Z-Solitons and Gradient Z-Solitons on α-Cosymplectic Manifolds

Abstract

In this paper, we study Z-solitons and gradient Z-solitons on $\alpha$-cosymplectic manifolds. The soliton structure is defined by the generalized tensor $Z=S+\beta g$, where S denotes the Ricci tensor, g the metric tensor, and $\beta$ a smooth function. We investigate the geometric implications of Z-solitons under various curvature conditions, with a focus on the interplay between the Z-tensor and the Q-curvature tensor, as well as the case of Z-recurrent $\alpha$-cosymplectic manifolds. Our classification results establish that such manifolds can be Einstein, $\eta$-Einstein, or of constant curvature. Finally, we construct a concrete five-dimensional example of an $\alpha$-cosymplectic manifold that admits a Z-soliton structure, thereby illustrating the theoretical framework.

1. Introduction

Geometric flows, especially the Ricci flow, have become a central topic in modern differential geometry due to their profound applications in topology, global analysis, and mathematical physics. The Ricci flow was first introduced by Hamilton [1] and later became a pivotal tool for investigating Thurston’s geometrization conjecture. The Ricci flow deforms the Riemannian metric $g^M$ (representing the geometry of the manifold M) in the direction of its Ricci curvature, aiming to produce a metric of constant curvature (e.g., spherical). In this context, a self-similar solution to the Ricci flow is known as a Ricci soliton. A Ricci soliton can be viewed as a fixed point of the Ricci flow dynamics and serves as a natural generalization of an Einstein metric. On a Riemannian manifold $(M,g^M)$, a smooth vector field V defines a Ricci soliton if it satisfies the condition [2]

$${\displaystyle \frac{1}{2}}{\mathcal{L}}_V{g}^M+S^M+\lambda g^M=0,$$

(1)

where ${\mathcal{L}}_V{g}^M$ denotes the Lie derivative of the metric $g^M$ in the direction of V, $S^M$ is the Ricci tensor, and $\lambda$ is a scalar. A Ricci soliton is classified as shrinking, steady, or expanding if $\lambda <0$, $\lambda =0$, or $\lambda >0$, respectively. A Ricci soliton $(M,g^M)$ is said to be trivial if the soliton vector field $\xi$ is Killing, i.e. ${\mathcal{L}}_{\xi }{g}^M=0$, in which case $(M,g^M)$ reduces to an Einstein manifold. This observation shows that Ricci solitons naturally extend the concept of Einstein metrics. If the soliton vector field V is the gradient of a smooth potential function $-f$, then $(M,g^M)$ is called a gradient Ricci soliton, and Equation (1) reduces to

$$\nabla \nabla f+S^M+\lambda g^M=0.$$

Metrics satisfying (1) are often called quasi-Einstein metrics and have significant applications in physics [3,4]. Compact Ricci solitons [5] emerge as fixed points of the Ricci flow

$${\displaystyle \frac{\partial g^M}{\partial t}}=-2S^M,$$

when projected from the space of metrics onto the quotient modulo diffeomorphisms and scaling. Ricci solitons play a central role in the study of singularities of the Ricci flow as well as in the classification problems of Riemannian manifolds [6]. Furthermore, Ricci solitons and their generalizations have been extensively studied on various almost contact metric structures, such as Sasakian, Kenmotsu, and $\alpha$-cosymplectic manifolds, leading to numerous rigidity theorems and explicit examples [7,8,9,10,11,12,13,14,15,16,17].

The concept of a Z-tensor was first introduced by Mantica and Molinari [18], who investigated the properties of weakly Z-symmetric manifolds. A symmetric $(0,2)$-tensor Z is called a Z-tensor [19] if it satisfies the relation

$$Z=S^M+\beta g^M,$$

(2)

where $S^M$ and $g^M$ denote the Ricci and metric tensors of a smooth manifold M, respectively, and $\beta$ is a scalar function. From (2), it follows that an n-dimensional Z-flat manifold is necessarily Einstein, with $\beta =-\frac{r}{n}$, where r is the scalar curvature; morever, r is constant when $n>2$. If $\nabla Z=0$, the manifold reduces to a Z-symmetric manifold. Furthermore, if ${\nabla }_j{Z}_{kl}={\lambda }_j{Z}_{kl}$ (i.e., the manifold is Z-recurrent), then it specializes to a generalized Ricci recurrent manifold [20]. In recent years, the study of Z-tensor fields has attracted considerable attention, with significant contributions in [21,22,23,24,25].

A Riemannian metric $g^M$ is said to define a Z-soliton if it satisfies

$${\displaystyle \frac{1}{2}}{\mathcal{L}}_V{g}^M+Z+\lambda g^M=0,$$

(3)

where V is a smooth vector field on M and $\lambda$ is a constant. A vector field V is called solenoidal (or divergence-free) if its divergence vanishes, i.e., $\mathrm{div}\left(V\right)=0$. If V is the gradient of a smooth function f, the soliton is referred to as a gradient Z-soliton, in which case (3) reduces to

$${\nabla }^{2}f+Z+\lambda g^M=0,$$

(4)

where ${\nabla }^{2}f$ denotes the Hessian of f [24,26].

