CL-Transformation on 3-Dimensional Quasi Sasakian Manifolds and Their Ricci Soliton

Abstract

This paper explores the geometry of 3-dimensional quasi Sasakian manifolds under CL-transformations. We construct both infinitesimal and $CL$-transformation and demonstrate that the former does not necessarily yield projective killing vector fields. A novel invariant tensor, termed the $CL$-curvature tensor, is introduced and shown to remain invariant under $CL$-transformations. Utilizing this tensor, we characterize $CL$-flat, $CL$-symmetric, $CL$-$\phi$ symmetric and $CL$-$\phi$ recurrent structures on such manifolds by mean of differential equations. Furthermore, we investigate conditions under which a Ricci soliton exists on a CL-transformed quasi Sasakian manifold, revealing that under flat curvature, the structure becomes Einstein. These findings contribute to the understanding of curvature dynamics and soliton theory within the context of contact metric geometry.

1. Introduction

A curve on a unit sphere that intersects meridians at a fixed angle is called a loxodrome. Loxodromes were mainly used in navigation and are usually called rhumb lines. Tashiro and Tachibana [1] introduced the notion of C-loxodrome in an almost contact manifold with an affine connection, which is an extension of the loxodrome. C-loxodrome is a loxodrome that intersects the geodesic trajectories of the characteristic vector field on a Sasakian manifold at a constant angle. It is important to note that a conformal transformation locally preserves angles but not necessarily lengths. It also preserves the shape of small figures but neither their size nor curvature, and the conformal curvature tensor is the invariant of such a transformation [2]. In 1963, $CL$-transformation was introduced by Tashiro and Tachibana [1] on a Sasakian manifold, in which they observed that the angle between two C-loxodromes is constant under a $CL$-transformation, which is where the name ‘$CL$’ originated from. The consistency of the $CL$-transformation depends on the particular features of the manifold. Koto and Nagao [3] obtained a tensor field on a Sasakian manifold, which is invariant under a $CL$-transformation. Also, in 1966, the infinitesimal $CL$-transformation on compact Sasakian manifolds was studied by Takamatsu and Mizusawa [4]. In 1963, $CL$-transformation was introduced by Tashiro and Tachibana [1] on a88, Matsumoto and Mihai [5] obtained a tensor field on a Sasakian manifold, which is invariant under a $CL$-transformation. Also, in 1966, the infinitesimal $CL$-transformation on compact Sasakian manifolds was studied by Takamatsu and Mizusawa [4] and they observed that such transformation is necessarily projective. Furthermore, in 1988, Matsumoto and Mihai [5] studied an invariant tensor field under a $CL$-t-transformation on an LP-Sasakian manifold and found numerous other interesting results. Recently, in 2014, Shaikh and Ahmad [6] carried out an investigation on infinitesimal $CL$-transformation and $CL$-transformation on Lorentzian concircular structure manifolds. They obtained a new invariant tensor field and interesting results.

The notion of a Ricci soliton can be viewed as a natural expansion of Einstein manifolds. The concept of Ricci flow was first introduced by Hamilton [7] with the objective of establishing a canonical metric on a smooth manifold. Ricci flow has proven to be a potent tool in the examination of Riemannian manifolds, particularly those with positive curvature. The Ricci flow, which is defined as a differential equation governing the evolution of metrics on a Riemannian manifold, can be expressed as $\frac{\partial g}{\partial t}=-2\ddot{S}$. Essentially, a Ricci soliton can be understood as a form of Ricci flow that undergoes transformations exclusively through a one-parameter group of diffeomorphisms and scaling. Perelman [8,9] utilized Ricci flow to validate the Poincare conjecture. A Ricci soliton $(g,V_0,\gamma )$ on a Riemannian manifold $(M^{(2n+1)},g)$ is characterized by differential equations [10,11,12], such as the following:

$$L_{V_0}g+2\ddot{S}+2\lambda g=0,$$

(1)

in which $L_{V_0}$ is the Lie derivative along $V_0$ on M, S denotes the Ricci tensor, and $\lambda$ represents a constant. The Ricci soliton is categorized as shrinking, steady, or expanding based on whether $\lambda <0$, $\lambda =0$ or $\lambda >0$, respectively. Sharma [13] conducted an in-depth investigation into Ricci solitons in contact geometry, while Ghosh et al. [14] explored gradient Ricci solitons on non-Sasakian $(\kappa ,\mu )$-contact manifolds. Shigeo Sasaki [15] first coined the Sasakian manifold in 1960, and since then, numerous geometers studied the Ricci solion of Sasakian manifold, which can be seen in [11,16,17,18,19,20]. Recently, several authors studied the Ricci solitons of 3-dimensional quasi Sasakian manifolds, which can be seen in [21,22,23,24,25]. Motivated by their studies, we carried out a new investigation regarding the Ricci soliton of the $CL$-transformation on 3-dimensional quasi Sasakian manifolds, as detailed in Section 7.

A quasi Sasakian manifold is a natural extension of a Sasakian manifold, a concept introduced by Blair [26] to unify Sasakian and cosymplectic structures. The characteristics of quasi Sasakian manifolds have been explored by several researchers, including Gonzalez and Chinea [27], Kanemaki [28,29], De and Sarkar [30], De et al. [31,32,33], Chaturvedi et al. [34], and Turan et al. [35], among others. In a 3-dimensional quasi Sasakian manifold, Olszak [36] defined the structure function $\beta$ and used it to establish the necessary and sufficient conditions for the manifold to be conformally flat [37]. He further demonstrated that if the manifold is conformally flat with a constant $\beta$, then either (a) the manifold is locally a product of R and a 2-dimensional Kählerian space with constant Gauss curvature (the cosymplectic case) or (b) the manifold has constant positive curvature (the non-cosymplectic case, where the quasi Sasakian structure is homothetic to a Sasakian structure).

We organized this paper as follows: Section 1 presents the history and introduction of $CL$-transformation and briefly outlines notable contributions by various geometers. Section 2 covers the preliminaries of 3-dimensional quasi Sasakian manifolds and highlights their key properties. In Section 3 and Section 4, we examine the infinitesimal $CL$-transformation and $CL$-transformation on a 3-dimensional quasi Sasakian manifold. Our results show that an infinitesimal $CL$-transformation does not represent a projective killing vector field unless $\alpha =0$ and $\ddot{r}=6{\beta }^2$. Additionally, we identify a new invariant tensor field A under $CL$-transformation. Using this invariant tensor A, we investigate the $CL$-flat and $CL$-symmetric conditions on a 3-dimensional quasi Sasakian manifold in Section 5. In Section 6, we study the $CL$-$\phi$ symmetric and $CL$-$\phi$ recurrent conditions on 3-dimensional quasi Sasakian manifolds. Finally, in Section 7, we explore the Ricci soliton under the $CL$-transformation of a 3-dimensional quasi Sasakian manifold, and present concluding remarks in Section 8.

