Ricci–Yamabe Solitons on Sasakian Manifolds with the Generalized Tanaka–Webster Connection

Abstract

In this article, we analyze some curvature restrictions satisfying by the concircular curvature tensor in $(2n+1)$-dimensional Sasakian manifolds with the generalized Tanaka–Webster connection $\overline{\nabla }$ admitting Ricci–Yamabe solitons. Finally, we give an example of three-dimensional Sasakian manifolds which verifies some of our findings.

1. Introduction

In real life, a fractal is a general phenomenon [1]. Recently, the concept of fractal theory has been used in engineering techniques and the latest technologies, such as biological engineering, ocean engineering, material science, space science, etc. A solitary wave is a wave that propagates without any temporal development in size or shape when viewed in the reference frame moving with the wave’s group velocity. The solitary wave came fundamentally from the famous Korteweg de Vries (KdV) equation [2]. These waves arise from several points of view, including the intensity of light in optical fibers and the elevation of the surface of water. A distinct set of nonlinear dynamics in a given medium is formed by each type of solitary wave. The interaction of the solitary wave is an important and fascinating phenomenon for both mathematical and physical reasons. From the physical point of view, such type of interactions have arisen in many disciplines such as optics, Josephson junctions, and water waves. The fractal solitary theory [3] establishes a connection between the fractal dimensions and the motion dynamics of the solitary wave by allowing the fractal geometry to handle the properties of the solitary wave. By using fractal vibration theory, Ji et al. [4] studied the transverse vibration of a porous concrete and obtained fractal solitons.

Hamilton [5,6] proposed the notion of Yamabe flow, which deforms g (the metric of a Riemannian manifold $\mathcal{M}$) through the relation $r\left(t\right)g\left(t\right)+{\displaystyle \frac{\partial }{\partial t}}g\left(t\right)=0,g\left(0\right)=g_0$, where $r\left(t\right)$ denotes the scalar curvature. For two-dimensional manifolds, the Yamabe flow and the Ricci flow, $2\mathcal{S}\left(g\left(t\right)\right)+{\displaystyle \frac{\partial }{\partial t}}g\left(t\right)=0$, are equivalent, where $\mathcal{S}$ denotes the Ricci tensor. Meanwhile, for the dimension n ($n>2$) case, the Ricci and Yamabe flows are not identical because the conformal class of $g\left(t\right)$ is generally preserved by the Yamabe flow only.

In [7], the authors defined an advanced class of geometric flows named the Ricci–Yamabe flow of type $(\vartheta ,\rho )$ by the equation

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial }{\partial t}}g\left(t\right)+2\vartheta \mathcal{S}\left(g\left(t\right)\right)+\rho r\left(t\right)g\left(t\right)=0,g\left(0\right)=g_0\end{array}$$

(1)

for some $\vartheta$ and $\rho$ scalars.

A solution of (1) is called a Ricci–Yamabe soliton $(g,\mathcal{F},\delta ,\vartheta ,\rho )$ if it depends only on one parameter group of diffeomorphism and scaling. An $\mathcal{M}$ is said to have an RYS if

$$\begin{array}{c}\hfill \left({\pounds }_{\mathcal{F}}g\right)({\mathcal{U}}_1,{\mathcal{V}}_1)+2\vartheta \mathcal{S}({\mathcal{U}}_1,{\mathcal{V}}_1)+(2\delta -\rho r)g({\mathcal{U}}_1,{\mathcal{V}}_1)=0,\end{array}$$

(2)

where ${\pounds }_{\mathcal{F}}$ denotes the Lie derivative operator along the smooth vector field $\mathcal{F}$ on $\mathcal{M}$, and $\delta \in \mathbb{R}$. An RYS is said to be a Yamabe soliton [6], a Ricci soliton [5], an $\omega$-Einstein soliton [8], or an Einstein soliton [9] if $\vartheta =0, \rho =1$, $\vartheta =1, \rho =0$, $\vartheta =1, \rho =-2\omega$, or $\vartheta =1, \rho =-1$, respectively.

Geometric flows such as the Ricci flow, Ricci–Bourguignon flow, Yamabe flow, and Ricci–Yamabe flow have been utilized to deal many prominent problems, such as parameterization in graphics, wireless sensor networking, cancer detection in medical imaging, deformable surface registration in vision, and manifold spline construction in geometric modeling. These flows have also been utilized in theoretical physics, especially to study the geometry of spacetime in general relativity. In addition, these flows have been applied to understand the large-scale structure of the universe and the behavior of black holes. Due to the widespread uses of Ricci solitons and their generalizations in various fields of medical science, engineering, science, computer science, etc., they have become a highly popular and significant area of research (see [10,11,12]).

The geometry of contact manifolds has become a very useful branch of differential geometry due to its widespread applications in various areas such as physics, the theory of relativity, cosmology, engineering, and many others. In 1960, an odd-dimensional counterpart of Kahler manifolds was proposed by Sasaki [13].

The Tanaka–Webster connection [14,15,16] is a canonical affine connection on non-degenerate, pseudo-Hermitian $CR$-submanifolds. For contact metric manifolds, the generalized Tanaka–Webster connection was defined by Tanno [17]. These connections coincide if the associated $CR$-structure is integrable.

We recommend the papers ([18,19,20,21,22,23,24,25,26,27]) and those cited in the text for more detailed studies in the context of contact Riemannian geometry.

Motivated by the above studies, we study Ricci–Yamabe solitons $(g,\zeta ,\delta ,\vartheta ,\rho )$ in ($2n+1$)-dimensional Sasakian manifolds ${\mathcal{M}}_S^{2n+1}$ with a generalized Tanaka–Webster connection $\overline{\nabla }$, which we denote by $({\mathcal{M}}_S^{2n+1},g,\zeta ,\delta ,\vartheta ,\rho ,\overline{\nabla })$.

The work is presented as follows: A brief introduction of Sasakian manifolds is given in Section 2. Section 3 is divided into four subsections: in Section 3.1, we deduce some relations between concircular curvature tensors with respect to ∇ and $\overline{\nabla }$ on ${\mathcal{M}}_S^{2n+1}$; in Section 3.2, we study $(g,\zeta ,\delta ,\vartheta ,\rho )$ in ${\mathcal{M}}_S^{2n+1}$ with respect to $\overline{\nabla }$; Section 3.3 deals with the study of $({\mathcal{M}}_S^{2n+1},g,\zeta ,\delta ,\vartheta ,\rho ,\overline{\nabla })$ satisfying $\zeta$-concircularly flat and $\psi$-concircularly semisymmetric conditions; Section 3.4 is devoted to the study of $({\mathcal{M}}_S^{2n+1},g,\zeta ,\delta ,\vartheta ,\rho ,\overline{\nabla })$ satisfying $\overline{C}(\zeta ,{\mathcal{U}}_1)·\overline{S}=0$. Finally, we construct examples of ${\mathcal{M}}_S^3$ to validate some of our findings.

2. Preliminaries

Let $\mathcal{M}$ (of dimension $(2n+1)$) be an almost contact metric manifold with an almost contact metric structure $(\psi ,\zeta ,\eta ,g)$, where $\psi$ is a $(1,1)$ tensor field, $\zeta$ is a vector field, $\eta$ is a 1-form, and g is a Riemannian metric, fulfilling [28]

$${\psi }^2{\mathcal{U}}_1=-{\mathcal{U}}_1+\eta \left({\mathcal{U}}_1\right)\zeta ,\eta \left(\zeta \right)=1,\psi \zeta =0,\eta \left(\psi {\mathcal{U}}_1\right)=0,$$

(3)

$$g(\psi {\mathcal{U}}_1,\psi {\mathcal{V}}_1)=g({\mathcal{U}}_1,{\mathcal{V}}_1)-\eta \left({\mathcal{U}}_1\right)\eta \left({\mathcal{V}}_1\right),g({\mathcal{U}}_1,\zeta )=\eta \left({\mathcal{U}}_1\right),$$

(4)

$$g({\mathcal{U}}_1,\psi {\mathcal{V}}_1)=-g(\psi {\mathcal{U}}_1,{\mathcal{V}}_1),$$

(5)

for any vector fields ${\mathcal{U}}_1$, ${\mathcal{V}}_1$ on $\mathcal{M}$.

A normal contact metric manifold is named a Sasakian manifold. The manifold $\mathcal{M}$ fulfilling (3)–(5) is Sasakian if and only if [29,30]

$$\left({\nabla }_{{\mathcal{U}}_1}\psi \right){\mathcal{V}}_1=g({\mathcal{U}}_1,{\mathcal{V}}_1)\zeta -\eta \left({\mathcal{V}}_1\right){\mathcal{U}}_1,$$

(6)

for any ${\mathcal{U}}_1,{\mathcal{V}}_1$ on $\mathcal{M}$, where ∇ refers to the Levi-Civita connection of the Riemannian metric.

From the above relations, it follows that

$${\nabla }_{{\mathcal{U}}_1}\zeta =-\psi {\mathcal{U}}_1,$$

(7)

$$\left({\nabla }_{{\mathcal{U}}_1}\eta \right){\mathcal{V}}_1=g({\mathcal{U}}_1,\psi {\mathcal{V}}_1),$$

(8)

for any vector fields ${\mathcal{U}}_1,{\mathcal{V}}_1$ on $\mathcal{M}$, the above relations are the covariant derivative of $\zeta$ and $\eta$ along ${\mathcal{U}}_1$.

