)\zeta +\eta \left(X\right)\phi Z]+a_1\eta \left(\mathcal{X}\right)\phi \left(QZ\right)=0.\hfill \end{array}$$

(53)

Interchanging X by $\phi X$ in (53) and using (1), (3), we obtain

$$\begin{array}{ccc}\hfill S(Z,X)& =& -{\displaystyle \frac{[2a_0+(5n-7)a_1+2a_2(r+3n^2-7n+4)]}{a_1}}g(Z,X)\hfill \\ & & -{\displaystyle \frac{2[a_0+(3n-4)a_1+a_2(r+3n^2-7n+4)]}{a_1}}\eta \left(Z\right)\eta \left(X\right),a_1\ne 0.\hfill \end{array}$$

(54)

Thus, we have the following result:

Theorem 4.

An ${\left(LPK\right)}_n$-manifold along with a semi-symmetric metric connection satisfying condition φ-generalized $\overline{B}$-semi-symmetric, then the manifold is an η-Einstein manifold of the form (54).

6. (LPK)_n-Manifold Satisfying the Curvature Condition $\overline{\mathit{B}}.\mathit{Q}=\mathbf{0}$

In this section, we study the ${\left(LPK\right)}_n$-manifold satisfying ($\overline{B}(Y,Z).Q)X=0$. This implies

$$\overline{B}(Y,Z)QX-Q\left(\overline{B}(Y,Z)X\right)=0.$$

(55)

Put $Z=\zeta$ in (55), we have

$$\overline{B}(Y,\zeta )QX-Q\left(\overline{B}(Y,\zeta )X\right)=0,$$

which in view of (32) turns to

$$\begin{array}{c}(2a_0+(5n-7)a_1+2a_2(r+3n^2-7n+4))(n-1)[\eta \left(X\right)Y-g(Y,X)\zeta ]\hfill \\ -2(a_0+(2n-3)a_1+a_2(r+3n^2-7n+4))[\eta \left(X\right)QY+S(Y,X)\zeta ]\\ \hfill -a_{1}S(Y,QX)\zeta -a_1\eta \left(X\right)Q\left(QY\right)=0.\end{array}$$

(56)

Performing the inner product of (56) with $\zeta$, we find

$$\begin{array}{c}{a}_{1}S(Y,QX)+2(a_0+(2n-3)a_1+a_2(r+3n^2-7n+4))S(Y,X)\hfill \\ \hfill -(2a_0+(5n-7)a_1+2a_2(r+3n^2-7n+4))(n-1)g(Y,X)=0.\end{array}$$

(57)

Contracting (57) over Y and X, we obtain

$$\begin{array}{c}{a}_1\sum _{i=1}^{n}S(e_i,Q{e}_i)+2(a_0+(2n-3)a_1+a_2(r+3n^2-7n+4))\sum _{i=1}^{n}S(e_i,e_i)\hfill \\ \hfill -(2a_0+(5n-7)a_1+2a_2(r+3n^2-7n+4))(n-1)\sum _{i=1}^{n}g(e_i,e_i)=0,\end{array}$$

$$\begin{array}{c}2a_2{r}^2+2(a_0+(2n-3)a_1+2{(n-1)}^2{a}_2)r\hfill \\ \hfill -n(n-1)(2a_0+(5n-7)a_1+2(3n^2-7n+4)a_2)+a_{1}tr{Q}^2=0.\end{array}$$

(58)

The relation (58) is quadratic in r. Let $a=2a_2$, $b=2(a_0+(2n-3)a_1+2{(n-1)}^2{a}_2)$ and $c=-n(n-1)(2a_0+(5n-7)a_1+2(3n^2-7n+4)a_2)+a_{1}tr{Q}^2$. After applying the well-known formula $r={\displaystyle \frac{-b\pm \sqrt{b^2-4ac}}{2a}}$, it yields the two values of a scalar curvature. Thus, we have the following result:

Theorem 5.

