Codazzi Tensors and the Quasi-Statistical Structure Associated with Affine Connections on Three-Dimensional Lorentzian Lie Groups

Abstract

In this paper, we classify three-dimensional Lorentzian Lie groups on which Ricci tensors associated with Bott connections, canonical connections and Kobayashi–Nomizu connections are Codazzi tensors associated with these connections. We also classify three-dimensional Lorentzian Lie groups with the quasi-statistical structure associated with Bott connections, canonical connections and Kobayashi–Nomizu connections.

1. Introduction

In [1], Andrzej and Shen studied some geometric and topological consequences of the existence of a non-trivial Codazzi tensor on a Riemannian manifold. They also introduced Codazzi tensors associated with any linear connections. Bourguignon obtained the results of this type and gave the proof of the existence of such a strong constraint of the tensor on the curvature operator in [2]. In [3], Dajczer and Tojeiro found the correspondence between the Ribaucour transformation of a submanifold and Codazzi tensor exchanged with its second fundamental form. In [4], the authors defined a Codazzi tensor on conformally symmetric space, and characterized the Einstein manifold and constant sectional curvature manifold by inequalities between certain functions of this tensor.

In [5], Merton and Gabe discussed the classification of Codazzi tensors with exactly two eigenfunctions on a Riemannian manifold of three or more dimensions. In [6], Blaga and Nannicini considered the statistical structure on a smooth manifold with a torsion-free affine connection, and they also gave the definition of the quasi-statistical structure, which is the generalization of the statistical structure. Wang gave algebraic Ricci solitons and affine Ricci solitons associated with canonical connections and Kobayashi–Nomizu connections on three-dimensional Lorentzian Lie groups, respectively in [7,8]. In [2,9], the authors gave the definition of the Bott connection.

In [1], Andrzej and Shen showed that the existence of nontrivial Codazzi tensors on Riemannian manifolds induces some geometric and topological results. We believe that our study of affine Codazzi tensors will induce some geometric and topological results in affine geometry, and we are prepared to study this question in the future. The classification of Class B manifolds: $(({\nabla }_X^L\rho )(Y,Z)=({\nabla }_Y^L\rho )(X,Z))$ in three-dimensional Lie groups is given by Calvaruso in [10]. Our research shows that the Ricci tensors of Bott connections, canonical connections and Kobayashi–Nomizu connections are Codazzi tensors can be used as an affine parallel to the above results in [10]. Class B manifolds are widely used in differential geometry classifications, and one can find more examples in [11]. We believe our results are useful for the classification of affine Lie groups. Blaga and Nannicini considered the statistical structure on a smooth manifold with a torsion-free affine connection, and they also gave the definition of the quasi-statistical structure, which is the generalization of the statistical structure in [6]. Blaga and Nannicini proved that any quasi-statistical structure on M, defined by a symmetric or skew-symmetric tensor, induces the generalized quasi-statistical structures and the generalized dual quasi-statistical connection on $TM⨁T^{*}M$ in [12]. In this paper, we classify three-dimensional Lorentzian Lie groups on which Ricci tensors associated with Bott connections, canonical connections and Kobayashi–Nomizu connections are Codazzi tensors associated with these connections. We also classify three-dimensional Lorentzian Lie groups with the quasi-statistical structure associated with Bott connections, canonical connections and Kobayashi–Nomizu connections.

In Section 2, we classify three-dimensional Lorentzian Lie groups on which Ricci tensors and Kobayashi–Nomizu connections are Codazzi tensors associated with Bott connections. In Section 3, we classify three-dimensional Lorentzian Lie groups with the quasi-statistical structure associated with Bott connections. In Section 4, we classify three-dimensional Lorentzian Lie groups on which Ricci tensors associated with canonical connections and Kobayashi–Nomizu connections are Codazzi tensors associated with canonical connections and Kobayashi–Nomizu connections. In Section 5, we classify three-dimensional Lorentzian Lie groups with the quasi-statistical structure associated with canonical connections and Kobayashi–Nomizu connections.

2. Codazzi Tensors Associated with Bott Connections on Three-Dimensional Lorentzian Lie Groups

Let ${\{G_i\}}_{i=1,\dots ,7}$, denote the connected, simply connected three-dimensional Lie group equipped with a left-invariant Lorentzian metric g and having Lie algebra ${\{{\mathfrak{g}}_i\}}_{i=1,\dots ,7}$ and let ${\nabla }^L$ be the Levi–Civita connection of $G_i$. Next, we recall the definition of the Bott connection ${\nabla }^B$. Let M be a smooth manifold, and let $TM=span\{{\tilde{e}}_1,{\tilde{e}}_2,{\tilde{e}}_3\}$, then take the distribution $D=span\{{\tilde{e}}_1,{\tilde{e}}_2\}$ and $D^{\perp }=span\{{\tilde{e}}_3\}$.

The definition of the Bott connection ${\nabla }^B$ is given as follows: (see [2,9])

$$\begin{array}{c}\hfill {\nabla }_X^{B}Y=\left\{\begin{array}{cc}{\pi }_D({\nabla }_X^{L}Y),\hfill & X,Y\in {\mathsf{\Gamma }}^{\infty }(D)\hfill \\ {\pi }_D([X,Y]),\hfill & X\in {\mathsf{\Gamma }}^{\infty }(D^{\perp }),Y\in {\mathsf{\Gamma }}^{\infty }(D)\hfill \\ {\pi }_{D^{\perp }}([X,Y]),\hfill & X\in {\mathsf{\Gamma }}^{\infty }(D),Y\in {\mathsf{\Gamma }}^{\infty }(D^{\perp })\hfill \\ {\pi }_{D^{\perp }}({\nabla }_X^{L}Y),\hfill & X,Y\in {\mathsf{\Gamma }}^{\infty }(D^{\perp }),\hfill \end{array}\right.\end{array}$$

(1)

where ${\pi }_D$(resp. ${\pi }_D^{\perp }$) is the projection on D (resp. $D^{\perp }$).

We define

$$R^B(X,Y)Z={\nabla }_X^B{\nabla }_Y^{B}Z-{\nabla }_Y^B{\nabla }_X^{B}Z-{\nabla }_{[X,Y]}^{B}Z.$$

(2)

The Ricci tensor of $(G_i,g)$ associated with the Bott connection ${\nabla }^B$ is defined by

$${\rho }^B(X,Y)=-g(R^B(X,{\tilde{e}}_1)Y,{\tilde{e}}_1)-g(R^B(X,{\tilde{e}}_2)Y,{\tilde{e}}_2)+g(R^B(X,{\tilde{e}}_3)Y,{\tilde{e}}_3),$$

(3)

where ${\tilde{e}}_1,{\tilde{e}}_2,{\tilde{e}}_3$ is a pseudo-orthonormal basis, with ${\tilde{e}}_3$ timelike.

Let

$${\tilde{\rho }}^B(X,Y)=\frac{{\rho }^B(X,Y)+{\rho }^B(Y,X)}{2}.$$

(4)

Let $\omega$ be a (0,2) tensor field, then we define:

$$({\nabla }_X\omega )(Y,Z):=X[\omega (Y,Z)]-\omega ({\nabla }_{X}Y,Z)-\omega (Y,{\nabla }_{X}Z),$$

(5)

for arbitrary vector fields $X,Y,Z$.

Definition 1.

([1], p. 17) Let M be a smooth manifold endowed with a linear connection ∇, and the tensor fields ω is called a Codazzi tensor on $(M,\nabla )$, if it satisfies

$$f(X,Y,Z)=({\nabla }_X\omega )(Y,Z)-({\nabla }_Y\omega )(X,Z)=0,$$

(6)

where f is $C^{\infty }(M)$-linear for $X,Y,Z$.

Proposition 1.

The tensor ω is called a Codazzi tensor on$(M,\nabla )$if and only if

$$f(X,Y,Z)=-f(Y,X,Z)$$

(7)

Then, we have that $\omega$ is a Codazzi tensor on ${(G_i,\nabla )}_{i=1,2\dots ,7}$ if and only if the following three equations hold:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}f({\tilde{e}}_1,{\tilde{e}}_2,{\tilde{e}}_j)=0\hfill \\ f({\tilde{e}}_1,{\tilde{e}}_3,{\tilde{e}}_j)=0\hfill \\ f({\tilde{e}}_2,{\tilde{e}}_3,{\tilde{e}}_j)=0,\hfill \end{array}\right.\end{array}$$

(8)

where $1\le j\le 3$.

2.1. Codazzi Tensors of $G_1$

By [13], we have the following Lie algebra of $G_1$ which satisfies

$$[{\tilde{e}}_1,{\tilde{e}}_2]=\alpha {\tilde{e}}_1-\beta {\tilde{e}}_3,[{\tilde{e}}_1,{\tilde{e}}_3]=-\alpha {\tilde{e}}_1-\beta {\tilde{e}}_2,[{\tilde{e}}_2,{\tilde{e}}_3]=\beta {\tilde{e}}_1+\alpha {\tilde{e}}_2+\alpha {\tilde{e}}_3,\alpha \ne 0,$$

(9)

where ${\tilde{e}}_1,{\tilde{e}}_2,{\tilde{e}}_3$ is a pseudo-orthonormal basis, with ${\tilde{e}}_3$ timelike.

Lemma 1.

The Bott connection${\nabla }^B$of$G_1$is given by

$$\begin{array}{cc}\hfill & {\nabla }_{{\tilde{e}}_1}^B{\tilde{e}}_1=-\alpha {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_1}^B{\tilde{e}}_2=\alpha {\tilde{e}}_1,{\nabla }_{{\tilde{e}}_1}^B{\tilde{e}}_3=0,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_2}^B{\tilde{e}}_1=0,{\nabla }_{{\tilde{e}}_2}^B{\tilde{e}}_2=0,{\nabla }_{{\tilde{e}}_2}^B{\tilde{e}}_3=\alpha {\tilde{e}}_3,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_3}^B{\tilde{e}}_1=\alpha {\tilde{e}}_1+\beta {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_3}^B{\tilde{e}}_2=-\beta {\tilde{e}}_1-\alpha {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_3}^B{\tilde{e}}_3=0.\hfill \end{array}$$

(10)

Lemma 2.

The curvature$R^B$of the Bott connection${\nabla }^B$of$(G_1,g)$is given by

$$\begin{array}{cc}\hfill & R^B({\tilde{e}}_1,{\tilde{e}}_2){\tilde{e}}_1=\alpha \beta {\tilde{e}}_1+({\alpha }^2+{\beta }^2){\tilde{e}}_2,R^B({\tilde{e}}_1,{\tilde{e}}_2){\tilde{e}}_2=-({\alpha }^2+{\beta }^2){\tilde{e}}_1-\alpha \beta {\tilde{e}}_2,R^B({\tilde{e}}_1,{\tilde{e}}_2){\tilde{e}}_3=0,\hfill \\ \hfill & R^B({\tilde{e}}_1,{\tilde{e}}_3){\tilde{e}}_1=-3{\alpha }^2{\tilde{e}}_2,R^B({\tilde{e}}_1,{\tilde{e}}_3){\tilde{e}}_2=-{\alpha }^2{\tilde{e}}_1,R^B({\tilde{e}}_1,{\tilde{e}}_3){\tilde{e}}_3=\alpha \beta {\tilde{e}}_3,\hfill \\ \hfill & R^B({\tilde{e}}_2,{\tilde{e}}_3){\tilde{e}}_1=-{\alpha }^2{\tilde{e}}_1,R^B({\tilde{e}}_2,{\tilde{e}}_3){\tilde{e}}_2={\alpha }^2{\tilde{e}}_2,R^B({\tilde{e}}_2,{\tilde{e}}_3){\tilde{e}}_3=-{\alpha }^2{\tilde{e}}_3.\hfill \end{array}$$

(11)

By (3), we have

$$\begin{array}{cc}\hfill & {\rho }^B({\tilde{e}}_1,{\tilde{e}}_1)=-({\alpha }^2+{\beta }^2),{\rho }^B({\tilde{e}}_1,{\tilde{e}}_2)=\alpha \beta ,{\rho }^B({\tilde{e}}_1,{\tilde{e}}_3)=-\alpha \beta ,\hfill \\ \hfill & {\rho }^B({\tilde{e}}_2,{\tilde{e}}_1)=\alpha \beta ,{\rho }^B({\tilde{e}}_2,{\tilde{e}}_2)=-({\alpha }^2+{\beta }^2),{\rho }^B({\tilde{e}}_2,{\tilde{e}}_3)={\alpha }^2,\hfill \\ \hfill & {\rho }^B({\tilde{e}}_3,{\tilde{e}}_1)={\rho }^B({\tilde{e}}_3,{\tilde{e}}_2)={\rho }^B({\tilde{e}}_3,{\tilde{e}}_3)=0.\hfill \end{array}$$

(12)

Then,

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^B({\tilde{e}}_1,{\tilde{e}}_1)=-({\alpha }^2+{\beta }^2),{\tilde{\rho }}^B({\tilde{e}}_1,{\tilde{e}}_2)=\alpha \beta ,{\tilde{\rho }}^B({\tilde{e}}_1,{\tilde{e}}_3)=-\frac{\alpha \beta }{2},\hfill \\ \hfill & {\tilde{\rho }}^B({\tilde{e}}_2,{\tilde{e}}_2)=-({\alpha }^2+{\beta }^2),{\tilde{\rho }}^B({\tilde{e}}_2,{\tilde{e}}_3)=\frac{{\alpha }^2}{2},{\tilde{\rho }}^B({\tilde{e}}_3,{\tilde{e}}_3)=0.\hfill \end{array}$$

(13)

By (5), we have

$$\begin{array}{cc}\hfill & ({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_1)=0,({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_1)=0,({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_2)=-2{\alpha }^2\beta ,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_2)=0,({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_3)=\frac{{\alpha }^2\beta }{2},({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_3)=\frac{{\alpha }^2\beta }{2},\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_1)=\frac{{\alpha }^3}{2},({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_1)=2{\alpha }^3,({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_2)=\frac{{\alpha }^2\beta }{2},\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_2)=0,({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_3)=0,({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_3)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_1)=\frac{{\alpha }^2\beta }{2},({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_1)=0,({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_2)=-\frac{{\alpha }^2}{2},\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_2)=-2{\alpha }^2,({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_3)=0,({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_3)=\frac{\alpha }{2}({\alpha }^2-{\beta }^2).\hfill \end{array}$$

(14)

Then, if ${\tilde{\rho }}^B$ is a Codazzi tensor on $(G_1,{\nabla }^B)$, by (6) and (7), we have the following three equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\alpha \beta =0\hfill \\ \alpha =0\hfill \\ \alpha ({\alpha }^2-{\beta }^2)=0.\hfill \end{array}\right.\end{array}$$

(15)

By solving (15), we get $\alpha =0$, and there is a contradiction. So,

Theorem 1.

${\tilde{\rho }}^B$is not a Codazzi tensor on$(G_1,{\nabla }^B)$.

2.2. Codazzi Tensors of $G_2$

By [13], we have the following Lie algebra of $G_2$ which satisfies

$$[{\tilde{e}}_1,{\tilde{e}}_2]=\gamma {\tilde{e}}_2-\beta {\tilde{e}}_3,[{\tilde{e}}_1,{\tilde{e}}_3]=-\beta {\tilde{e}}_2-\gamma {\tilde{e}}_3,[{\tilde{e}}_2,{\tilde{e}}_3]=\alpha {\tilde{e}}_1,\gamma \ne 0,$$

(16)

where ${\tilde{e}}_1,{\tilde{e}}_2,{\tilde{e}}_3$ is a pseudo-orthonormal basis, with ${\tilde{e}}_3$ timelike.

Lemma 3.

The Bott connection${\nabla }^B$of$G_2$is given by

$$\begin{array}{cc}\hfill & {\nabla }_{{\tilde{e}}_1}^B{\tilde{e}}_1=0,{\nabla }_{{\tilde{e}}_1}^B{\tilde{e}}_2=0,{\nabla }_{{\tilde{e}}_1}^B{\tilde{e}}_3=-\gamma {\tilde{e}}_3,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_2}^B{\tilde{e}}_1=-\gamma {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_2}^B{\tilde{e}}_2=\gamma {\tilde{e}}_1,{\nabla }_{{\tilde{e}}_2}^B{\tilde{e}}_3=0,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_3}^B{\tilde{e}}_1=\beta {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_3}^B{\tilde{e}}_2=-\alpha {\tilde{e}}_1,{\nabla }_{{\tilde{e}}_3}^B{\tilde{e}}_3=0.\hfill \end{array}$$

(17)

Lemma 4.