In an ${\mathcal{M}}_S^{2n+1}$, the following relations holds [31,32]:

$$\mathcal{R}({\mathcal{U}}_1,{\mathcal{V}}_1)\zeta =\eta \left({\mathcal{V}}_1\right){\mathcal{U}}_1-\eta \left({\mathcal{U}}_1\right){\mathcal{V}}_1,$$

(9)

$$S({\mathcal{U}}_1,\zeta )=2n\eta \left({\mathcal{U}}_1\right)\iff Q\zeta =2n\zeta ,$$

(10)

$$S(\psi {\mathcal{U}}_1,\psi {\mathcal{V}}_1)=S({\mathcal{U}}_1,{\mathcal{V}}_1)-2n\eta \left({\mathcal{U}}_1\right)\eta \left({\mathcal{V}}_1\right),$$

(11)

$$\eta \left(\mathcal{R}({\mathcal{U}}_1,{\mathcal{V}}_1){\mathcal{W}}_1\right)=g({\mathcal{V}}_1,{\mathcal{W}}_1)\eta \left({\mathcal{U}}_1\right)-g({\mathcal{U}}_1,{\mathcal{W}}_1)\eta \left({\mathcal{V}}_1\right),$$

(12)

for any ${\mathcal{U}}_1$, ${\mathcal{V}}_1$, and ${\mathcal{W}}_1$ on ${\mathcal{M}}_S^{2n+1}$, where $\mathcal{Q}$ and $\mathcal{R}$ are the Ricci operator and the curvature tensor in ${\mathcal{M}}_S^{2n+1}$ with respect to ∇, respectively.

Definition 1

([33]). A manifold $\mathcal{M}$ is called an η-Einstein manifold if its Ricci tensor $\mathcal{S}$ is expressed as

$$\mathcal{S}({\mathcal{U}}_1,{\mathcal{V}}_1)=pg({\mathcal{U}}_1,{\mathcal{V}}_1)+q\eta \left({\mathcal{U}}_1\right)\eta \left({\mathcal{V}}_1\right),$$

(13)

where p and q are scalar functions on $\mathcal{M}$. In particular, if $q=0$, then (13) is named an Einstein manifold.

Now, putting ${\mathcal{U}}_1={\mathcal{V}}_1=\zeta$ in (13) and using (3), (4), and (10), we have

$$2n=p+q.$$

(14)

By contracting (13) over ${\mathcal{U}}_1$ and ${\mathcal{V}}_1$, we also have

$$r=(2n+1)p+q.$$

(15)

From (14) and (15), it follows that $p={\displaystyle \frac{r}{2n}}-1$ and $q=(2n+1)-{\displaystyle \frac{r}{n}}$. Thus, (13) takes the form

$$\mathcal{S}({\mathcal{U}}_1,{\mathcal{V}}_1)=({\displaystyle \frac{r}{2n}}-1)g({\mathcal{U}}_1,{\mathcal{V}}_1)+(2n+1-{\displaystyle \frac{r}{n}})\eta \left({\mathcal{U}}_1\right)\eta \left({\mathcal{V}}_1\right).$$

(16)

Equation (16) is the expression of the Ricci tensor in terms of scalar curvature.

From (16), we have

$$\mathcal{Q}{\mathcal{U}}_1=({\displaystyle \frac{r}{2n}}-1){\mathcal{U}}_1+(2n+1-{\displaystyle \frac{r}{n}})\eta \left({\mathcal{U}}_1\right)\zeta .$$

(17)

By differentiating (17) covariantly along ${\mathcal{V}}_1$ and using (7), (8), (17), we have

$$\left({\nabla }_{{\mathcal{V}}_1}\mathcal{Q}\right){\mathcal{U}}_1={\displaystyle \frac{{\mathcal{V}}_1\left(r\right)}{2n}}{\mathcal{U}}_1+(2n+1-{\displaystyle \frac{r}{n}})(g({\mathcal{V}}_1,\psi {\mathcal{U}}_1)\zeta -\eta \left({\mathcal{U}}_1\right)\psi {\mathcal{V}}_1)-{\displaystyle \frac{{\mathcal{V}}_1\left(r\right)}{2n}}\eta \left({\mathcal{U}}_1\right)\zeta .$$

(18)

By contracting (18) over ${\mathcal{V}}_1$, we infer

$${\mathcal{U}}_1\left(r\right)={\displaystyle \frac{\zeta \left(r\right)}{n-1}}\eta \left({\mathcal{U}}_1\right).$$

(19)

Through this equation, we set a relation between the covariant derivative of r along ${\mathcal{U}}_1$ and $\zeta$.

Now, by putting ${\mathcal{U}}_1=\zeta$ in (19) and using (3), we find $\zeta \left(r\right)=0$. Thus, from (19), it follows that ${\mathcal{U}}_1\left(r\right)=0$, i.e., r is constant. Thus we can state the following proposition:

Proposition 1.

A $(2n+1)$ dimensional η-Einstein Sasakian manifold possesses constant scalar curvature.

In an $\mathcal{M}$, the connections $\overline{\nabla }$ and ∇ are related by [17]

$${\overline{\nabla }}_{{\mathcal{U}}_1}{\mathcal{V}}_1={\nabla }_{{\mathcal{U}}_1}{\mathcal{V}}_1+g({\mathcal{U}}_1,\psi {\mathcal{V}}_1)\zeta +\eta \left({\mathcal{V}}_1\right)\psi {\mathcal{U}}_1-\eta \left({\mathcal{U}}_1\right)\psi {\mathcal{V}}_1,$$

(20)

for all ${\mathcal{U}}_1,{\mathcal{V}}_1$ on $\mathcal{M}$, where $\psi$ is a $(1,1)$ tensor field and $\eta$ is a 1-form.

3. Main Results

3.1. Concircular Curvature Tensor in ${\mathcal{M}}_S^{2n+1}$ with Respect to $\overline{\nabla }$

In this subsection, we first mention the following proposition:

Proposition 2.

Let the curvature tensor, the Ricci tensor, the scalar curvature, and the Ricci operator in an ${\mathcal{M}}_S^{2n+1}$ with respect to $\overline{\nabla }$ be denoted by $\overline{\mathcal{R}}$, $\overline{\mathcal{S}}$, $\overline{r}$, and $\overline{\mathcal{Q}}$, respectively. Then, we have [34]

$$\begin{gathered} \overline{\mathcal{R}}({\mathcal{U}}_1,{\mathcal{V}}_1){\mathcal{W}}_1=\mathcal{R}({\mathcal{U}}_1,{\mathcal{V}}_1){\mathcal{W}}_1+[g({\mathcal{U}}_1,{\mathcal{W}}_1)\eta \left({\mathcal{V}}_1\right)-g({\mathcal{V}}_1,{\mathcal{W}}_1)\eta \left({\mathcal{U}}_1\right)]\zeta -g({\mathcal{V}}_1,\psi {\mathcal{W}}_1)\psi {\mathcal{U}}_1 \\ +g({\mathcal{U}}_1,\psi {\mathcal{W}}_1)\psi {\mathcal{V}}_1+2g({\mathcal{V}}_1,\psi {\mathcal{U}}_1)\psi {\mathcal{W}}_1-\eta \left({\mathcal{V}}_1\right)\eta \left({\mathcal{W}}_1\right){\mathcal{U}}_1+\eta \left({\mathcal{U}}_1\right)\eta \left({\mathcal{W}}_1\right){\mathcal{V}}_1, \end{gathered}$$

(21)

$$\overline{\mathcal{R}}({\mathcal{U}}_1,{\mathcal{V}}_1)\zeta =\overline{\mathcal{R}}(\zeta ,{\mathcal{V}}_1){\mathcal{W}}_1=\eta \left(\overline{\mathcal{R}}({\mathcal{U}}_1,{\mathcal{V}}_1){\mathcal{W}}_1\right)=0,$$

(22)

$$\overline{\mathcal{S}}({\mathcal{V}}_1,{\mathcal{W}}_1)=\mathcal{S}({\mathcal{V}}_1,{\mathcal{W}}_1)-2g({\mathcal{V}}_1,{\mathcal{W}}_1)-2(n-1)\eta \left({\mathcal{V}}_1\right)\eta \left({\mathcal{W}}_1\right),$$

(23)

$$\overline{\mathcal{S}}({\mathcal{V}}_1,\zeta )=0,$$

(24)

$$\overline{\mathcal{Q}}{\mathcal{V}}_1=\mathcal{Q}{\mathcal{V}}_1-2{\mathcal{V}}_1-2(n-1)\eta \left({\mathcal{V}}_1\right)\zeta ,\overline{\mathcal{Q}}\zeta =0,$$

(25)

$$\overline{r}=r-6n,$$

(26)

for all vector fields ${\mathcal{U}}_1,{\mathcal{V}}_1,$ on ${\mathcal{M}}_S^{2n+1}$; the symbols used are defined in Section 2.