An ${\left(LPK\right)}_n$-manifold along with a semi-symmetric metric connection satisfying $\overline{B}$.Q = 0 gives the scalar curvature in the quadratic form described in (58).

7. (LPK)_n-Manifold Satisfying the Curvature Condition $\overline{\mathit{B}}.\overline{\mathit{Q}}=\mathbf{0}$

In this section, we study the ${\left(LPK\right)}_n$-manifold satisfying ($\overline{B}(Y,Z).\overline{Q})X=0$. This implies

$$\overline{B}(Y,Z)\overline{Q}X-\overline{Q}\left(\overline{B}(Y,Z)X\right)=0.$$

(59)

Put $Z=\zeta$ in (59), we have

$$\overline{B}(Y,\zeta )\overline{Q}X-\overline{Q}\left(\overline{B}(Y,\zeta )X\right)=0.$$

(60)

In view of (32), (60) takes the form

$$\begin{array}{c}-2(a_0+(3n-5)a_1+a_2(r+3n^2-7n+4))[S(Y,X)\zeta +\eta \left(X\right)QY]\hfill \\ -(n-3)(2a_0+(5n-7)a_1+2a_2(r+3n^2-7n+4))[\eta \left(X\right)Y+g(Y,X)\zeta ]\\ -8(n-2)(a_0+(3n-4)a_1+a_2(r+3n^2-7n+4))\eta \left(X\right)\eta \left(Y\right)\zeta \\ \hfill -a_{1}S(Y,QX)\zeta -a_1\eta \left({\mathcal{X}}_1\right)Q\left(QY\right)=0,\end{array}$$

which by taking the inner product with $\zeta$ becomes

$$\begin{array}{c}{a}_{1}S(Y,QX)+2(a_0+(3n-5)a_1+(r+3n^2-7n+4)a_2)S(Y,X)\hfill \\ +(n-3)(2a_0+(5n-7)a_1+2(r+3n^2-7n+4)a_2)g(Y,X)\\ \hfill +4(n-2)(a_0+(3n-4)a_1+(r+3n^2-7n+4)a_2)\eta \left(X\right)\eta \left(Y\right)=0.\end{array}$$

(61)

Contracting (61) over Y and X, we obtain

$$\begin{array}{c}{a}_1\sum _{i=1}^{n}S(e_i,Q{e}_i)+2(a_0+(3n-5)a_1+(r+3n^2-7n+4)a_2)\sum _{i=1}^{n}S(e_i,e_i)\hfill \\ +(n-3)(2a_0+(5n-7)a_1+2(r+3n^2-7n+4)a_2)\sum _{i=1}^{n}g(e_i,e_i)\\ \hfill +4(n-2)(a_0+(3n-4)a_1+(r+3n^2-7n+4)a_2)\sum _{i=1}^{n}g(e_i,\zeta )g(e_i,\zeta )=0,\end{array}$$

$$\begin{array}{c}{a}_{1}tr{Q}^2+2(a_0+(3n-5)a_1+(r+3n^2-7n+4)a_2)r\hfill \\ +n(n-3)(2a_0+(5n-7)a_1+2(r+3n^2-7n+4)a_2)\\ \hfill -4(n-2)(a_0+(3n-4)a_1+(r+3n^2-7n+4)a_2)=0.\end{array}$$

We arrange the above relation as follows:

$$\begin{array}{c}2a_2{r}^2+2(a_0+(3n-5)a_1+4(n-1)(n-2)a_2)r+(5n^3-34n^2+61n-32)a_1\hfill \\ \hfill +2(n-1)(n-4)(a_0+(3n^2-7n+4))a_2+a_{1}tr{Q}^2=0.\end{array}$$

(62)

The relation (62) is quadratic in r. Let $a=2a_2$, $b=2(a_0+(2n-3)a_1+2{(n-1)}^2{a}_2)$, and $c=-n(n-1)(2a_0+(5n-7)a_1+2(3n^2-7n+4)a_2)+a_{1}tr{Q}^2$. After applying the well-known formula $r={\displaystyle \frac{-b\pm \sqrt{b^2-4ac}}{2a}}$, it yields the two values of the scalar curvature. Thus, we have the following result:

Theorem 6.