In a parallel development, the Q-curvature tensor has emerged as an important tool in differential geometry. Introduced in [27], it is defined on a $(2n+1)$-dimensional Riemannian manifold by

$$Q(q_x,q_y)q_w=R^M(q_x,q_y)q_w-{\displaystyle \frac{\pi }{2n}}\left[g^M(q_y,q_w)q_x-g^M(q_x,q_w)q_y\right],$$

(5)

where $R^M$ denotes the Riemannian curvature tensor, and $\pi$ is an arbitrary scalar function. In particular, recall that the concircular curvature tensor C of $(M,g^M)$ is defined by

$$C(q_x,q_y)q_w=R^M(q_x,q_y)q_w-{\displaystyle \frac{r}{2n(2n+1)}}\left[g^M(q_y,q_w)q_x-g^M(q_x,q_w)q_y\right],$$

(6)

where r denotes the scalar curvature. Notably, if $\pi =\frac{r}{(2n+1)}$, then the Q-curvature tensor reduces to the concircular curvature tensor C. This relation will be used subsequently [28,29,30].

On the other hand, contact geometry has emerged as a highly influential branch of differential geometry, with significant applications in geometric optics, thermodynamics, geometric quantization, and the mechanics of dynamical systems with time-dependent Hamiltonians. In recent years, the study of contact Riemannian manifolds and their associated structures has received considerable attention. Within this framework, cosymplectic manifolds form an important subclass of almost contact manifolds. They were first introduced by Goldberg and Yano [31] in 1969.

A cosymplectic manifold is defined as a smooth manifold of dimension $2n+1$ equipped with a closed 1-form $\eta$ and a closed 2-form $\omega$, such that the condition $\eta \wedge {\omega }^n$ defines a volume form. The most basic examples of almost cosymplectic manifolds are given by the product of an almost Kähler manifold with either the real line $\mathbb{R}$ or the unit circle $S^1$ [32]. For a detailed exposition of cosymplectic geometry, its interconnections with other areas of mathematics (particularly geometric mechanics), and its applications in physics, we refer the reader to [33].

In 2005, Kim and Pak [34] defined almost $\alpha$-cosymplectic manifolds by combining almost $\alpha$-Kenmotsu and almost cosymplectic structures. A normal almost $\alpha$-cosymplectic manifold is called an $\alpha$-cosymplectic manifold. Their structural properties and curvature conditions have been studied in various works [15,35,36,37,38,39,40,41].

The aim of this paper is to study the geometry of Z-solitons and gradient Z-solitons on $\alpha$-cosymplectic manifolds. We first recall the necessary definitions and preliminary results. In the main part, we establish several characterization theorems for Z-solitons under various curvature constraints, including $Q(\xi ,q_x)·Z=0$, $Q(\xi ,q_x)·Q=0$, $\left(\xi {\wedge }_Z{q}_x\right)·Q=0$, and $Z(q_x,\xi )·R^M=0$. We also examine Z-recurrent $\alpha$-cosymplectic manifolds, leading to rigidity results. Moreover, we show that, depending on the algebraic relations among the curvature components, such manifolds can be classified as Einstein, $\eta$-Einstein, or of constant curvature. Finally, we construct an explicit five-dimensional example admitting a Z-soliton structure, thereby illustrating the theoretical findings of the paper.

2. Preliminaries

Let M be a differentiable manifold of dimension $2n+1$ endowed with a triple $(\psi ,\xi ,\eta )$, where $\psi$ is a $(1,1)$-tensor field, $\xi$ is a vector field, and $\eta$ is a 1-form on M, satisfying [42]

$${\psi }^2=-I+\eta \otimes \xi ,\eta \left(\xi \right)=1.$$

(7)

From these relations, it follows that

$$\psi \xi =0,\eta \circ \psi =0,\mathrm{rank}\left(\psi \right)=2n.$$

If M admits a Riemannian metric $g^M$ such that, for all vector fields $q_x,q_y\in \chi \left(M\right)$,

$$g^M(\psi q_x,\psi q_y)=g^M(q_x,q_y)-\eta \left(q_x\right)\eta \left(q_y\right),$$

(8)

$$g^M(q_x,\psi q_y)=-g^M(\psi q_x,q_y),g^M(q_x,\xi )=\eta \left(q_x\right),$$

(9)

then $(\psi ,\xi ,\eta ,g^M)$ is said to define an almost contact metric structure.

The associated fundamental 2-form $\Phi$ is defined by

$$\Phi (q_x,q_y)=g^M(\psi q_x,q_y),\mathrm{for}\mathrm{all}{q}_x,q_y\in \chi \left(M\right).$$

An almost contact metric manifold $(M,\psi ,\xi ,\eta ,g^M)$ is characterized as follows:

• It is almost cosymplectic [31] if $d\eta =0$ and $d\Phi =0$.
• It is normal if its Nijenhuis tensor, defined as

  $$N_{\psi }(q_x,q_y)=[\psi q_x,\psi q_y]-\psi [\psi q_x,q_y]-\psi [q_x,\psi q_y]+{\psi }^2[q_x,q_y]+2d\eta (q_x,q_y)\xi ,$$

  vanishes identically for all $q_x,q_y$. In this case, the almost cosymplectic manifold is called a cosymplectic manifold.
• It is called almost α-Kenmotsu if $d\eta =0$ and $d\Phi =2\alpha \eta \wedge \Phi$ with $\alpha \ne 0$ constant.