The concircular curvature tensor $\mathcal{C}$ on an ${\mathcal{M}}_S^{2n+1}$ with respect to ∇ is defined by [35]

$$\mathcal{C}({\mathcal{U}}_1,{\mathcal{V}}_1){\mathcal{W}}_1=\mathcal{R}({\mathcal{U}}_1,{\mathcal{V}}_1){\mathcal{W}}_1-{\displaystyle \frac{r}{2n(2n+1)}}[g({\mathcal{V}}_1,{\mathcal{W}}_1){\mathcal{U}}_1-g({\mathcal{U}}_1,{\mathcal{W}}_1){\mathcal{V}}_1],$$

(27)

for all ${\mathcal{U}}_1,{\mathcal{V}}_1,{\mathcal{W}}_1$ on ${\mathcal{M}}_S^{2n+1}$.

Now, the concircular curvature tensor $\overline{\mathcal{C}}$ on an ${\mathcal{M}}_S^{2n+1}$ with respect to $\overline{\nabla }$ is given by

$$\overline{\mathcal{C}}({\mathcal{U}}_1,{\mathcal{V}}_1){\mathcal{W}}_1=\overline{\mathcal{R}}({\mathcal{U}}_1,{\mathcal{V}}_1){\mathcal{W}}_1-{\displaystyle \frac{\overline{r}}{2n(2n+1)}}[g({\mathcal{V}}_1,{\mathcal{W}}_1){\mathcal{U}}_1-g({\mathcal{U}}_1,{\mathcal{W}}_1){\mathcal{V}}_1],$$

(28)

for all ${\mathcal{U}}_1,{\mathcal{V}}_1,{\mathcal{W}}_1$ on ${\mathcal{M}}_S^{2n+1}$.

Taking the inner product of (28) with ${\mathcal{Z}}_1$, we have

$$\begin{array}{ccc}\hfill \overline{\mathcal{C}}({\mathcal{U}}_1,{\mathcal{V}}_1,{\mathcal{W}}_1,{\mathcal{Z}}_1)& =& \overline{\mathcal{R}}({\mathcal{U}}_1,{\mathcal{V}}_1,{\mathcal{W}}_1,{\mathcal{Z}}_1)\hfill \\ & -& {\displaystyle \frac{\overline{r}}{2n(2n+1)}}[g({\mathcal{V}}_1,{\mathcal{W}}_1)g({\mathcal{U}}_1,{\mathcal{Z}}_1)-g({\mathcal{U}}_1,{\mathcal{W}}_1)g({\mathcal{V}}_1,{\mathcal{Z}}_1)],\hfill \end{array}$$

(29)

where $g(\overline{\mathcal{C}}({\mathcal{U}}_1,{\mathcal{V}}_1){\mathcal{W}}_1,{\mathcal{Z}}_1)=\overline{\mathcal{C}}({\mathcal{U}}_1,{\mathcal{V}}_1,{\mathcal{W}}_1,{\mathcal{Z}}_1)$ and $g(\overline{\mathcal{R}}({\mathcal{U}}_1,{\mathcal{V}}_1){\mathcal{W}}_1,{\mathcal{Z}}_1)=\overline{\mathcal{R}}({\mathcal{U}}_1,{\mathcal{V}}_1,$${\mathcal{W}}_1,{\mathcal{Z}}_1)$.

By interchanging ${\mathcal{U}}_1$ and ${\mathcal{V}}_1$ in (29), we have

$$\begin{array}{ccc}\hfill \overline{\mathcal{C}}({\mathcal{V}}_1,{\mathcal{U}}_1,{\mathcal{W}}_1,{\mathcal{Z}}_1)& =& \overline{\mathcal{R}}({\mathcal{V}}_1,{\mathcal{U}}_1,{\mathcal{W}}_1,{\mathcal{Z}}_1)\hfill \end{array}$$

(30)

$$\begin{array}{ccc}& -& {\displaystyle \frac{\overline{r}}{2n(2n+1)}}[g({\mathcal{U}}_1,{\mathcal{W}}_1)g({\mathcal{V}}_1,{\mathcal{Z}}_1)-g({\mathcal{V}}_1,{\mathcal{W}}_1)g({\mathcal{U}}_1,{\mathcal{Z}}_1)].\hfill \end{array}$$

(31)

Adding (29) and (30) leads to

$$\overline{\mathcal{C}}({\mathcal{U}}_1,{\mathcal{V}}_1,{\mathcal{W}}_1,{\mathcal{Z}}_1)+\overline{\mathcal{C}}({\mathcal{V}}_1,{\mathcal{U}}_1,{\mathcal{W}}_1,{\mathcal{Z}}_1)=\overline{\mathcal{R}}({\mathcal{U}}_1,{\mathcal{V}}_1,{\mathcal{W}}_1,{\mathcal{Z}}_1)+\overline{\mathcal{R}}({\mathcal{V}}_1,{\mathcal{U}}_1,{\mathcal{W}}_1,{\mathcal{Z}}_1).$$

(32)

In view of (21) and the fact that $\mathcal{R}({\mathcal{U}}_1,{\mathcal{V}}_1,{\mathcal{W}}_1,{\mathcal{Z}}_1)+\mathcal{R}({\mathcal{V}}_1,{\mathcal{U}}_1,{\mathcal{W}}_1,{\mathcal{Z}}_1)=0$, (32) reduces to $\overline{\mathcal{C}}({\mathcal{U}}_1,{\mathcal{V}}_1,{\mathcal{W}}_1,{\mathcal{Z}}_1)+\overline{\mathcal{C}}({\mathcal{V}}_1,{\mathcal{U}}_1,{\mathcal{W}}_1,{\mathcal{Z}}_1)=0$.

Next, interchanging ${\mathcal{W}}_1$ and ${\mathcal{Z}}_1$ in (29), we have

$$\begin{array}{ccc}\hfill \overline{\mathcal{C}}({\mathcal{U}}_1,{\mathcal{V}}_1,{\mathcal{Z}}_1,{\mathcal{W}}_1)& =& \overline{\mathcal{R}}({\mathcal{U}}_1,{\mathcal{V}}_1,{\mathcal{Z}}_1,{\mathcal{W}}_1)\hfill \\ & -& {\displaystyle \frac{\overline{r}}{2n(2n+1)}}[g({\mathcal{V}}_1,{\mathcal{Z}}_1)g({\mathcal{U}}_1,{\mathcal{W}}_1)-g({\mathcal{U}}_1,{\mathcal{Z}}_1)g({\mathcal{V}}_1,{\mathcal{W}}_1)].\hfill \end{array}$$

(33)

By adding (29) and (33), we have

$$\overline{\mathcal{C}}({\mathcal{U}}_1,{\mathcal{V}}_1,{\mathcal{W}}_1,{\mathcal{Z}}_1)+\overline{\mathcal{C}}({\mathcal{U}}_1,{\mathcal{V}}_1,{\mathcal{Z}}_1,{\mathcal{W}}_1)=\overline{\mathcal{R}}({\mathcal{U}}_1,{\mathcal{V}}_1,{\mathcal{W}}_1,{\mathcal{Z}}_1)+\overline{\mathcal{R}}({\mathcal{U}}_1,{\mathcal{V}}_1,{\mathcal{Z}}_1,{\mathcal{W}}_1),$$

(34)

By using (21) and the fact that $\mathcal{R}({\mathcal{U}}_1,{\mathcal{V}}_1,{\mathcal{W}}_1,{\mathcal{Z}}_1)+\mathcal{R}({\mathcal{U}}_1,{\mathcal{V}}_1,{\mathcal{Z}}_1,{\mathcal{W}}_1)=0$, (34) reduces to $\overline{\mathcal{C}}({\mathcal{U}}_1,{\mathcal{V}}_1,{\mathcal{W}}_1,{\mathcal{Z}}_1)+\overline{\mathcal{C}}({\mathcal{U}}_1,{\mathcal{V}}_1,{\mathcal{Z}}_1,{\mathcal{W}}_1)=0$.

Now interchanging the pair of slots in (29), we have

$$\begin{array}{ccc}\hfill \overline{\mathcal{C}}({\mathcal{W}}_1,{\mathcal{Z}}_1,{\mathcal{U}}_1,{\mathcal{V}}_1)& =& \overline{\mathcal{R}}({\mathcal{W}}_1,{\mathcal{Z}}_1,{\mathcal{U}}_1,{\mathcal{V}}_1)\hfill \\ & -& {\displaystyle \frac{\overline{r}}{2n(2n+1)}}[g({\mathcal{Z}}_1,{\mathcal{U}}_1)g({\mathcal{W}}_1,{\mathcal{V}}_1)-g({\mathcal{W}}_1,{\mathcal{U}}_1)g({\mathcal{Z}}_1,{\mathcal{V}}_1)].\hfill \end{array}$$

(35)

By subtracting (35) from (29), we have

$$\overline{\mathcal{C}}({\mathcal{U}}_1,{\mathcal{V}}_1,{\mathcal{W}}_1,{\mathcal{Z}}_1)-\overline{\mathcal{C}}({\mathcal{W}}_1,{\mathcal{Z}}_1,{\mathcal{U}}_1,{\mathcal{V}}_1)=\overline{\mathcal{R}}({\mathcal{U}}_1,{\mathcal{V}}_1,{\mathcal{W}}_1,{\mathcal{Z}}_1)-\overline{\mathcal{R}}({\mathcal{W}}_1,{\mathcal{Z}}_1,{\mathcal{U}}_1,{\mathcal{V}}_1)$$

(36)

In view of (21) and the fact that $\mathcal{R}({\mathcal{U}}_1,{\mathcal{V}}_1,{\mathcal{W}}_1,{\mathcal{Z}}_1)-\mathcal{R}({\mathcal{W}}_1,{\mathcal{Z}}_1,{\mathcal{U}}_1,{\mathcal{V}}_1)=0$, (36) reduces to $\overline{\mathcal{C}}({\mathcal{U}}_1,{\mathcal{V}}_1,{\mathcal{W}}_1,{\mathcal{Z}}_1)-\overline{\mathcal{C}}({\mathcal{W}}_1,{\mathcal{Z}}_1,{\mathcal{U}}_1,{\mathcal{V}}_1)=0$.