2. Preliminaries

Let M be a connected almost contact metric manifold of odd dimensional $(2n+1)$ with an almost contact metric structure $(\phi ,\xi ,\eta ,g)$, where $\phi$ is a tensor field of type (1, 1), $\xi$ is a vector field, $\eta$ is a 1-form, and g is a Riemannian metric on M. The structure satisfies [38,39,40]

$$\begin{array}{c}\hfill {\phi }^2\left(X_0\right)=-X_0+\eta \left(X_0\right)\xi ,\eta \left(\xi \right)=1,\eta \left(\phi X_0\right)=0,\phi \xi =0,\end{array}$$

(2)

$$g(\phi X_0,Y_0)=-g(X_0,\phi Y_0),$$

(3)

$$g(\phi X_0,\phi Y_0)=g(X_0,Y_0)-\eta \left(X_0\right)\eta \left(Y_0\right),$$

(4)

for all vector fields $X_0$ and $Y_0$ on M. Blair [26] was the first who introduced quasi Sasakian manifold, stating that an odd dimensional differentiable manifold M, endowed with an almost contact metric structure $(\phi ,\xi ,\eta ,g)$, is said to be a quasi Sasakian manifold if the almost contact structure $(\phi ,\xi ,\eta ,g)$ is normal and the fundamental 2-form $\mathrm{\Phi }$ defined by $\mathrm{\Phi }(X_0,Y_0)=g(X_0,\phi Y_0)$ is closed, i.e., $d\mathrm{\Phi }=0$. Specifically, a 3-dimensional almost contact metric manifold M is quasi Sasakian if [41]

$${\ddot{\nabla }}_{X_0}\xi =-\beta \phi X_0,$$

(5)

for some certain function $\beta$ on M, where $\xi \beta =0$, and $\ddot{\nabla }$ is the Levi-Civita connection on M. A 3-dimensional quasi Sasakian manifold is cosymplectic if and only if $\beta =0$. If $\beta$ is a constant, the manifold is referred to as a $\beta$-Sasakian manifold, and $\beta =1$ gives the Sasakian structure. Throughout this paper, it is assumed that $\beta$ is constant. For a 3-dimensional quasi Sasakian manifold, the following holds [36,38,42]:

$$\begin{array}{c}\hfill \left({\ddot{\nabla }}_{X_0}\phi \right)Y_0=-\beta (g(X_0,Y_0)\xi -\eta \left(Y_0\right)X_0),\end{array}$$

(6)

$$\left({\ddot{\nabla }}_{X_0}\eta \right)Y_0=-\beta g(\phi X_0,Y_0),$$

(7)

$$\begin{array}{cc}\hfill \ddot{R}(X_0,Y_0)Z_0& =\left({\displaystyle \frac{\ddot{r}}{2}}-2{\beta }^2\right)(g(Y_0,Z_0)X_0-g(X_0,Z_0)Y_0)\hfill \\ \hfill & +\left(3{\beta }^2-{\displaystyle \frac{\ddot{r}}{2}}\right)(g(Y_0,Z_0)\eta \left(X_0\right)\xi \hfill \\ \hfill & -g(X_0,Z_0)\eta \left(Y_0\right)\xi +\eta \left(Y_0\right)\eta \left(Z_0\right)X_0\hfill \\ \hfill & -\eta \left(X_0\right)\eta \left(Z_0\right)Y_0),\hfill \end{array}$$

(8)

$$\ddot{S}(X_0,Y_0)=\left({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2\right)g(X_0,Y_0)-\left(3{\beta }^2-{\displaystyle \frac{\ddot{r}}{2}}\right)\eta \left(X_0\right)\eta \left(Y_0\right),$$

(9)

$$\eta \left(\ddot{R}(X_0,Y_0)Z_0\right)={\beta }^2\left(g(Y_0,Z_0)\eta \left(X_0\right)-g(X_0,Z_0)\eta \left(Y_0\right)\right),$$

(10)

$$\begin{array}{c}\hfill \ddot{R}(\xi ,Y_0)Z_0={\beta }^2\left(g(Y_0,Z_0)\xi -\eta \left(Z_0\right)Y_0\right),\end{array}$$

(11)

$$\ddot{S}(X_0,\xi )=(\ddot{r}-4{\beta }^2)\eta \left(X_0\right),$$

(12)

$$\left(L_{\xi }g\right)(X_0,Y_0)=g({\ddot{\nabla }}_{X_0}\xi ,Y_0)+g(X_0,{\ddot{\nabla }}_{Y_0}\xi ),$$

(13)

for all $X_0,Y_0,Z_0$ on M.

Definition 1. 

A quasi Sasakian manifold is termed an almost η-Einstein if its non-vanishing Ricci tensor $\ddot{S}$ is expressed as

$$\ddot{S}(X_0,Y_0)=ag(X_0,Y_0)+b\eta \left(X_0\right)\eta \left(Y_0\right),$$

where a and b are smooth functions on the manifold M, and if a and b are constants, then it would be called an η-Einstein manifold. Also, if $b=0$, then the manifold is called an Einstein manifold [38].

From now on, we will denote a 3-dimensional quasi Sasakian manifold as a 3-D quasi Sasakian manifold.

3. Infinitesimal CL-Transformation on 3-D Quasi Sasakian Manifold

Definition 2. 

A vector field $V_0$ on a 3-D quasi Sasakian manifold M is said to be an infinitesimal CL-transformation [4] if it satisfies

$$L_{V_0}\left(\begin{array}{c}h\\ ji\end{array}\right)={\rho }_j{\delta }_i^h+{\rho }_i{\delta }_j^h+\alpha ({\eta }_j{\phi }_i^h+{\eta }_i{\phi }_j^h)$$

(14)

for a certain constant α, where ${\rho }_i$ is the component of a 1-form ρ, $L_{V_0}$ denotes the Lie derivative with respect to vector field $V_0$ and $\left(\begin{array}{c}h\\ ji\end{array}\right)$ is the Christoffel symbol of the Riemannian metric g.

Proposition 1. 

If $V_0$ is an infinitesimal CL-transformation on a 3-D quasi Sasakian manifold, then the 1-form ρ is closed.

Proof. 

Contracting Equation (14) with h and j, we can observe that ${\rho }_i$ is a gradient. Hence, the 1-form $\rho$ is closed. □

Theorem 1. 

If $V_0$ is an infinitesimal CL-transformation on a 3-D quasi Sasakian manifold M and the scalar curvature $r=6{\beta }^2$, then the relation

$${\beta }^2\left(L_{V_0}g\right)(Y_0,Z_0)=-\left({\ddot{\nabla }}_{Y_0}\rho \right)\left(Z_0\right)+\alpha \left\{\beta (g(Y_0,Z_0)-\eta \left(Y_0\right)\eta \left(Z_0\right))\right\}$$

(15)

holds for any vector fields $Y_0$ and $Z_0$ on M, where α is constant.

Proof. 

It is known from [43,44] that

$$L_{V_0}{\ddot{R}}_{kji}^h={\ddot{\nabla }}_k{L}_{V_0}\left(\begin{array}{c}h\\ ji\end{array}\right)-{\ddot{\nabla }}_j{L}_{V_0}\left(\begin{array}{c}h\\ ki\end{array}\right).$$

(16)

Employing Equations (6), (7), and (14) in the above equation, we get

$$\begin{array}{cc}\hfill \left(L_{V_0}\ddot{R}\right)(X_0,Y_0)Z_0& =\left({\ddot{\nabla }}_{X_0}\rho \right)\left(Z_0\right)Y_0-\left({\ddot{\nabla }}_{Y_0}\rho \right)\left(Z_0\right)X_0\hfill \\ \hfill & +\alpha \beta [\left(g(\phi Y_0,Z_0)\phi X_0-g(\phi X_0,Z_0)\phi Y_0\right)\hfill \\ \hfill & -2g(\phi X_0,Y_0)\phi Z_0-(\eta \left(Y_0\right)g(X_0,Z_0)\hfill \\ \hfill & -\eta \left(X_0\right)g(Y_0,Z_0)\left)\xi \right],\hfill \end{array}$$

(17)

for any vector fields $X_0,Y_0$, and $Z_0$ on M. Operating $\eta$ in the above equation, we get

$$\begin{array}{cc}\hfill \eta \left(\left(L_{V_0}\ddot{R}\right)(X_0,Y_0)Z_0\right)& =\left({\ddot{\nabla }}_{X_0}\rho \right)\left(Z_0\right)\eta \left(Y_0\right)-\left({\ddot{\nabla }}_{Y_0}\rho \right)\left(Z_0\right)\eta \left(X_0\right)\hfill \\ \hfill & -\alpha \beta \left[g(X_0,Z_0)\eta \left(Y_0\right)-g(Y_0,Z_0)\eta \left(X_0\right)\right].\hfill \end{array}$$

(18)