Almost $\alpha$-cosymplectic manifolds were introduced by Kim and Pak [34] as a unifying structure that generalizes both almost $\alpha$-Kenmotsu and almost cosymplectic manifolds, and are defined by

$$d\eta =0,d\Phi =2\alpha \eta \wedge \Phi ,\alpha \in \mathbb{R}.$$

If the structure is normal, the manifold is termed an $\alpha$-cosymplectic manifold. In particular, $\alpha =0$ corresponds to cosymplectic manifolds, whereas $\alpha \ne 0$ yields $\alpha$-Kenmotsu manifolds.

On an $\alpha$-cosymplectic manifold [41], the following relations hold:

$$\left({\nabla }_{q_x}\psi \right)q_y=\alpha \left(g^M(\psi q_x,q_y)\xi -\eta \left(q_y\right)\psi q_x\right),$$

which further implies

$${\nabla }_{q_x}\xi =-\alpha {\psi }^2{q}_x=\alpha \left[q_x-\eta \left(q_x\right)\xi \right].$$

(10)

Moreover, the following curvature relations hold for all $q_x,q_y\in \chi \left(M\right)$:

$$\begin{array}{cc}\hfill \eta \left(R^M(q_x,q_y)q_w\right)& ={\alpha }^2\left(\eta \left(q_y\right)g^M(q_x,q_w)-\eta \left(q_x\right)g^M(q_y,q_w)\right),\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill R^M(q_x,q_y)\xi & ={\alpha }^2\left(\eta \left(q_x\right)q_y-\eta \left(q_y\right)q_x\right),\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill R^M(q_x,\xi )q_y& ={\alpha }^2\left(\eta \left(q_y\right)q_x-g^M(q_x,q_y)\xi \right),\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill R^M(\xi ,q_x)q_y& ={\alpha }^2\left(\eta \left(q_y\right)q_x-g^M(q_x,q_y)\xi \right),\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill S^M(q_x,\xi )& =-2n{\alpha }^2\eta \left(q_x\right),\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill Z(\xi ,q_y)& =(\beta -2n{\alpha }^2)\eta \left(q_y\right),\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill Z(\xi ,\xi )& =\beta -2n{\alpha }^2,\hfill \end{array}$$

(17)

where $R^M$, $S^M$, and Z denote the Riemannian curvature tensor, the Ricci tensor, and the Z-tensor, respectively.

By considering (5) together with (12)–(14), we deduce

$$\begin{array}{cc}\hfill Q(q_x,q_y)\xi & =\left({\alpha }^2+{\displaystyle \frac{\pi }{2n}}\right)\left[\eta \left(q_x\right)q_y-\eta \left(q_y\right)q_x\right],\hspace{1em}\hspace{1em}\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill Q(\xi ,q_y)q_w& =\left({\alpha }^2+{\displaystyle \frac{\pi }{2n}}\right)\left[\eta \left(q_w\right)q_y-g^M(q_y,q_w)\xi \right],\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill Q(q_x,\xi )q_w& =\left({\alpha }^2+{\displaystyle \frac{\pi }{2n}}\right)\left[g^M(q_x,q_w)\xi -\eta \left(q_w\right)q_x\right].\hfill \end{array}$$

(20)

Definition 1.

An α-cosymplectic manifold is said to be anη-Einstein manifold if its Ricci tensor satisfies

$$S^M(q_x,q_y)=a{g}^M(q_x,q_y)+b\eta \left(q_x\right)\eta \left(q_y\right),$$

for some smooth functions such as a and b. In particular, if $b=0$, the manifold reduces to an Einstein manifold.

3. $\mathit{Z}$-Soliton and Gradient $\mathit{Z}$-Soliton on $\mathit{\alpha }$-Cosymplectic Manifolds

Theorem 1.

Let $(M,g^M)$ be an α-cosymplectic manifold with structure $(\psi ,\xi ,\eta )$. If $(M,g^M)$ admits a Z-soliton, then

(i) 

The Z-soliton is expanding if $\beta <2n{\alpha }^2$;

(ii) 

The Z-soliton is shrinking if $\beta >2n{\alpha }^2$;

(iii) 

The Z-soliton is steady if $\beta =2n{\alpha }^2$.

Proof. 

We consider a Z-soliton with $V=\xi$ on an $\alpha$-cosymplectic manifold. Then, from (3) we have

$$L_{\xi }{g}^M(q_x,q_y)+2Z(q_x,q_y)+2\lambda g^M(q_x,q_y)=0.$$

(21)

Using (8) and (10), we find

$$L_{\xi }{g}^M(q_x,q_y)=2\alpha g^M(\psi q_x,\psi q_y).$$

(22)

Now using (2) and (22) in (21), we obtain

$$S^M(q_x,q_y)=(-\beta -\alpha -\lambda )g^M(q_x,q_y)+\alpha \eta \left(q_x\right)\eta \left(q_y\right).$$

(23)

Substituting $q_y=\xi$ and using (15), we have

$$(2n{\alpha }^2-\beta -\lambda )\eta \left(q_x\right)=0,$$

(24)

which implies that

$$\lambda =2n{\alpha }^2-\beta .$$

Therefore, depending on the value of $\beta$, the Z-soliton is expanding, shrinking, or steady, as stated. □

3.1. $\alpha$-Cosymplectic Manifolds Admitting $Q(\xi ,q_x)·Z=0$

Theorem 2.