Thus, from the above discussion we can state the following:

Theorem 1.

In an ${\mathcal{M}}_S^{2n+1}$ admitting $\overline{\nabla }$, the following relations hold:

$\left(i\right)\overline{\mathcal{C}}({\mathcal{U}}_1,{\mathcal{V}}_1,{\mathcal{W}}_1,{\mathcal{Z}}_1)+\overline{\mathcal{C}}({\mathcal{V}}_1,{\mathcal{U}}_1,{\mathcal{W}}_1,{\mathcal{Z}}_1)=0,$

$\left(ii\right)\overline{\mathcal{C}}({\mathcal{U}}_1,{\mathcal{V}}_1,{\mathcal{W}}_1,{\mathcal{Z}}_1)+\overline{\mathcal{C}}({\mathcal{U}}_1,{\mathcal{V}}_1,{\mathcal{Z}}_1,{\mathcal{W}}_1)=0$,

$\left(iii\right)\overline{\mathcal{C}}({\mathcal{U}}_1,{\mathcal{V}}_1,{\mathcal{W}}_1,{\mathcal{Z}}_1)-\overline{\mathcal{C}}({\mathcal{W}}_1,{\mathcal{Z}}_1,{\mathcal{U}}_1,{\mathcal{V}}_1)=0$

for any ${\mathcal{U}}_1,{\mathcal{V}}_1,{\mathcal{W}}_1,{\mathcal{Z}}_1$ on ${\mathcal{M}}_S^{2n+1}$.

Next, we proceed with our study for the equivalence of $\zeta$-concircularly flatnesses with respect to ∇ and $\overline{\nabla }$.

In view of (21), (26)–(28) can be written as

$$\begin{array}{ccc}\hfill \overline{\mathcal{C}}({\mathcal{U}}_1,{\mathcal{V}}_1){\mathcal{W}}_1& =& \mathcal{C}({\mathcal{U}}_1,{\mathcal{V}}_1){\mathcal{W}}_1+[g({\mathcal{U}}_1,{\mathcal{W}}_1)\eta \left({\mathcal{V}}_1\right)-g({\mathcal{V}}_1,{\mathcal{W}}_1)\eta \left({\mathcal{U}}_1\right)]\zeta -g({\mathcal{V}}_1,\psi {\mathcal{W}}_1)\psi {\mathcal{U}}_1\hfill \\ & +& g({\mathcal{U}}_1,\psi {\mathcal{W}}_1)\psi {\mathcal{V}}_1+2g({\mathcal{V}}_1,\psi {\mathcal{U}}_1)\psi {\mathcal{W}}_1-\eta \left({\mathcal{V}}_1\right)\eta \left({\mathcal{W}}_1\right){\mathcal{U}}_1+\eta \left({\mathcal{U}}_1\right)\eta \left({\mathcal{W}}_1\right){\mathcal{V}}_1\hfill \\ & -& {\displaystyle \frac{3}{(2n+1)}}[g({\mathcal{V}}_1,{\mathcal{W}}_1){\mathcal{U}}_1-g({\mathcal{U}}_1,{\mathcal{W}}_1){\mathcal{V}}_1],\hfill \end{array}$$

(37)

where $\mathcal{C}({\mathcal{U}}_1,{\mathcal{V}}_1){\mathcal{W}}_1$ is given by (27).

Now, by putting ${\mathcal{W}}_1=\zeta$ in (37) and using (3), we find

$$\overline{\mathcal{C}}({\mathcal{U}}_1,{\mathcal{V}}_1)\zeta =\mathcal{C}({\mathcal{U}}_1,{\mathcal{V}}_1)\zeta -{\displaystyle \frac{2(n-1)}{(2n+1)}}[\eta \left({\mathcal{V}}_1\right){\mathcal{U}}_1-\eta \left({\mathcal{U}}_1\right){\mathcal{V}}_1].$$

(38)

Since, in general, $\eta \left({\mathcal{V}}_1\right){\mathcal{U}}_1-\eta \left({\mathcal{U}}_1\right){\mathcal{V}}_1=\mathcal{R}({\mathcal{U}}_1,{\mathcal{V}}_1)\zeta \ne 0$ in an ${\mathcal{M}}_S^{2n+1}$, then we have the following result:

Theorem 2.

In an ${\mathcal{M}}_S^{2n+1}$, ζ-concircularly flatnesses with respect to ∇ and $\overline{\nabla }$ are not equivalent.

If ${\mathcal{U}}_1$ and ${\mathcal{V}}_1$ are orthogonal to $\zeta$, then (38) reduces to $\overline{\mathcal{C}}({\mathcal{U}}_1,{\mathcal{V}}_1)\zeta =\mathcal{C}({\mathcal{U}}_1,{\mathcal{V}}_1)\zeta$. Thus, we have the following:

Corollary 1.

In an ${\mathcal{M}}_S^{2n+1}$, ζ-concircularly flatnesses with respect to ∇ and $\overline{\nabla }$ are equivalent, if ${\mathcal{U}}_1$ and ${\mathcal{V}}_1$ are orthogonal to ζ.

3.2. $({\mathcal{M}}_S^{2n+1},g,\mathcal{F},\delta ,\vartheta ,\rho ,\overline{\nabla })$

In this section we study ${\mathcal{M}}_S^{2n+1}$ with respect to $\overline{\nabla }$ admitting $(g,\mathcal{F},\delta ,\vartheta ,\rho )$.

Let the metric of an ${\mathcal{M}}_S^{2n+1}$ with respect to $\overline{\nabla }$ be $(g,\mathcal{F},\delta ,\vartheta ,\rho )$, then we write (2) as

$$\begin{array}{c}\hfill \left({\overline{\pounds }}_{\mathcal{F}}g\right)({\mathcal{U}}_1,{\mathcal{V}}_1)+2\vartheta \overline{\mathcal{S}}({\mathcal{U}}_1,{\mathcal{V}}_1)+(2\delta -\rho \overline{r})g({\mathcal{U}}_1,{\mathcal{V}}_1)=0,\end{array}$$

(39)

for any ${\mathcal{U}}_1$, ${\mathcal{V}}_1$ on ${\mathcal{M}}_S^{2n+1}$.

The Lie derivative of g with respect to $\overline{\nabla }$ along $\mathcal{F}$ is given by

$$\begin{array}{c}\hfill \left({\overline{\pounds }}_{\mathcal{F}}g\right)({\mathcal{U}}_1,{\mathcal{V}}_1)=g({\overline{\nabla }}_{{\mathcal{U}}_1}\mathcal{F},{\mathcal{V}}_1)+g({\mathcal{U}}_1,{\overline{\nabla }}_{{\mathcal{V}}_1}\mathcal{F}).\end{array}$$

(40)

By using (20) in (40), it follows that

$$\begin{array}{c}\hfill \left({\overline{\pounds }}_{\mathcal{F}}g\right)({\mathcal{U}}_1,{\mathcal{V}}_1)=g({\nabla }_{{\mathcal{U}}_1}\mathcal{F},{\mathcal{V}}_1)+g({\mathcal{U}}_1,{\nabla }_{{\mathcal{V}}_1}\mathcal{F})=\left({\pounds }_{\mathcal{F}}g\right)({\mathcal{U}}_1,{\mathcal{V}}_1),\end{array}$$

(41)

where (5) being used. Equation (41) is the relation between the Lie derivatives of g along $\mathcal{F}$ with respect to ∇ and $\overline{\nabla }$.

Now, in view of (23), (26), and (41), (39) takes the form

$$\begin{array}{ccc}\hfill \left({\pounds }_{\mathcal{F}}g\right)({\mathcal{U}}_1,{\mathcal{V}}_1)& =& \{(\rho -{\displaystyle \frac{\vartheta }{n}})r+(6\vartheta -6\rho n-2\delta )\}g({\mathcal{U}}_1,{\mathcal{V}}_1)\hfill \\ & & +\vartheta ({\displaystyle \frac{r}{n}}-6)\eta \left({\mathcal{U}}_1\right)\eta \left({\mathcal{V}}_1\right).\hfill \end{array}$$

(42)

By putting ${\mathcal{U}}_1={\mathcal{V}}_1=\zeta$ in (42) and using (3), we have

$$\begin{array}{c}\hfill \left({\pounds }_{\mathcal{F}}g\right)(\zeta ,\zeta )=\rho (r-6n)-2\delta .\end{array}$$

(43)

The Lie derivative of $g(\zeta ,\zeta )=1$ along $\mathcal{F}$ leads to

$$\begin{array}{c}\hfill \left({\pounds }_{\mathcal{F}}g\right)(\zeta ,\zeta )=-2\eta \left({\pounds }_V\zeta \right).\end{array}$$

(44)

Thus, from (43) and (44), we deduce

$$\begin{array}{c}\hfill \eta \left({\pounds }_{\mathcal{F}}\zeta \right)=\delta -{\displaystyle \frac{\rho }{2}}(r-6n).\end{array}$$

(45)

Thus, we state the following result:

Theorem 3.