First, we take the Lie derivative of Equation (10) with respect to the vector field $V_0$. Then, using the equation stated above and interchanging $X_0$ and $Y_0$ with $Y_0$ and $\xi$, respectively, we get

$$\begin{array}{cc}\hfill {\beta }^2\left(L_{V_0}g\right)(Y_0,Z_0)& =\left({\beta }^2\left(L_{V_0}g\right)(\xi ,Z_0)+\left({\ddot{\nabla }}_{\xi }\rho \right)\left(Z_0\right)\right)\eta \left(Y_0\right)-\left({\ddot{\nabla }}_{Y_0}\rho \right)\left(Z_0\right)\hfill \\ \hfill & +\alpha \beta \left[g(Y_0,Z_0)-\eta \left(Y_0\right)\eta \left(Z_0\right)\right].\hfill \end{array}$$

(19)

Again, interchanging $Y_0$ and $Z_0$ in the above equation, we have

$$\begin{array}{cc}\hfill {\beta }^2\left(L_{V_0}g\right)(Z_0,Y_0)& =\left({\beta }^2\left(L_{V_0}g\right)(\xi ,Y_0)+\left({\ddot{\nabla }}_{\xi }\rho \right)\left(Y_0\right)\right)\eta \left(Z_0\right)-\left({\ddot{\nabla }}_{Z_0}\rho \right)\left(Y_0\right)\hfill \\ \hfill & +\alpha \beta \left[g(Z_0,Y_0)-\eta \left(Z_0\right)\eta \left(Y_0\right)\right].\hfill \end{array}$$

(20)

Subtracting Equation (20) from Equation (19), we get

$$\begin{array}{c}\hfill \left({\beta }^2\left(L_{V_0}g\right)(\xi ,Z_0)+\left({\ddot{\nabla }}_{\xi }\rho \right)\left(Z_0\right)\right)\eta \left(Y_0\right)=\left({\beta }^2\left(L_{V_0}g\right)(\xi ,Y_0)+\left({\ddot{\nabla }}_{\xi }\rho \right)\left(Y_0\right)\right)\eta \left(Z_0\right).\end{array}$$

(21)

Setting $Y_0=\xi$ in the above equation, we get

$$\begin{array}{c}\hfill {\beta }^2\left(L_{V_0}g\right)(\xi ,Z_0)+\left({\ddot{\nabla }}_{\xi }\rho \right)\left(Z_0\right)=\left({\beta }^2\left(L_{V_0}g\right)(\xi ,\xi )+\left({\ddot{\nabla }}_{\xi }\rho \right)\left(\xi \right)\right)\eta \left(Z_0\right).\end{array}$$

(22)

From Equations (22) and (19), we get

$$\begin{array}{cc}\hfill {\beta }^2\left(L_{V_0}g\right)(Y_0,Z_0)& =\left({\beta }^2\left(L_{V_0}g\right)(\xi ,\xi )+\left({\ddot{\nabla }}_{\xi }\rho \right)\left(\xi \right)\right)\eta \left(Y_0\right)\eta \left(Z_0\right)-\left({\ddot{\nabla }}_{Y_0}\rho \right)\left(Z_0\right)\hfill \\ \hfill & +\alpha \beta \left[g(Y_0,Z_0)-\eta \left(Y_0\right)\eta \left(Z_0\right)\right].\hfill \end{array}$$

(23)

Now, we take the inner product of Equation (17) with a vector field $W_0$ on M. Then, by contracting along the vector fields $X_0$ and $W_0$, we obtain

$$\left(L_{V_0}\ddot{S}\right)(Y_0,Z_0)=-2\left({\ddot{\nabla }}_{Y_0}\rho \right)\left(Z_0\right)+\alpha \beta \left[g(Y_0,Z_0)-\eta \left(Y_0\right)\eta \left(Z_0\right)\right].$$

(24)

Setting $Y_0=\xi$ in the above equation, we have

$$\left(L_{V_0}\ddot{S}\right)(\xi ,Z_0)=-2\left({\ddot{\nabla }}_{\xi }\rho \right)\left(Z_0\right).$$

(25)

Taking Lie derivative of (12) with respect to the vector field $V_0$ and using the above equation we get

$$-2\left({\ddot{\nabla }}_{\xi }\rho \right)\left(Z_0\right)=(\ddot{r}-4{\beta }^2)\left(L_{V_0}g\right)(Z_0,\xi ).$$

(26)

Replacing $Z_0=\xi$ in the above equation, we get

$$-2\left({\ddot{\nabla }}_{\xi }\rho \right)\left(\xi \right)=(\ddot{r}-4{\beta }^2)\left(L_{V_0}g\right)(\xi ,\xi ).$$

(27)

Employing the above equation in Equation (23), we get

$$\begin{array}{cc}\hfill {\beta }^2\left(L_{V_0}g\right)(Y_0,Z_0)& =\left[-{\displaystyle \frac{(\ddot{r}-4{\beta }^2)}{2}}\left(L_{V_0}g\right)(\xi ,\xi )+{\beta }^2\left(L_{V_0}g\right)(\xi ,\xi )\right]\eta \left(Y_0\right)\eta \left(Z_0\right)\hfill \\ \hfill & -\left({\ddot{\nabla }}_{Y_0}\rho \right)\left(Z_0\right)+\alpha \beta \left[g(Y_0,Z_0)-\eta \left(Y_0\right)\eta \left(Z_0\right)\right].\hfill \end{array}$$

(28)

Again, setting $\ddot{r}=6{\beta }^2$, we get

$$\begin{array}{c}\hfill {\beta }^2\left(L_{V_0}g\right)(Y_0,Z_0)=-\left({\ddot{\nabla }}_{Y_0}\rho \right)\left(Z_0\right)+\alpha \beta \left[g(Y_0,Z_0)-\eta \left(Y_0\right)\eta \left(Z_0\right)\right].\end{array}$$

(29)

This completes the proof. □

From (29), we can state the following:

Theorem 2. 

An infinitesimal CL-transformation $V_0$ on a 3-D quasi Sasakian manifold M is not a projective killing vector field unless $\alpha =0$ and $\ddot{r}=6{\beta }^2$.

Corollary 1. 

Any infinitesimal CL-transformation $V_0$ on a 3-D quasi Sasakian manifold M is not necessarily a projective killing vector field.

4. CL-Transformation on a 3-D Quasi Sasakian Manifold

Definition 3. 

A transformation f on a 3-D quasi Sasakian manifold M with structure $(\phi ,\xi ,\eta ,g)$ is said to be a CL-transformation if the Levi-Civita connection $\ddot{\nabla }$ and a symmetric affine connection ${\ddot{\nabla }}^f$ induced from $\ddot{\nabla }$ by f are related by [3]

$$\begin{array}{c}\hfill {\ddot{\nabla }}_{X_0}^f{Y}_0={\ddot{\nabla }}_{X_0}{Y}_0+\rho \left(X_0\right)Y_0+\rho \left(Y_0\right)X_0+\alpha \{\eta \left(X_0\right)\phi Y_0+\eta \left(Y_0\right)\phi X_0\},\end{array}$$

(30)

where ρ is a 1-form and α is a constant.

We represent the geometric objects ${\ddot{R}}^f$ and ${\ddot{S}}^f$ as curvature tensor and Ricci tensor, respectively, with respect to the affine connection ${\ddot{\nabla }}^f$.