Let $(M,g^M)$ be an α-cosymplectic manifold with structure $(\psi ,\xi ,\eta )$. If $(M,g^M)$ satisfies the condition $Q(\xi ,q_x)·Z=0$, then the manifold is an Einstein manifold, provided that ${\alpha }^2\ne -{\displaystyle \frac{\pi }{2n}}$.

Proof. 

The condition $Q(\xi ,q_x)·Z=0$ on $(M,g^M)$ implies that

$$Z(Q(\xi ,q_x)q_y,q_w)+Z(q_y,Q(\xi ,q_x)q_w)=0.$$

(25)

Using (5), (14), (16), and (17) in (25), we obtain

$$\left({\alpha }^2+{\displaystyle \frac{\pi }{2n}}\right)\left[\eta \left(q_y\right)Z(q_x,q_w)-(\beta -2n{\alpha }^2)\eta \left(q_w\right)g^M(q_x,q_y)+\eta \left(q_w\right)Z(q_x,q_y)-\eta \left(q_y\right)g^M(q_x,q_w)\right]=0.$$

(26)

Putting $q_y=\xi$ in (26) and using (2), (9), and (16), we find

$$\left({\alpha }^2+{\displaystyle \frac{\pi }{2n}}\right)\left[S^M(q_x,q_w)+(\beta -1)g^M(q_x,q_w)\right]=0.$$

(27)

This implies that either ${\alpha }^2=-{\displaystyle \frac{\pi }{2n}}$, or $S^M(q_x,q_w)=(1-\beta )g^M(q_x,q_y)$. If ${\alpha }^2\ne -{\displaystyle \frac{\pi }{2n}}$, then we obtain

$$S^M(q_x,q_y)=(1-\beta )g^M(q_x,q_y).$$

(28)

Therefore, $(M,g^M)$ is an Einstein manifold under the given condition. □

Corollary 1.

Let $(M,g^M)$ be an α-cosymplectic manifold with structure $(\psi ,\xi ,\eta )$. If $(M,g^M)$ satisfies the condition $C(\xi ,q_x)·Z=0$, then the manifold is an Einstein manifold, provided that ${\alpha }^2\ne -{\displaystyle \frac{r}{2n(2n+1)}}$.

Proof. 

Using the relations (5) and (6) together with the computations in the proof of Theorem 2, the conclusion follows immediately. □

Corollary 2.

Let $(M,g^M)$ be an α-cosymplectic manifold admitting a Z-soliton with soliton vector field $V.$ If the condition $Q(\xi ,q_x)·Z=0$ holds and V is solenoidal (i.e., $\mathrm{div}\left(V\right)=0$), then the Z-soliton is shrinking.

Proof. 

From (2), (3), and (28), we have

$${\displaystyle \frac{1}{2}}{L}_V{g}^M(q_x,q_y)+(1+\lambda )g^M(q_x,q_y)=0.$$

(29)

Taking $q_x=q_y=e_i$ in (29) and summing over i$(1\le i\le 2n+1)$, we obtain

$${\displaystyle \frac{1}{2}}{L}_V{g}^M(e_i,e_i)+(1+\lambda )g^M(e_i,e_i)=0.$$

(30)

which is equivalent to

$$\mathrm{div}\left(V\right)+(1+\lambda )(2n+1)=0.$$

(31)

If V is solenoidal, that is, if $\mathrm{div}\left(V\right)=0$, then (31) reduces to

$$\lambda =-1.$$

(32)

Therefore, the soliton is shrinking. □

Corollary 3.

Let $(M,g^M)$ be an α-cosymplectic manifold admitting a gradient Z-soliton with soliton potential function $f.$ If the condition $Q(\xi ,q_x)·Z=0$ holds, then the potential function f satisfies the Laplace equation

$$\Delta f=-(1+\lambda )(2n+1),$$

provided that ${\alpha }^2\ne -{\displaystyle \frac{\pi }{2n}}$.

Proof. 

If we take $V=\mathrm{grad}\left(f\right)$, then from (31) we obtain

$$\Delta f=-(1+\lambda )(2n+1),$$

(33)

where $\Delta f$ is the Laplacian of f. □

3.2. $\alpha$-Cosymplectic Manifolds Satisfying $Q(\xi ,q_x).Q=0$

Theorem 3.

If an α-cosymplectic manifold $(M,g^M)$ that admits a Z-tensor satisfies the condition $Q(\xi ,q_x).Q=0$, then the manifold has constant curvature ${\alpha }^2.$

Proof. 