Let an ${\mathcal{M}}_S^{2n+1}$ with respect to $\overline{\nabla }$ admit $(g,\mathcal{F},\delta ,\vartheta ,\rho )$; then, ${\pounds }_{\mathcal{F}}\zeta$ is orthogonal to ζ if and only if $\delta ={\displaystyle \frac{\rho }{2}}(r-6n).$

Now we discuss a particular case, i.e., $\mathcal{F}=\zeta$, then (39) takes the form

$$\begin{array}{c}\hfill \left({\overline{\pounds }}_{\zeta }g\right)({\mathcal{U}}_1,{\mathcal{V}}_1)+2\vartheta \overline{\mathcal{S}}({\mathcal{U}}_1,{\mathcal{V}}_1)+(2\delta -\rho \overline{r})g({\mathcal{U}}_1,{\mathcal{V}}_1)=0.\end{array}$$

(46)

Since ${\overline{\nabla }}_{{\mathcal{U}}_1}\zeta =0$, then we find

$$\begin{array}{c}\hfill \left({\overline{\pounds }}_{\zeta }g\right)({\mathcal{U}}_1,{\mathcal{V}}_1)=g({\overline{\nabla }}_{{\mathcal{U}}_1}\zeta ,{\mathcal{V}}_1)+g({\mathcal{U}}_1,{\overline{\nabla }}_{{\mathcal{V}}_1}\zeta )=0,\end{array}$$

(47)

which shows that the Lie derivative of g along $\zeta$ with respect to $\overline{\nabla }$ vanishes.

Thus, from (46) and (47), it follows that

$$\begin{array}{c}\hfill \overline{S}({\mathcal{U}}_1,{\mathcal{V}}_1)=-{\displaystyle \frac{1}{\vartheta }}(\delta -{\displaystyle \frac{\rho \overline{r}}{2}})g({\mathcal{U}}_1,{\mathcal{V}}_1),\vartheta \ne 0,\end{array}$$

(48)

which is the equation of the Einstein manifold.

Now, by putting ${\mathcal{U}}_1={\mathcal{V}}_1=\zeta$ in (48), and then using (3) and (24), we obtain

$$\begin{array}{c}\hfill \delta ={\displaystyle \frac{\rho \overline{r}}{2}},\vartheta \ne 0.\end{array}$$

(49)

Thus, from (48) and (49), we state the following theorem:

Theorem 4.

Let an ${\mathcal{M}}_S^{2n+1}$ with respect to $\overline{\nabla }$ admit $(g,\zeta ,\delta ,\vartheta ,\rho )$ and ${\mathcal{M}}_S^{2n+1}$ be an Einstein manifold; the soliton constant δ is given by $\delta ={\displaystyle \frac{\rho \overline{r}}{2}}$, $\vartheta \ne 0.$

Now, we have the following corollary:

Corollary 2.

Let an ${\mathcal{M}}_S^{2n+1}$ with respect to $\overline{\nabla }$ admit $(g,\zeta ,\delta ,\vartheta ,\rho )$. Then, we have

Values of $\rho$	Values of $\overline{r}$	Values of $\delta$	Conditions for the $(g,\zeta ,\delta ,\vartheta ,\rho )$ to be expanding, steady, or shrinking
$\rho >0$	$\left(i\right)\overline{r}>0$ $\left(ii\right)\overline{r}=0$ $\left(iii\right)\overline{r}<0$	$\left(i\right)\delta >0$ $\left(ii\right)\delta =0$ $\left(iii\right)\delta <0$	$\left(i\right)$ expanding $\left(ii\right)$ steady $\left(iii\right)$ shrinking
$\rho =0$	$\overline{r}>0$, $=0$ or $<0$	$\delta =0$	steady
$\rho <0$	$\left(i\right)\overline{r}>0$ $\left(ii\right)\overline{r}=0$ $\left(iii\right)\overline{r}<0$	$\left(i\right)\delta <0$ $\left(ii\right)\delta =0$ $\left(iii\right)\delta >0$	$\left(i\right)$ shrinking $\left(ii\right)$ steady $\left(iii\right)$ expanding

Now, let an ${\mathcal{M}}_S^{2n+1}$ with respect to $\overline{\nabla }$ admit $(g,\mathcal{F},\delta ,\vartheta ,\rho )$ such that $\mathcal{F}$ is pointwise collinear with $\zeta$, i.e., $\mathcal{F}=b\zeta$, where b is a function. The, (39) holds and thus we have

$$\begin{gathered} bg({\overline{\nabla }}_{{\mathcal{U}}_1}\zeta ,{\mathcal{V}}_1)+\left({\mathcal{U}}_{1}b\right)\eta \left({\mathcal{V}}_1\right)+bg({\mathcal{U}}_1,{\overline{\nabla }}_{{\mathcal{V}}_1}\zeta )+\left({\mathcal{V}}_{1}b\right)\eta \left({\mathcal{U}}_1\right) \\ +2\vartheta \overline{\mathcal{S}}({\mathcal{U}}_1,{\mathcal{V}}_1)+(2\delta -\rho \overline{r})g({\mathcal{U}}_1,{\mathcal{V}}_1)=0. \end{gathered}$$

By using ${\overline{\nabla }}_{{\mathcal{U}}_1}\zeta =0$, the above equation takes the form

$$\left({\mathcal{U}}_{1}b\right)\eta \left({\mathcal{V}}_1\right)+\left({\mathcal{V}}_{1}b\right)\eta \left({\mathcal{U}}_1\right)+2\vartheta \overline{\mathcal{S}}({\mathcal{U}}_1,{\mathcal{V}}_1)+(2\delta -\rho \overline{r})g({\mathcal{U}}_1,{\mathcal{V}}_1)=0.$$

(50)

Now, taking ${\mathcal{V}}_1=\zeta$ in (50), and then using (3) and (24), we have

$$\left({\mathcal{U}}_{1}b\right)+\left(\zeta b\right)\eta \left({\mathcal{U}}_1\right)+(2\delta -\rho \overline{r})\eta \left({\mathcal{U}}_1\right)=0.$$

(51)

Again putting ${\mathcal{U}}_1=\zeta$ in (51) and using (3), we find

$$\left(\zeta b\right)=-(\delta -{\displaystyle \frac{\rho \overline{r}}{2}}).$$

(52)

Combining the relations (51) and (52), we obtain

$$db=-[\delta -{\displaystyle \frac{\rho \overline{r}}{2}}]\eta .$$

(53)

Now, applying d on (53) leads to

$$(\delta -{\displaystyle \frac{\rho \overline{r}}{2}})d\eta =0.$$

(54)

Since $d\eta \ne 0$, it follows from (54) that

$$\delta ={\displaystyle \frac{\rho \overline{r}}{2}}.$$

(55)

By using (55) in (53), we find $db=0\Rightarrow b$ is a constant. Therefore, (50) turns into

$$\overline{S}({\mathcal{U}}_1,{\mathcal{V}}_1)=-{\displaystyle \frac{1}{\vartheta }}(\delta -{\displaystyle \frac{\rho \overline{r}}{2}})g({\mathcal{U}}_1,{\mathcal{V}}_1),\vartheta \ne 0,$$

(56)

which is the equation of the Einstein manifold.

Thus, we have the following result:

Theorem 5.

Let an ${\mathcal{M}}_S^{2n+1}$ with respect to $\overline{\nabla }$ admit $(g,\mathcal{F},\delta ,\mu ,\vartheta ,\rho )$ such that $\mathcal{F}$ is pointwise collinear with ζ; then, $\mathcal{F}$ is a constant multiple of ζ and ${\mathcal{M}}_S^{2n+1}$ is an Einstein manifold with respect to $\overline{\nabla }$. Moreover, the soliton constant δ is given by $\delta ={\displaystyle \frac{\rho \overline{r}}{2}}$, $\vartheta \ne 0.$

3.3. $({\mathcal{M}}_S^{2n+1},g,\zeta ,\delta ,\vartheta ,\rho ,\overline{\nabla })$ Satisfying $\zeta$-Concircularly Flat and $\psi$-Concircularly Semisymmetric Conditions

First, we consider that an ${\mathcal{M}}_S^{2n+1}$ with respect to $\overline{\nabla }$ admitting $(g,\zeta ,\delta ,\vartheta ,\rho )$ is $\zeta$-concircularly flat, i.e., $\overline{C}({\mathcal{U}}_1,{\mathcal{V}}_1)\zeta =0$. Then, from (28), it follows that

$$\overline{R}({\mathcal{U}}_1,{\mathcal{V}}_1)\zeta =-{\displaystyle \frac{\overline{r}}{2n(2n+1)}}[\eta \left({\mathcal{V}}_1\right){\mathcal{U}}_1-\eta \left({\mathcal{U}}_1\right){\mathcal{V}}_1],$$

where $\overline{R}$ and $\overline{r}$ indicate the curvature tensor and the scalar curvature tensor with respect to $\overline{\nabla }$.