The curvature$R^B$of the Bott connection${\nabla }^B$of$(G_2,g)$is given by

$$\begin{array}{cc}\hfill & R^B({\tilde{e}}_1,{\tilde{e}}_2){\tilde{e}}_1=({\beta }^2+{\gamma }^2){\tilde{e}}_2,R^B({\tilde{e}}_1,{\tilde{e}}_2){\tilde{e}}_2=-({\gamma }^2+\alpha \beta ){\tilde{e}}_1,R^B({\tilde{e}}_1,{\tilde{e}}_2){\tilde{e}}_3=0,\hfill \\ \hfill & R^B({\tilde{e}}_1,{\tilde{e}}_3){\tilde{e}}_1=0,R^B({\tilde{e}}_1,{\tilde{e}}_3){\tilde{e}}_2=\gamma (\alpha -\beta ){\tilde{e}}_1,R^B({\tilde{e}}_1,{\tilde{e}}_3){\tilde{e}}_3=0,\hfill \\ \hfill & R^B({\tilde{e}}_2,{\tilde{e}}_3){\tilde{e}}_1=\gamma (\beta -\alpha ){\tilde{e}}_1,R^B({\tilde{e}}_2,{\tilde{e}}_3){\tilde{e}}_2=\gamma (\alpha -\beta ){\tilde{e}}_2,R^B({\tilde{e}}_2,{\tilde{e}}_3){\tilde{e}}_3=\alpha \gamma {\tilde{e}}_3.\hfill \end{array}$$

(18)

By (3), we have

$$\begin{array}{cc}\hfill & {\rho }^B({\tilde{e}}_1,{\tilde{e}}_1)=-({\beta }^2+{\gamma }^2),{\rho }^B({\tilde{e}}_1,{\tilde{e}}_2)=0,{\rho }^B({\tilde{e}}_1,{\tilde{e}}_3)=0,\hfill \\ \hfill & {\rho }^B({\tilde{e}}_2,{\tilde{e}}_1)=0,{\rho }^B({\tilde{e}}_2,{\tilde{e}}_2)=-({\gamma }^2+\alpha \beta ),{\rho }^B({\tilde{e}}_2,{\tilde{e}}_3)=-\alpha \gamma ,\hfill \\ \hfill & {\rho }^B({\tilde{e}}_3,{\tilde{e}}_1)={\rho }^B({\tilde{e}}_3,{\tilde{e}}_2)={\rho }^B({\tilde{e}}_3,{\tilde{e}}_3)=0.\hfill \end{array}$$

(19)

Then,

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^B({\tilde{e}}_1,{\tilde{e}}_1)=-({\beta }^2+{\gamma }^2),{\tilde{\rho }}^B({\tilde{e}}_1,{\tilde{e}}_2)=0,{\tilde{\rho }}^B({\tilde{e}}_1,{\tilde{e}}_3)=0,\hfill \\ \hfill & {\tilde{\rho }}^B({\tilde{e}}_2,{\tilde{e}}_2)=-({\gamma }^2+\alpha \beta ),{\tilde{\rho }}^B({\tilde{e}}_2,{\tilde{e}}_3)=-\frac{\alpha \gamma }{2},{\tilde{\rho }}^B({\tilde{e}}_3,{\tilde{e}}_3)=0.\hfill \end{array}$$

(20)

By (5), we have

$$\begin{array}{cc}\hfill & ({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_1)=0,({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_1)=0,({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_2)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_2)=\gamma ({\beta }^2-\alpha \beta ),({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_3)=-\frac{\alpha {\gamma }^2}{2},({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_3)=-\frac{\alpha {\gamma }^2}{2},\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_1)=0,({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_1)=0,({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_2)=-\frac{\alpha {\gamma }^2}{2},\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_2)={\gamma }^2(\beta -\alpha ),({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_3)=0,({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_3)=\frac{\alpha \beta \gamma }{2},\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_1)=-\frac{\alpha {\gamma }^2}{2},({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_1)={\gamma }^2(\beta -\alpha ),({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_2)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_2)=0,({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_3)=0,({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_3)=0.\hfill \end{array}$$

(21)

Then, if ${\tilde{\rho }}^B$ is a Codazzi tensor on $(G_2,{\nabla }^B)$, by (6) and (7), we have the following three equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\gamma \beta (\beta -\alpha )=0\hfill \\ \alpha \beta \gamma =0\hfill \\ \gamma (\alpha -2\beta )=0.\hfill \end{array}\right.\end{array}$$

(22)

By solving (22), we get $\alpha =\beta =0$, and this condition does not hold. Therefore,

Theorem 2.

${\tilde{\rho }}^B$is not a Codazzi tensor on$(G_2,{\nabla }^B)$.

2.3. Codazzi Tensors of $G_3$

By [13], we have the following Lie algebra of $G_3$ which satisfies

$$[{\tilde{e}}_1,{\tilde{e}}_2]=-\gamma {\tilde{e}}_3,[{\tilde{e}}_1,{\tilde{e}}_3]=-\beta {\tilde{e}}_2,[{\tilde{e}}_2,{\tilde{e}}_3]=\alpha {\tilde{e}}_1,$$

(23)

where ${\tilde{e}}_1,{\tilde{e}}_2,{\tilde{e}}_3$ is a pseudo-orthonormal basis, with ${\tilde{e}}_3$ timelike.

Lemma 5.

The Bott connection${\nabla }^B$of$G_3$is given by

$$\begin{array}{cc}\hfill & {\nabla }_{{\tilde{e}}_1}^B{\tilde{e}}_1=0,{\nabla }_{{\tilde{e}}_1}^B{\tilde{e}}_2=0,{\nabla }_{{\tilde{e}}_1}^B{\tilde{e}}_3=-\gamma {\tilde{e}}_3,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_2}^B{\tilde{e}}_1=0,{\nabla }_{{\tilde{e}}_2}^B{\tilde{e}}_2=0,{\nabla }_{{\tilde{e}}_2}^B{\tilde{e}}_3=0,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_3}^B{\tilde{e}}_1=\beta {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_3}^B{\tilde{e}}_2=-\alpha {\tilde{e}}_1,{\nabla }_{{\tilde{e}}_3}^B{\tilde{e}}_3=0.\hfill \end{array}$$

(24)

Lemma 6.

The curvature$R^B$of the Bott connection${\nabla }^B$of$(G_3,g)$is given by

$$\begin{array}{cc}\hfill & R^B({\tilde{e}}_1,{\tilde{e}}_2){\tilde{e}}_1=\beta \gamma {\tilde{e}}_2,R^B({\tilde{e}}_1,{\tilde{e}}_2){\tilde{e}}_2=-\alpha \gamma {\tilde{e}}_1,R^B({\tilde{e}}_1,{\tilde{e}}_2){\tilde{e}}_3=0,\hfill \\ \hfill & R^B({\tilde{e}}_1,{\tilde{e}}_3){\tilde{e}}_1=0,R^B({\tilde{e}}_1,{\tilde{e}}_3){\tilde{e}}_2=0,R^B({\tilde{e}}_1,{\tilde{e}}_3){\tilde{e}}_3=0,\hfill \\ \hfill & R^B({\tilde{e}}_2,{\tilde{e}}_3){\tilde{e}}_1=0,R^B({\tilde{e}}_2,{\tilde{e}}_3){\tilde{e}}_2=0,R^B({\tilde{e}}_2,{\tilde{e}}_3){\tilde{e}}_3=0.\hfill \end{array}$$

(25)

By (3), we have

$$\begin{array}{cc}\hfill & {\rho }^B({\tilde{e}}_1,{\tilde{e}}_1)=-\beta \gamma ,{\rho }^B({\tilde{e}}_1,{\tilde{e}}_2)=0,{\rho }^B({\tilde{e}}_1,{\tilde{e}}_3)=0,\hfill \\ \hfill & {\rho }^B({\tilde{e}}_2,{\tilde{e}}_1)=0,{\rho }^B({\tilde{e}}_2,{\tilde{e}}_2)=-\alpha \gamma ,{\rho }^B({\tilde{e}}_2,{\tilde{e}}_3)=0,\hfill \\ \hfill & {\rho }^B({\tilde{e}}_3,{\tilde{e}}_1)={\rho }^B({\tilde{e}}_3,{\tilde{e}}_2)={\rho }^B({\tilde{e}}_3,{\tilde{e}}_3)=0.\hfill \end{array}$$

(26)

Then,

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^B({\tilde{e}}_1,{\tilde{e}}_1)=-\beta \gamma ,{\tilde{\rho }}^B({\tilde{e}}_1,{\tilde{e}}_2)={\tilde{\rho }}^B({\tilde{e}}_1,{\tilde{e}}_3)=0,\hfill \\ \hfill & {\tilde{\rho }}^B({\tilde{e}}_2,{\tilde{e}}_2)=-\alpha \gamma ,{\tilde{\rho }}^B({\tilde{e}}_2,{\tilde{e}}_3)={\tilde{\rho }}^B({\tilde{e}}_3,{\tilde{e}}_3)=0.\hfill \end{array}$$

(27)

By (5), we have

$$\begin{array}{cc}\hfill & ({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_j)=({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_j)=({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_j)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_j)=({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_j)=({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_j)=0,\hfill \end{array}$$

(28)

where $1\le j\le 3$.

Then, we obtain

Theorem 3.

${\tilde{\rho }}^B$is a Codazzi tensor on$(G_3,{\nabla }^B)$.

2.4. Codazzi Tensors of $G_4$

By [13], we have the following Lie algebra of $G_4$ which satisfies

$$[{\tilde{e}}_1,{\tilde{e}}_2]=-{\tilde{e}}_2+(2\eta -\beta ){\tilde{e}}_3,\eta =\pm 1,[{\tilde{e}}_1,{\tilde{e}}_3]=-\beta {\tilde{e}}_2+{\tilde{e}}_3,[{\tilde{e}}_2,{\tilde{e}}_3]=\alpha {\tilde{e}}_1,$$

(29)

where ${\tilde{e}}_1,{\tilde{e}}_2,{\tilde{e}}_3$ is a pseudo-orthonormal basis, with ${\tilde{e}}_3$ timelike.

Lemma 7.

The Bott connection${\nabla }^B$of$G_4$is given by

$$\begin{array}{cc}\hfill & {\nabla }_{{\tilde{e}}_1}^B{\tilde{e}}_1=0,{\nabla }_{{\tilde{e}}_1}^B{\tilde{e}}_2=0,{\nabla }_{{\tilde{e}}_1}^B{\tilde{e}}_3={\tilde{e}}_3,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_2}^B{\tilde{e}}_1={\tilde{e}}_2,{\nabla }_{{\tilde{e}}_2}^B{\tilde{e}}_2=-{\tilde{e}}_1,{\nabla }_{{\tilde{e}}_2}^B{\tilde{e}}_3=0,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_3}^B{\tilde{e}}_1=\beta {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_3}^B{\tilde{e}}_2=-\alpha {\tilde{e}}_1,{\nabla }_{{\tilde{e}}_3}^B{\tilde{e}}_3=0.\hfill \end{array}$$

(30)

Lemma 8.

The curvature$R^B$of the Bott connection${\nabla }^B$of$(G_4,g)$is given by

$$\begin{array}{cc}\hfill & R^B({\tilde{e}}_1,{\tilde{e}}_2){\tilde{e}}_1={(\beta -\eta )}^2{\tilde{e}}_2,R^B({\tilde{e}}_1,{\tilde{e}}_2){\tilde{e}}_2=(2\alpha \eta -\alpha \beta -1){\tilde{e}}_1,R^B({\tilde{e}}_1,{\tilde{e}}_2){\tilde{e}}_3=0,\hfill \\ \hfill & R^B({\tilde{e}}_1,{\tilde{e}}_3){\tilde{e}}_1=0,R^B({\tilde{e}}_1,{\tilde{e}}_3){\tilde{e}}_2=(\alpha -\beta ){\tilde{e}}_1,R^B({\tilde{e}}_1,{\tilde{e}}_3){\tilde{e}}_3=0,\hfill \\ \hfill & R^B({\tilde{e}}_2,{\tilde{e}}_3){\tilde{e}}_1=(\alpha -\beta ){\tilde{e}}_1,R^B({\tilde{e}}_2,{\tilde{e}}_3){\tilde{e}}_2=(\beta -\alpha ){\tilde{e}}_2,R^B({\tilde{e}}_2,{\tilde{e}}_3){\tilde{e}}_3=-\alpha {\tilde{e}}_3.\hfill \end{array}$$

(31)

By (3), we have

$$\begin{array}{cc}\hfill & {\rho }^B({\tilde{e}}_1,{\tilde{e}}_1)=-{(\beta -\eta )}^2,{\rho }^B({\tilde{e}}_1,{\tilde{e}}_2)=0,{\rho }^B({\tilde{e}}_1,{\tilde{e}}_3)=0,\hfill \\ \hfill & {\rho }^B({\tilde{e}}_2,{\tilde{e}}_1)=(2\alpha \eta -\alpha \beta -1),{\rho }^B({\tilde{e}}_2,{\tilde{e}}_2)=\alpha ,{\rho }^B({\tilde{e}}_2,{\tilde{e}}_3)=0,\hfill \\ \hfill & {\rho }^B({\tilde{e}}_3,{\tilde{e}}_1)={\rho }^B({\tilde{e}}_3,{\tilde{e}}_2)={\rho }^B({\tilde{e}}_3,{\tilde{e}}_3)=0.\hfill \end{array}$$

(32)

Then,

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^B({\tilde{e}}_1,{\tilde{e}}_1)=-{(\beta -\eta )}^2,{\tilde{\rho }}^B({\tilde{e}}_1,{\tilde{e}}_2)=0,{\tilde{\rho }}^B({\tilde{e}}_1,{\tilde{e}}_3)=0,\hfill \\ \hfill & {\tilde{\rho }}^B({\tilde{e}}_2,{\tilde{e}}_2)=(2\alpha \eta -\alpha \beta -1),{\tilde{\rho }}^B({\tilde{e}}_2,{\tilde{e}}_3)=\frac{\alpha }{2},{\tilde{\rho }}^B({\tilde{e}}_3,{\tilde{e}}_3)=0.\hfill \end{array}$$

(33)

By (5), we have

$$\begin{array}{cc}\hfill & ({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_1)=0,({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_1)=0,({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_2)=\alpha \beta +2\beta \eta -2\alpha \eta -{\beta }^2,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_2)=-\frac{\alpha }{2},({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_3)=-\frac{\alpha }{2},({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_3)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_1)=0,({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_1)=0,({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_2)=-\frac{\alpha }{2},\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_2)=\beta -\alpha ,({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_3)=0,({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_3)=-\frac{\alpha \beta }{2},\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_1)=-\frac{\alpha }{2},({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_1)=\beta -\alpha ,({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_2)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_2)=0,({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_3)=0,({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_3)=0.\hfill \end{array}$$

(34)

Then, if ${\tilde{\rho }}^B$ is a Codazzi tensor on $(G_4,{\nabla }^B)$, by (6) and (7), we have the following three equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{(\beta -\eta )}^2+2\alpha \eta -\alpha \beta -1=0\hfill \\ \alpha -2\beta =0\hfill \\ \alpha \beta =0.\hfill \end{array}\right.\end{array}$$

(35)

By solving (35), it turns out that

Theorem 4.

${\tilde{\rho }}^B$is a Codazzi tensor on$(G_4,{\nabla }^B)$if and only if$\alpha =\beta =0$.

2.5. Codazzi Tensors of $G_5$

By [13], we have the following Lie algebra of $G_5$ which satisfies

$$[{\tilde{e}}_1,{\tilde{e}}_2]=0,[{\tilde{e}}_1,{\tilde{e}}_3]=\alpha {\tilde{e}}_1+\beta {\tilde{e}}_2,[{\tilde{e}}_2,{\tilde{e}}_3]=\gamma {\tilde{e}}_1+\delta {\tilde{e}}_2,\alpha +\delta \ne 0,\alpha \gamma +\beta \delta =0,$$

(36)

where ${\tilde{e}}_1,{\tilde{e}}_2,{\tilde{e}}_3$ is a pseudo-orthonormal basis, with ${\tilde{e}}_3$ timelike.

Lemma 9.

The Bott connection ${\nabla }^B$ of $G_5$ is given by

$$\begin{array}{cc}\hfill & {\nabla }_{{\tilde{e}}_1}^B{\tilde{e}}_1=0,{\nabla }_{{\tilde{e}}_1}^B{\tilde{e}}_2=0,{\nabla }_{{\tilde{e}}_1}^B{\tilde{e}}_3=0,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_2}^B{\tilde{e}}_1=0,{\nabla }_{{\tilde{e}}_2}^B{\tilde{e}}_2=0,{\nabla }_{{\tilde{e}}_2}^B{\tilde{e}}_3=0,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_3}^B{\tilde{e}}_1=-\alpha {\tilde{e}}_1-\beta {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_3}^B{\tilde{e}}_2=-\gamma {\tilde{e}}_1-\delta {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_3}^B{\tilde{e}}_3=0.\hfill \end{array}$$

(37)

Lemma 10.

The curvature$R^B$of the Bott connection${\nabla }^B$of$(G_5,g)$is given by

$$\begin{array}{c}\hfill R^B({\tilde{e}}_i,{\tilde{e}}_j){\tilde{e}}_k=0,\end{array}$$

(38)

for any$(i,j,k)$.

By (3), we have

$$\begin{array}{c}\hfill {\rho }^B({\tilde{e}}_i,{\tilde{e}}_j)=0,\end{array}$$

(39)

then,

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^B({\tilde{e}}_i,{\tilde{e}}_j)=0,\hfill \end{array}$$

(40)

for any pairs $(i,j)$.

By (5), we have

$$\begin{array}{cc}\hfill & ({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_j)=({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_j)=({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_j)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_j)=({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_j)=({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_j)=0,\hfill \end{array}$$

(41)

where $1\le j\le 3$.

So,

Theorem 5.

${\tilde{\rho }}^B$is a Codazzi tensor on$(G_5,{\nabla }^B)$.

2.6. Codazzi Tensors of $G_6$

By [13], we have the following Lie algebra of $G_6$ which satisfies

$$[{\tilde{e}}_1,{\tilde{e}}_2]=\alpha {\tilde{e}}_2+\beta {\tilde{e}}_3,[{\tilde{e}}_1,{\tilde{e}}_3]=\gamma {\tilde{e}}_2+\delta {\tilde{e}}_3,[{\tilde{e}}_2,{\tilde{e}}_3]=0,\alpha +\delta \ne 0,\alpha \gamma -\beta \delta =0,$$

(42)

where ${\tilde{e}}_1,{\tilde{e}}_2,{\tilde{e}}_3$ is a pseudo-orthonormal basis, with ${\tilde{e}}_3$ timelike.

Lemma 11.

The Bott connection${\nabla }^B$of$G_6$is given by

$$\begin{array}{cc}\hfill & {\nabla }_{{\tilde{e}}_1}^B{\tilde{e}}_1=0,{\nabla }_{{\tilde{e}}_1}^B{\tilde{e}}_2=0,{\nabla }_{{\tilde{e}}_1}^B{\tilde{e}}_3=\delta {\tilde{e}}_3,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_2}^B{\tilde{e}}_1=-\alpha {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_2}^B{\tilde{e}}_2=\alpha {\tilde{e}}_1,{\nabla }_{{\tilde{e}}_2}^B{\tilde{e}}_3=0,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_3}^B{\tilde{e}}_1=-\gamma {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_3}^B{\tilde{e}}_2=0,{\nabla }_{{\tilde{e}}_3}^B{\tilde{e}}_3=0.\hfill \end{array}$$

(43)

Lemma 12.