If f is a $CL$-transformal on a 3-D quasi Sasakian manifold M, then by virtue of (30), (2), (6), and (7), the curvature tensor ${\ddot{R}}^f(X_0,Y_0)Z_0$ of the connection ${\ddot{\nabla }}^f$ is given by

$$\begin{array}{cc}\hfill {\ddot{R}}^f(X_0,Y_0)Z_0& =\ddot{R}(X_0,Y_0)Z_0+\{B(X_0,Y_0)-B(Y_0,X_0)\}{Z}_0+B(X_0,Z_0)Y_0\hfill \\ \hfill & -B(Y_0,Z_0)X_0-\alpha [\beta \{g(\phi Y_0,Z_0)\phi X_0-g(\phi X_0,Z_0)\phi Y_0\}\hfill \\ \hfill & -\{\beta g(Y_0,Z_0)\eta \left(X_0\right)-\beta g(X_0,Z_0)\eta \left(Y_0\right)\}\xi ],\hfill \end{array}$$

(31)

for any vector field $X_0,Y_0$, and $Z_0$ on M, where the tensor field $B(X_0,Y_0)$ is defined by

$$\begin{array}{cc}\hfill B(X_0,Y_0)& =\left({\ddot{\nabla }}_{X_0}\rho \right)\left(Y_0\right)-\rho \left(X_0\right)\rho \left(Y_0\right)-{\alpha }^2\eta \left(X_0\right)\eta \left(Y_0\right)-\alpha \{\eta \left(X_0\right)\rho \left(\phi Y_0\right)\hfill \\ \hfill & +\eta \left(Y_0\right)\rho \left(\phi X_0\right)\}.\hfill \end{array}$$

(32)

From (31), we get

$$\begin{array}{cc}\hfill g\left({\ddot{R}}^f(X_0,Y_0)Z_0,U_0\right)& =g\left(\ddot{R}(X_0,Y_0)Z_0,U_0\right)+\{B(X_0,Y_0)\hfill \\ \hfill & -B(Y_0,X_0)\}g(Z_0,U_0)+B(X_0,Z_0)g(Y_0,U_0)\hfill \\ \hfill & -B(Y_0,Z_0)g(X_0,U_0)-\alpha \left[\beta \right\{g(\phi Y_0,Z_0)g(\phi X_0,U_0)\hfill \\ \hfill & -g(\phi X_0,Z_0)g(\phi Y_0,U_0)\}-\beta \{g(Y_0,Z_0)\eta \left(X_0\right)\hfill \\ \hfill & -g(X_0,Z_0)\eta \left(Y_0\right)\}g(\xi ,U_0)],\hfill \end{array}$$

(33)

where $U_0$ is any vector field on M. By contracting Equation (33) first over $X_0$ and $U_0$, and then over $Z_0$ and $U_0$, and subsequently adding the resulting expressions, we follow the method outlined in [3]. This allows us to verify that $B(X_0,Y_0)$ is symmetric. Hence, Equation (32) implies that the 1-form $\rho$ is closed.

Theorem 3. 

Let A be the tensor field of $(1,3)$-type, given by

$$\begin{array}{cc}\hfill A(X_0,Y_0)Z_0& =\ddot{R}(X_0,Y_0)Z_0-{\displaystyle \frac{1}{2}}[\left(\ddot{S}(Y_0,Z_0)X_0-\ddot{S}(X_0,Z_0)Y_0\right)\hfill \\ \hfill & -({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)g(Y_0,Z_0)\eta \left(X_0\right)\xi +({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)g(X_0,Z_0)\eta \left(Y_0\right)\xi \hfill \\ \hfill & +({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)g(\phi Y_0,Z_0)\phi X_0-({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)g(\phi X_0,Z_0)\phi Y_0]\hfill \\ \hfill & -{\beta }^2\left[g(Y_0,Z_0)\eta \left(X_0\right)\xi -g(X_0,Z_0)\eta \left(Y_0\right)\xi \right]+{\beta }^2[g(\phi Y_0,Z_0)\phi X_0\hfill \\ \hfill & -g(\phi X_0,Z_0)\phi Y_0],\hfill \end{array}$$

(34)

where $X_0$ and $Y_0$ are vector fields on M. This tensor field A is invariant under CL-transformation and is said to be the CL-curvature tensor field on a 3-D quasi Sasakian manifold M.

Proof. 

Assume that f is a $CL$-transformation on a 3-D quasi Sasakian manifold M. Then, the relations (30)–(33) hold. Since the tensor $B(X_0,Y_0)$ is symmetric, Equation (31) can be rewritten as

$$\begin{array}{cc}\hfill {\ddot{R}}^f(X_0,Y_0)Z_0& =\ddot{R}(X_0,Y_0)Z_0+B(X_0,Z_0)Y_0-B(Y_0,Z_0)X_0\hfill \\ \hfill & -\alpha [\beta \{g(\phi Y_0,Z_0)\phi X_0-g(\phi X_0,Z_0)\phi Y_0\}-\beta \{g(Y_0,Z_0)\eta \left(X_0\right)\hfill \\ \hfill & -g(X_0,Z_0)\eta \left(Y_0\right)\left\}\xi \right],\hfill \end{array}$$

(35)

which yields

$$\begin{array}{c}\hfill 2B(Y_0,Z_0)=\ddot{S}(Y_0,Z_0)-{\ddot{S}}^f(Y_0,Z_0).\end{array}$$

(36)

Substituting Equation (36) into (35), we get

$$\begin{array}{cc}\hfill {\ddot{R}}^f(X_0,Y_0)Z_0& =\ddot{R}(X_0,Y_0)Z_0+{\displaystyle \frac{1}{2}}\ddot{S}(X_0,Z_0)Y_0-{\displaystyle \frac{1}{2}}{\ddot{S}}^f(X_0,Z_0)Y_0\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}\ddot{S}(Y_0,Z_0)X_0+{\displaystyle \frac{1}{2}}{\ddot{S}}^f(Y_0,Z_0)X_0-\alpha \left[\beta \right\{g(\phi Y_0,Z_0)\phi X_0\hfill \\ \hfill & -g(\phi X_0,Z_0)\phi Y_0\}-\beta \{g(Y_0,Z_0)\eta \left(X_0\right)-g(X_0,Z_0)\eta \left(Y_0\right)\}\xi ].\hfill \end{array}$$

(37)

and

$$\begin{array}{cc}\hfill {\ddot{P}}^f(X_0,Y_0)Z_0& =\ddot{P}(X_0,Y_0)Z_0-\alpha [\beta \{g(\phi Y_0,Z_0)\phi X_0-g(\phi X_0,Z_0)\phi Y_0\}\hfill \\ \hfill & -\beta \{g(Y_0,Z_0)\eta \left(X_0\right)-g(X_0,Z_0)\eta \left(Y_0\right)\}\xi ],\hfill \end{array}$$

(38)

where $\ddot{P}$ is the projective curvature tensor given by [2]

$$\begin{array}{c}\hfill \ddot{P}(X_0,Y_0)Z_0=\ddot{R}(X_0,Y_0)Z_0-{\displaystyle \frac{1}{2}}\{\ddot{S}(Y_0,Z_0)X_0-\ddot{S}(X_0,Z_0)Y_0\}.\end{array}$$

(39)

Replacing $X_0$ to $\xi$ in Equation (38), we have

$$\begin{array}{c}\hfill {\ddot{P}}^f(\xi ,Y_0)Z_0=\ddot{P}(\xi ,Y_0)Z_0+\alpha \beta \left[g(Y_0,Z_0)-\eta \left(Y_0\right)\eta \left(Z_0\right)\right]\xi .\end{array}$$

(40)

Operating $\eta$ on both sides of the above equation, we get

$$\begin{array}{c}\hfill \eta \left({\ddot{P}}^f(\xi ,Y_0)Z_0\right)-\eta \left(\ddot{P}(\xi ,Y_0)Z_0\right)=\alpha \beta \left[g(Y_0,Z_0)-\eta \left(Y_0\right)\eta \left(Z_0\right)\right].\end{array}$$

(41)

Using (41) in (38), we get

$$\begin{array}{cc}\hfill {\ddot{P}}^f(X_0,Y_0)Z_0& =\ddot{P}(X_0,Y_0)Z_0-\alpha \beta \left[g(\phi Y_0,Z_0)\phi X_0-g(\phi X_0,Z_0)\phi Y_0\right]\hfill \\ \hfill & +\eta \left\{{\ddot{P}}^f(\xi ,Y_0)Z_0\right\}\eta \left(X_0\right)\xi -\eta \left\{\ddot{P}(\xi ,Y_0)Z_0\right\}\eta \left(X_0\right)\xi \hfill \\ \hfill & -\eta \left\{{\ddot{P}}^f(\xi ,X_0)Z_0\right\}\eta \left(Y_0\right)\xi +\eta \left\{\ddot{P}(\xi ,X_0)Z_0\right\}\eta \left(Y_0\right)\xi ,\hfill \end{array}$$

(42)

and

$$H^f(X_0,Y_0)Z_0=H(X_0,Y_0)Z_0-\alpha \left[\beta \{g(\phi Y_0,Z_0)\phi X_0-g(\phi X_0,Z_0)\phi Y_0\}\right],$$

(43)

where

$$H(X_0,Y_0)Z_0=\ddot{P}(X_0,Y_0)Z_0-\eta \left\{\ddot{P}(\xi ,Y_0)Z_0\right\}\eta \left(X_0\right)\xi +\eta \left\{\ddot{P}(\xi ,X_0)Z_0\right\}\eta \left(Y_0\right)\xi .$$

(44)

$H^f(X_0,Y_0)Z_0$ is also defined in a similar way.