In view of (22), the above equation turns to

$${\displaystyle \frac{\overline{r}}{2n(2n+1)}}[\eta \left({\mathcal{V}}_1\right){\mathcal{U}}_1-\eta \left({\mathcal{U}}_1\right){\mathcal{V}}_1]=0.$$

(57)

This gives either $\overline{r}=0$ or $\eta \left({\mathcal{V}}_1\right){\mathcal{U}}_1-\eta \left({\mathcal{U}}_1\right){\mathcal{V}}_1=0$. Since (in general) $\eta \left({\mathcal{V}}_1\right){\mathcal{U}}_1-\eta \left({\mathcal{U}}_1\right){\mathcal{V}}_1\ne 0$ in an ${\mathcal{M}}_S^{2n+1}$, by using $\overline{r}=0$ in (49), it follows that

$$\delta =0.$$

(58)

This implies that the soliton $(g,\zeta ,\delta ,\vartheta ,\rho ,\overline{\nabla })$ is steady.

Thus, we state the following result:

Theorem 6.

If an ${\mathcal{M}}_S^{2n+1}$ is ζ-concircularly flat and admits $(g,\zeta ,\delta ,\vartheta ,\rho ,\overline{\nabla })$, then the soliton $(g,\zeta ,\delta ,\vartheta ,\rho ,\overline{\nabla })$ is steady.

Secondly, we consider that an ${\mathcal{M}}_S^{2n+1}$ with respect to $\overline{\nabla }$ admitting $(g,\zeta ,\delta ,\vartheta ,\rho )$ is $\psi$-concircularly semisymmetric, i.e., $\overline{\mathcal{C}}·\psi =0$. This implies

$$(\overline{\mathcal{C}}({\mathcal{U}}_1,{\mathcal{V}}_1)·\psi ){\mathcal{W}}_1=\overline{\mathcal{C}}({\mathcal{U}}_1,{\mathcal{V}}_1)\psi {\mathcal{W}}_1-\psi \overline{\mathcal{C}}({\mathcal{U}}_1,{\mathcal{V}}_1){\mathcal{W}}_1=0,$$

(59)

for any ${\mathcal{U}}_1,{\mathcal{V}}_1,{\mathcal{W}}_1$ on ${\mathcal{M}}_S^{2n+1}$, where $\overline{\mathcal{C}}$ is the concircular curvature tensor with respect to $\overline{\nabla }$.

From (28), we find

$$\overline{\mathcal{C}}({\mathcal{U}}_1,{\mathcal{V}}_1)\psi {\mathcal{W}}_1=\overline{\mathcal{R}}({\mathcal{U}}_1,{\mathcal{V}}_1)\psi {\mathcal{W}}_1-{\displaystyle \frac{\overline{r}}{2n(2n+1)}}[g({\mathcal{V}}_1,\psi {\mathcal{W}}_1){\mathcal{U}}_1-g({\mathcal{U}}_1,\psi {\mathcal{W}}_1){\mathcal{V}}_1],$$

(60)

and

$$\psi \overline{\mathcal{C}}({\mathcal{U}}_1,{\mathcal{V}}_1){\mathcal{W}}_1=\psi \overline{\mathcal{R}}({\mathcal{U}}_1,{\mathcal{V}}_1){\mathcal{W}}_1-{\displaystyle \frac{\overline{r}}{2n(2n+1)}}[g({\mathcal{V}}_1,{\mathcal{W}}_1)\psi {\mathcal{U}}_1-g({\mathcal{U}}_1,{\mathcal{W}}_1)\psi {\mathcal{V}}_1].$$

(61)

By using (60) and (61) in (59), we have

$$\begin{array}{ccc}\hfill & & \overline{\mathcal{R}}({\mathcal{U}}_1,{\mathcal{V}}_1)\psi {\mathcal{W}}_1-\psi \overline{\mathcal{R}}({\mathcal{U}}_1,{\mathcal{V}}_1){\mathcal{W}}_1\hfill \\ & & -{\displaystyle \frac{\overline{r}}{2n(2n+1)}}[g({\mathcal{V}}_1,\psi {\mathcal{W}}_1){\mathcal{U}}_1-g({\mathcal{U}}_1,\psi {\mathcal{W}}_1){\mathcal{V}}_1-g({\mathcal{V}}_1,{\mathcal{W}}_1)\psi {\mathcal{U}}_1+g({\mathcal{U}}_1,{\mathcal{W}}_1)\psi {\mathcal{V}}_1]=0.\hfill \end{array}$$

(62)

Taking ${\mathcal{U}}_1=\zeta$ in (62), and then using (3), (4) and (22), we have

$${\displaystyle \frac{\overline{r}}{2n(2n+1)}}(\eta \left({\mathcal{W}}_1\right)\psi {\mathcal{U}}_1+g({\mathcal{U}}_1,\psi {\mathcal{W}}_1)\zeta )=0.$$

(63)

Now, taking the inner product of (63) with $\zeta$ and using (3) leads to

$${\displaystyle \frac{\overline{r}}{2n(2n+1)}}g({\mathcal{U}}_1,\psi {\mathcal{W}}_1)=0.$$

This gives either $\overline{r}=0$ or $g({\mathcal{U}}_1,\psi {\mathcal{W}}_1)=0.$ However, $g({\mathcal{U}}_1,\psi {\mathcal{W}}_1)\ne 0,$; therefore, $\overline{r}=0$ and, using in (49), we infer

$$\delta =0.$$

(64)

This shows that the soliton $(g,\zeta ,\delta ,\vartheta ,\rho ,\overline{\nabla })$ is steady.

Thus, we state the following result:

Theorem 7.

If an ${\mathcal{M}}_S^{2n+1}$ is ψ-concircularly semisymmetric and admits $(g,\zeta ,\delta ,\vartheta ,\rho ,\overline{\nabla })$, then the soliton $(g,\zeta ,\delta ,\vartheta ,\rho ,\overline{\nabla })$ is steady.

3.4. $({\mathcal{M}}_S^{2n+1},g,\zeta ,\delta ,\vartheta ,\rho ,\overline{\nabla })$ Satisfying $\overline{C}(\zeta ,{\mathcal{U}}_1)·\overline{\mathcal{S}}=0$

This subsection deals with ${\mathcal{M}}_S^{2n+1}$ admitting $(g,\zeta ,\delta ,\vartheta ,\rho )$ and satisfying $\overline{\mathcal{C}}(\zeta ,{\mathcal{U}}_1)·\overline{\mathcal{S}}=0$. This implies

$$\overline{\mathcal{S}}(\overline{\mathcal{C}}(\zeta ,C){\mathcal{V}}_1,{\mathcal{W}}_1)+\overline{\mathcal{S}}({\mathcal{V}}_1,\overline{\mathcal{C}}(\zeta ,{\mathcal{U}}_1){\mathcal{W}}_1)=0,$$

(65)

for any ${\mathcal{U}}_1,{\mathcal{V}}_1,{\mathcal{W}}_1$ on ${\mathcal{M}}_S^{2n+1}$, where $\overline{\mathcal{S}}$ is the Ricci tensor with respect to $\overline{\nabla }$.

Replacing ${\mathcal{U}}_1$ by $\zeta$, ${\mathcal{V}}_1$ by ${\mathcal{U}}_1$ and ${\mathcal{W}}_1$ by ${\mathcal{V}}_1$ in (28) and using (22), we have

$$\overline{\mathcal{C}}(\zeta ,{\mathcal{U}}_1){\mathcal{V}}_1=-{\displaystyle \frac{\overline{r}}{2n(2n+1)}}[g({\mathcal{U}}_1,{\mathcal{V}}_1)\zeta -\eta \left({\mathcal{V}}_1\right){\mathcal{U}}_1].$$

(66)

By virtue of (66), (65) takes the form

$${\displaystyle \frac{\overline{r}}{2n(2n+1)}}[\eta \left({\mathcal{V}}_1\right)\overline{\mathcal{S}}({\mathcal{U}}_1,{\mathcal{W}}_1)-g({\mathcal{U}}_1,{\mathcal{V}}_1)\overline{\mathcal{S}}(\zeta ,{\mathcal{W}}_1)+\eta \left({\mathcal{W}}_1\right)\overline{\mathcal{S}}({\mathcal{U}}_1,{\mathcal{V}}_1)-g({\mathcal{U}}_1,{\mathcal{W}}_1)\overline{\mathcal{S}}({\mathcal{V}}_1,\zeta )]=0.$$

By using (24), the above equation reduces to

$${\displaystyle \frac{\overline{r}}{2n(2n+1)}}[\eta \left({\mathcal{V}}_1\right)\overline{\mathcal{S}}({\mathcal{U}}_1,{\mathcal{W}}_1)+\eta \left({\mathcal{W}}_1\right)\overline{\mathcal{S}}({\mathcal{U}}_1,{\mathcal{V}}_1)]=0.$$

(67)

Taking ${\mathcal{W}}_1=\zeta$ in (67), and then using (3) and (24), we obtain

$${\displaystyle \frac{\overline{r}}{2n(2n+1)}}\overline{\mathcal{S}}({\mathcal{U}}_1,{\mathcal{V}}_1)=0.$$

Thus, we have either $\overline{r}=0$, or $\overline{\mathcal{S}}({\mathcal{U}}_1,{\mathcal{V}}_1)=0.$ By using $\overline{r}=0$ in (49), we find

$$\delta =0.$$

(68)

The relation $\delta =0$ indicates that the soliton $(g,\zeta ,\delta ,\vartheta ,\rho ,\overline{\nabla })$ is steady.