The condition $(Q(\xi ,q_x).Q)(q_y,q_w)q_u=0$ on $(M,g^M)$ implies that

$$\begin{array}{cc}\hfill & Q(\xi ,q_x)Q(q_y,q_w)q_u-Q(Q(\xi ,q_x)q_y,q_w)q_u\hfill \\ \hfill & -Q(q_y,Q(\xi ,q_x)q_w)q_u-Q(q_y,q_w)Q(\xi ,q_x)q_u=0.\hfill \end{array}$$

(34)

Also from (5), (7), (18), (19) and (20), we have (for brevity, $A=({\alpha }^2+\frac{\pi }{2n})$),

$$\begin{array}{cc}\hfill Q(\xi ,q_x)Q(q_y,q_w)q_u& =A\left[\eta \left(Q(q_y,q_w)q_u\right)q_x-g^M(q_x,Q(q_y,q_w)q_u)\xi \right],\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill Q(Q(\xi ,q_x)q_y,q_w)q_u& =A[\eta \left(q_y\right)Q(q_x,q_w)q_u\hfill \\ \hfill & -A\eta \left(q_u\right)g^M(q_x,q_y)q_w+A{g}^M(q_x,q_y)g^M(q_u,q_w)],\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill Q(q_y,Q(\xi ,q_x)q_w)q_u& =A[\eta \left(q_w\right)Q(q_y,q_x)q_u\hfill \\ \hfill & -A{g}^M(q_x,q_w)g^M(q_y,q_u)+A\eta \left(q_u\right)g^M(q_x,q_w)q_y],\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill Q(q_y,q_w)Q(\xi ,q_x)q_u& =A[\eta \left(q_u\right)Q(q_y,q_w)q_x\hfill \\ \hfill & -A\eta \left(q_y\right)g^M(q_x,q_u)q_w+A\eta \left(q_w\right)g^M(q_x,q_u)q_y].\hfill \end{array}$$

(38)

Using (35), (36), (37) and (38) in (34), then taking $q_u=\xi$ we obtain

$$Q(q_y,q_w)q_x=A\left[g^M(q_x,q_y)q_w-g^M(q_x,q_w)q_y\right].$$

By considering Equation (5), we have

$$R^M(q_y,q_w)q_x={\alpha }^2\left[g^M(q_x,q_y)q_w-g^M(q_x,q_w)q_y\right].$$

(39)

From last equation, the manifold is of constant curvature ${\alpha }^2.$ □

Corollary 4.

Let $(M,g^M)$ be an α-cosymplectic manifold that admits a Z-soliton with soliton vector field V. If the condition $Q(\xi ,q_x)·Q=0$ holds and V is solenoidal (i.e., $\mathrm{div}\left(V\right)=0$), then the soliton is expanding, steady, or shrinking according to $\beta <2n{\alpha }^2$, $\beta =2n{\alpha }^2$, or $\beta >2n{\alpha }^2$.

Proof. 

Taking the inner product of Equation (39) with $q_u$, we obtain

$$g^M(R^M(q_y,q_w)q_x,q_u)={\alpha }^2\left[g^M(q_x,q_y)g^M(q_w,q_u)-g^M(q_x,q_w)g^M(q_y,q_u)\right].$$

(40)

We take $q_y=q_u=e_i$, where $\left\{e_i\right\}$ is an orthonormal basis of $T_P\left(M\right)$ for $i=1,2,\dots ,2n+1$, to infer

$$S^M(q_x,q_w)=-2n{\alpha }^2{g}^M(q_x,q_w).$$

(41)

Again from (2), (3), and (41), we have

$${\displaystyle \frac{1}{2}}{L}_V{g}^M(q_x,q_w)+(\beta -2n{\alpha }^2+\lambda )g^M(q_x,q_w)=0.$$

(42)

Taking $q_x=q_w=e_i$ in (42), where $\left\{e_i\right\}$ is an orthonormal basis of $T_P\left(M\right)$ for $i=1,2,\dots ,2n+1$, we obtain

$${\displaystyle \frac{1}{2}}{L}_V{g}^M(e_i,e_i)+(\beta -2n{\alpha }^2+\lambda )g^M(e_i,e_i)=0,$$

(43)

which is equivalent to

$$\mathrm{div}\left(V\right)+(\beta -2n{\alpha }^2+\lambda )(2n+1)=0.$$

(44)

If V is solenoidal, that is, if $\mathrm{div}\left(V\right)=0$, then (44) reduces to

$$\lambda =2n{\alpha }^2-\beta .$$

(45)

Therefore, the soliton is expanding, steady, or shrinking according as $\beta <2n{\alpha }^2$, $\beta =2n{\alpha }^2$, or $\beta >2n{\alpha }^2$. This completes the proof of Corollary. □

Corollary 5.

Let $(M,g^M)$ be an α-cosymplectic manifold that admits a gradient Z-soliton with soliton potential function f. If the condition $Q(\xi ,q_x)·Q=0$ holds, then the potential function f satisfies the Laplace equation

$$\Delta f=-(\beta -2n{\alpha }^2+\lambda )(2n+1).$$

Proof. 

If we take $V=\mathrm{grad}\left(f\right)$, then from (44), we obtain

$$\Delta f=-(\beta -2n{\alpha }^2+\lambda )(2n+1),$$

(46)

where $\Delta f$ is the Laplacian of f. □

3.3. $\alpha$-Cosymplectic Manifolds Satisfying $\left(\xi {\wedge }_Z{q}_x\right).Q=0$

Theorem 4.