Taking the inner product of Equation (43) with vector field $W_0$, we get

$$\begin{array}{cc}\hfill g\left(H^f(X_0,Y_0)Z_0,W_0\right)& =g\left(H(X_0,Y_0)Z_0,W_0\right)-\alpha \beta g(\phi Y_0,Z_0)g(\phi X_0,W_0)\hfill \\ \hfill & +\alpha \beta g(\phi X_0,Z_0)g(\phi Y_0,W_0).\hfill \end{array}$$

(45)

Using Equation (41) in the above equation, we get

$$\begin{array}{cc}\hfill g\left(H^f(X_0,Y_0)Z_0,W_0\right)& =g\left(H(X_0,Y_0)Z_0,W_0\right)\hfill \\ \hfill & -\eta \left\{{\ddot{P}}^f(\xi ,\phi Y_0)Z_0\right\}g(\phi X_0,W_0)\hfill \\ \hfill & +\eta \left\{\ddot{P}(\xi ,\phi Y_0)Z_0\right\}g(\phi X_0,W_0)\hfill \\ \hfill & +\eta \left\{{\ddot{P}}^f(\xi ,\phi X_0)Z_0\right\}g(\phi Y_0,W_0)\hfill \\ \hfill & -\eta \left\{\ddot{P}(\xi ,\phi X_0)Z_0\right\}g(\phi Y_0,W_0).\hfill \end{array}$$

(46)

It can be written as

$$g\left(L^f(X_0,Y_0)Z_0,W_0\right)=g\left(L(X_0,Y_0)Z_0,W_0\right),$$

(47)

where the tensor field L is defined by

$$\begin{array}{cc}\hfill g\left(L(X_0,Y_0)Z_0,W_0\right)& =g\left(H(X_0,Y_0)Z_0,W_0\right)+\eta \left\{\ddot{P}(\xi ,\phi Y_0)Z_0\right\}g(\phi X_0,W_0)\hfill \\ \hfill & -\eta \left\{\ddot{P}(\xi ,\phi X_0)Z_0\right\}g(\phi Y_0,W_0),\hfill \end{array}$$

(48)

and $L^f$ is also defined similarly. Using Equations (10), (12), (39) and (48), we have

$$g\left(L(X_0,Y_0)Z_0,W_0\right)=g\left(A(X_0,Y_0)Z_0,W_0\right),$$

(49)

that is, $L=A$, and similarly, $L^f=A^f$. Hence, we obtain

$$g\left(A^f(X_0,Y_0)Z_0,W_0\right)=g\left(A(X_0,Y_0)Z_0,W_0\right).$$

(50)

This completes the proof. □

This tensor field A on a 3-D quasi Sasakian manifold M is invariant under a $CL$-transformation and is said to be the $CL$-curvature tensor field on M.

Corollary 2. 

The CL-curvature tensor field on a 3-D cosymplectic quasi Sasakian manifold M is given by

$$\begin{array}{cc}\hfill A(X_0,Y_0)Z_0& =\ddot{R}(X_0,Y_0)Z_0-{\displaystyle \frac{1}{2}}[\left(\ddot{S}(Y_0,Z_0)X_0-\ddot{S}(X_0,Z_0)Y_0\right)\hfill \\ \hfill & -{\displaystyle \frac{\ddot{r}}{2}}g(Y_0,Z_0)\eta \left(X_0\right)\xi +{\displaystyle \frac{\ddot{r}}{2}}g(X_0,Z_0)\eta \left(Y_0\right)\xi \hfill \\ \hfill & +{\displaystyle \frac{\ddot{r}}{2}}g(\phi Y_0,Z_0)\phi X_0-{\displaystyle \frac{\ddot{r}}{2}}g(\phi X_0,Z_0)\phi Y_0].\hfill \end{array}$$

(51)

Proof. 

By setting $\beta =0$ in Equation (34), we obtain the result. □

5. CL-Flat and CL-Symmetric on a 3-D Quasi Sasakian Manifold

Definition 4. 

A 3-D quasi Sasakian manifold M is said to be CL-flat if the CL-curvature tensor field A of type $(1,3)$ vanishes identically on M.

The $CL$-flat manifold was introduced by Kota and Nagao in [3] for a Sasakian manifold.

From (34), we have

$$\begin{array}{cc}\hfill \ddot{R}(X_0,Y_0)Z_0& ={\displaystyle \frac{1}{2}}[\left(\ddot{S}(Y_0,Z_0)X_0-\ddot{S}(X_0,Z_0)Y_0\right)\hfill \\ \hfill & -({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)g(Y_0,Z_0)\eta \left(X_0\right)\xi +({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)g(X_0,Z_0)\eta \left(Y_0\right)\xi \hfill \\ \hfill & +({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)g(\phi Y_0,Z_0)\phi X_0-({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)g(\phi X_0,Z_0)\phi Y_0]\hfill \\ \hfill & +{\beta }^2\left[g(Y_0,Z_0)\eta \left(X_0\right)\xi -g(X_0,Z_0)\eta \left(Y_0\right)\xi \right]-{\beta }^2[g(\phi Y_0,Z_0)\phi X_0\hfill \\ \hfill & -g(\phi X_0,Z_0)\phi Y_0].\hfill \end{array}$$

(52)

Taking the inner product of the above equation with $U_0$ and contracting over $Y_0$ and $Z_0$, we get

$$\begin{array}{c}\hfill \ddot{S}(X_0,U_0)={\displaystyle \frac{1}{3}}\left[3{\beta }^2-2\ddot{r}\right]g(X_0,U_0)+{\displaystyle \frac{1}{3}}\left[9{\beta }^2-2\ddot{r}\right]\eta \left(X_0\right)\eta \left(U_0\right).\end{array}$$

(53)

Hence, from Equation (53), we have the following theorem:

Theorem 4. 

A 3-D quasi Sasakian manifold M is said to be CL-flat if the Ricci tensor of M satisfies Equation (53).

With the help of Theorem 4, we have the following corollaries.

Corollary 3. 

A CL-flat 3-D quasi Sasakian manifold M is said to be an η-Einstein manifold if $\ddot{r}$ is constant and $\ddot{r}\ne {\displaystyle \frac{9}{2}}{\beta }^2$.

Corollary 4. 

If $\ddot{r}={\displaystyle \frac{9}{2}}{\beta }^2$, then a CL-flat 3-D quasi Sasakian manifold M is an Einstein manifold.

Theorem 5. 

A 3-D cosymplectic quasi Sasakian manifold M is said to be CL-flat if the Ricci tensor of M satisfies

$$\begin{array}{c}\hfill \ddot{S}(X_0,U_0)=-{\displaystyle \frac{2}{3}}\ddot{r}g(X_0,U_0)-{\displaystyle \frac{2}{3}}\ddot{r}\eta \left(X_0\right)\eta \left(U_0\right).\end{array}$$

(54)

Proof. 

By setting $\beta =0$ in Equation (53), we obtain the result. □

With the help of Theorem 5, we have the following corollary.

Corollary 5. 

A CL-flat 3-D cosymplectic quasi Sasakian manifold is said to be η-Einstein manifold if $\ddot{r}$ is constant.

Theorem 6. 

In a CL-flat 3-D quasi Sasakian manifold M, the scalar curvature of the manifold is constant if and only if $\ddot{r}={\displaystyle \frac{3}{2}}{\beta }^2$.

Proof. 