Thus, we state the following theorem:

Theorem 8.

If $({\mathcal{M}}_S^{2n+1},g,\zeta ,\delta ,\vartheta ,\rho ,\overline{\nabla })$ satisfies $\overline{\mathcal{C}}(\zeta ,{\mathcal{U}}_1)·\overline{\mathcal{S}}=0$. Then, either the soliton is steady or the manifold is Ricci flat with respect to $\overline{\nabla }$.

Example 1.

We consider an ${\mathcal{M}}^3=\left\{(u,v,w)\in R^3:w\ne 0\right\}$, where $(u,v,w)$ are the common coordinates in $R^3$. Let ${ϵ}_1,$${ϵ}_2$ and ${ϵ}_3$ be the vector fields on ${\mathcal{M}}^3$ given by

$${ϵ}_1={\displaystyle \frac{\partial }{\partial u}}-v{\displaystyle \frac{\partial }{\partial w}},{ϵ}_2=2{\displaystyle \frac{\partial }{\partial v}},{ϵ}_3={\displaystyle \frac{\partial }{\partial w}}=\zeta ,$$

which are linearly independent at each point of ${\mathcal{M}}^3$. Let the Riemannian metric g be defined by

$$g({ϵ}_i,{ϵ}_j)=1,1\le i=j\le 3;g({ϵ}_i,{ϵ}_j)=0,otherwise.$$

Let the 1-form $\eta$ on ${\mathcal{M}}^3$ be defined by $\eta \left({\mathcal{U}}_1\right)=g({\mathcal{U}}_1,{ϵ}_3)=g({\mathcal{U}}_1,\zeta )$ for all ${\mathcal{U}}_1$ on ${\mathcal{M}}^3$. Let $\psi$, a $(1,1)$ tensor field on ${\mathcal{M}}^3$, be defined by

$$\psi {ϵ}_1={ϵ}_2,\psi {ϵ}_2=-{ϵ}_1,\psi {ϵ}_3=0.$$

The linearity property of $\psi$ and g yields

$$\eta \left(\zeta \right)=g(\zeta ,\zeta )=1,{\psi }^2{\mathcal{U}}_1=-{\mathcal{U}}_1+\eta \left({\mathcal{U}}_1\right)\zeta ,\eta \left(\psi {\mathcal{U}}_1\right)=0,$$

$$g({\mathcal{U}}_1,\zeta )=\eta \left({\mathcal{U}}_1\right),g(\psi {\mathcal{U}}_1,\psi {\mathcal{V}}_1)=g({\mathcal{U}}_1,{\mathcal{V}}_1)-\eta \left({\mathcal{U}}_1\right)\eta \left({\mathcal{V}}_1\right)$$

for all ${\mathcal{U}}_1,{\mathcal{V}}_1$ on ${\mathcal{M}}^3$.

Now, by easy computations, we obtain the following values of the Lie brackets

$$[{ϵ}_1,{ϵ}_2]=2{ϵ}_3,[{ϵ}_1,{ϵ}_3]=0,[{ϵ}_2,{ϵ}_3]=0.$$

By using Koszul’s formula, the following values of the Lie derivatives are obtained:

$$\begin{gathered} {\nabla }_{{ϵ}_1}{ϵ}_1=0,{\nabla }_{{ϵ}_2}{ϵ}_1=-{ϵ}_3,{\nabla }_{{ϵ}_3}{ϵ}_1=-{ϵ}_2,{\nabla }_{{ϵ}_1}{ϵ}_2={ϵ}_3,{\nabla }_{{ϵ}_2}{ϵ}_2=0, \\ {\nabla }_{{ϵ}_3}{ϵ}_2={ϵ}_1,{\nabla }_{{ϵ}_1}{ϵ}_3=-{ϵ}_2,{\nabla }_{{ϵ}_2}{ϵ}_3={ϵ}_1,{\nabla }_{{ϵ}_3}{ϵ}_3=0. \end{gathered}$$

(69)

Also, it can be easily verified that

$${\nabla }_{{\mathcal{U}}_1}\zeta =-\psi {\mathcal{U}}_1\mathrm{and}\left({\nabla }_{{\mathcal{U}}_1}\psi \right){\mathcal{V}}_1=g({\mathcal{U}}_1,{\mathcal{V}}_1)\zeta -\eta \left({\mathcal{V}}_1\right){\mathcal{U}}_1.$$

Thus, the manifold ${\mathcal{M}}^3$ is a Sasakian manifold.

By using (69) in (20), we obtain the following values of the Lie derivatives:

$$\begin{gathered} {\overline{\nabla }}_{{ϵ}_1}{ϵ}_1=0,{\overline{\nabla }}_{{ϵ}_2}{ϵ}_1=0,{\overline{\nabla }}_{{ϵ}_3}{ϵ}_1=-2{ϵ}_2,{\overline{\nabla }}_{{ϵ}_1}{ϵ}_2=0,{\nabla }_{{ϵ}_2}{ϵ}_2=0, \\ {\overline{\nabla }}_{{ϵ}_3}{ϵ}_2=2{ϵ}_1,{\overline{\nabla }}_{{ϵ}_1}{ϵ}_3=0,{\overline{\nabla }}_{{ϵ}_2}{ϵ}_3=0,{\overline{\nabla }}_{{ϵ}_3}{ϵ}_3=0. \end{gathered}$$

(70)

The curvature tensor in terms of Lie derivative is defined by

$$\mathcal{R}({\mathcal{U}}_1,{\mathcal{V}}_1){\mathcal{W}}_1={\nabla }_{{\mathcal{U}}_1}{\nabla }_{{\mathcal{V}}_1}{\mathcal{W}}_1-{\nabla }_{{\mathcal{V}}_1}{\nabla }_{{\mathcal{U}}_1}{\mathcal{W}}_1-{\nabla }_{[{\mathcal{U}}_1,{\mathcal{V}}_1]}{\mathcal{W}}_1.$$

(71)

By using (69) and (70) in (71), we can easily obtain the following components of $\mathcal{R}$ and $\overline{\mathcal{R}}$:

$$\begin{gathered} \mathcal{R}({ϵ}_1,{ϵ}_2){ϵ}_1=3{ϵ}_2,\mathcal{R}({ϵ}_1,{ϵ}_2){ϵ}_2=-3{ϵ}_1,\mathcal{R}({ϵ}_1,{ϵ}_2){ϵ}_3=0, \\ \mathcal{R}({ϵ}_2,{ϵ}_3){ϵ}_1=0,\mathcal{R}({ϵ}_2,{ϵ}_3){ϵ}_2=-{ϵ}_3,\mathcal{R}({ϵ}_2,{ϵ}_3){ϵ}_3={ϵ}_2 \\ \mathcal{R}({ϵ}_1,{ϵ}_3){ϵ}_1=-{ϵ}_3,\mathcal{R}({ϵ}_1,{ϵ}_3){ϵ}_2=0,\mathcal{R}({ϵ}_1,{ϵ}_3){ϵ}_3={ϵ}_1 \end{gathered}$$

(72)

and

$$\begin{gathered} \overline{\mathcal{R}}({ϵ}_1,{ϵ}_2){ϵ}_1=4{ϵ}_2,\overline{\mathcal{R}}({ϵ}_1,{ϵ}_2){ϵ}_2=-4{ϵ}_1,\overline{\mathcal{R}}({ϵ}_1,{ϵ}_2){ϵ}_3=0, \\ \overline{\mathcal{R}}({ϵ}_2,{ϵ}_3){ϵ}_1=0,\overline{\mathcal{R}}({ϵ}_2,{ϵ}_3){ϵ}_2=0,\overline{\mathcal{R}}({ϵ}_2,{ϵ}_3){ϵ}_3=0, \\ \overline{\mathcal{R}}({ϵ}_1,{ϵ}_3){ϵ}_1=0,\overline{\mathcal{R}}({ϵ}_1,{ϵ}_3){ϵ}_2=0,\overline{\mathcal{R}}({ϵ}_1,{ϵ}_3){ϵ}_3=0, \end{gathered}$$

(73)

repectively.