An ${\left(LPK\right)}_n$-manifold along with semi-symmetric metric connection satisfying $\overline{B}.\overline{Q}$ = 0 gives the scalar curvature in quadratic form as described in Equation (62).

8. Example

We consider the 3-dimensional manifold $M=\{(x_1,x_2,z)\in {\mathbb{R}}^3:z>0\}$, where ($x_1,x_2,z$) are the standard coordinates in ${\mathbb{R}}^3$. Let $e_1$, $e_2$, and $e_3$ be the vector fields on M defined by

$$e_1=z{\displaystyle \frac{\partial }{\partial x_1}},e_2=z{\displaystyle \frac{\partial }{\partial x_2}},e_3=z{\displaystyle \frac{\partial }{\partial z}}=\zeta ,$$

which are linearly independent at each point p of M. Let g be the Lorentzian metric defined by

$$g(e_i,e_i)=1,\mathrm{for}i=1,2\mathrm{and}g(e_3,e_3)=-1,$$

$$g(e_i,e_j)=0,\mathrm{for}i\ne j,1\le i,j\le 3.$$

Let $\eta$ be the 1-form defined by $\eta \left(X\right)$ = $g(X,e_3)$ = $g(X,\zeta )$ for all $X\in \chi \left(M\right)$, and let $\phi$ be the (1, 1)-tensor field defined by

$$\phi e_1=-e_2,\phi e_2=-e_1,\phi e_3=0.$$

(63)

By applying linearity of $\phi$ and g, we have

$$\eta \left(\zeta \right)=g(\zeta ,\zeta )=-1,{\phi }^{2}X=X+\eta \left(X\right)\zeta ,$$

$$\mathrm{and},g(\varphi X,\varphi Y)=g(X,Y)+\eta \left(X\right)\eta \left(Y\right),$$

for all $X,Y\in \chi \left(M\right)$. Thus, for $e_3=\zeta$, the structure ($\phi ,\zeta ,\eta ,g$) defines a Lorentzian almost para-contact metric structure on M. Then, we have

$$[e_i,e_j]=0,\mathrm{if}i\ne j,\mathrm{and}i,j=1,2,$$

(64)

$$[e_i,e_3]=-e_i,\mathrm{for}i=1,2.$$

(65)

By using the well-known Koszul’s formula, we find

$${\nabla }_{e_1}{e}_1=-e_3,{\nabla }_{e_1}{e}_2=0,{\nabla }_{e_1}{e}_3=-e_1,$$

$${\nabla }_{e_2}{e}_1=0,{\nabla }_{e_2}{e}_2=-e_3,{\nabla }_{e_2}{e}_3=-e_2,$$

$${\nabla }_{e_3}{e}_1=0,{\nabla }_{e_3}{e}_2=0,{\nabla }_{e_3}{e}_3=0.$$

Now, let

$$X=\sum _{i=1}^3{X}^i{e}_i=X^1{e}_1+X^2{e}_2+X^3{e}_3,$$

$$Y=\sum _{j=1}^3{Y}^j{e}_j=Y^1{e}_1+Y^2{e}_2+Y^3{e}_3,$$

$$Z=\sum _{k=1}^3{Z}^k{e}_k=Z^1{e}_1+Z^2{e}_2+Z^3{e}_3,$$

for all $X,Y,Z\in \chi \left(M\right)$. Also, one can easily verify that

$${\nabla }_X\xi =-X-\eta \left(X\right)\xi \mathrm{and}\left({\nabla }_X\phi \right)Y=-g(\phi X,Y)\zeta -\eta \left(Y\right)\phi X.$$

Therefore, the manifold is a Lorentzian para-Kenmotsu manifold.