We have

$$\begin{array}{c}\hfill \left({\ddot{\nabla }}_{B_0}\ddot{S}\right)(X_0,U_0)={\ddot{\nabla }}_{B_0}\ddot{S}(X_0,U_0)-S({\ddot{\nabla }}_{B_0}{X}_0,U_0)-\ddot{S}(X_0,{\ddot{\nabla }}_{B_0}{U}_0),\end{array}$$

(55)

where $B_0,U_0$, and $X_0$ are vector fields on M. So, taking an orthonormal frame field on Equation (53) and contracting along $X_0$ and $U_0$, we get

$$\begin{array}{c}\hfill d\ddot{r}\left(B_0\right)=3{\beta }^2-2\ddot{r}.\end{array}$$

(56)

□

Theorem 7. 

In a CL-flat 3-D cosymplectic quasi Sasakian manifold M, the scalar curvature of the manifold is constant if and only if $\ddot{r}=0$.

Proof. 

Setting $\beta =0$ in Equation (56), we obtain the result. □

Definition 5. 

A 3-D quasi Sasakian manifold M is said to be CL-symmetric if

$$\left({\ddot{\nabla }}_{U_0}A\right)(X_0,Y_0)Z_0=0,$$

for all vector fields $U_0,X_0,Y_0$, and $Z_0$ on M.

Differentiating Equation (34) covariantly with respect to vector field $U_0$ and using differential Equations (5)–(7), we get

$$\begin{array}{cc}\hfill \left({\ddot{\nabla }}_{U_0}A\right)& (X_0,Y_0)Z_0=\left({\ddot{\nabla }}_{U_0}\ddot{R}\right)(X_0,Y_0)Z_0-{\displaystyle \frac{1}{2}}[\left({\ddot{\nabla }}_{U_0}\ddot{S}\right)(Y_0,Z_0)X_0\hfill \\ \hfill & -\left({\ddot{\nabla }}_{U_0}\ddot{S}\right)(X_0,Z_0)Y_0-{\displaystyle \frac{1}{2}}(d\ddot{r}\left(U_0\right)g(Y_0,Z_0)\eta \left(X_0\right)\hfill \\ \hfill & -d\ddot{r}\left(U_0\right)g(X_0,Z_0)\eta \left(Y_0\right))\xi +\beta ({\displaystyle \frac{1}{2}}-{\beta }^2)g(Y_0,Z_0)g(\phi U_0,X_0)\xi \hfill \\ \hfill & +\beta ({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)g(Y_0,Z_0)\eta \left(X_0\right)\phi U_0-\beta ({\displaystyle \frac{1}{2}}-{\beta }^2)g(X_0,Z_0)g(\phi U_0,Y_0)\xi \hfill \\ \hfill & -\beta ({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)g(X_0,Z_0)\eta \left(Y_0\right)\phi U_0\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}d\ddot{r}\left(U_0\right)\left(g(\phi Y_0,Z_0)\phi X_0-g(\phi X_0,Z_0)\phi Y_0\right)\hfill \\ \hfill & -\beta ({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)g(\phi Y_0,Z_0)g(U_0,X_0)\xi +\beta ({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)g(\phi Y_0,Z_0)\eta \left(X_0\right)U_0\hfill \\ \hfill & +\beta ({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)g(\phi X_0,Z_0)g(U_0,Y_0)\xi -\beta ({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)g(\phi X_0,Z_0)\eta \left(Y_0\right)U_0]\hfill \\ \hfill & +{\beta }^3\left[\right(g(Y_0,Z_0)g(\phi U_0,X_0)-g(X_0,Z_0)g(\phi U_0,Y_0)\hfill \\ \hfill & -g(\phi Y_0,Z_0)g(U_0,X_0)\hfill \\ \hfill & +g(\phi X_0,Z_0)g(U_0,Y_0))\xi +\left(g(Y_0,Z_0)\eta \left(X_0\right)-g(X_0,Z_0)\eta \left(Y_0\right)\right)\phi U_0\hfill \\ \hfill & +\left(g(\phi Y_0,Z_0)\eta \left(X_0\right)-g(\phi X_0,Z_0)\eta \left(Y_0\right)\right)U_0].\hfill \end{array}$$

(57)

We consider the condition that our manifold is a $CL$-symmetric, i.e., $\left({\ddot{\nabla }}_{U_0}A\right)(X_0,Y_0)Z_0=0$; then, Equation (57) becomes

$$\begin{array}{cc}\hfill \left({\ddot{\nabla }}_{U_0}\ddot{R}\right)& (X_0,Y_0,Z_0,W_0)={\displaystyle \frac{1}{2}}[\left({\ddot{\nabla }}_{U_0}\ddot{S}\right)(Y_0,Z_0)g(X_0,W_0)\hfill \\ \hfill & -\left({\ddot{\nabla }}_{U_0}\ddot{S}\right)(X_0,Z_0)g(Y_0,W_0)\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}\left(d\ddot{r}\left(U_0\right)g(Y_0,Z_0)\eta \left(X_0\right)-d\ddot{r}\left(U_0\right)g(X_0,Z_0)\eta \left(Y_0\right)\right)g(\xi ,W_0)\hfill \\ \hfill & +\beta ({\displaystyle \frac{1}{2}}-{\beta }^2)g(Y_0,Z_0)g(\phi U_0,X_0)g(\xi ,W_0)\hfill \\ \hfill & +\beta ({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)g(Y_0,Z_0)\eta \left(X_0\right)g(\phi U_0,W_0)\hfill \\ \hfill & -\beta ({\displaystyle \frac{1}{2}}-{\beta }^2)g(X_0,Z_0)g(\phi U_0,Y_0)g(\xi ,W_0)\hfill \\ \hfill & -\beta ({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)g(X_0,Z_0)\eta \left(Y_0\right)g(\phi U_0,W_0)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}d\ddot{r}\left(U_0\right)\left(g(\phi Y_0,Z_0)g(\phi X_0,W_0)-g(\phi X_0,Z_0)g(\phi Y_0,W_0)\right)\hfill \\ \hfill & -\beta ({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)g(\phi Y_0,Z_0)g(U_0,X_0)g(\xi ,W_0)\hfill \\ \hfill & +\beta ({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)g(\phi Y_0,Z_0)\eta \left(X_0\right)g(U_0,W_0)\hfill \\ \hfill & +\beta ({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)g(\phi X_0,Z_0)g(U_0,Y_0)g(\xi ,W_0)\hfill \\ \hfill & -\beta ({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)g(\phi X_0,Z_0)\eta \left(Y_0\right)g(U_0,W_0)]\hfill \\ \hfill & -{\beta }^3\left[\right(g(Y_0,Z_0)g(\phi U_0,X_0)-g(X_0,Z_0)g(\phi U_0,Y_0)\hfill \\ \hfill & -g(\phi Y_0,Z_0)g(U_0,X_0)+g(\phi X_0,Z_0)g(U_0,Y_0))g(\xi ,W_0)\hfill \\ \hfill & +\left(g(Y_0,Z_0)\eta \left(X_0\right)-g(X_0,Z_0)\eta \left(Y_0\right)\right)g(\phi U_0,W_0)\hfill \\ \hfill & +\left(g(\phi Y_0,Z_0)\eta \left(X_0\right)-g(\phi X_0,Z_0)\eta \left(Y_0\right)\right)g(U_0,W_0)].\hfill \end{array}$$

(58)

Contracting the above equation with $Y_0$ and $Z_0$, we get

$$\begin{array}{cc}\hfill \left({\ddot{\nabla }}_{U_0}\ddot{S}\right)& (X_0,W_0)={\displaystyle \frac{1}{2}}[d\ddot{r}\left(U_0\right)g(X_0,W_0)-\left({\ddot{\nabla }}_{U_0}\ddot{S}\right)(X_0,W_0)\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}\left(3d\ddot{r}\left(U_0\right)\eta \left(X_0\right)\eta \left(W_0\right)-d\ddot{r}\left(U_0\right)\eta \left(X_0\right)\eta \left(W_0\right)\right)\hfill \\ \hfill & +3\beta ({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)g(\phi U_0,X_0)\eta \left(W_0\right)\hfill \\ \hfill & +3\beta ({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)g(\phi U_0,W_0)\eta \left(X_0\right)-\beta ({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)g(X_0,\phi U_0)\eta \left(W_0\right)\hfill \\ \hfill & -\beta ({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)\eta \left(X_0\right)g(\phi U_0,W_0)+{\displaystyle \frac{1}{2}}d\ddot{r}\left(U_0\right)(-g({\phi }^2{X}_0,W_0))\hfill \\ \hfill & +\beta ({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)g(\phi X_0,U_0)\eta \left(W_0\right)]-{\beta }^3[3g(\phi U_0,X_0)\eta \left(W_0\right)\hfill \\ \hfill & -g(X_0,\phi U_0)\eta \left(W_0\right)+g(\phi X_0,U_0)\eta \left(W_0\right)+3\eta \left(X_0\right)g(\phi U_0,W_0)\hfill \\ \hfill & -\eta \left(X_0\right)g(\phi U_0,W_0)],\hfill \end{array}$$

(59)

which yields

$$d\ddot{r}\left(U_0\right)=0.$$

(60)

Hence, we can state the following theorem.