From (72) and (73), we obtain the following components of $\mathcal{S}$ and $\overline{\mathcal{S}}$, respectively:

$$\mathcal{S}({ϵ}_1,{ϵ}_1)=-2,\mathcal{S}({ϵ}_2,{ϵ}_2)=-2,\mathcal{S}({ϵ}_3,{ϵ}_3)=2\Rightarrow r=-2,$$

and

$$\overline{\mathcal{S}}({ϵ}_1,{ϵ}_1)=-4,\overline{\mathcal{S}}({ϵ}_2,{ϵ}_2)=-4,\overline{\mathcal{S}}({ϵ}_3,{ϵ}_3)=0,\Rightarrow \overline{r}=-8.$$

Now, taking ${\mathcal{U}}_1={\mathcal{V}}_1={ϵ}_3$ in (48), we have

$$\begin{array}{c}\hfill \overline{\mathcal{S}}({ϵ}_3,{ϵ}_3)=-{\displaystyle \frac{1}{\vartheta }}(\delta -{\displaystyle \frac{\rho \overline{r}}{2}})g({ϵ}_3,{ϵ}_3),\vartheta \ne 0.\end{array}$$

(74)

By using the values $\overline{\mathcal{S}}({ϵ}_3,{ϵ}_3)=0$, $\overline{r}=-8$ and $g({ϵ}_3,{ϵ}_3)=1$ in (74), we obtain $\delta =-4\rho$. Thus, for $\overline{r}=-8$, $(g,\zeta ,\delta ,\vartheta ,\rho )$ is shrinking ($\delta <0$) if $\rho >0$, steady ($\delta =0$) if $\rho =0$, and expanding ($\delta >0$) if $\rho <0$. This verifies Corollary 2 for the negative values of $\overline{r}$.

Example 2.

We consider an ${\mathcal{M}}^3=\left\{(u,v,w)\in R^3:w\ne 0\right\}$, where $(u,v,w)$ are the common coordinates in $R^3$. Let ${ϵ}_1,$${ϵ}_2$ and ${ϵ}_3$ be the vector fields on ${\mathcal{M}}^3$ given by

$${ϵ}_1=e^w({\displaystyle \frac{\partial }{\partial u}}+{\displaystyle \frac{\partial }{\partial v}}),{ϵ}_2=-e^w({\displaystyle \frac{\partial }{\partial u}}-{\displaystyle \frac{\partial }{\partial v}}),{ϵ}_3={\displaystyle \frac{\partial }{\partial w}}=\zeta ,$$

which are linearly independent at each point of ${\mathcal{M}}^3$. Let the Riemannian metric g be defined by

$$g({ϵ}_i,{ϵ}_j)=1,1\le i=j\le 3;g({ϵ}_i,{ϵ}_j)=0,otherwise.$$

Let the 1-form $\eta$ on ${\mathcal{M}}^3$ be defined by $\eta \left({\mathcal{U}}_1\right)=g({\mathcal{U}}_1,{ϵ}_3)=g({\mathcal{U}}_1,\zeta )$ for all ${\mathcal{U}}_1$ on ${\mathcal{M}}^3$. Let $\psi$, a $(1,1)$ tensor field on ${\mathcal{M}}^3$, be defined by

$$\psi {ϵ}_1={ϵ}_1,\psi {ϵ}_2={ϵ}_2,\psi {ϵ}_3=0.$$

Then, we have

$${\psi }^2{\mathcal{U}}_1=-{\mathcal{U}}_1+\eta \left({\mathcal{U}}_1\right)\zeta ,g(\psi {\mathcal{U}}_1,\psi {\mathcal{V}}_1)=g({\mathcal{U}}_1,{\mathcal{V}}_1)-\eta \left({\mathcal{U}}_1\right)\eta \left({\mathcal{V}}_1\right)$$

for all ${\mathcal{U}}_1,{\mathcal{V}}_1$ on ${\mathcal{M}}^3$.

Thus, for ${ϵ}_3=\zeta$, ${\mathcal{M}}^3(\psi ,\zeta ,\eta ,g)$ defines an almost contact metric structure on ${\mathcal{M}}^3$.

Now, using easy computations, we obtain the following values of the Lie brackets:

$$[{ϵ}_1,{ϵ}_2]=0,[{ϵ}_1,{ϵ}_3]=-{ϵ}_1,[{ϵ}_2,{ϵ}_3]=-{ϵ}_2.$$

By using Koszul’s formula, we can easily compute the following values of the Lie derivatives:

$$\begin{gathered} {\nabla }_{{ϵ}_1}{ϵ}_1={ϵ}_3,{\nabla }_{{ϵ}_2}{ϵ}_1=0,{\nabla }_{{ϵ}_3}{ϵ}_1=0,{\nabla }_{{ϵ}_1}{ϵ}_2=0,{\nabla }_{{ϵ}_2}{ϵ}_2={ϵ}_3, \\ {\nabla }_{{ϵ}_3}{ϵ}_2=0,{\nabla }_{{ϵ}_1}{ϵ}_3=-{ϵ}_1,{\nabla }_{{ϵ}_2}{ϵ}_3=-{ϵ}_2,{\nabla }_{{ϵ}_3}{ϵ}_3=0. \end{gathered}$$

(75)

It can also be easily verified that ${\nabla }_{{\mathcal{U}}_1}\zeta =-\psi {\mathcal{U}}_1.$ Thus, the manifold ${\mathcal{M}}^3$ is a Sasakian manifold.

By using (75) in (20), we compute the following values of the Lie derivatives:

$$\begin{gathered} {\overline{\nabla }}_{{ϵ}_1}{ϵ}_1=2{ϵ}_3,{\overline{\nabla }}_{{ϵ}_2}{ϵ}_1=0,{\overline{\nabla }}_{{ϵ}_3}{ϵ}_1=-{ϵ}_1,{\overline{\nabla }}_{{ϵ}_1}{ϵ}_2=0,{\nabla }_{{ϵ}_2}{ϵ}_2=2{ϵ}_3, \\ {\overline{\nabla }}_{{ϵ}_3}{ϵ}_2=-{ϵ}_2,{\overline{\nabla }}_{{ϵ}_1}{ϵ}_3=0,{\overline{\nabla }}_{{ϵ}_2}{ϵ}_3=0,{\overline{\nabla }}_{{ϵ}_3}{ϵ}_3=0. \end{gathered}$$

(76)

By using (75) and (76) in (71), we can easily obtain the following components of $\mathcal{R}$ and $\overline{\mathcal{R}}$

$$\begin{gathered} \mathcal{R}({ϵ}_1,{ϵ}_2){ϵ}_1={ϵ}_2,\mathcal{R}({ϵ}_1,{ϵ}_2){ϵ}_2=-{ϵ}_1,\mathcal{R}({ϵ}_1,{ϵ}_2){ϵ}_3=0, \\ \mathcal{R}({ϵ}_2,{ϵ}_3){ϵ}_1=0,\mathcal{R}({ϵ}_2,{ϵ}_3){ϵ}_2={ϵ}_3,\mathcal{R}({ϵ}_2,{ϵ}_3){ϵ}_3=-{ϵ}_2 \\ \mathcal{R}({ϵ}_1,{ϵ}_3){ϵ}_1={ϵ}_3,\mathcal{R}({ϵ}_1,{ϵ}_3){ϵ}_2=0,\mathcal{R}({ϵ}_1,{ϵ}_3){ϵ}_3=-{ϵ}_1, \end{gathered}$$

(77)

and

$$\begin{gathered} \overline{\mathcal{R}}({ϵ}_1,{ϵ}_2){ϵ}_1=0,\overline{\mathcal{R}}({ϵ}_1,{ϵ}_2){ϵ}_2=0,\overline{\mathcal{R}}({ϵ}_1,{ϵ}_2){ϵ}_3=0, \\ \overline{\mathcal{R}}({ϵ}_2,{ϵ}_3){ϵ}_1=0,\overline{\mathcal{R}}({ϵ}_2,{ϵ}_3){ϵ}_2=0,\overline{\mathcal{R}}({ϵ}_2,{ϵ}_3){ϵ}_3=0, \\ \overline{\mathcal{R}}({ϵ}_1,{ϵ}_3){ϵ}_1=0,\overline{\mathcal{R}}({ϵ}_1,{ϵ}_3){ϵ}_2=0,\overline{\mathcal{R}}({ϵ}_1,{ϵ}_3){ϵ}_3=0, \end{gathered}$$

(78)

repectively.

From (77) and (78), we calculate the followings components of $\mathcal{S}$ and $\overline{\mathcal{S}}$, respectively:

$$\mathcal{S}({ϵ}_1,{ϵ}_1)=\mathcal{S}({ϵ}_2,{ϵ}_2)=\mathcal{S}({ϵ}_3,{ϵ}_3)=2,\Rightarrow r=6,$$

(79)

and

$$\overline{\mathcal{S}}({ϵ}_1,{ϵ}_1)=\overline{\mathcal{S}}({ϵ}_2,{ϵ}_2)=\overline{\mathcal{S}}({ϵ}_3,{ϵ}_3)=0,\Rightarrow \overline{r}=0.$$

(80)

With the help of Equations (77)–(80), we can easily verify Proposition 2.

Moreover, using (80) in (48), it follows that $\delta =0$, for all $\rho \in \mathbb{R}$ and $\vartheta \ne 0$. Thus, $(g,\zeta ,\delta ,\vartheta ,\rho )$ is steady. This verifies Corollary 2 for the values $\overline{r}=0$ and $\rho (\in \mathbb{R})$.