If an α-cosymplectic manifold $(M,g^M)$ that admits a Z-tensor satisfies the condition $(\left(\xi {\wedge }_Z{q}_x\right)·Q)=0$, then the manifold is an η-Einstein manifold, provided that ${\alpha }^2\ne -{\displaystyle \frac{\pi }{2n}}$.

Proof. 

Let the condition $(\left(\xi {\wedge }_Z{q}_x\right)·Q)(q_y,q_w)q_u=0$ hold on $(M,g^M)$. Then we have

$$\begin{array}{cc}\hfill & Z\left(q_x,Q(q_y,q_w)q_u\right)\xi -Z\left(\xi ,Q(q_y,q_w)q_u\right)q_x-Z(q_x,q_y)Q(\xi ,q_w)q_u\hfill \\ \hfill & +Z(\xi ,q_y)Q(q_x,q_w)q_u-Z(q_x,q_w)Q(q_y,\xi )q_u+Z(\xi ,q_w)Q(q_y,q_x)q_u\hfill \\ \hfill & -Z(q_x,q_u)Q(q_y,q_w)\xi +Z(\xi ,q_u)Q(q_y,q_w)q_x=0.\hfill \end{array}$$

(47)

Using (2) and (16) in (47), we get

$$\begin{array}{cc}\hfill & S^M\left(q_x,Q(q_y,q_w)q_u\right)\xi +\beta g^M\left(q_x,Q(q_y,q_w)q_u\right)\xi +\left(2n{\alpha }^2-\beta \right)\eta \left(Q(q_y,q_w)q_u\right)q_x\hfill \\ \hfill & -S^M(q_x,q_y)Q(\xi ,q_w)q_u-\beta g^M(q_x,q_y)Q(\xi ,q_w)q_u+\left(2n{\alpha }^2-\beta \right)\eta \left(q_y\right)Q(q_x,q_w)q_u\hfill \\ \hfill & -S^M(q_x,q_w)Q(q_y,\xi )q_u-\beta g^M(q_x,q_w)Q(q_y,\xi )q_u+\left(2n{\alpha }^2-\beta \right)\eta \left(q_w\right)Q(q_y,q_x)q_u\hfill \\ \hfill & -S^M(q_x,q_u)Q(q_y,q_w)\xi -\beta g^M(q_x,q_u)Q(q_y,q_w)\xi +\left(2n{\alpha }^2-\beta \right)\eta \left(q_u\right)Q(q_y,q_w)q_x=0.\hfill \end{array}$$

(48)

Taking the inner product of (48) with ξ and using (18), (19), and (20), we obtain

$$\begin{array}{cc}\hfill & S^M\left(q_x,Q(q_y,q_w)q_u\right)+\beta g^M\left(q_x,Q(q_y,q_w)q_u\right)+\left(2n{\alpha }^2-\beta \right)\eta \left(Q(q_y,q_w)q_u\right)\eta \left(q_x\right)\hfill \\ \hfill & -\beta \eta \left(q_w\right)\eta \left(q_u\right)g^M(q_x,q_y)+\left(2n{\alpha }^2-\beta \right)\eta \left(q_y\right)\eta \left(Q(q_x,q_w)q_u\right)\hfill \\ \hfill & -\left({\alpha }^2+\frac{\pi }{2n}\right)g^M(q_y,q_u)S^M(q_x,q_w)+\left({\alpha }^2+\frac{\pi }{2n}\right)\eta \left(q_y\right)\eta \left(q_u\right)S^M(q_x,q_w)\hfill \\ \hfill & -\left({\alpha }^2+\frac{\pi }{2n}\right)\beta g^M(q_x,q_w)g^M(q_y,q_u)+\left(2n{\alpha }^2-\beta \right)\eta \left(q_u\right)\eta \left(Q(q_y,q_w)q_x\right)=0.\hfill \end{array}$$

(49)

Taking $q_u=\xi$ in (49) and using (9), we obtain

$$\begin{array}{cc}\hfill & \left({\alpha }^2+\frac{\pi }{2n}\right)\eta \left(q_y\right)S^M(q_x,q_w)-\left({\alpha }^2+\frac{\pi }{2n}\right)\eta \left(q_w\right)S^M(q_x,q_y)\hfill \\ \hfill & -\left({\alpha }^2+\frac{\pi }{2n}\right)\beta \eta \left(q_w\right)g^M(q_x,q_y)-\beta \eta \left(q_w\right)g^M(q_x,q_y)=0.\hfill \end{array}$$

(50)

Then, substituting $q_w=\xi$, we have

$$S^M(q_x,q_y)=\left({\displaystyle \frac{\beta }{{\alpha }^2+\frac{\pi }{2n}}}+1\right)g^M(q_x,q_y)+2n{\alpha }^2\eta \left(q_x\right)\eta \left(q_y\right).$$

Thus, the manifold is η-Einstein, in case ${\alpha }^2\ne -\frac{\pi }{2n}$. □

Corollary 6.

Let $(M,g^M)$ be an α-cosymplectic manifold with structure $(\psi ,\xi ,\eta )$. If $(M,g^M)$ satisfies the condition $(\left(\xi {\wedge }_Z{q}_x\right)·C)=0$, then $(M,g^M)$ is an η-Einstein manifold, provided that ${\alpha }^2\ne -{\displaystyle \frac{r}{2n(2n+1)}}$.