The curvature$R^B$of the Bott connection${\nabla }^B$of$(G_6,g)$is given by

$$\begin{array}{cc}\hfill & R^B({\tilde{e}}_1,{\tilde{e}}_2){\tilde{e}}_1=({\alpha }^2+\beta \gamma ){\tilde{e}}_2,R^B({\tilde{e}}_1,{\tilde{e}}_2){\tilde{e}}_2=-{\alpha }^2{\tilde{e}}_1,R^B({\tilde{e}}_1,{\tilde{e}}_2){\tilde{e}}_3=0,\hfill \\ \hfill & R^B({\tilde{e}}_1,{\tilde{e}}_3){\tilde{e}}_1=\gamma (\alpha +\delta ){\tilde{e}}_2,R^B({\tilde{e}}_1,{\widehat{e}}_3){\tilde{e}}_2=-\alpha \gamma {\tilde{e}}_1,R^B({\tilde{e}}_1,{\tilde{e}}_3){\tilde{e}}_3=0,\hfill \\ \hfill & R^B({\tilde{e}}_2,{\tilde{e}}_3){\tilde{e}}_1=-\alpha \gamma {\tilde{e}}_1,R^B({\tilde{e}}_2,{\tilde{e}}_3){\tilde{e}}_2=\alpha \gamma {\tilde{e}}_2,R^B({\tilde{e}}_2,{\tilde{e}}_3){\tilde{e}}_3=0.\hfill \end{array}$$

(44)

By (3), we have

$$\begin{array}{cc}\hfill & {\rho }^B({\tilde{e}}_1,{\tilde{e}}_1)=-({\alpha }^2+\beta \gamma ),{\rho }^B({\tilde{e}}_1,{\tilde{e}}_2)={\rho }^B({\tilde{e}}_1,{\tilde{e}}_3)=0,\hfill \\ \hfill & {\rho }^B({\tilde{e}}_2,{\tilde{e}}_1)=0,{\rho }^B({\tilde{e}}_2,{\tilde{e}}_2)=-{\alpha }^2,{\rho }^B({\tilde{e}}_2,{\tilde{e}}_3)=0,\hfill \\ \hfill & {\rho }^B({\tilde{e}}_3,{\tilde{e}}_1)={\rho }^B({\tilde{e}}_3,{\tilde{e}}_2)={\rho }^B({\tilde{e}}_3,{\tilde{e}}_3)=0.\hfill \end{array}$$

(45)

Then,

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^B({\tilde{e}}_1,{\tilde{e}}_1)=-({\alpha }^2+\beta \gamma ),{\tilde{\rho }}^B({\tilde{e}}_1,{\tilde{e}}_2)={\tilde{\rho }}^B({\tilde{e}}_1,{\tilde{e}}_3)=0,\hfill \\ \hfill & {\tilde{\rho }}^B({\tilde{e}}_2,{\tilde{e}}_2)=-{\alpha }^2,{\tilde{\rho }}^B({\tilde{e}}_2,{\tilde{e}}_3)=0,{\tilde{\rho }}^B({\tilde{e}}_3,{\tilde{e}}_3)=0.\hfill \end{array}$$

(46)

By (5), we have

$$\begin{array}{cc}\hfill & ({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_1)=({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_1)=({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_2)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_2)=\alpha \beta \gamma ,({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_3)=0,({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_3)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_1)=({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_1)=({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_2)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_2)=-{\alpha }^2\gamma ,({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_3)=0,({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_3)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_1)=0,({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_1)=-{\alpha }^2\gamma ,({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_2)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_2)=({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_3)=({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_3)=0.\hfill \end{array}$$

(47)

Then, if ${\tilde{\rho }}^B$ is a Codazzi tensor on $(G_6,{\nabla }^B)$, by (6) and (7), we have the following two equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\alpha \beta \gamma =0\hfill \\ \alpha \gamma =0.\hfill \end{array}\right.\end{array}$$

(48)

By solving (48), we obtain

Theorem 6.

${\tilde{\rho }}^B$is a Codazzi tensor on$(G_6,{\nabla }^B)$if and only if

$$\begin{array}{ccc}& & (1)\alpha =\beta =0,\delta \ne 0;\hfill \\ & & (2)\alpha \ne 0,\gamma =\beta \delta =0.\hfill \end{array}$$

2.7. Codazzi Tensors of $G_7$

By [13], we have the following Lie algebra of $G_7$ which satisfies

$$[{\tilde{e}}_1,{\tilde{e}}_2]=-\alpha {\tilde{e}}_1-\beta {\tilde{e}}_2-\beta {\tilde{e}}_3,[{\tilde{e}}_1,{\tilde{e}}_3]=\alpha {\tilde{e}}_1+\beta {\tilde{e}}_2+\beta {\tilde{e}}_3,[{\tilde{e}}_2,{\tilde{e}}_3]=\gamma {\tilde{e}}_1+\delta {\tilde{e}}_2+\delta {\tilde{e}}_3,\alpha +\delta \ne 0,\alpha \gamma =0,$$

(49)

where ${\tilde{e}}_1,{\tilde{e}}_2,{\tilde{e}}_3$ is a pseudo-orthonormal basis, with ${\tilde{e}}_3$ timelike.

Lemma 13.

The Bott connection${\nabla }^B$of$G_7$is given by

$$\begin{array}{cc}\hfill & {\nabla }_{{\tilde{e}}_1}^B{\tilde{e}}_1=\alpha {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_1}^B{\tilde{e}}_2=-\alpha {\tilde{e}}_1,{\nabla }_{{\tilde{e}}_1}^B{\tilde{e}}_3=\beta {\tilde{e}}_3,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_2}^B{\tilde{e}}_1=\beta {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_2}^B{\tilde{e}}_2=-\beta {\tilde{e}}_1,{\nabla }_{{\tilde{e}}_2}^B{\tilde{e}}_3=\delta {\tilde{e}}_3,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_3}^B{\tilde{e}}_1=-\alpha {\tilde{e}}_1-\beta {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_3}^B{\tilde{e}}_2=-\gamma {\tilde{e}}_1-\delta {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_3}^B{\tilde{e}}_3=0.\hfill \end{array}$$

(50)

Lemma 14.

The curvature$R^B$of the Bott connection${\nabla }^B$of$(G_7,g)$is given by

$$\begin{array}{cc}\hfill & R^B({\tilde{e}}_1,{\tilde{e}}_2){\tilde{e}}_1={\alpha }^2{\tilde{e}}_2-\alpha \beta {\tilde{e}}_1,R^B({\tilde{e}}_1,{\tilde{e}}_2){\tilde{e}}_2=-({\alpha }^2+{\beta }^2+\beta \gamma ){\tilde{e}}_1-\beta \delta {\tilde{e}}_2,\hfill \\ \hfill & R^B({\tilde{e}}_1,{\tilde{e}}_2){\tilde{e}}_3=\beta (\alpha -\delta ){\tilde{e}}_3,R^B({\tilde{e}}_1,{\tilde{e}}_3){\tilde{e}}_1=\alpha (2\beta +\gamma ){\tilde{e}}_1+(\alpha \delta -2{\alpha }^2){\tilde{e}}_2,\hfill \\ \hfill & R^B({\tilde{e}}_1,{\tilde{e}}_3){\tilde{e}}_2=(\alpha \delta +{\beta }^2+\beta \gamma ){\tilde{e}}_1+(\beta \delta -\alpha \beta -\alpha \gamma ){\tilde{e}}_2,R^B({\tilde{e}}_1,{\tilde{e}}_3){\tilde{e}}_3=-\beta (\alpha +\delta ){\tilde{e}}_3,\hfill \\ \hfill & R^B({\tilde{e}}_2,{\tilde{e}}_3){\tilde{e}}_1=({\beta }^2+\beta \gamma +\alpha \delta ){\tilde{e}}_1+(\beta \delta -\alpha \beta -\alpha \gamma ){\tilde{e}}_2,\hfill \\ \hfill & R^B({\tilde{e}}_2,{\tilde{e}}_3){\tilde{e}}_2=(2\beta \delta +\delta \gamma +\alpha \gamma -\alpha \beta ){\tilde{e}}_1+({\delta }^2-{\beta }^2-\beta \gamma ){\tilde{e}}_2,R^B({\tilde{e}}_2,{\tilde{e}}_3){\tilde{e}}_3=-(\beta \gamma +{\delta }^2){\tilde{e}}_3.\hfill \end{array}$$

(51)

By (3), we have

$$\begin{array}{cc}\hfill & {\rho }^B({\tilde{e}}_1,{\tilde{e}}_1)=-{\alpha }^2,{\rho }^B({\tilde{e}}_1,{\tilde{e}}_2)=\beta \delta ,{\rho }^B({\tilde{e}}_1,{\tilde{e}}_3)=\beta (\alpha +\delta ),\hfill \\ \hfill & {\rho }^B({\tilde{e}}_2,{\tilde{e}}_1)=-\alpha \beta ,{\rho }^B({\tilde{e}}_2,{\tilde{e}}_2)=-({\alpha }^2+{\beta }^2+\beta \gamma ),{\rho }^B({\tilde{e}}_2,{\tilde{e}}_3)=(\beta \gamma +{\delta }^2),\hfill \\ \hfill & {\rho }^B({\tilde{e}}_3,{\tilde{e}}_1)=\beta (\alpha +\delta ),{\rho }^B({\tilde{e}}_3,{\tilde{e}}_2)=\delta (\alpha +\delta ),{\rho }^B({\tilde{e}}_3,{\tilde{e}}_3)=0.\hfill \end{array}$$

(52)

Then,

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^B({\tilde{e}}_1,{\tilde{e}}_1)=-{\alpha }^2,{\tilde{\rho }}^B({\tilde{e}}_1,{\tilde{e}}_2)=\frac{\beta (\delta -\alpha )}{2},{\tilde{\rho }}^B({\tilde{e}}_1,{\tilde{e}}_3)=\delta (\alpha +\delta ),\hfill \\ \hfill & {\tilde{\rho }}^B({\tilde{e}}_2,{\tilde{e}}_2)=-({\alpha }^2+{\beta }^2+\beta \gamma ),{\tilde{\rho }}^B({\tilde{e}}_2,{\tilde{e}}_3)={\delta }^2+\frac{\beta \gamma +\alpha \delta }{2},{\tilde{\rho }}^B({\tilde{e}}_3,{\tilde{e}}_3)=0.\hfill \end{array}$$

(53)

By (5), we have

$$\begin{array}{cc}\hfill & ({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_1)=\alpha ({\beta }^2+\beta \gamma ),({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_1)={\beta }^2(\alpha -\delta ),({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_2)=\alpha \beta (\delta -\alpha ),\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_2)={\beta }^2(\beta +\gamma ),({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_3)={\alpha }^2\beta +\frac{\alpha \beta \delta -{\beta }^2\gamma }{2}-\beta {\delta }^2,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_3)=-(2\beta {\delta }^2+\frac{{\beta }^2\gamma +3\alpha \beta \delta }{2}),({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_1)=-(\alpha {\beta }^2+{\beta }^2\delta +\alpha {\delta }^2+\frac{\alpha \beta \gamma +{\alpha }^2\delta }{2}),\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_1)={\beta }^2\delta -\alpha {\beta }^2-2{\alpha }^3,({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_2)={\alpha }^2\beta +\frac{\alpha \beta \delta }{2}-\beta {\delta }^2-\frac{{\beta }^2\gamma }{2},\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_2)=\frac{\beta ({\delta }^2-3{\alpha }^2)}{2}-{\beta }^3-{\beta }^2\gamma -{\alpha }^2\gamma ,({\nabla }_{{\tilde{e}}_1}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_3)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_1,{\tilde{e}}_3)={\alpha }^2\beta +\frac{3\alpha \beta \delta }{2}+\beta {\delta }^2+\frac{{\beta }^2\gamma }{2},({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_1)=-(2\beta {\delta }^2+\frac{3\alpha \beta \delta +{\beta }^2\gamma }{2}),\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_1)=\frac{\beta {\delta }^2-3{\alpha }^2\beta }{2}-{\alpha }^2\gamma -{\beta }^3-{\beta }^2\gamma ,({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_2)={\beta }^2(\alpha +\delta )-{\delta }^3-\frac{\beta \delta \gamma +\alpha {\delta }^2}{2},\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_2)=-(\alpha \beta \gamma +\beta \delta \gamma +2{\alpha }^2\delta +2{\beta }^2\delta ),({\nabla }_{{\tilde{e}}_2}^B{\tilde{\rho }}^B)({\tilde{e}}_3,{\tilde{e}}_3)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^B{\tilde{\rho }}^B)({\tilde{e}}_2,{\tilde{e}}_3)=\alpha \beta \gamma +{\delta }^3+\frac{3\beta \delta \gamma +\alpha {\delta }^2}{2}.\hfill \end{array}$$

(54)

Then, if ${\tilde{\rho }}^B$ is a Codazzi tensor on $(G_7,{\nabla }^B)$, by (6) and (7), we have the following nine equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\beta (\alpha \gamma +\beta \delta )=0\hfill \\ \beta (\alpha \delta -{\alpha }^2-{\beta }^2-\beta \gamma )=0\hfill \\ \beta (\alpha +\delta )=0\hfill \\ 4{\alpha }^3-4{\beta }^2\delta -2\alpha {\delta }^2-(\alpha \beta \gamma +{\alpha }^2\delta )=0\hfill \\ 5{\alpha }^2\beta +\alpha \beta \delta +{\beta }^2\gamma -3\beta {\delta }^2+2({\alpha }^2\gamma +{\beta }^3)=0\hfill \\ \beta (2{\alpha }^2+6\alpha \delta +2{\delta }^2+\beta \gamma )=0\hfill \\ 3{\alpha }^2\beta -3\alpha \beta \delta +{\beta }^2\gamma -5\beta {\delta }^2+2({\beta }^3+{\alpha }^2\gamma )=0\hfill \\ \beta \delta \gamma -\alpha {\delta }^2+2(\alpha \beta \gamma +2{\alpha }^2\delta +\alpha {\beta }^2+3{\beta }^2\delta -{\delta }^3)=0\hfill \\ 2(\alpha \beta \gamma +{\delta }^3)+3\beta \delta \gamma +\alpha {\delta }^2=0.\hfill \end{array}\right.\end{array}$$

(55)

By solving (55), we obtain $\alpha =\delta =0$, and the result is wrong. So,

Theorem 7.

${\tilde{\rho }}^B$is not a Codazzi tensor on$(G_7,{\nabla }^B)$.

3. Quasi-Statistical Structure Associated with Bott Connections on Three-Dimensional Lorentzian Lie Groups

The torsion tensor of $(G_i,g,{\nabla }^B)$ is defined by

$$T^B(X,Y)={\nabla }_X^{B}Y-{\nabla }_Y^{B}X-[X,Y].$$

(56)

Next, we recall the quasi-statistical structure.

Definition 2

([6]). Let M be a smooth manifold endowed with a linear connection ∇, and a tensor field ω. Then $(M,\nabla ,\omega )$ is called a quasi-statistical structure, if it satisfies

$$\tilde{f}(X,Y,Z)=({\nabla }_X\omega )(Y,Z)-({\nabla }_Y\omega )(X,Z)+\omega (T(X,Y),Z)=0,$$

(57)

where $\tilde{f}$ is $C^{\infty }(M)$-linear for $X,Y,Z$.

Proposition 2.

$(M,\nabla ,\omega )$is called a quasi-statistical structure and is called a Codazzi tensor on$(M,\nabla )$if and only if

$$\tilde{f}(X,Y,Z)=-\tilde{f}(Y,X,Z).$$

(58)

Then we have that ${(G_i,{\nabla }^B,\omega )}_{i=1,2,\dots ,7}$ is a quasi-statistical structure if and only if the following three equations hold:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\tilde{f}({\tilde{e}}_1,{\tilde{e}}_2,{\tilde{e}}_j)=0\hfill \\ \tilde{f}({\tilde{e}}_1,{\tilde{e}}_3,{\tilde{e}}_j)=0\hfill \\ \tilde{f}({\tilde{e}}_2,{\tilde{e}}_3,{\tilde{e}}_j)=0,\hfill \end{array}\right.\end{array}$$

(59)

where $1\le j\le 3$.

For $(G_1,{\nabla }^B)$, we have

$$\begin{array}{c}\hfill T^B({\tilde{e}}_1,{\tilde{e}}_2)=\beta {\tilde{e}}_3,T^B({\tilde{e}}_1,{\tilde{e}}_3)=T^B({\tilde{e}}_2,{\tilde{e}}_3)=0,\end{array}$$

(60)

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^B(T^B({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_1)=-\frac{\alpha {\beta }^2}{2},{\tilde{\rho }}^B(T^B({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_2)=\frac{\alpha {\beta }^2}{2},{\tilde{\rho }}^B(T^B({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_3)=0,\hfill \\ \hfill & {\tilde{\rho }}^B(T^B({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_j)={\tilde{\rho }}^B(T^B({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_j)=0,\hfill \end{array}$$

(61)

where $1\le j\le 3$.

Then, if $(G_1,{\nabla }^B,{\tilde{\rho }}^B)$ is a quasi-statistical structure, by (57) and (58), we have the following three equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\alpha \beta =0\hfill \\ \alpha =0\hfill \\ \alpha ({\alpha }^2-{\beta }^2)=0.\hfill \end{array}\right.\end{array}$$

(62)

By solving (62), we obtain $\alpha =0$, and there is a contradiction. So,

Theorem 8.

$(G_1,{\nabla }^B,{\tilde{\rho }}^B)$is not a quasi-statistical structure.

For $(G_2,{\nabla }^B)$, we have

$$\begin{array}{c}\hfill T^B({\tilde{e}}_1,{\tilde{e}}_2)=\beta {\tilde{e}}_3,T^B({\tilde{e}}_1,{\tilde{e}}_3)=T^B({\tilde{e}}_2,{\tilde{e}}_3)=0,\end{array}$$

(63)

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^B(T^B({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_1)=0,{\tilde{\rho }}^B(T^B({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_2)=-\frac{\alpha \beta \gamma }{2},{\tilde{\rho }}^B(T^B({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_3)=0,\hfill \\ \hfill & {\tilde{\rho }}^B(T^B({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_j)={\tilde{\rho }}^B(T^B({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_j)=0,\hfill \end{array}$$

(64)

where $1\le j\le 3$.

Then, if $(G_2,{\nabla }^B,{\tilde{\rho }}^B)$ is a quasi-statistical structure, by (57) and (58), we have the following two equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\alpha \beta \gamma =0\hfill \\ \gamma (\alpha -2\beta )=0.\hfill \end{array}\right.\end{array}$$

(65)

By solving (65), it turns out that

Theorem 9.

$(G_2,{\nabla }^B,{\tilde{\rho }}^B)$is a quasi-statistical structure if and only if$\alpha =\beta =0,\gamma \ne 0$.

For $(G_3,{\nabla }^B)$, we have

$$\begin{array}{c}\hfill T^B({\tilde{e}}_1,{\tilde{e}}_2)=\gamma {\tilde{e}}_3,T^B({\tilde{e}}_1,{\tilde{e}}_3)=T^B({\tilde{e}}_2,{\tilde{e}}_3)=0,\end{array}$$

(66)

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^B(T^B({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_j)={\tilde{\rho }}^B(T^B({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_j)={\tilde{\rho }}^B(T^B({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_j)=0,\hfill \end{array}$$

(67)

where $1\le j\le 3$.