Theorem 8. 

In a CL-symmetric 3-D quasi Sasakian manifold, the scalar curvature is found to be constant.

6. CL-$\mathit{\phi }$ Symmetric and CL-$\mathit{\phi }$ Recurrent on a 3-D Quasi Sasakian Manifold

Definition 6. 

A 3-dimensional quasi Sasakian manifold M is said to be CL-φ symmetric if

$${\phi }^2\left(\left({\ddot{\nabla }}_{U_0}A\right)(X_0,Y_0)Z_0\right)=0,$$

(61)

for all vector fields $U_0,X_0,Y_0$, and $Z_0$ on M.

If the vector fields $U_0,X_0,Y_0$, and $Z_0$ are orthonormal with vector field $\xi$ or $U_0,X_0,Y_0$, and $Z_0$ are horizontal vector fields, then Equation (60) is referred to as locally $CL$-$\phi$ symmetric. So, using Equations (2) and (61), we have

$$-\left({\ddot{\nabla }}_{U_0}A\right)(X_0,Y_0)Z_0+\eta \left(\left({\ddot{\nabla }}_{U_0}A\right)(X_0,Y_0)Z_0\right)\xi =0,$$

which gives

$$\begin{array}{cc}\hfill \left({\ddot{\nabla }}_{U_0}A\right)& (X_0,Y_0)Z_0=g\left(\left({\ddot{\nabla }}_{U_0}A\right)(X_0,Y_0)Z_0,\xi \right)\xi \hfill \\ \hfill & ={\displaystyle \frac{\ddot{r}}{2}}\beta g(Y_0,Z_0)g(\phi U_0,X_0)\xi -{\displaystyle \frac{\ddot{r}}{2}}\beta g(X_0,Z_0)g(\phi U_0,Y_0)\xi \hfill \\ \hfill & -{\displaystyle \frac{1}{2}}d\ddot{r}\left(U_0\right)\eta \left(X_0\right)\eta \left(Y_0\right)\eta \left(Z_0\right)\xi +{\displaystyle \frac{\ddot{r}}{2}}\beta g(\phi U_0,Y_0)\eta \left(X_0\right)\eta \left(Z_0\right)\xi \hfill \\ \hfill & -{\displaystyle \frac{1}{2}}d\ddot{r}\left(U_0\right)\beta g(\phi U_0,X_0)\eta \left(Y_0\right)\eta \left(Z_0\right)\xi -{\displaystyle \frac{1}{2}}[\left({\ddot{\nabla }}_{U_0}\ddot{S}\right)(Y_0,Z_0)\eta \left(X_0\right)\xi \hfill \\ \hfill & -\left({\ddot{\nabla }}_{U_0}\ddot{S}\right)(X_0,Z_0)\eta \left(Y_0\right)\xi -{\displaystyle \frac{1}{2}}(d\ddot{r}\left(U_0\right)g(Y_0,Z_0)\eta \left(X_0\right)\hfill \\ \hfill & -d\ddot{r}\left(U_0\right)g(X_0,Z_0)\eta \left(Y_0\right))\xi +\beta ({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)g(Y_0,Z_0)g(\phi U_0,X_0)\xi \hfill \\ \hfill & -\beta ({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)g(X_0,Z_0)g(\phi U_0,Y_0)\xi -\beta ({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)g(\phi Y_0,Z_0)g(U_0,X_0)\xi \hfill \\ \hfill & +\beta ({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)g(\phi Y_0,Z_0)\eta \left(X_0\right)\xi +\beta ({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)g(\phi X_0,Z_0)g(U_0,Y_0)\xi \hfill \\ \hfill & -\beta ({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)g(\phi X_0,Z_0)\eta \left(Y_0\right)\eta \left(U_0\right)\xi ]+{\beta }^3[(g(Y_0,Z_0)g(\phi U_0,X_0)\hfill \\ \hfill & -g(X_0,Z_0)g(\phi U_0,Y_0)-g(\phi Y_0,Z_0)g(U_0,X_0)+g(\phi X_0,Z_0)g(U_0,Y_0))\xi \hfill \\ \hfill & +g(\phi Y_0,Z_0)\eta \left(X_0\right)\eta \left(U_0\right)\xi -g(\phi X_0,Z_0)\eta \left(Y_0\right)\eta \left(U_0\right)\xi ].\hfill \end{array}$$

(62)

Hence, we can conclude the following theorems.

Theorem 9. 

A 3-D quasi Sasakian manifold is said to be CL-φ symmetric if and only if Equation (62) is satisfied.

Also, we can obtain a locally symmetric condition by considering the vector fields $U_0,X_0,Y_0$, and $Z_0$ to be orthogonal to vector field $\xi$. So, Equation (62) becomes

$$\begin{array}{cc}\hfill \left({\ddot{\nabla }}_{U_0}A\right)& (X_0,Y_0)Z_0={\displaystyle \frac{\ddot{r}}{2}}\beta \left(g(Y_0,Z_0)g(\phi U_0,X_0)\xi -g(X_0,Z_0)g(\phi U_0,Y_0)\xi \right)\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}[\beta ({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)\left(g(Y_0,Z_0)g(\phi U_0,X_0)\xi -g(X_0,Z_0)g(\phi U_0,Y_0)\xi \right)\hfill \\ \hfill & -\beta ({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)\left(g(\phi Y_0,Z_0)g(U_0,X_0)\xi -g(\phi X_0,Z_0)g(U_0,Y_0)\xi \right)]\hfill \\ \hfill & +{\beta }^3[g(Y_0,Z_0)g(\phi U_0,X_0)-g(X_0,Z_0)g(\phi U_0,Y_0)\hfill \\ \hfill & -g(\phi Y_0,Z_0)g(U_0,X_0)+g(\phi X_0,Z_0)g(U_0,Y_0)]\xi .\hfill \end{array}$$

(63)

Hence, we can have the following.

Theorem 10. 

A 3-D quasi Sasakian manifold M is said to be locally CL-φ symmetric if and only if the Equation (63) is satisfied.

Theorem 11. 

A 3-D cosymplectic quasi Sasakian manifold is said to be CL-φ symmetric if and only if

$$\begin{array}{cc}\hfill \left({\ddot{\nabla }}_{U_0}A\right)& (X_0,Y_0)Z_0=-{\displaystyle \frac{1}{2}}d\ddot{r}\left(U_0\right)\eta \left(X_0\right)\eta \left(Y_0\right)\eta \left(Z_0\right)\xi -{\displaystyle \frac{1}{2}}[\left({\ddot{\nabla }}_{U_0}\ddot{S}\right)(Y_0,Z_0)\eta \left(X_0\right)\xi \hfill \\ \hfill & -\left({\ddot{\nabla }}_{U_0}\ddot{S}\right)(X_0,Z_0)\eta \left(Y_0\right)\xi -{\displaystyle \frac{1}{2}}(d\ddot{r}\left(U_0\right)g(Y_0,Z_0)\eta \left(X_0\right)\hfill \\ \hfill & -d\ddot{r}\left(U_0\right)g(X_0,Z_0)\eta \left(Y_0\right)\left)\xi \right].\hfill \end{array}$$

(64)

Proof. 