Proof. 

Using the relations (5) and (6) together with the computations in the proof of the previous theorem, the conclusion follows immediately. □

3.4. $\alpha$-Cosymplectic Manifolds Satisfying $Z(q_x,\xi ).R^M=0$

Theorem 5.

Let $(M,g^M)$ be an α-cosymplectic manifold with Riemannian curvature tensor $R^M.$ If $(M,g^M)$ admits a Z-tensor and satisfies the condition $Z(q_x,\xi ).R^M=0$, then $(M,g^M)$ is an Einstein manifold.

Proof. 

We suppose that $(M,g^M)$ satisfies the condition

$$(Z(q_x,\xi ).R^M)(q_y,q_w)q_u=0,$$

(51)

which implies that

$$(Z(q_x,\xi ).R^M)(q_y,q_w)q_u=(\left(q_x{\wedge }_Z\xi \right).R^M)(q_y,q_w)q_u,$$

(52)

where the endomorphism $\left(q_x{\wedge }_Z{q}_u\right)q_w$ is defined as

$$\left(q_x{\wedge }_Z{q}_u\right)q_w=Z(q_y,q_w)q_x-Z(q_x,q_w)q_y.$$

(53)

Now, from (52) we have

$$\begin{array}{ccc}\hfill (Z(q_x,\xi ).R^M)(q_y,q_w)q_u& =& \left(\left(q_x{\wedge }_Z\xi \right)R^M\right)(q_y,q_w)q_u-R^M(\left(q_x{\wedge }_Z\xi \right)q_y,q_w)q_u\hfill \\ & & -R^M(q_y,\left(q_x{\wedge }_Z\xi \right)q_w)q_u-R^M(q_y,q_w)\left(q_x{\wedge }_Z\xi \right)q_u.\hfill \end{array}$$

(54)

Also, in view of (51), (53), and (54) we get

$$\begin{array}{c}Z(\xi ,R^M(q_y,q_w)q_u)q_x-Z(q_x,R^M(q_y,q_w)q_u)\xi -Z(\xi ,q_y)R^M(q_x,q_w)q_u\\ +Z(q_x,q_y)R^M(\xi ,q_w)q_u-Z(\xi ,q_w)R^M(q_y,q_x)q_u+Z(q_x,q_w)R^M(q_y,\xi )q_u\\ -Z(\xi ,q_u)R^M(q_y,q_w)q_x+Z(q_x,q_u)R^M(q_y,q_w)\xi =0.\end{array}$$

(55)

Using (2), (12), (13), (14), and (16) in (55) and then taking inner product with $\xi$, we obtain

$$\begin{array}{c}{S}^M(q_x,R^M(q_y,q_w)q_u)+\beta g^M(q_x,R^M(q_y,q_w)q_u)-\eta \left(q_w\right)\eta \left(q_u\right)Z(q_x,q_y)\\ +g^M(q_w,q_u)Z(q_x,q_w)-{\alpha }^2{g}^M(q_x,q_u)Z(q_x,q_u)+{\alpha }^2\eta \left(q_x\right)\eta \left(q_u\right)Z(q_x,q_w)\\ +{\alpha }^2\beta \eta \left(q_u\right)\eta \left(q_y\right)g^M(q_w,q_x)-{\alpha }^2\beta \eta \left(q_w\right)\eta \left(q_u\right)g^M(q_y,q_x)=0.\end{array}$$

(56)

Putting $q_w=\xi$ in (56), we get

$$\eta \left(q_u\right)S^M(q_x,q_y)-2n{\alpha }^4\eta \left(q_u\right)g^M(q_x,q_y)=0$$

(57)

Again, putting $q_u=\xi$ in (57), we find

$$S^M(q_x,q_y)=2n{\alpha }^4{g}^M(q_x,q_y).$$

(58)

Therefore, the manifold is an Einstein manifold. □

Corollary 7.

Let $(M,g^M)$ be an α-cosymplectic manifold that admits a Z-soliton. If $(M,g^M)$ satisfies the condition $Z(q_x,\xi )·R^M=0$, where V is solenoidal, then the soliton is expanding, steady, or shrinking according as $\beta <-2n{\alpha }^4$, $\beta =-2n{\alpha }^4$, or $\beta >-2n{\alpha }^4$.

Proof. 

From (2), (3), and (58), we have

$${\displaystyle \frac{1}{2}}{L}_V{g}^M(q_x,q_w)+[2n{\alpha }^4+\beta +\lambda ]g^M(q_x,q_w)=0.$$

(59)

Taking $q_x=q_w=e_i$ in (59), where $\left\{e_i\right\}$ is an orthonormal basis of $T_P\left(M\right)$ for $i=1,2,\dots ,2n+1$, we obtain

$${\displaystyle \frac{1}{2}}{L}_V{g}^M(e_i,e_i)+[2n{\alpha }^4+\beta +\lambda ]g^M(e_i,e_i)=0,$$

(60)

which is equivalent to

$$\mathrm{div}\left(V\right)+[2n{\alpha }^4+\beta +\lambda ](2n+1)=0.$$

(61)

If V is solenoidal, that is, if $\mathrm{div}\left(V\right)=0$, then (61) reduces to

$$\lambda =-2n{\alpha }^4-\beta .$$

(62)

Therefore, the soliton is expanding, steady, or shrinking according as $\beta <-2n{\alpha }^4$, $\beta =-2n{\alpha }^4$, or $\beta >-2n{\alpha }^4$. □

Corollary 8.