Similarly, we can obtain

Theorem 10.

$(G_3,{\nabla }^B,{\tilde{\rho }}^B)$is a quasi-statistical structure.

For $(G_4,{\nabla }^B)$, we have

$$\begin{array}{c}\hfill T^B({\tilde{e}}_1,{\tilde{e}}_2)=(\beta -2\eta ){\tilde{e}}_3,T^B({\tilde{e}}_1,{\tilde{e}}_3)=T^B({\tilde{e}}_2,{\tilde{e}}_3)=0,\end{array}$$

(68)

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^B(T^B({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_1)=0,{\tilde{\rho }}^B(T^B({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_2)=\frac{\alpha (\beta -2\eta )}{2},{\tilde{\rho }}^B(T^B({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_3)=0,\hfill \\ \hfill & {\tilde{\rho }}^B(T^B({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_j)={\tilde{\rho }}^B(T^B({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_j)=0,\hfill \end{array}$$

(69)

where $1\le j\le 3$.

Then, if $(G_4,{\nabla }^B,{\tilde{\rho }}^B)$ is a quasi-statistical structure, by (57) and (58), we have the following three equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\beta }^2-2\beta \eta +\alpha \eta =0\hfill \\ \alpha -2\beta =0\hfill \\ \alpha \beta =0.\hfill \end{array}\right.\end{array}$$

(70)

By solving (70), it turns out that

Theorem 11.

$(G_4,{\nabla }^B,{\tilde{\rho }}^B)$is a quasi-statistical structure if and only if$\alpha =\beta =0$.

For $(G_5,{\nabla }^B)$, we have

$$\begin{array}{c}\hfill T^B({\tilde{e}}_1,{\tilde{e}}_2)=T^B({\tilde{e}}_1,{\tilde{e}}_3)=T^B({\tilde{e}}_2,{\tilde{e}}_3)=0,\end{array}$$

(71)

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^B(T^B({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_j)={\tilde{\rho }}^B(T^B({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_j)={\tilde{\rho }}^B(T^B({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_j)=0,\hfill \end{array}$$

(72)

where $1\le j\le 3$.

Similarly, we obtain

Theorem 12.

$(G_5,{\nabla }^B,{\tilde{\rho }}^B)$is a quasi-statistical structure.

For $(G_6,{\nabla }^B)$, we have

$$\begin{array}{c}\hfill T^B({\tilde{e}}_1,{\tilde{e}}_2)=-\beta {\tilde{e}}_3,T^B({\tilde{e}}_1,{\tilde{e}}_3)=T^B({\tilde{e}}_2,{\tilde{e}}_3)=0,\end{array}$$

(73)

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^B(T^B({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_1)={\tilde{\rho }}^B(T^B({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_j)={\tilde{\rho }}^B(T^B({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_j)=0,\hfill \end{array}$$

(74)

where $1\le j\le 3$.

Then, if $(G_6,{\nabla }^B,{\tilde{\rho }}^B)$ is a quasi-statistical structure, by (57) and (58), we have the following two equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\alpha \beta \gamma =0\hfill \\ \alpha \gamma =0.\hfill \end{array}\right.\end{array}$$

(75)

By solving (75), we get

Theorem 13.

$(G_6,{\nabla }^B,{\tilde{\rho }}^B)$is a quasi-statistical structure if and only if

$$\begin{array}{ccc}& & (1)\alpha =\beta =0,\delta \ne 0;\hfill \\ & & (2)\alpha \ne 0,\gamma =\beta \delta =0.\hfill \end{array}$$

For $(G_7,{\nabla }^B)$, we have

$$\begin{array}{c}\hfill T^B({\tilde{e}}_1,{\tilde{e}}_2)=\beta {\tilde{e}}_3,T^B({\tilde{e}}_1,{\tilde{e}}_3)=T^B({\tilde{e}}_2,{\tilde{e}}_3)=0,\end{array}$$

(76)

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^B(T^B({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_1)={\beta }^2(\alpha +\delta ),{\tilde{\rho }}^B(T^B({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_2)=\beta {\delta }^2+\frac{\alpha \beta \delta +{\beta }^2\gamma }{2},{\tilde{\rho }}^B(T^B({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_3)=0,\hfill \\ \hfill & {\tilde{\rho }}^B(T^B({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_j)={\tilde{\rho }}^B(T^B({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_j)=0,\hfill \end{array}$$

(77)

where $1\le j\le 3$.

Then, if $(G_7,{\nabla }^B,{\tilde{\rho }}^B)$ is a quasi-statistical structure, by (57) and (58), we have the following nine equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\beta (\alpha \gamma +\alpha \beta +2\beta \delta )=0\hfill \\ 2\beta (\alpha \delta -{\alpha }^2-{\beta }^2-\beta \gamma +{\delta }^2)+{\beta }^2\gamma +\alpha \beta \delta =0\hfill \\ \beta {(\alpha +\delta )}^2=0\hfill \\ 4{\alpha }^3-4{\beta }^2\delta -2\alpha {\delta }^2-\alpha (\beta \gamma +\alpha \delta )=0\hfill \\ 5{\alpha }^2\beta +\alpha \beta \delta +{\beta }^2\gamma -3\beta {\delta }^2+2{\alpha }^2\gamma +2{\beta }^3=0\hfill \\ 2\beta ({\alpha }^2+3\alpha \delta +{\delta }^2)+{\beta }^2\gamma =0\hfill \\ 3{\alpha }^2\beta -3\alpha \beta \delta +{\beta }^2\gamma -5\beta {\delta }^2+2{\beta }^3+2{\alpha }^2\gamma =0\hfill \\ \beta \delta \gamma -\alpha {\delta }^2+2\alpha \beta \gamma +4{\alpha }^2\delta +2\alpha {\beta }^2+6{\beta }^2\delta -2{\delta }^3=0\hfill \\ 2\alpha \beta \gamma +2{\delta }^3+3\beta \delta \gamma +\alpha {\delta }^2=0.\hfill \end{array}\right.\end{array}$$

(78)

By solving (78), we get $\alpha =\delta =0$, and this condition does not hold. So,

Theorem 14.

$(G_7,{\nabla }^B,{\tilde{\rho }}^B)$is not a quasi-statistical structure.

4. Codazzi Tensors Associated with Canonical Connections and Kobayashi–Nomizu Connections on Three-Dimensional Lorentzian Lie Groups

By [14], we define canonical connections and Kobayashi–Nomizu connections as follows:

$$\begin{array}{c}\hfill {\nabla }_X^{c}Y={\nabla }_X^{L}Y-\frac{1}{2}({{\nabla }^L}_{X}J)JY,\end{array}$$

(79)

$$\begin{array}{c}\hfill {\nabla }_X^{k}Y={\nabla }_X^{c}Y-\frac{1}{4}[({{\nabla }^L}_{Y}J)JX-({{\nabla }^L}_{JY}J)X],\end{array}$$

(80)

where J is a product structure on ${\{G_i\}}_{i=1,2\dots ,7}$ by $J{\tilde{e}}_1={\tilde{e}}_1,J{\tilde{e}}_2={\tilde{e}}_2,J{\tilde{e}}_3=-{\tilde{e}}_3$.

4.1. Codazzi Tensors of $G_1$

Lemma 15

([7]). The canonical connection ${\nabla }^c$ of $G_1$ is given by

$$\begin{array}{cc}\hfill & {\nabla }_{{\tilde{e}}_1}^c{\tilde{e}}_1=-\alpha {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_1}^c{\tilde{e}}_2=\alpha {\tilde{e}}_1,{\nabla }_{{\tilde{e}}_1}^c{\tilde{e}}_3=0,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_2}^c{\tilde{e}}_1={\nabla }_{{\tilde{e}}_2}^c{\tilde{e}}_2={\nabla }_{{\tilde{e}}_2}^c{\tilde{e}}_3=0,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_3}^c{\tilde{e}}_1=\frac{\beta }{2}{\tilde{e}}_2,{\nabla }_{{\tilde{e}}_3}^c{\tilde{e}}_2=-\frac{\beta }{2}{\tilde{e}}_1,{\nabla }_{{\tilde{e}}_3}^c{\tilde{e}}_3=0.\hfill \end{array}$$

(81)

Then,

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^c({\tilde{e}}_1,{\tilde{e}}_1)=-({\alpha }^2+\frac{{\beta }^2}{2}),{\tilde{\rho }}^c({\tilde{e}}_1,{\tilde{e}}_2)=0,{\tilde{\rho }}^c({\tilde{e}}_1,{\tilde{e}}_3)=\frac{\alpha \beta }{4},\hfill \\ \hfill & {\tilde{\rho }}^c({\tilde{e}}_2,{\tilde{e}}_2)=-({\alpha }^2+\frac{{\beta }^2}{2}),{\tilde{\rho }}^c({\tilde{e}}_2,{\tilde{e}}_3)=\frac{{\alpha }^2}{2},{\tilde{\rho }}^c({\tilde{e}}_3,{\tilde{e}}_3)=0.\hfill \end{array}$$

(82)

By (5), we have

$$\begin{array}{cc}\hfill & ({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_1)=({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_1)=({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_2)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_2)=0,({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_3)=-\frac{\alpha \beta }{4},({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_3)=-\frac{\alpha \beta }{4},\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_1)=\frac{{\alpha }^3}{2},({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_1)=0,({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_2)=\frac{{\alpha }^2\beta }{2},\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_2)=({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_3)=0,({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_3)=-\frac{{\alpha }^2\beta }{4},\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_1)=({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_1)=0,({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_2)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_2)=-2{\alpha }^2,({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_3)=0,({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_3)=\frac{\alpha {\beta }^2}{8}.\hfill \end{array}$$

(83)

Then, if ${\tilde{\rho }}^c$ is a Codazzi tensor on $(G_1,{\nabla }^c)$, by (6) and (7), we have the following two equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\alpha =0\hfill \\ \alpha \beta =0.\hfill \end{array}\right.\end{array}$$

(84)

By solving (84), we get $\alpha =0$, and the result is not true. So,

Theorem 15.

${\tilde{\rho }}^c$is not a Codazzi tensor on$(G_1,{\nabla }^c)$.

Lemma 16

([7]). The Kobayashi–Nomizu connection ${\nabla }^k$ of $G_1$ is given by

$$\begin{array}{cc}\hfill & {\nabla }_{{\tilde{e}}_1}^k{\tilde{e}}_1=-\alpha {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_1}^k{\tilde{e}}_2=\alpha {\tilde{e}}_1,{\nabla }_{{\tilde{e}}_1}^k{\tilde{e}}_3=0,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_2}^k{\tilde{e}}_1=0,{\nabla }_{{\tilde{e}}_2}^k{\tilde{e}}_2=0,{\nabla }_{{\tilde{e}}_2}^k{\tilde{e}}_3=\alpha {\tilde{e}}_3,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_3}^k{\tilde{e}}_1=\alpha {\tilde{e}}_1+\beta {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_3}^k{\tilde{e}}_2=-\beta {\tilde{e}}_1-\alpha {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_3}^k{\tilde{e}}_3=0.\hfill \end{array}$$

(85)

Then,

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^k({\tilde{e}}_1,{\tilde{e}}_1)=-({\alpha }^2+{\beta }^2),{\tilde{\rho }}^k({\tilde{e}}_1,{\tilde{e}}_2)=\alpha \beta ,{\tilde{\rho }}^k({\tilde{e}}_1,{\tilde{e}}_3)=-\frac{\alpha \beta }{2},\hfill \\ \hfill & {\tilde{\rho }}^k({\tilde{e}}_2,{\tilde{e}}_2)=-({\alpha }^2+{\beta }^2),{\tilde{\rho }}^k({\tilde{e}}_2,{\tilde{e}}_3)=\frac{{\alpha }^2}{2},{\tilde{\rho }}^k({\tilde{e}}_3,{\tilde{e}}_3)=0.\hfill \end{array}$$

(86)

By (5), we have

$$\begin{array}{cc}\hfill & ({\nabla }_{{\tilde{e}}_1}^k{\tilde{\rho }}^k)({\tilde{e}}_2,{\tilde{e}}_1)=0,({\nabla }_{{\tilde{e}}_2}^k{\tilde{\rho }}^k)({\tilde{e}}_1,{\tilde{e}}_1)=0,({\nabla }_{{\tilde{e}}_1}^k{\tilde{\rho }}^k)({\tilde{e}}_2,{\tilde{e}}_2)=-2{\alpha }^2\beta ,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_2}^k{\tilde{\rho }}^k)({\tilde{e}}_1,{\tilde{e}}_2)=0,({\nabla }_{{\tilde{e}}_1}^k{\tilde{\rho }}^k)({\tilde{e}}_2,{\tilde{e}}_3)=\frac{{\alpha }^2\beta }{2},({\nabla }_{{\tilde{e}}_2}^k{\tilde{\rho }}^k)({\tilde{e}}_1,{\tilde{e}}_3)=\frac{{\alpha }^2\beta }{2},\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_1}^k{\tilde{\rho }}^k)({\tilde{e}}_3,{\tilde{e}}_1)=\frac{{\alpha }^3}{2},({\nabla }_{{\tilde{e}}_3}^k{\tilde{\rho }}^k)({\tilde{e}}_1,{\tilde{e}}_1)=2{\alpha }^3,({\nabla }_{{\tilde{e}}_1}^k{\tilde{\rho }}^k)({\tilde{e}}_3,{\tilde{e}}_2)=\frac{{\alpha }^2\beta }{2},\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^k{\tilde{\rho }}^k)({\tilde{e}}_1,{\tilde{e}}_2)=0,({\nabla }_{{\tilde{e}}_1}^k{\tilde{\rho }}^k)({\tilde{e}}_3,{\tilde{e}}_3)=0,({\nabla }_{{\tilde{e}}_3}^k{\tilde{\rho }}^k)({\tilde{e}}_1,{\tilde{e}}_3)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_2}^k{\tilde{\rho }}^k)({\tilde{e}}_3,{\tilde{e}}_1)=\frac{{\alpha }^2\beta }{2},({\nabla }_{{\tilde{e}}_3}^k{\tilde{\rho }}^k)({\tilde{e}}_2,{\tilde{e}}_1)=0,({\nabla }_{{\tilde{e}}_2}^k{\tilde{\rho }}^k)({\tilde{e}}_3,{\tilde{e}}_2)=-\frac{{\alpha }^3}{2},\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^k{\tilde{\rho }}^k)({\tilde{e}}_2,{\tilde{e}}_2)=-2{\alpha }^3,({\nabla }_{{\tilde{e}}_2}^k{\tilde{\rho }}^k)({\tilde{e}}_3,{\tilde{e}}_3)=0,({\nabla }_{{\tilde{e}}_3}^k{\tilde{\rho }}^k)({\tilde{e}}_2,{\tilde{e}}_3)=\frac{\alpha }{2}({\alpha }^2-{\beta }^2).\hfill \end{array}$$

(87)

Then, if ${\tilde{\rho }}^k$ is a Codazzi tensor on $(G_1,{\nabla }^k)$, by (6) and (7), we have the following three equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\alpha \beta =0\hfill \\ \alpha =0\hfill \\ \alpha ({\beta }^2-{\alpha }^2)=0.\hfill \end{array}\right.\end{array}$$

(88)

By solving (88), we get $\alpha =0$, and the result does not match the condition. So,

Theorem 16.

${\tilde{\rho }}^k$is not a Codazzi tensor on$(G_1,{\nabla }^k)$.

4.2. Codazzi Tensors of $G_2$

Lemma 17

([7]). The canonical connection ${\nabla }^c$ of $G_2$ is given by

$$\begin{array}{cc}\hfill & {\nabla }_{{\tilde{e}}_1}^c{\tilde{e}}_1=0,{\nabla }_{{\tilde{e}}_1}^c{\tilde{e}}_2=\alpha {\tilde{e}}_1,{\nabla }_{{\tilde{e}}_1}^c{\tilde{e}}_3=0,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_2}^c{\tilde{e}}_1=-\gamma {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_2}^c{\tilde{e}}_2=\gamma {\tilde{e}}_1,{\nabla }_{{\tilde{e}}_2}^c{\tilde{e}}_3=0,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_3}^c{\tilde{e}}_1=\frac{\alpha }{2}{\tilde{e}}_2,{\nabla }_{{\tilde{e}}_3}^c{\tilde{e}}_2=-\frac{\alpha }{2}{\tilde{e}}_1,{\nabla }_{{\tilde{e}}_3}^c{\tilde{e}}_3=0.\hfill \end{array}$$

(89)

Then,

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^c({\tilde{e}}_1,{\tilde{e}}_1)=-({\gamma }^2+\frac{\alpha \beta }{2}),{\tilde{\rho }}^c({\tilde{e}}_1,{\tilde{e}}_2)=0,{\tilde{\rho }}^c({\tilde{e}}_1,{\tilde{e}}_3)=0,\hfill \\ \hfill & {\tilde{\rho }}^c({\tilde{e}}_2,{\tilde{e}}_2)=-({\gamma }^2+\frac{\alpha \beta }{2}),{\tilde{\rho }}^c({\tilde{e}}_2,{\tilde{e}}_3)=\gamma (\frac{\beta }{2}-\frac{\alpha }{4}),{\tilde{\rho }}^c({\tilde{e}}_3,{\tilde{e}}_3)=0.\hfill \end{array}$$

(90)

By (5), we have

$$\begin{array}{cc}\hfill & ({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_1)=({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_1)=({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_2)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_2)=({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_3)=0,({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_3)={\gamma }^2(\frac{\alpha }{4}-\frac{\beta }{2}),\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_1)=({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_1)=({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_2)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_2)=({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_3)=({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_3)=\frac{\alpha \gamma }{4}(\frac{\alpha }{2}-\beta ),\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_1)={\gamma }^2(\frac{\beta }{2}-\frac{\alpha }{4}),({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_1)=({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_2)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_2)=({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_3)=({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_3)=0.\hfill \end{array}$$

(91)

Then, if ${\tilde{\rho }}^c$ is a Codazzi tensor on $(G_2,{\nabla }^c)$, by (6) and (7), we have the following two equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\gamma (\alpha -2\beta )=0\hfill \\ \alpha \gamma (\alpha -2\beta )=0.\hfill \end{array}\right.\end{array}$$

(92)

By solving (92), we obtain

Theorem 17.

${\tilde{\rho }}^c$is a Codazzi tensor on$(G_2,{\nabla }^c)$if and only if$\gamma \ne 0,\alpha =2\beta$.