• From the above results, we can easily obtain the non-vanishing components of the curvature tensor as follows:

$$R(e_1,e_2)e_1=-e_2,R(e_1,e_2)e_2=e_1,R(e_1,e_3)e_1=-e_3,$$

$$R(e_1,e_3)e_3=e_1,R(e_2,e_3)e_2=-e_3,R(e_2,e_3)e_3=-e_2,$$

from which it is clear that

$$R(X,Y)Z=g(Y,Z)X-g(X,Z)Y.$$

(66)

Thus, the manifold is of a constant curvature. Also, we calculate the Ricci tensors as follows:

$$S(e_1,e_1)=S(e_2,e_2)=2,S(e_3,e_3)=-2.$$

Hence, we find

$$r=S(e_1,e_1)+S(e_2,e_2)-S(e_3,e_3)=6.$$

By contracting (22), it follows that

$$S(Y,Z)=2g(Y,Z),r=6,$$

(67)

which are same as the values of Ricci tensor and scalar curvature.

• The relation between the semi-symmetric metric connection $\overline{\nabla }$ and the Levi-Civita connection ∇ has been given by:

$${\overline{\nabla }}_{X}Y={\nabla }_{X}Y+\eta \left(Y\right)X-g(X,Y)\zeta .$$

(68)

Using this relation, we find:

$${\overline{\nabla }}_{e_1}{e}_1=-2e_3,{\overline{\nabla }}_{e_1}{e}_2=0,{\overline{\nabla }}_{e_1}{e}_3=0,$$

$${\overline{\nabla }}_{e_2}{e}_1=0,{\overline{\nabla }}_{e_2}{e}_2=-2e_3,{\overline{\nabla }}_{e_2}{e}_3=0,$$

$${\overline{\nabla }}_{e_3}{e}_1=0,{\overline{\nabla }}_{e_3}{e}_2=0,{\overline{\nabla }}_{e_3}{e}_3=-2e_3,$$

Also, one can easily verify that

$${\overline{\nabla }}_X\xi =-2X-2\eta \left(X\right)\xi \mathrm{and}\left({\overline{\nabla }}_X\phi \right)Y=-2g(\phi X,Y)\zeta -2\eta \left(Y\right)\varphi X.$$

Therefore, this shows that the manifold is a Lorentzian para-Kenmotsu manifold along a semi-symmetric metric connection.

• From the above results, we obtain the non-vanishing components of the curvature tensor as:

$$\overline{R}(e_1,e_2)e_1=-4e_2,\overline{R}(e_1,e_2)e_2=4e_1,\overline{R}(e_1,e_3)e_1=-2e_3,$$

$$\overline{R}(e_1,e_3)e_3=-2e_1,\overline{R}(e_2,e_3)e_2=-2e_3,\overline{R}(e_2,e_3)e_3=-2e_2,$$

from which it is clear that

$$\begin{array}{ccc}\hfill \overline{R}(X,Y)Z& =& R(X,Y)Z+3g(Y,Z)X-3g(X,Z)Y+2\eta \left(Y\right)\eta \left(Z\right)X\hfill \\ & & -2\eta \left(X\right)\eta \left(Z\right)Y+2g(Y,Z)\eta \left(X\right)\zeta -2g(X,Z)\eta \left(Y\right)\zeta .\hfill \end{array}$$

(69)

Also, we calculate the Ricci tensors along the semi-symmetric metric connection, as follows:

$$\overline{S}(e_1,e_1)=\overline{S}(e_2,e_2)=6,\overline{S}(e_3,e_3)=-4.$$

Hence, we find

$$\overline{r}=\overline{S}(e_1,e_1)+\overline{S}(e_2,e_2)-\overline{S}(e_3,e_3)=16.$$

By contracting (69), it follows that

$$\overline{S}(X,Y)=6g(X,Y)+2\eta \left(X\right)\eta \left(Y\right).$$

(70)

By using (26) in (70), we find $S(X,Y)=2g(X,Y)$. Hence, $QX=2X$, which implies that ${\phi }^2\left(\left({\overline{\nabla }}_{W}Q\right)X\right)=0$. Thus, we observe that the manifold under consideration is $\phi$-Ricci symmetric with respect to $\overline{\nabla }$.