Setting $\beta =0$ in Equation (62), we obtain the result. □

With the help of Theorem 11, we can have the following corollary.

Corollary 6. 

A locally CL-φ symmetric 3-D cosymplectic quasi Sasakian manifold M is a CL-symmetric manifold.

Proof. 

Setting $\beta =0$ in Equation (63), we get

$$\begin{array}{c}\hfill \left({\ddot{\nabla }}_{U_0}A\right)(X_0,Y_0)Z_0=0.\end{array}$$

(65)

□

Definition 7. 

A 3-D quasi Sasakian manifold M is said to be CL-φ recurrent if

$${\phi }^2\left(\left({\ddot{\nabla }}_{U_0}A\right)(X_0,Y_0)Z_0\right)=D\left(U_0\right)A(X_0,Y_0)Z_0,$$

(66)

where D is a 1-form.

By the use of Equation (2) in Equation (66), we have

$$\begin{array}{c}\hfill -\left({\ddot{\nabla }}_{U_0}A\right)(X_0,Y_0)Z_0+g\left(\left({\ddot{\nabla }}_{U_0}A\right)(X_0,Y_0)Z_0,\xi \right)\xi =D\left(U_0\right)A(X_0,Y_0)Z_0.\end{array}$$

(67)

Using Equations (34) and (57) and contracting along with vector field $X_0$, we get the following differential equation:

$$\begin{array}{cc}\hfill -\left({\ddot{\nabla }}_{U_0}{A}^s\right)(Y_0,Z_0)& =-d\ddot{r}\left(U_0\right)\eta \left(Y_0\right)\eta \left(Z_0\right)-{\displaystyle \frac{1}{2}}[\left({\ddot{\nabla }}_{U_0}\ddot{S}\right)(Y_0,Z_0)\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}d\ddot{r}\left(U_0\right)\eta \left(Y_0\right)\eta \left(Z_0\right)+\beta \ddot{r}\eta \left(Y_0\right)g(\phi U_0,Z_0)\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}d\ddot{r}\left(U_0\right)g(Y_0,Z_0)-\beta ({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)\eta \left(Z_0\right)g(\phi U_0,Y_0)\hfill \\ \hfill & -\beta ({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)\eta \left(U_0\right)g(\phi Y_0,Z_0)+\beta ({\displaystyle \frac{\ddot{r}}{2}}-{\beta }^2)g(\phi Y_0,Z_0)]\hfill \\ \hfill & +{\beta }^3\eta \left(Z_0\right)g(\phi U_0,Y_0).\hfill \end{array}$$

(68)

Once again, contracting along the vector fields $Y_0$ and $Z_0$, we get

$$d{A}^r\left(U_0\right)=0,$$

(69)

where $A^s$ and $A^r$ are the $CL$-Ricci tensor and $CL$-scalar curvature, respectively. So, we can conclude the following theorem.

Theorem 12. 

A 3-D quasi Sasakian manifold is said to be CL-φ recurrent if and only if the CL-scalar curvature is constant.

7. Ricci Soliton on the CL Transformation of a 3-D Quasi Sasakian Manifold

In this section, in order to find the condition of the Ricci soliton, we consider the curvature of the 3-D quasi Sasakian manifold to be flat, i.e., $\ddot{R}(X_0,Y_0)Z_0=0$. So, Equation (34) becomes

$$\begin{array}{cc}\hfill A(X_0,Y_0)Z_0& =-{\displaystyle \frac{1}{2}}[{\beta }^{2}g(Y_0,Z_0)\eta \left(X_0\right)\xi -{\beta }^{2}g(X_0,Z_0)\eta \left(Y_0\right)\xi \hfill \\ \hfill & -{\beta }^{2}g(\phi Y_0,Z_0)\phi X_0+{\beta }^{2}g(\phi X_0,Z_0)\phi Y_0]\hfill \\ \hfill & -{\beta }^2\left[g(Y_0,Z_0)\eta \left(X_0\right)\xi -g(X_0,Z_0)\eta \left(Y_0\right)\xi \right]\hfill \\ \hfill & +{\beta }^2\left[g(\phi Y_0,Z_0)\phi X_0-g(\phi X_0,Z_0)\phi Y_0\right].\hfill \end{array}$$

(70)

By contraction along the vector field $X_0$, we get

$$A^s(Y_0,Z_0)=-{\displaystyle \frac{1}{2}}{\beta }^{2}g(Y_0,Z_0),$$

(71)

where $A^s$ is the $CL$-Ricci tensor. From [7], we know the expression of the Ricci soliton as

$$\begin{array}{c}\hfill L_{V_0}g(X_0,Y_0)+2\ddot{S}(X_0,Y_0)+2\lambda g(X_0,Y_0)=0.\end{array}$$

(72)

Here, We can have two conditions regarding vector field $V_0$, which are $V_0\in Span\xi$ and $V_0\perp \xi$. We concentrate only on the first one, i.e., $V_0=\xi$. So, in our case, we can write it as

$$\begin{array}{c}\hfill L_{\xi }g(X_0,Y_0)+2A^s(X_0,Y_0)+2\lambda g(X_0,Y_0)=0.\end{array}$$

(73)

Using differential Equations (3) and (5), we get

$$\begin{array}{cc}\hfill L_{\xi }g(X_0,Y_0)& =g({\ddot{\nabla }}_{X_0}\xi ,Y_0)+g(X_0,{\ddot{\nabla }}_{Y_0}\xi )\hfill \\ \hfill & =0.\hfill \end{array}$$

(74)

Using Equation (74) in (73), we get

$$A^s(Y_0,Z_0)=-\lambda g(Y_0,Z_0).$$

(75)

Theorem 13. 

If $(g,\xi ,\lambda )$ is a Ricci soliton on the CL-transformation of 3-D quasi Sasakian manifold, with the condition that the curvature of the 3-D quasi Sasakian manifold is flat, then the CL-transformation on the 3-D quasi Sasakian manifold is an Einstein manifold.

Also, by the use of Equation (71) in (75), we get

$$\lambda ={\displaystyle \frac{1}{2}}{\beta }^2.$$

(76)

Hence, we have the following corollaries.

Corollary 7. 

In the CL-transformation of 3-D quasi Sasakian manifold, with the condition that the curvature of the 3-D quasi Sasakian manifold is flat, the Ricci soliton $(g,\xi ,\lambda )$ of CL-transformation on the 3-D quasi Sasakian manifold is found to be shrinking, steady, or expanding when ${\beta }^2<0$, ${\beta }^2=0$, or ${\beta }^2>0$, respectively.

Corollary 8. 

In the CL-transformation of 3-D cosymplectic quasi Sasakian manifold, with the condition that the curvature of the 3-D cosymplectic quasi Sasakian manifold is flat, the Ricci soliton $(g,\xi ,\lambda )$ of CL-transformation on the 3-D cosymplectic quasi Sasakian manifold is found to be steady.

8. Conclusions

The study of CL-transformations on 3-dimensional quasi Sasakian manifolds has revealed rich geometric behavior and introduced new tools for exploring curvature and symmetry within contact geometry. Notably, the invariant tensor field defined under these transformations offers potential for deeper structural analysis that extends beyond classical frameworks.

Looking forward, future work could focus on exploring the interaction of CL-transformations with other geometric flows, such as generalized Ricci or Yamabe solitons, especially in higher dimensions or under alternative connection types. Another promising direction is the application of the CL-curvature tensor in the classification of more general classes of contact metric manifolds. Additionally, integrating this framework into modern theoretical models, such as $f\left(R\right)$ or $f(R,T)$ gravity theories, or manifolds with semi-symmetric or non-metric connections may uncover further geometric and physical insights. These paths offer a fertile ground for continued research on the intersection of differential geometry and mathematical physics.