Lemma 18

([7]). The Kobayashi–Nomizu connection ${\nabla }^k$ of $G_2$ is given by

$$\begin{array}{cc}\hfill & {\nabla }_{{\tilde{e}}_1}^k{\tilde{e}}_1=0,{\nabla }_{{\tilde{e}}_1}^k{\tilde{e}}_2=0,{\nabla }_{{\tilde{e}}_1}^k{\tilde{e}}_3=-\gamma {\tilde{e}}_3,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_2}^k{\tilde{e}}_1=-\gamma {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_2}^k{\tilde{e}}_2=\gamma {\tilde{e}}_1,{\nabla }_{{\tilde{e}}_2}^k{\tilde{e}}_3=0,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_3}^k{\tilde{e}}_1=\beta {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_3}^k{\tilde{e}}_2=-\alpha {\tilde{e}}_1,{\nabla }_{{\tilde{e}}_3}^k{\tilde{e}}_3=0.\hfill \end{array}$$

(93)

Then,

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^k({\tilde{e}}_1,{\tilde{e}}_1)=-({\gamma }^2+{\beta }^2),{\tilde{\rho }}^k({\tilde{e}}_1,{\tilde{e}}_2)=0,{\tilde{\rho }}^k({\tilde{e}}_1,{\tilde{e}}_3)=0,\hfill \\ \hfill & {\tilde{\rho }}^k({\tilde{e}}_2,{\tilde{e}}_2)=-({\gamma }^2+\alpha \beta ),{\tilde{\rho }}^k({\tilde{e}}_2,{\tilde{e}}_3)=-\frac{\alpha \gamma }{2},{\tilde{\rho }}^k({\tilde{e}}_3,{\tilde{e}}_3)=0.\hfill \end{array}$$

(94)

By (5), we have

$$\begin{array}{cc}\hfill & ({\nabla }_{{\tilde{e}}_1}^k{\tilde{\rho }}^k)({\tilde{e}}_2,{\tilde{e}}_1)=0,({\nabla }_{{\tilde{e}}_2}^k{\tilde{\rho }}^k)({\tilde{e}}_1,{\tilde{e}}_1)=0,({\nabla }_{{\tilde{e}}_1}^k{\tilde{\rho }}^k)({\tilde{e}}_2,{\tilde{e}}_2)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_2}^k{\tilde{\rho }}^k)({\tilde{e}}_1,{\tilde{e}}_2)=\beta \gamma (\beta -\alpha ),({\nabla }_{{\tilde{e}}_1}^k{\tilde{\rho }}^k)({\tilde{e}}_2,{\tilde{e}}_3)=-\frac{\alpha {\gamma }^2}{2},({\nabla }_{{\tilde{e}}_2}^k{\tilde{\rho }}^k)({\tilde{e}}_1,{\tilde{e}}_3)=-\frac{\alpha {\gamma }^2}{2},\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_1}^k{\tilde{\rho }}^k)({\tilde{e}}_3,{\tilde{e}}_1)=0,({\nabla }_{{\tilde{e}}_3}^k{\tilde{\rho }}^k)({\tilde{e}}_1,{\tilde{e}}_1)=0,({\nabla }_{{\tilde{e}}_1}^k{\tilde{\rho }}^k)({\tilde{e}}_3,{\tilde{e}}_2)=-\frac{\alpha {\gamma }^2}{2},\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^k{\tilde{\rho }}^k)({\tilde{e}}_1,{\tilde{e}}_2)={\gamma }^2(\beta -\alpha ),({\nabla }_{{\tilde{e}}_1}^k{\tilde{\rho }}^k)({\tilde{e}}_3,{\tilde{e}}_3)=0,({\nabla }_{{\tilde{e}}_3}^k{\tilde{\rho }}^k)({\tilde{e}}_1,{\tilde{e}}_3)=\frac{\alpha \beta \gamma }{2},\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_2}^k{\tilde{\rho }}^k)({\tilde{e}}_3,{\tilde{e}}_1)=-\frac{\alpha {\gamma }^2}{2},({\nabla }_{{\tilde{e}}_3}^k{\tilde{\rho }}^k)({\tilde{e}}_2,{\tilde{e}}_1)={\gamma }^2(\beta -\alpha ),({\nabla }_{{\tilde{e}}_2}^k{\tilde{\rho }}^k)({\tilde{e}}_3,{\tilde{e}}_2)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^k{\tilde{\rho }}^k)({\tilde{e}}_2,{\tilde{e}}_2)=0,({\nabla }_{{\tilde{e}}_2}^k{\tilde{\rho }}^k)({\tilde{e}}_3,{\tilde{e}}_3)=0,({\nabla }_{{\tilde{e}}_3}^k{\tilde{\rho }}^k)({\tilde{e}}_2,{\tilde{e}}_3)=0.\hfill \end{array}$$

(95)

Then, if ${\tilde{\rho }}^k$ is a Codazzi tensor on $(G_2,{\nabla }^k)$, by (6) and (7), we have the following three equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\beta \gamma (\alpha -\beta )=0\hfill \\ \gamma (\alpha -2\beta )=0\hfill \\ \alpha \beta \gamma =0.\hfill \end{array}\right.\end{array}$$

(96)

The above equation induces the following theorem:

Theorem 18.

${\tilde{\rho }}^k$is a Codazzi tensor on$(G_2,{\nabla }^k)$if and only if$\gamma \ne 0,\alpha =\beta =0$.

4.3. Codazzi Tensors of $G_3$

Lemma 19

([7]). The canonical connection ${\nabla }^c$ of $G_3$ is given by

$$\begin{array}{cc}\hfill & {\nabla }_{{\tilde{e}}_1}^c{\tilde{e}}_1=0,{\nabla }_{{\tilde{e}}_1}^c{\tilde{e}}_2=0,{\nabla }_{{\tilde{e}}_1}^c{\tilde{e}}_3=0,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_2}^c{\tilde{e}}_1=0,{\nabla }_{{\tilde{e}}_2}^c{\tilde{e}}_2=0,{\nabla }_{{\tilde{e}}_2}^c{\tilde{e}}_3=0,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_3}^c{\tilde{e}}_1=m_3{\tilde{e}}_2,{\nabla }_{{\tilde{e}}_3}^c{\tilde{e}}_2=-m_3{\tilde{e}}_1,{\nabla }_{{\tilde{e}}_3}^c{\tilde{e}}_3=0,\hfill \end{array}$$

(97)

where

$$\begin{array}{c}\hfill m_1=\frac{\alpha -\beta -\gamma }{2},m_2=\frac{\alpha -\beta +\gamma }{2},m_3=\frac{\alpha +\beta -\gamma }{2}.\end{array}$$

(98)

Then,

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^c({\tilde{e}}_1,{\tilde{e}}_1)=-m_3\gamma ,{\tilde{\rho }}^c({\tilde{e}}_1,{\tilde{e}}_2)={\tilde{\rho }}^c({\tilde{e}}_1,{\tilde{e}}_3)=0,\hfill \\ \hfill & {\tilde{\rho }}^c({\tilde{e}}_2,{\tilde{e}}_2)=-m_3\gamma ,{\tilde{\rho }}^c({\tilde{e}}_2,{\tilde{e}}_3)={\tilde{\rho }}^c({\tilde{e}}_3,{\tilde{e}}_3)=0.\hfill \end{array}$$

(99)

By (5), we have

$$\begin{array}{cc}\hfill & ({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_j)=({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_j)=({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_j)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_j)=({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_j)=({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_j)=0,\hfill \end{array}$$

(100)

where $1\le j\le 3$.

Obviously, the following theorem holds:

Theorem 19.

${\tilde{\rho }}^c$is a Codazzi tensor on$(G_3,{\nabla }^c)$.

Lemma 20

([7]). The Kobayashi–Nomizu connection ${\nabla }^k$ of $G_3$ is given by

$$\begin{array}{cc}\hfill & {\nabla }_{{\tilde{e}}_1}^k{\tilde{e}}_1=0,{\nabla }_{{\tilde{e}}_1}^k{\tilde{e}}_2=0,{\nabla }_{{\tilde{e}}_1}^k{\tilde{e}}_3=0,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_2}^k{\tilde{e}}_1=0,{\nabla }_{{\tilde{e}}_2}^k{\tilde{e}}_2=0,{\nabla }_{{\tilde{e}}_2}^k{\tilde{e}}_3=0,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_3}^k{\tilde{e}}_1=(m_3-m_1){\tilde{e}}_2,{\nabla }_{{\tilde{e}}_3}^k{\tilde{e}}_2=-(m_2+m_3){\tilde{e}}_1,{\nabla }_{{\tilde{e}}_3}^k{\tilde{e}}_3=0,\hfill \end{array}$$

(101)

where

$$\begin{array}{c}\hfill m_1=\frac{\alpha -\beta -\gamma }{2},m_2=\frac{\alpha -\beta +\gamma }{2},m_3=\frac{\alpha +\beta -\gamma }{2}.\end{array}$$

(102)

Then,

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^k({\tilde{e}}_1,{\tilde{e}}_1)=\gamma (m_1-m_3),{\tilde{\rho }}^k({\tilde{e}}_1,{\tilde{e}}_2)={\tilde{\rho }}^k({\tilde{e}}_1,{\tilde{e}}_3)=0,\hfill \\ \hfill & {\tilde{\rho }}^k({\tilde{e}}_2,{\tilde{e}}_2)=-\gamma (m_2+m_3),{\tilde{\rho }}^k({\tilde{e}}_2,{\tilde{e}}_3)={\tilde{\rho }}^k({\tilde{e}}_3,{\tilde{e}}_3)=0.\hfill \end{array}$$

(103)

By (5), we have

$$\begin{array}{cc}\hfill & ({\nabla }_{{\tilde{e}}_1}^k{\tilde{\rho }}^k)({\tilde{e}}_2,{\tilde{e}}_j)=({\nabla }_{{\tilde{e}}_2}^k{\tilde{\rho }}^k)({\tilde{e}}_1,{\tilde{e}}_j)=({\nabla }_{{\tilde{e}}_1}^k{\tilde{\rho }}^k)({\tilde{e}}_3,{\tilde{e}}_j)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^k{\tilde{\rho }}^k)({\tilde{e}}_1,{\tilde{e}}_j)=({\nabla }_{{\tilde{e}}_2}^k{\tilde{\rho }}^k)({\tilde{e}}_3,{\tilde{e}}_j)=({\nabla }_{{\tilde{e}}_3}^k{\tilde{\rho }}^k)({\tilde{e}}_2,{\tilde{e}}_j)=0,\hfill \end{array}$$

(104)

where $1\le j\le 3$.

Therefore, it turns out

Theorem 20.

${\tilde{\rho }}^k$is a Codazzi tensor on$(G_3,{\nabla }^k)$.

4.4. Codazzi Tensors of $G_4$

Lemma 21

([7]). The canonical connection ${\nabla }^c$ of $G_4$ is given by

$$\begin{array}{cc}\hfill & {\nabla }_{{\tilde{e}}_1}^c{\tilde{e}}_1=0,{\nabla }_{{\tilde{e}}_1}^c{\tilde{e}}_2=0,{\nabla }_{{\tilde{e}}_1}^c{\tilde{e}}_3=0,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_2}^c{\tilde{e}}_1={\tilde{e}}_2,{\nabla }_{{\tilde{e}}_2}^c{\tilde{e}}_2=-{\tilde{e}}_1,{\nabla }_{{\tilde{e}}_2}^c{\tilde{e}}_3=0,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_3}^c{\tilde{e}}_1=n_3{\tilde{e}}_2,{\nabla }_{{\tilde{e}}_3}^c{\tilde{e}}_2=-n_3{\tilde{e}}_1,{\nabla }_{{\tilde{e}}_3}^c{\tilde{e}}_3=0.\hfill \end{array}$$

(105)

where

$$\begin{array}{c}\hfill n_1=\frac{\alpha }{2}+\eta -\beta ,n_2=\frac{\alpha }{2}-\eta ,n_3=\frac{\alpha }{2}+\eta .\end{array}$$

(106)

Then,

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^c({\tilde{e}}_1,{\tilde{e}}_1)=(2\eta -\beta )n_3-1,{\tilde{\rho }}^c({\tilde{e}}_1,{\tilde{e}}_2)=0,{\tilde{\rho }}^c({\tilde{e}}_1,{\tilde{e}}_3)=0,\hfill \\ \hfill & {\tilde{\rho }}^c({\tilde{e}}_2,{\tilde{e}}_2)=(2\eta -\beta )n_3-1,{\tilde{\rho }}^c({\tilde{e}}_2,{\tilde{e}}_3)=(\frac{n_3-\beta }{2},{\tilde{\rho }}^c({\tilde{e}}_3,{\tilde{e}}_3)=0.\hfill \end{array}$$

(107)

By (5), we have

$$\begin{array}{cc}\hfill & ({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_1)=({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_1)=({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_2)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_2)=({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_3)=({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_3)=\frac{\beta -n_3}{2},\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_1)=({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_1)=({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_2)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_2)=({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_3)=0,({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_3)=\frac{n_3(n_3-\beta )}{2},\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_1)=\frac{\beta -n_3}{2},({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_1)=({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_2)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_2)=0,({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_3)=({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_3)=0.\hfill \end{array}$$

(108)

Then, if ${\tilde{\rho }}^c$ is a Codazzi tensor on $(G_4,{\nabla }^c)$, by (6) and (7), we have the following two equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\beta -n_3=0\hfill \\ n_3(\beta -n_3)=0.\hfill \end{array}\right.\end{array}$$

(109)

By solving (109), we get

Theorem 21.

${\tilde{\rho }}^c$is a Codazzi tensor on$(G_4,{\nabla }^c)$if and only if$\frac{\alpha }{2}+\eta -\beta =0$.

Lemma 22

([7]). The Kobayashi–Nomizu connection ${\nabla }^k$ of $G_4$ is given by

$$\begin{array}{cc}\hfill & {\nabla }_{{\tilde{e}}_1}^k{\tilde{e}}_1=0,{\nabla }_{{\tilde{e}}_1}^k{\tilde{e}}_2=0,{\nabla }_{{\tilde{e}}_1}^k{\tilde{e}}_3={\tilde{e}}_3,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_2}^k{\tilde{e}}_1={\tilde{e}}_2,{\nabla }_{{\tilde{e}}_2}^k{\tilde{e}}_2=-{\tilde{e}}_1,{\nabla }_{{\tilde{e}}_2}^k{\tilde{e}}_3=0,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_3}^k{\tilde{e}}_1=(n_3-n_1){\tilde{e}}_2,{\nabla }_{{\tilde{e}}_3}^k{\tilde{e}}_2=-(n_2+n_3){\tilde{e}}_1,{\nabla }_{{\tilde{e}}_3}^k{\tilde{e}}_3=0,\hfill \end{array}$$

(110)

where

$$\begin{array}{c}\hfill n_1=\frac{\alpha }{2}+\eta -\beta ,n_2=\frac{\alpha }{2}-\eta ,n_3=\frac{\alpha }{2}+\eta .\end{array}$$

(111)

Then,

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^k({\tilde{e}}_1,{\tilde{e}}_1)=-[1+(\beta -2\eta )(n_3-n_1)],{\tilde{\rho }}^k({\tilde{e}}_1,{\tilde{e}}_2)=0,{\tilde{\rho }}^k({\tilde{e}}_1,{\tilde{e}}_3)=0,\hfill \\ \hfill & {\tilde{\rho }}^k({\tilde{e}}_2,{\tilde{e}}_2)=-[1+(\beta -2\eta )(n_2+n_3)],{\tilde{\rho }}^k({\tilde{e}}_2,{\tilde{e}}_3)=\frac{\alpha +n_3-n_1-\beta }{2},{\tilde{\rho }}^k({\tilde{e}}_3,{\tilde{e}}_3)=0.\hfill \end{array}$$

(112)

By (5), we have

$$\begin{array}{cc}\hfill & ({\nabla }_{{\tilde{e}}_1}^k{\tilde{\rho }}^k)({\tilde{e}}_2,{\tilde{e}}_1)=({\nabla }_{{\tilde{e}}_2}^k{\tilde{\rho }}^k)({\tilde{e}}_1,{\tilde{e}}_1)=({\nabla }_{{\tilde{e}}_1}^k{\tilde{\rho }}^k)({\tilde{e}}_2,{\tilde{e}}_2)=0,({\nabla }_{{\tilde{e}}_2}^k{\tilde{\rho }}^k)({\tilde{e}}_1,{\tilde{e}}_2)=(n_1+n_2)(\beta -2\eta ),\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_1}^k{\tilde{\rho }}^k)({\tilde{e}}_2,{\tilde{e}}_3)=\frac{n_1+\beta -\alpha -n_3}{2},({\nabla }_{{\tilde{e}}_2}^k{\tilde{\rho }}^k)({\tilde{e}}_1,{\tilde{e}}_3)=\frac{n_1+\beta -\alpha -n_3}{2},\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_1}^k{\tilde{\rho }}^k)({\tilde{e}}_3,{\tilde{e}}_1)=({\nabla }_{{\tilde{e}}_3}^k{\tilde{\rho }}^k)({\tilde{e}}_1,{\tilde{e}}_1)=0,({\nabla }_{{\tilde{e}}_1}^k{\tilde{\rho }}^k)({\tilde{e}}_3,{\tilde{e}}_2)=\frac{n_1+\beta -\alpha -n_3}{2},\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^k{\tilde{\rho }}^k)({\tilde{e}}_1,{\tilde{e}}_2)=-(n_1+n_2),({\nabla }_{{\tilde{e}}_1}^k{\tilde{\rho }}^k)({\tilde{e}}_3,{\tilde{e}}_3)=0,({\nabla }_{{\tilde{e}}_3}^k{\tilde{\rho }}^k)({\tilde{e}}_1,{\tilde{e}}_3)=(n_3-n_1)\frac{n_1+\beta -\alpha -n_3}{2},\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_2}^k{\tilde{\rho }}^k)({\tilde{e}}_3,{\tilde{e}}_1)=\frac{n_1+\beta -\alpha -n_3}{2},({\nabla }_{{\tilde{e}}_3}^k{\tilde{\rho }}^k)({\tilde{e}}_2,{\tilde{e}}_1)=-(n_1+n_2),({\nabla }_{{\tilde{e}}_2}^k{\tilde{\rho }}^k)({\tilde{e}}_3,{\tilde{e}}_2)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^k{\tilde{\rho }}^k)({\tilde{e}}_2,{\tilde{e}}_2)=({\nabla }_{{\tilde{e}}_2}^k{\tilde{\rho }}^k)({\tilde{e}}_3,{\tilde{e}}_3)=({\nabla }_{{\tilde{e}}_3}^k{\tilde{\rho }}^k)({\tilde{e}}_2,{\tilde{e}}_3)=0.\hfill \end{array}$$

(113)

Then, if ${\tilde{\rho }}^k$ is a Codazzi tensor on $(G_4,{\nabla }^k)$, by (6) and (7), we have the following three equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}(2\eta -\beta )(n_1+n_2)=0\hfill \\ 3n_1+\beta -\alpha -n_3+2n_2=0\hfill \\ (n_3-n_1)\alpha +n_3-n_1-\beta =0.\hfill \end{array}\right.\end{array}$$

(114)

The above equations induce the following theorem:

Theorem 22.

${\tilde{\rho }}^k$is a Codazzi tensor on$(G_4,{\nabla }^k)$if and only if$\alpha =\beta =0$.

4.5. Codazzi Tensors of $G_5$

Lemma 23

([7]). The canonical connection ${\nabla }^c$ of $G_5$ is given by

$$\begin{array}{cc}\hfill & {\nabla }_{{\tilde{e}}_1}^c{\tilde{e}}_1=0,{\nabla }_{{\tilde{e}}_1}^c{\tilde{e}}_2=0,{\nabla }_{{\tilde{e}}_1}^c{\tilde{e}}_3=0,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_2}^c{\tilde{e}}_1=0,{\nabla }_{{\tilde{e}}_2}^c{\tilde{e}}_2=0,{\nabla }_{{\tilde{e}}_2}^c{\tilde{e}}_3=0,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_3}^c{\tilde{e}}_1=\frac{\gamma -\beta }{2}{\tilde{e}}_2,{\nabla }_{{\tilde{e}}_3}^c{\tilde{e}}_2=\frac{\beta -\gamma }{2}{\tilde{e}}_1,{\nabla }_{{\tilde{e}}_3}^c{\tilde{e}}_3=0.\hfill \end{array}$$

(115)

Then,

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^c({\tilde{e}}_1,{\tilde{e}}_1)={\tilde{\rho }}^c({\tilde{e}}_1,{\tilde{e}}_2)={\tilde{\rho }}^c({\tilde{e}}_1,{\tilde{e}}_3)=0,\hfill \\ \hfill & {\tilde{\rho }}^c({\tilde{e}}_2,{\tilde{e}}_2)={\tilde{\rho }}^c({\tilde{e}}_2,{\tilde{e}}_3)={\tilde{\rho }}^c({\tilde{e}}_3,{\tilde{e}}_3)=0.\hfill \end{array}$$

(116)

By (5), we have

$$\begin{array}{cc}\hfill & ({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_j)=({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_j)=({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_j)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_j)=({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_j)=({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_j)=0,\hfill \end{array}$$

(117)

where $1\le j\le 3$.

Then, we get

Theorem 23.

${\tilde{\rho }}^c$is a Codazzi tensor on$(G_5,{\nabla }^c)$.

Lemma 24

([7]). The Kobayashi–Nomizu connection ${\nabla }^k$ of $G_5$ is given by

$$\begin{array}{cc}\hfill & {\nabla }_{{\tilde{e}}_1}^k{\tilde{e}}_1=0,{\nabla }_{{\tilde{e}}_1}^k{\tilde{e}}_2=0,{\nabla }_{{\tilde{e}}_1}^k{\tilde{e}}_3=0,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_2}^k{\tilde{e}}_1=0,{\nabla }_{{\tilde{e}}_2}^k{\tilde{e}}_2=0,{\nabla }_{{\tilde{e}}_2}^k{\tilde{e}}_3=0,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_3}^k{\tilde{e}}_1=-\alpha {\tilde{e}}_1-\beta {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_3}^k{\tilde{e}}_2=-\gamma {\tilde{e}}_1-\delta {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_3}^k{\tilde{e}}_3=0.\hfill \end{array}$$

(118)

Then,

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^k({\tilde{e}}_1,{\tilde{e}}_1)={\tilde{\rho }}^k({\tilde{e}}_1,{\tilde{e}}_2)={\tilde{\rho }}^k({\tilde{e}}_1,{\tilde{e}}_3)=0,\hfill \\ \hfill & {\tilde{\rho }}^k({\tilde{e}}_2,{\tilde{e}}_2)={\tilde{\rho }}^k({\tilde{e}}_2,{\tilde{e}}_3)={\tilde{\rho }}^k({\tilde{e}}_3,{\tilde{e}}_3)=0.\hfill \end{array}$$

(119)

Then, we get

Theorem 24.

${\tilde{\rho }}^k$is a Codazzi tensor on$(G_5,{\nabla }^k)$.

4.6. Codazzi Tensors of $G_6$

Lemma 25

([7]). The canonical connection ${\nabla }^c$ of $G_6$ is given by

$$\begin{array}{cc}\hfill & {\nabla }_{{\tilde{e}}_1}^c{\tilde{e}}_1=0,{\nabla }_{{\tilde{e}}_1}^c{\tilde{e}}_2=\alpha {\tilde{e}}_1,{\nabla }_{{\tilde{e}}_1}^c{\tilde{e}}_3=0,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_2}^c{\tilde{e}}_1=-\alpha {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_2}^c{\tilde{e}}_2=\alpha {\tilde{e}}_1,{\nabla }_{{\tilde{e}}_2}^c{\tilde{e}}_3=0,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_3}^c{\tilde{e}}_1=\frac{\beta -\gamma }{2}{\tilde{e}}_2,{\nabla }_{{\tilde{e}}_3}^c{\tilde{e}}_2=-\frac{\beta -\gamma }{2}{\tilde{e}}_1,{\nabla }_{{\tilde{e}}_3}^c{\tilde{e}}_3=0.\hfill \end{array}$$

(120)

Then,

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^c({\tilde{e}}_1,{\tilde{e}}_1)=\frac{1}{2}\beta (\beta -\gamma )-{\alpha }^2,{\tilde{\rho }}^c({\tilde{e}}_1,{\tilde{e}}_2)=0,{\tilde{\rho }}^c({\tilde{e}}_1,{\tilde{e}}_3)=0,\hfill \\ \hfill & {\tilde{\rho }}^c({\tilde{e}}_2,{\tilde{e}}_2)=\frac{1}{2}\beta (\beta -\gamma )-{\alpha }^2,{\tilde{\rho }}^c({\tilde{e}}_2,{\tilde{e}}_3)=\frac{1}{2}[-\alpha \gamma +\frac{1}{2}\delta (\beta -\gamma )],{\tilde{\rho }}^c({\tilde{e}}_3,{\tilde{e}}_3)=0.\hfill \end{array}$$

(121)

By (5), we have

$$\begin{array}{cc}\hfill & ({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_1)=0,({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_1)=0,({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_2)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_2)=0,({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_3)=0,({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_3)=\frac{\alpha }{2}[-\alpha \gamma +\frac{1}{2}\delta (\beta -\gamma )],\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_1)=0,({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_1)=0,({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_2)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_2)=0,({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_3)=0,({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_3)=\frac{\gamma -\beta }{4}[-\alpha \gamma +\frac{1}{2}\delta (\beta -\gamma )],\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_1)=\frac{\alpha }{2}[-\alpha \gamma +\frac{1}{2}\delta (\beta -\gamma )],({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_1)=0,({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_2)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_2)=0,({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_3)=0,({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_3)=0.\hfill \end{array}$$

(122)

Then, if ${\tilde{\rho }}^c$ is a Codazzi tensor on $(G_6,{\nabla }^c)$, by (6) and (7), we have the following two equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\alpha [-2\alpha \gamma +\delta (\beta -\gamma )]=0\hfill \\ (\beta -\gamma )[-2\alpha \gamma +\delta (\beta -\gamma )]=0.\hfill \end{array}\right.\end{array}$$

(123)

By solving (123), we get

Theorem 25.

${\tilde{\rho }}^c$is a Codazzi tensor on$(G_6,{\nabla }^c)$if and only if

$$\begin{array}{ccc}& & (1)\alpha =\beta =\gamma =0,\delta \ne 0;\hfill \\ & & (2)\alpha \ne 0,\beta =\gamma =0,\alpha +\delta \ne 0;\hfill \\ & & (3)\alpha \ne 0,\delta =\gamma =0.\hfill \end{array}$$

Lemma 26

([7]). The Kobayashi–Nomizu connection ${\nabla }^k$ of $G_6$ is given by

$$\begin{array}{cc}\hfill & {\nabla }_{{\tilde{e}}_1}^k{\tilde{e}}_1=0,{\nabla }_{{\tilde{e}}_1}^k{\tilde{e}}_2=0,{\nabla }_{{\tilde{e}}_1}^k{\tilde{e}}_3=\delta {\tilde{e}}_3,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_2}^k{\tilde{e}}_1=-\alpha {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_2}^k{\tilde{e}}_2=\alpha {\tilde{e}}_1,{\nabla }_{{\tilde{e}}_2}^k{\tilde{e}}_3=0,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_3}^k{\tilde{e}}_1=-\gamma {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_3}^k{\tilde{e}}_2=0,{\nabla }_{{\tilde{e}}_3}^k{\tilde{e}}_3=0.\hfill \end{array}$$

(124)

Then,

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^k({\tilde{e}}_1,{\tilde{e}}_1)=-({\alpha }^2+\beta \gamma ),{\tilde{\rho }}^k({\tilde{e}}_1,{\tilde{e}}_2)=0,{\tilde{\rho }}^k({\tilde{e}}_1,{\tilde{e}}_3)=0,\hfill \\ \hfill & {\tilde{\rho }}^k({\tilde{e}}_2,{\tilde{e}}_2)=-{\alpha }^2,{\tilde{\rho }}^k({\tilde{e}}_2,{\tilde{e}}_3)=-\frac{\alpha \gamma }{2},{\tilde{\rho }}^k({\tilde{e}}_3,{\tilde{e}}_3)=0.\hfill \end{array}$$

(125)

By (5), we have

$$\begin{array}{cc}\hfill & ({\nabla }_{{\tilde{e}}_1}^k{\tilde{\rho }}^k)({\tilde{e}}_2,{\tilde{e}}_1)=({\nabla }_{{\tilde{e}}_2}^k{\tilde{\rho }}^k)({\tilde{e}}_1,{\tilde{e}}_1)=({\nabla }_{{\tilde{e}}_1}^k{\tilde{\rho }}^k)({\tilde{e}}_2,{\tilde{e}}_2)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_2}^k{\tilde{\rho }}^k)({\tilde{e}}_1,{\tilde{e}}_2)=\alpha \beta \gamma ,({\nabla }_{{\tilde{e}}_1}^k{\tilde{\rho }}^k)({\tilde{e}}_2,{\tilde{e}}_3)=({\nabla }_{{\tilde{e}}_2}^k{\tilde{\rho }}^k)({\tilde{e}}_1,{\tilde{e}}_3)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_1}^k{\tilde{\rho }}^k)({\tilde{e}}_3,{\tilde{e}}_1)=({\nabla }_{{\tilde{e}}_3}^k{\tilde{\rho }}^k)({\tilde{e}}_1,{\tilde{e}}_1)=0,({\nabla }_{{\tilde{e}}_1}^k{\tilde{\rho }}^k)({\tilde{e}}_3,{\tilde{e}}_2)=-\frac{\alpha {\gamma }^2}{2},\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^k{\tilde{\rho }}^k)({\tilde{e}}_1,{\tilde{e}}_2)=-{\alpha }^2\gamma ,({\nabla }_{{\tilde{e}}_1}^k{\tilde{\rho }}^k)({\tilde{e}}_3,{\tilde{e}}_3)=({\nabla }_{{\tilde{e}}_3}^k{\tilde{\rho }}^k)({\tilde{e}}_1,{\tilde{e}}_3)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_2}^k{\tilde{\rho }}^k)({\tilde{e}}_3,{\tilde{e}}_1)=-{\alpha }^2\gamma ,({\nabla }_{{\tilde{e}}_3}^k{\tilde{\rho }}^k)({\tilde{e}}_2,{\tilde{e}}_1)=({\nabla }_{{\tilde{e}}_2}^k{\tilde{\rho }}^k)({\tilde{e}}_3,{\tilde{e}}_2)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^k{\tilde{\rho }}^k)({\tilde{e}}_2,{\tilde{e}}_2)=({\nabla }_{{\tilde{e}}_2}^k{\tilde{\rho }}^k)({\tilde{e}}_3,{\tilde{e}}_3)=({\nabla }_{{\tilde{e}}_3}^k{\tilde{\rho }}^k)({\tilde{e}}_2,{\tilde{e}}_3)=0.\hfill \end{array}$$

(126)

Then, if ${\tilde{\rho }}^k$ is a Codazzi tensor on $(G_6,{\nabla }^k)$, by (6) and (7), we have the following two equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\alpha \beta \gamma =0\hfill \\ {\alpha }^2\gamma =0.\hfill \end{array}\right.\end{array}$$

(127)

By solving (127), we get

Theorem 26.

${\tilde{\rho }}^k$is a Codazzi tensor on$(G_6,{\nabla }^k)$if and only if

$$\begin{array}{ccc}& & (1)\alpha =\beta =0,\delta \ne 0;\hfill \\ & & (2)\alpha \ne 0,\beta \delta =\gamma =0,\alpha +\delta \ne 0.\hfill \end{array}$$

4.7. Codazzi Tensors of $G_7$

Lemma 27

([7]). The canonical connection ${\nabla }^c$ of $G_7$ is given by

$$\begin{array}{cc}\hfill & {\nabla }_{{\tilde{e}}_1}^c{\tilde{e}}_1=\alpha {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_1}^c{\tilde{e}}_2=-\alpha {\tilde{e}}_1,{\nabla }_{{\tilde{e}}_1}^c{\tilde{e}}_3=0,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_2}^c{\tilde{e}}_1=\beta {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_2}^c{\tilde{e}}_2=-\beta {\tilde{e}}_1,{\nabla }_{{\tilde{e}}_2}^c{\tilde{e}}_3=0,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_3}^c{\tilde{e}}_1=(\frac{\gamma }{2}-\beta ){\tilde{e}}_2,{\nabla }_{{\tilde{e}}_3}^c{\tilde{e}}_2=(\beta -\frac{\gamma }{2}){\tilde{e}}_1,{\nabla }_{{\tilde{e}}_3}^c{\tilde{e}}_3=0.\hfill \end{array}$$

(128)

Then,

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^c({\tilde{e}}_1,{\tilde{e}}_1)=-({\alpha }^2+\frac{\beta \gamma }{2}),{\tilde{\rho }}^c({\tilde{e}}_1,{\tilde{e}}_2)=0,{\tilde{\rho }}^c({\tilde{e}}_1,{\tilde{e}}_3)=-\frac{1}{2}(\alpha \gamma +\frac{\delta \gamma }{2}),\hfill \\ \hfill & {\tilde{\rho }}^c({\tilde{e}}_2,{\tilde{e}}_2)=-({\alpha }^2+\frac{\beta \gamma 2}{2}),{\tilde{\rho }}^c({\tilde{e}}_2,{\tilde{e}}_3)=\frac{1}{2}({\alpha }^2+\frac{\beta \gamma }{2}),{\tilde{\rho }}^c({\tilde{e}}_3,{\tilde{e}}_3)=0.\hfill \end{array}$$

(129)

By (5), we have

$$\begin{array}{cc}\hfill & ({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_1)=({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_1)=({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_2)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_2)=0,({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_3)=-\frac{\alpha }{2}(\alpha \gamma +\frac{\beta \gamma }{2}),({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_3)=-\frac{\beta }{2}({\alpha }^2+\frac{\beta \gamma }{2}),\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_1)=-\frac{\alpha }{2}({\alpha }^2+\frac{\beta \gamma }{2}),({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_1)=0,({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_2)=-\frac{\alpha }{2}(\alpha \gamma +\frac{\delta \gamma }{2}),\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_2)=({\nabla }_{{\tilde{e}}_1}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_3)=0,({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_1,{\tilde{e}}_3)=(\beta -\frac{\gamma }{2})(\frac{{\alpha }^2}{2}+\frac{\beta \gamma }{4}),\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_1)=-\frac{\beta }{2}({\alpha }^2+\frac{\beta \gamma }{2}),({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_1)=0,({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_2)=-\frac{\beta }{2}(\alpha \gamma +\frac{\beta \gamma }{2}),\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_2)=-2{\alpha }^2,({\nabla }_{{\tilde{e}}_2}^c{\tilde{\rho }}^c)({\tilde{e}}_3,{\tilde{e}}_3)=0,({\nabla }_{{\tilde{e}}_3}^c{\tilde{\rho }}^c)({\tilde{e}}_2,{\tilde{e}}_3)=\frac{1}{2}(\beta -\frac{\gamma }{2})(\alpha \gamma +\frac{\delta \gamma }{2}).\hfill \end{array}$$

(130)

Then, if ${\tilde{\rho }}^c$ is a Codazzi tensor on $(G_7,{\nabla }^c)$, by (6) and (7), we have the following seven equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\beta (2{\alpha }^2+\beta \gamma )-\alpha (2\alpha \gamma +\delta \gamma )=0\hfill \\ \alpha (2{\alpha }^2+\beta \gamma )=0\hfill \\ \alpha \gamma (2\alpha +\delta )=0\hfill \\ (\gamma -2\beta )(2{\alpha }^2+\beta \gamma )=0\hfill \\ \beta (2{\alpha }^2+\beta \gamma )=0\hfill \\ \beta \gamma (2\alpha +\delta )=0\hfill \\ (\gamma -2\beta )(2\alpha \gamma +\delta \gamma )=0.\hfill \end{array}\right.\end{array}$$

(131)

The above equations induce the following theorem:

Theorem 27.

${\tilde{\rho }}^c$is a Codazzi tensor on$(G_7,{\nabla }^c)$if and only if$\alpha =\gamma =0,\delta \ne 0$.

Lemma 28

([7]). The Kobayashi–Nomizu connection ${\nabla }^k$ of $G_7$ is given by

$$\begin{array}{cc}\hfill & {\nabla }_{{\tilde{e}}_1}^k{\tilde{e}}_1=\alpha {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_1}^k{\tilde{e}}_2=-\alpha {\tilde{e}}_1,{\nabla }_{{\tilde{e}}_1}^k{\tilde{e}}_3=\beta {\tilde{e}}_3,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_2}^k{\tilde{e}}_1=\beta {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_2}^k{\tilde{e}}_2=-\beta {\tilde{e}}_1,{\nabla }_{{\tilde{e}}_2}^k{\tilde{e}}_3=\delta {\tilde{e}}_3,\hfill \\ \hfill & {\nabla }_{{\tilde{e}}_3}^k{\tilde{e}}_1=-\alpha {\tilde{e}}_1-\beta {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_3}^k{\tilde{e}}_2=-\gamma {\tilde{e}}_1-\delta {\tilde{e}}_2,{\nabla }_{{\tilde{e}}_3}^k{\tilde{e}}_3=0.\hfill \end{array}$$

(132)

Then,

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^k({\tilde{e}}_1,{\tilde{e}}_1)=-{\alpha }^2,{\tilde{\rho }}^k({\tilde{e}}_1,{\tilde{e}}_2)=\frac{\beta }{2}(\delta -\alpha ),{\tilde{\rho }}^k({\tilde{e}}_1,{\tilde{e}}_3)=\beta (\alpha +\delta ),\hfill \\ \hfill & {\tilde{\rho }}^k({\tilde{e}}_2,{\tilde{e}}_2)=-({\alpha }^2+{\beta }^2+\beta \gamma ),{\tilde{\rho }}^k({\tilde{e}}_2,{\tilde{e}}_3)=\frac{\alpha \delta +\beta \gamma +2{\delta }^2}{2},{\tilde{\rho }}^k({\tilde{e}}_3,{\tilde{e}}_3)=0.\hfill \end{array}$$

(133)

By (5), we have

$$\begin{array}{cc}\hfill & ({\nabla }_{{\tilde{e}}_1}^k{\tilde{\rho }}^k)({\tilde{e}}_2,{\tilde{e}}_1)=\alpha \beta (\beta +\gamma ),({\nabla }_{{\tilde{e}}_2}^k{\tilde{\rho }}^k)({\tilde{e}}_1,{\tilde{e}}_1)={\beta }^2(\alpha -\delta ),({\nabla }_{{\tilde{e}}_1}^k{\tilde{\rho }}^k)({\tilde{e}}_2,{\tilde{e}}_2)=\alpha \beta (\delta -\alpha ),\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_2}^k{\tilde{\rho }}^k)({\tilde{e}}_1,{\tilde{e}}_2)={\beta }^2(\beta +\gamma ),({\nabla }_{{\tilde{e}}_1}^k{\tilde{\rho }}^k)({\tilde{e}}_2,{\tilde{e}}_3)={\alpha }^2\beta +\frac{\alpha \beta \delta -{\beta }^2\gamma }{2}-\beta {\delta }^2,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_2}^k{\tilde{\rho }}^k)({\tilde{e}}_1,{\tilde{e}}_3)=-2\beta {\delta }^2-\frac{{\beta }^2\gamma +3\alpha \beta \delta }{2},({\nabla }_{{\tilde{e}}_1}^k{\tilde{\rho }}^k)({\tilde{e}}_3,{\tilde{e}}_1)=-{\beta }^2(\alpha +\delta )-\frac{\alpha }{2}(\alpha \delta +\beta \gamma +2{\delta }^2),\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^k{\tilde{\rho }}^k)({\tilde{e}}_1,{\tilde{e}}_1)={\beta }^2(\delta -\alpha )-2{\alpha }^3,({\nabla }_{{\tilde{e}}_1}^k{\tilde{\rho }}^k)({\tilde{e}}_3,{\tilde{e}}_2)=-\frac{\beta }{2}({\alpha }^2+\beta \gamma +2{\delta }^2),\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^k{\tilde{\rho }}^k)({\tilde{e}}_1,{\tilde{e}}_2)=\frac{\beta ({\delta }^2-{\alpha }^2)}{2}-\beta ({\alpha }^2+{\beta }^2+\beta \gamma )-{\alpha }^2\gamma ,({\nabla }_{{\tilde{e}}_1}^k{\tilde{\rho }}^k)({\tilde{e}}_3,{\tilde{e}}_3)=0,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^k{\tilde{\rho }}^k)({\tilde{e}}_1,{\tilde{e}}_3)=\alpha \beta (\alpha +\delta )+\frac{\beta }{2}(\alpha \delta +\beta \gamma +2{\delta }^2),({\nabla }_{{\tilde{e}}_2}^k{\tilde{\rho }}^k)({\tilde{e}}_3,{\tilde{e}}_1)=-\beta \delta (\alpha +\delta )-\frac{\beta }{2}(\alpha \delta +\beta \gamma +2{\delta }^2),\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_3}^k{\tilde{\rho }}^k)({\tilde{e}}_2,{\tilde{e}}_1)=\frac{\beta ({\delta }^2-{\alpha }^2)}{2}-\beta ({\alpha }^2+{\beta }^2+\beta \gamma )-{\alpha }^2\gamma ,\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_2}^k{\tilde{\rho }}^k)({\tilde{e}}_3,{\tilde{e}}_2)={\beta }^2(\alpha +\delta )-\frac{\delta }{2}(\alpha \delta +\beta \gamma +2{\delta }^2),({\nabla }_{{\tilde{e}}_3}^k{\tilde{\rho }}^k)({\tilde{e}}_2,{\tilde{e}}_2)=\beta \gamma (\delta -\alpha )-2\delta ({\alpha }^2+{\beta }^2+\beta \gamma ),\hfill \\ \hfill & ({\nabla }_{{\tilde{e}}_2}^k{\tilde{\rho }}^k)({\tilde{e}}_3,{\tilde{e}}_3)=0,({\nabla }_{{\tilde{e}}_3}^k{\tilde{\rho }}^k)({\tilde{e}}_2,{\tilde{e}}_3)=\beta \gamma (\alpha +\delta )+\frac{\delta }{2}(\alpha \delta +\beta \gamma +2{\delta }^2).\hfill \end{array}$$

(134)

Then, if ${\tilde{\rho }}^k$ is a Codazzi tensor on $(G_7,{\nabla }^k)$, by (6) and (7), we have the following nine equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\beta (\alpha \gamma +\beta \delta )=0\hfill \\ \beta (\alpha \delta -{\alpha }^2-{\beta }^2-\beta \gamma )=0\hfill \\ \beta {(\alpha +\delta )}^2=0\hfill \\ 4{\alpha }^3-4{\beta }^2\delta -2\alpha {\delta }^2-(\alpha \beta \gamma +{\alpha }^2\delta )=0\hfill \\ \beta (2{\alpha }^2+{\beta }^2+\beta \gamma +{\delta }^2)+2{\alpha }^2\gamma =0\hfill \\ 2\alpha \beta (\alpha +\delta )+\beta (\alpha \delta +\beta \gamma +2{\delta }^2)=0\hfill \\ 2{\alpha }^2\gamma +2{\beta }^3+3{\alpha }^2\beta +{\beta }^2\gamma -3\alpha \beta \delta -5\beta {\delta }^2=0\hfill \\ 2(\alpha \beta \gamma -{\delta }^3+3{\beta }^2\delta +2{\alpha }^2\delta )-\beta \delta \gamma -\alpha {\delta }^2=0\hfill \\ 2(\alpha \beta \gamma +{\delta }^3)+3\beta \delta \gamma +\alpha {\delta }^2=0.\hfill \end{array}\right.\end{array}$$

(135)

By solving (135), we get $\alpha =\delta =0$, and there is a contradiction. So,

Theorem 28.

${\tilde{\rho }}^k$is not a Codazzi tensor on$(G_7,{\nabla }^k)$.

5. Quasi-Statistical Structure Associated with Canonical Connections and Kobayashi–Nomizu Connections on Three-Dimensional Lorentzian Lie Groups

The torsion tensor of $(G_i,g,{\nabla }^c)$ is defined by

$$T^c(X,Y)={\nabla }_X^{c}Y-{\nabla }_Y^{c}X-[X,Y]$$

(136)

The torsion tensor of $(G_i,g,{\nabla }^k)$ is defined by

$$T^k(X,Y)={\nabla }_X^{k}Y-{\nabla }_Y^{k}X-[X,Y]$$

(137)

Then, for $G_1$, we have

$$\begin{array}{c}\hfill T^c({\tilde{e}}_1,{\tilde{e}}_2)=\beta {\tilde{e}}_3,T^c({\tilde{e}}_1,{\tilde{e}}_3)=\alpha {\tilde{e}}_1+\frac{\beta }{2}{\tilde{e}}_2,T^c({\tilde{e}}_2,{\tilde{e}}_3)=-\frac{\beta }{2}{\tilde{e}}_1-\alpha {\tilde{e}}_2-\alpha {\tilde{e}}_3.\end{array}$$

(138)

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_1)=\frac{\alpha {\beta }^2}{4},{\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_2)=\frac{{\alpha }^2\beta }{2},{\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_3)=0,\hfill \\ \hfill & {\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_1)=-\alpha ({\alpha }^2+\frac{\beta }{2}),{\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_2)=-\frac{\beta }{2}({\alpha }^2+\frac{\beta }{2}),{\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_1)=\frac{{\alpha }^2\beta }{2},\hfill \\ \hfill & {\tilde{\rho }}^c(T^c({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_1)=\frac{\beta }{4}({\alpha }^2+{\beta }^2),{\tilde{\rho }}^c(T^c({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_1)=\frac{\alpha }{2}({\alpha }^2+{\beta }^2),{\tilde{\rho }}^c(T^c({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_1)=-\alpha (\frac{{\beta }^2}{8}+\frac{{\alpha }^2}{2}).\hfill \end{array}$$

(139)

Then, if $(G_1,{\nabla }^c,{\tilde{\rho }}^c)$ is a quasi-statistical structure, by (57) and (58), we have the following three equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\alpha \beta =0\hfill \\ \alpha ({\alpha }^2+{\beta }^2)=0\hfill \\ \beta ({\alpha }^2+{\beta }^2)=0.\hfill \end{array}\right.\end{array}$$

(140)

By solving (140), we get $\alpha =0$, and there is a contradiction. So,

Theorem 29.

$(G_1,{\nabla }^c,{\tilde{\rho }}^c)$is not a quasi-statistical structure.

Similarly,

$$\begin{array}{c}\hfill T^k({\tilde{e}}_1,{\tilde{e}}_2)=\beta {\tilde{e}}_3,T^k({\tilde{e}}_1,{\tilde{e}}_3)=T^k({\tilde{e}}_2,{\tilde{e}}_3)=0,\end{array}$$

(141)

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^k(T^k({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_1)=0,{\tilde{\rho }}^k(T^k({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_2)=\frac{{\alpha }^2\beta }{2},{\tilde{\rho }}^k(T^k({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_3)=0,\hfill \\ \hfill & {\tilde{\rho }}^k(T^k({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_j)=0,{\tilde{\rho }}^k(T^k({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_j)=0,\hfill \end{array}$$

(142)

where $1\le j\le 3$.

Then, if $(G_1,{\nabla }^k,{\tilde{\rho }}^k)$ is a quasi-statistical structure, by (57) and (58), we have the following three equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\alpha \beta =0\hfill \\ \alpha =0\hfill \\ \alpha ({\alpha }^2-{\beta }^2)=0.\hfill \end{array}\right.\end{array}$$

(143)

By solving (143), we get $\alpha =0$, and this condition does not hold. Therefore,

Theorem 30.

$(G_1,{\nabla }^k,{\tilde{\rho }}^k)$is not a quasi-statistical structure.

For $G_2$, we have

$$\begin{array}{c}\hfill T^c({\tilde{e}}_1,{\tilde{e}}_2)=\beta {\tilde{e}}_3,T^c({\tilde{e}}_1,{\tilde{e}}_3)=(\beta -\frac{\alpha }{2}){\tilde{e}}_2+\gamma {\tilde{e}}_3,T^c({\tilde{e}}_2,{\tilde{e}}_3)=-\frac{\alpha }{2}{\tilde{e}}_1.\end{array}$$

(144)

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_1)=0,{\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_2)=\beta \gamma (\frac{\beta }{2}-\frac{\gamma }{4}),{\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_3)=0,\hfill \\ \hfill & {\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_1)=0,{\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_2)=(\frac{\alpha }{4}-\frac{\beta }{2})(\alpha \beta +{\gamma }^2),\hfill \\ \hfill & {\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_1)=(\beta -\frac{\alpha }{2})(\frac{\beta \gamma }{2}-\frac{\alpha \gamma }{4}),{\tilde{\rho }}^c(T^c({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_1)=\frac{\alpha }{2}({\gamma }^2+\frac{\alpha \beta }{2}),\hfill \\ \hfill & {\tilde{\rho }}^c(T^c({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_1)=\frac{\alpha }{2}({\alpha }^2+{\beta }^2),{\tilde{\rho }}^c(T^c({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_3)=0.\hfill \end{array}$$

(145)

Then, if $(G_2,{\nabla }^c,{\tilde{\rho }}^c)$ is a quasi-statistical structure, by (57) and (58), we have the following four equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\beta \gamma (2\beta -\alpha )=0\hfill \\ \gamma (2\beta -\alpha )=0\hfill \\ (2\beta -\alpha )({\gamma }^2+\alpha \beta )=0\hfill \\ \alpha ({\gamma }^2+\alpha \beta )+2\beta {\gamma }^2=0.\hfill \end{array}\right.\end{array}$$

(146)

By solving (146), we obtain

Theorem 31.

$(G_2,{\nabla }^c,{\tilde{\rho }}^c)$is a quasi-statistical structure if and only if$\gamma \ne 0,\alpha =\beta =0$.

Similarly,

$$\begin{array}{c}\hfill T^k({\tilde{e}}_1,{\tilde{e}}_2)=\beta {\tilde{e}}_3,T^k({\tilde{e}}_1,{\tilde{e}}_3)=T^k({\tilde{e}}_2,{\tilde{e}}_3)=0,\end{array}$$

(147)

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^k(T^k({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_1)=0,{\tilde{\rho }}^k(T^k({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_2)=\frac{\alpha \beta \gamma }{2},{\tilde{\rho }}^k(T^k({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_3)=0,\hfill \\ \hfill & {\tilde{\rho }}^k(T^k({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_j)=0,{\tilde{\rho }}^k(T^k({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_j)=0,\hfill \end{array}$$

(148)

where $1\le j\le 3$.

Then, if $(G_2,{\nabla }^k,{\tilde{\rho }}^k)$ is a quasi-statistical structure, by (57) and (58), we have the following three equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\beta \gamma (2\beta -\alpha )=0\hfill \\ \gamma (2\beta -\alpha )=0\hfill \\ \alpha \beta \gamma =0.\hfill \end{array}\right.\end{array}$$

(149)

By solving (149), we get

Theorem 32.

$(G_2,{\nabla }^k,{\tilde{\rho }}^k)$is a quasi-statistical structure if and only if$\gamma \ne 0,\alpha =\beta =0$.

For $G_3$, we have

$$\begin{array}{c}\hfill T^c({\tilde{e}}_1,{\tilde{e}}_2)=\gamma {\tilde{e}}_3,T^c({\tilde{e}}_1,{\tilde{e}}_3)=(\beta -m_3){\tilde{e}}_2,T^c({\tilde{e}}_2,{\tilde{e}}_3)=(m_3-\alpha ){\tilde{e}}_1,\end{array}$$

(150)

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_1)=0,{\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_2)={\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_3)=0,{\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_1)=0,\hfill \\ \hfill & {\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_2)=\gamma m_3(m_3-\beta ),{\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_1)=\gamma m_3(\alpha -m_3),\hfill \\ \hfill & {\tilde{\rho }}^c(T^c({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_1)={\tilde{\rho }}^c(T^c({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_1)={\tilde{\rho }}^c(T^c({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_3)=0.\hfill \end{array}$$

(151)

Then, if $(G_3,{\nabla }^c,{\tilde{\rho }}^c)$ is a quasi-statistical structure, by (57) and (58), we have the following two equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{m}_3\gamma (m_3-\beta )=0\hfill \\ m_3\gamma (\alpha -m_3)=0.\hfill \end{array}\right.\end{array}$$

(152)

By solving (152), we obtain

Theorem 33.

$(G_3,{\nabla }^c,{\tilde{\rho }}^c)$is a quasi-statistical structure if and only if

$$\begin{array}{ccc}& & (1)\gamma =0,\hfill \\ & & (2)\gamma \ne 0,\alpha +\beta -\gamma =0.\hfill \end{array}$$

Similarly,

$$\begin{array}{c}\hfill T^k({\tilde{e}}_1,{\tilde{e}}_2)=\gamma {\tilde{e}}_3,T^k({\tilde{e}}_1,{\tilde{e}}_3)=T^k({\tilde{e}}_2,{\tilde{e}}_3)=0,\end{array}$$

(153)

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^k(T^k({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_j)={\tilde{\rho }}^k(T^k({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_j)={\tilde{\rho }}^k(T^k({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_j)=0,\hfill \end{array}$$

(154)

where $1\le j\le 3$.

Obviously, the following theorem holds:

Theorem 34.

$(G_3,{\nabla }^k,{\tilde{\rho }}^k)$is a quasi-statistical structure.

For $G_4$, we have

$$\begin{array}{c}\hfill T^c({\tilde{e}}_1,{\tilde{e}}_2)=(\beta -2\eta ){\tilde{e}}_3,T^c({\tilde{e}}_1,{\tilde{e}}_3)=(\beta -n_3){\tilde{e}}_2-{\tilde{e}}_3,T^c({\tilde{e}}_2,{\tilde{e}}_3)=(n_3-\alpha ){\tilde{e}}_1,\end{array}$$

(155)

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_1)=0,{\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_2)=\frac{(\beta -2\eta )(n_3-\beta )}{2},{\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_3)=0,\hfill \\ \hfill & {\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_1)=0,{\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_2)=(\beta -n_3)[(2\eta -\beta )n_3-\frac{1}{2}],\hfill \\ \hfill & {\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_1)=-\frac{{(n_3-\beta )}^2}{2},{\tilde{\rho }}^c(T^c({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_1)=(n_3-\alpha )[(2\eta -\beta )n_3-1],\hfill \\ \hfill & {\tilde{\rho }}^c(T^c({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_2)={\tilde{\rho }}^c(T^c({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_3)=0.\hfill \end{array}$$

(156)

Then, if $(G_4,{\nabla }^c,{\tilde{\rho }}^c)$ is a quasi-statistical structure, by (57) and (58), we have the following five equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}(\beta -2\eta )(n_3-\beta )=0\hfill \\ n_3-\beta =0\hfill \\ (\beta -n_3)[2(2\eta -\beta )n_3-1]=0\hfill \\ (2n_3-\beta )(n_3-\beta )=0\hfill \\ \beta -n_3+2(n_3-\alpha )[(2\eta -\beta )n_3-1]=0.\hfill \end{array}\right.\end{array}$$

(157)

By solving (157), we get

Theorem 35.

$(G_4,{\nabla }^c,{\tilde{\rho }}^c)$is a quasi-statistical structure if and only if

$$\begin{array}{ccc}& & (1)\alpha =\beta =2\eta ,\hfill \\ & & (2)\alpha =0,\beta =\eta .\hfill \end{array}$$

Similarly,

$$\begin{array}{c}\hfill T^k({\tilde{e}}_1,{\tilde{e}}_2)=(\beta -2\eta ){\tilde{e}}_3,T^k({\tilde{e}}_1,{\tilde{e}}_3)=T^k({\tilde{e}}_2,{\tilde{e}}_3)=0,\end{array}$$

(158)

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^k(T^k({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_1)=0,{\tilde{\rho }}^k(T^k({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_2)=\frac{\alpha (\beta -2\eta )}{2},{\tilde{\rho }}^k(T^k({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_3)=0,\hfill \\ \hfill & {\tilde{\rho }}^k(T^k({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_j)=0,{\tilde{\rho }}^k(T^k({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_j)=0,\hfill \end{array}$$

(159)

where $1\le j\le 3$.

Then, if $(G_4,{\nabla }^k,{\tilde{\rho }}^k)$ is a quasi-statistical structure, by (57) and (38), we have the following three equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}(2\eta -\beta )(\alpha -2\beta )=0\hfill \\ \alpha -2\beta =0\hfill \\ \alpha \beta =0.\hfill \end{array}\right.\end{array}$$

(160)

By solving (160), we get

Theorem 36.

$(G_4,{\nabla }^k,{\tilde{\rho }}^k)$is a quasi-statistical structure if and only if$\alpha =\beta =0$.

For $G_5$, we have

$$\begin{array}{c}\hfill T^c({\tilde{e}}_1,{\tilde{e}}_2)=0,T^c({\tilde{e}}_1,{\tilde{e}}_3)=-\alpha {\tilde{e}}_1-\frac{\beta +\gamma }{2}{\tilde{e}}_2,T^c({\tilde{e}}_2,{\tilde{e}}_3)=-\frac{\beta +\gamma }{2}{\tilde{e}}_1-\delta {\tilde{e}}_2,\end{array}$$

(161)

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_j)={\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_j)={\tilde{\rho }}^c(T^c({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_j)=0,\hfill \end{array}$$

(162)

where $1\le j\le 3$.

So,

Theorem 37.

$(G_5,{\nabla }^c,{\tilde{\rho }}^c)$is a quasi-statistical structure.

Similarly,

$$\begin{array}{c}\hfill T^k({\tilde{e}}_1,{\tilde{e}}_2)=T^k({\tilde{e}}_1,{\tilde{e}}_3)=T^k({\tilde{e}}_2,{\tilde{e}}_3)=0,\end{array}$$

(163)

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^k(T^k({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_j)={\tilde{\rho }}^k(T^k({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_j)={\tilde{\rho }}^k(T^k({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_j)=0,\hfill \end{array}$$

(164)

where $1\le j\le 3$.

Obviously,

Theorem 38.

$(G_5,{\nabla }^k,{\tilde{\rho }}^k)$is a quasi-statistical structure.

For $G_6$, we have

$$\begin{array}{c}\hfill T^c({\tilde{e}}_1,{\tilde{e}}_2)=-\beta {\tilde{e}}_3,T^c({\tilde{e}}_1,{\tilde{e}}_3)=-\frac{\beta +\gamma }{2}{\tilde{e}}_2-\delta {\tilde{e}}_3,T^c({\tilde{e}}_2,{\tilde{e}}_3)=\frac{\beta -\gamma }{2}{\tilde{e}}_1,\end{array}$$

(165)

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_1)=0,{\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_2)=\frac{\beta }{2}[\alpha \gamma -\frac{\delta }{2}(\beta -\gamma )],{\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_3)=0,\hfill \\ \hfill & {\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_1)=0,{\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_2)=\frac{\beta +\gamma }{2}[{\alpha }^2-\frac{1}{2}\beta (\beta -\gamma )]+\frac{\delta }{2}[\alpha \gamma -\frac{\delta }{2}(\beta -\gamma )],\hfill \\ \hfill & {\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_3)=\frac{\beta +\gamma }{4}[\alpha \gamma -\frac{\delta }{2}(\beta -\gamma )],{\tilde{\rho }}^c(T^c({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_1)=\frac{\beta -\gamma }{2}[\frac{\beta }{2}(\beta -\gamma )-{\alpha }^2],\hfill \\ \hfill & {\tilde{\rho }}^c(T^c({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_1)={\tilde{\rho }}^c(T^c({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_3)=0.\hfill \end{array}$$

(166)

Then, if $(G_6,{\nabla }^c,{\tilde{\rho }}^c)$ is a quasi-statistical structure, by (57) and (58), we have the following five equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\beta [2\alpha \gamma -\delta (\beta -\gamma )]=0\hfill \\ \alpha [2\alpha \gamma -\delta (\beta -\gamma )]=0\hfill \\ (\beta +\gamma )[2{\alpha }^2-\beta (\beta -\gamma )]+\delta [2\alpha \gamma -\delta (\beta -\gamma )]=0\hfill \\ \gamma [2\alpha \gamma -\delta (\beta -\gamma )]=0\hfill \\ \alpha [2\alpha \gamma -\delta (\beta -\gamma )]+(\beta -\gamma )[2{\alpha }^2-\beta (\beta -\gamma )]=0.\hfill \end{array}\right.\end{array}$$

(167)

By solving (167), we obtain

Theorem 39.

$(G_6,{\nabla }^c,{\tilde{\rho }}^c)$is a quasi-statistical structure if and only if

$$\begin{array}{ccc}& & (1)\alpha =\beta =\gamma =0,\delta \ne 0,\hfill \\ & & (2)\alpha \ne 0,\delta =\gamma =0,2{\alpha }^2={\beta }^2,\hfill \\ & & (3)\alpha \ne 0,\beta =\gamma =0.\hfill \end{array}$$

Similarly,

$$\begin{array}{c}\hfill T^k({\tilde{e}}_1,{\tilde{e}}_2)=-\beta {\tilde{e}}_3,T^k({\tilde{e}}_1,{\tilde{e}}_3)=T^k({\tilde{e}}_2,{\tilde{e}}_3)=0,\end{array}$$

(168)

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^k(T^k({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_j)={\tilde{\rho }}^k(T^k({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_j)={\tilde{\rho }}^k(T^k({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_j)=0,\hfill \end{array}$$

(169)

where $1\le j\le 3$.

Then, if $(G_6,{\nabla }^k,{\tilde{\rho }}^k)$ is a quasi-statistical structure, by (57) and (58), we have the following two equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\alpha \beta \gamma =0\hfill \\ {\alpha }^2\gamma =0.\hfill \end{array}\right.\end{array}$$

(170)

By solving (170), we obtain

Theorem 40.

$(G_6,{\nabla }^k,{\tilde{\rho }}^k)$is a quasi-statistical structure if and only if

$$\begin{array}{ccc}& & (1)\alpha =\beta =0,\delta \ne 0,\hfill \\ & & (2)\alpha \ne 0,\beta \delta =\gamma =0.\hfill \end{array}$$

For $G_7$, we have

$$\begin{array}{c}\hfill T^c({\tilde{e}}_1,{\tilde{e}}_2)=\beta {\tilde{e}}_3,T^c({\tilde{e}}_1,{\tilde{e}}_3)=-\alpha {\tilde{e}}_1-\frac{\gamma }{2}{\tilde{e}}_2-\beta {\tilde{e}}_3,T^c({\tilde{e}}_2,{\tilde{e}}_3)=-(\beta +\frac{\gamma }{2}){\tilde{e}}_1-\delta {\tilde{e}}_2-\delta {\tilde{e}}_3,\end{array}$$

(171)

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_1)=-\frac{\beta }{2}(\alpha \gamma +\frac{\delta \gamma }{2}),{\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_2)=\frac{\beta }{2}({\alpha }^2+\frac{\beta \gamma }{2}),{\tilde{\rho }}^B(T^B({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_3)=0,\hfill \\ \hfill & {\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_1)={\alpha }^3+\alpha \beta \gamma +\frac{\beta \delta \gamma }{4},{\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_2)=(\frac{{\alpha }^2}{2}+\frac{\beta \gamma }{4})(\gamma -\beta ),\hfill \\ \hfill & {\tilde{\rho }}^c(T^c({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_1)=\frac{\alpha \gamma (\alpha +\delta )}{4}-\frac{\beta {\gamma }^2}{8}),{\tilde{\rho }}^c(T^c({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_1)=(\beta +\frac{\gamma }{2})({\alpha }^2+\frac{\beta \gamma }{2})+\frac{\delta \gamma }{2}(\alpha +\frac{\delta }{2}),\hfill \\ \hfill & {\tilde{\rho }}^c(T^c({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_1)=\frac{\delta }{2}({\alpha }^2+\frac{\beta \gamma }{2}),{\tilde{\rho }}^c(T^c({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_3)=\frac{\alpha \beta \gamma -{\alpha }^2\delta }{2}+\frac{\alpha {\gamma }^2}{4}+\frac{\delta {\gamma }^2}{8}.\hfill \end{array}$$

(172)

Then, if $(G_7,{\nabla }^c,{\tilde{\rho }}^c)$ is a quasi-statistical structure, by (57) and (58), we have the following nine equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\beta \gamma (2\alpha +\delta )=0\hfill \\ \beta (2{\alpha }^2+\beta \gamma )=0\hfill \\ \alpha \gamma (2\alpha +\delta )=0\hfill \\ 2{\alpha }^3+3\alpha \beta \gamma +\beta \delta \gamma =0\hfill \\ (\gamma -\beta )(4{\alpha }^2+\beta \gamma )-\alpha \gamma (2\alpha +\delta )=0\hfill \\ (\gamma -2\beta )(2{\alpha }^2+\beta \gamma )+2(\alpha \delta \gamma +{\alpha }^2\gamma )-\beta {\gamma }^2=0\hfill \\ \delta \gamma (2\alpha +\delta )+(\gamma -\beta )(2{\alpha }^2+\beta \gamma )=0\hfill \\ \delta (2{\alpha }^2+\beta \gamma )-\beta \gamma (2\alpha +\delta )=0\hfill \\ 4(\alpha \beta \gamma -{\alpha }^2\delta )+2\alpha {\gamma }^2+\delta {\gamma }^2-(2\beta -\gamma )(2\alpha \gamma +\delta \gamma )=0.\hfill \end{array}\right.\end{array}$$

(173)

By solving (173), we obtain

Theorem 41.

$(G_7,{\nabla }^c,{\tilde{\rho }}^c)$is a quasi-statistical structure if and only if$\delta \ne 0,\alpha =\gamma =0$.

Similarly,

$$\begin{array}{c}\hfill T^k({\tilde{e}}_1,{\tilde{e}}_2)=\beta {\tilde{e}}_3,T^k({\tilde{e}}_1,{\tilde{e}}_3)=T^k({\tilde{e}}_2,{\tilde{e}}_3)=0,\end{array}$$

(174)

$$\begin{array}{cc}\hfill & {\tilde{\rho }}^k(T^k({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_1)={\beta }^2(\alpha +\delta ),{\tilde{\rho }}^k(T^k({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_2)=\frac{\beta }{2}(\alpha \delta +\beta \gamma +2{\delta }^2),{\tilde{\rho }}^k(T^k({\tilde{e}}_1,{\tilde{e}}_2),{\tilde{e}}_3)=0,\hfill \\ \hfill & {\tilde{\rho }}^k(T^k({\tilde{e}}_1,{\tilde{e}}_3),{\tilde{e}}_j)=0,{\tilde{\rho }}^k(T^k({\tilde{e}}_2,{\tilde{e}}_3),{\tilde{e}}_j)=0,\hfill \end{array}$$

(175)

where $1\le j\le 3$.

Then, if $(G_7,{\nabla }^k,{\tilde{\rho }}^k)$ is a quasi-statistical structure, by (57) and (58), we have the following nine equations:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\beta (\alpha \gamma +\beta \delta +2\beta \delta )=0\hfill \\ \beta (2{\delta }^2-2{\alpha }^2-2{\beta }^2+3\alpha \delta -\beta \gamma )=0\hfill \\ \beta (\alpha +\delta )=0\hfill \\ 4{\alpha }^3-2\alpha {\delta }^2-4{\beta }^2\delta -(\alpha \beta \gamma +{\alpha }^2\delta )=0\hfill \\ \beta (2{\alpha }^2+2{\beta }^2+\beta \gamma +{\delta }^2)+2{\alpha }^2\gamma =0\hfill \\ 2\alpha \beta (\alpha +\delta )+\beta (\beta \gamma +\alpha \delta +2{\delta }^2)=0\hfill \\ 2({\alpha }^2\gamma +{\beta }^3)+3{\alpha }^2\beta +{\beta }^2\gamma -3\alpha \beta \delta -5\beta {\delta }^2=0\hfill \\ 2(\alpha \beta \gamma -{\delta }^3+3{\beta }^2\delta +2{\alpha }^2\delta )+\beta \delta \gamma -\alpha {\delta }^2=0\hfill \\ 2(\alpha \beta \gamma +{\delta }^3)+3\beta \delta \gamma +\alpha {\delta }^2=0.\hfill \end{array}\right.\end{array}$$

(176)

By solving (176), we obtain $\alpha =\delta =0$, and there is a contradiction. So,

Theorem 42.

$(G_7,{\nabla }^k,{\tilde{\rho }}^k)$is not a quasi-statistical structure.

6. Conclusions

It is clearly shown in the above two tables that there are Codazzi tensors associated with affine connections on three-dimensional Lorentzian Lie groups and there is a quasi-statistical structure associated with affine connections on three-dimensional Lorentzian Lie groups.

Table 1. Codazzi tensors associated with affine connections on three-dimensional Lorentzian Lie groups.

Codazzi Tensors	Bott Connections	Canonical Connections	Kobayashi–Nomizu Connections
$G_1$	No solution	No solution	No solution
$G_2$	No solution	$\gamma \ne 0,\alpha =2\beta$	$\gamma \ne 0,\alpha =\beta =0$
$G_3$	Permanent establishment	Permanent establishment	Permanent establishment
$G_4$	$\alpha =\beta =0$	$\frac{\alpha }{2}+\eta -\beta =0$	$\alpha =\beta =0$
$G_5$	Permanent establishment	Permanent establishment	Permanent establishment
$G_6$	(1) $\alpha =\beta =0,\delta \ne 0$ (2) $\alpha \ne 0,\beta \delta =\gamma =0$	(1) $\alpha =\beta =\gamma =0,\delta \ne 0$ (2) $\alpha \ne 0,\beta =\gamma =0,\alpha +\delta \ne 0$ (3) $\alpha \ne 0,\delta =\gamma =0$	(1) $\alpha =\beta =0,\delta \ne 0$ (2) $\alpha \ne 0,\beta \delta =\gamma =0,\alpha +\delta \ne 0$
$G_7$	No solution	$\alpha =\gamma =0,\delta \ne 0$	No solution

shows the conditions that Ricci tensors associated with Bott connections, canonical connections and Kobayashi–Nomizu connections are Codazzi tensors associated with Bott connections, canonical connections and Kobayashi–Nomizu connections on ${\{G_i\}}_{i=1,\dots ,7}$. For Bott connections, there is a contradiction on $G_1,G_2,G_7$, there is a permanent establishment on $G_3,G_5$ and there are corresponding conditions on $G_4,G_6$. For canonical connections, there is a contradiction on $G_1$, there is a permanent establishment on $G_3,G_5$ and there are corresponding conditions on $G_2,G_4,G_6,G_7$. For Kobayashi–Nomizu connections, there is a contradiction on $G_1,G_7$, there is a permanent establishment on $G_3,G_5$ and there are corresponding conditions on $G_2,G_4,G_6$.

As is shown in

Table 2. The quasi-statistical structure associated with affine connections on three-dimensional Lorentzian Lie groups.

Quasi-Statistical Structure	Bott Connections	Canonical Connections	Kobayashi–Nomizu Connections
$G_1$	No solution	No solution	No solution
$G_2$	$\alpha =\beta =0,\gamma \ne 0$	$\gamma \ne 0,\alpha =\beta =0$	$\gamma \ne 0,\alpha =\beta =0$
$G_3$	(1) $\gamma =0$ (2) $\gamma \ne 0,\alpha +\beta -\gamma =0$	Permanent establishment	Permanent establishment
$G_4$	$\alpha =\beta =0$	(1) $\alpha =\beta =2\eta$ (2) $\alpha =0,\beta =\eta$	$\alpha =\beta =0$
$G_5$	Permanent establishment	Permanent establishment	Permanent establishment
$G_6$	(1) $\alpha =\beta =0,\delta \ne 0$ (2) $\alpha \ne 0,\beta \delta =\gamma =0$	(1) $\alpha =\beta =\gamma =0,\delta \ne 0$ (2) $\alpha \ne 0,\delta =\gamma =0,2{\alpha }^2={\beta }^2$ (3) $\alpha \ne 0,\beta =\gamma =0$	(1) $\alpha =\beta =0,\delta \ne 0$ (2) $\alpha \ne 0,\beta \delta =\gamma =0,\alpha +\delta \ne 0$
$G_7$	No solution	$\alpha =\gamma =0,\delta \ne 0$	No solution

, we can obtain ${\{G_i\}}_{i=1,\dots ,7}$ with the quasi-statistical structure associated with Bott connections, canonical connections and Kobayashi–Nomizu connections. For Bott connections, there is a contradiction on $G_1,G_7$, there is a permanent establishment on $G_5$ and there are corresponding conditions on $G_2,G_3,G_4,G_6$. For canonical connections, there is a contradiction on $G_1$, there is a permanent establishment on $G_3,G_5$ and there are corresponding conditions on $G_2,G_4,G_6,G_7$. For Kobayashi–Nomizu connections, there is a contradiction on $G_1,G_7$, there is a permanent establishment on $G_3,G_5$ and there are corresponding conditions on $G_2,G_4,G_6$.

The two tables have something in common in that there are three kinds of situations for three different affine connections on seven connected or simply connected three-dimensional Lorentzian Lie groups. The main difference is Bott connections. The results of no solution and constancy in Table 1 are more than those in Table 2. Therefore, we know that for three kinds of connections, torsion tensor has a stronger effect on Bott connections than the other two.