Let $(M,g^M)$ be an α-cosymplectic manifold that admits a gradient Z-soliton with soliton potential function f. If the condition $Z(q_x,\xi )·R^M=0$ holds, then the potential function f satisfies the Laplace equation

$$\Delta f=-[2n{\alpha }^4+\beta +\lambda ](2n+1).$$

Proof. 

If we take $V=\mathrm{grad}\left(f\right)$, then from (61) we obtain

$$\Delta f=-[2n{\alpha }^4+\beta +\lambda ](2n+1),$$

(63)

where $\Delta f$ is the Laplacian of f. □

3.5. Z-Recurrent $\alpha$-Cosymplectic Manifolds

Recall that a non-flat Riemannian or semi-Riemannian manifold $(M,g^M)$ is said to be a Z-recurrent manifold [20] if its Z-tensor satisfies the condition

$$\left({\nabla }_{q_x}Z\right)(q_y,q_w)=\eta \left(q_x\right)Z(q_y,q_w).$$

(64)

where $\eta$ is a non-zero 1-form on $M.$

Theorem 6.

In a Z-recurrent α-cosymplectic manifold, we have

$$q_x(\beta -2n{\alpha }^2)=\eta \left(q_x\right)(\beta -2n{\alpha }^2),$$

(65)

for all $q_x\in \chi \left(M\right)$.

Proof. 

Using Equation (64), we have

$$\left({\nabla }_{q_x}Z\right)(q_y,q_w)=q_{x}Z(q_y,q_w)-Z({\nabla }_{q_x}{q}_y,q_w)-Z(q_y,{\nabla }_{q_x}{q}_w).$$

(66)

With the help of (64) and (66) we obtain

$$q_{x}Z(q_y,q_w)-Z({\nabla }_{q_x}{q}_y,q_w)-Z(q_y,{\nabla }_{q_x}{q}_w)=\eta \left(q_x\right)Z(q_y,q_w).$$

(67)

By taking $q_y=q_w=\xi$ in (67) and using (7), (10), (16), and (17), we obtain

$$q_x(\beta -2n{\alpha }^2)=\eta \left(q_x\right)(\beta -2n{\alpha }^2).$$

(68)

A Z-recurrent manifold is Z-symmetric if and only if the 1-form $\eta$ is zero. □

Corollary 9.

In a Z-symmetric α-cosymplectic manifold, $\beta -2n{\alpha }^2$ is constant.

Corollary 10.

If an α-cosymplectic manifolds is Z-recurrent and if $\beta -2n{\alpha }^2$ is constant, then either $\beta -2n{\alpha }^2=0$ or, $(M,g^M)$ reduces to a Z-symmetric manifold.

3.6. Example

Consider the manifold

$$M=\{(x,y,z,u,v)\in {\mathbb{R}}^5\},$$

where $(x,y,z,u,v)$ are the standard coordinates in ${\mathbb{R}}^5$. We define the following vector fields on M:

$$q_1=e^{\alpha v}{\displaystyle \frac{\partial }{\partial x}},q_2=e^{\alpha v}{\displaystyle \frac{\partial }{\partial y}},q_3=e^{\alpha v}{\displaystyle \frac{\partial }{\partial z}},q_4=e^{\alpha v}{\displaystyle \frac{\partial }{\partial u}},q_5=-{\displaystyle \frac{\partial }{\partial v}},$$

which are linearly independent at each point of M.

We define the Riemannian metric $g^M$ by

$$g_{ij}^M=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}i=j,i,j\in \{1,2,3,4,5\},\hfill \\ 0,\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$$

Let $\eta$ be the 1-form defined by $\eta \left(q_x\right)=g^M(q_x,q_5)$ for any $q_x\in \chi \left(M\right)$. Define the $(1,1)$-tensor field $\psi$ by

$$\psi \left(q_1\right)=q_3,\psi \left(q_2\right)=q_4,\psi \left(q_3\right)=-q_1,\psi \left(q_4\right)=-q_2,\psi \left(q_5\right)=0.$$

These tensors satisfy the relations that characterize an almost contact metric structure

$$\eta \left(\xi \right)=1,{\psi }^2\left(q_x\right)=-q_x+\eta \left(q_x\right)\xi ,g^M(\psi q_x,\psi q_y)=g^M(q_x,q_y)-\eta \left(q_x\right)\eta \left(q_y\right),$$

where $\xi =q_5$ and $q_x,q_y\in \chi \left(M\right)$. Therefore, the structure $(M,\psi ,\xi ,\eta ,g^M)$ defines an almost contact metric manifold.

The Lie brackets are computed as

$$\begin{array}{cccc}\hfill [q_1,q_2]& =0,\hspace{1em}\hspace{1em}\hfill & \hfill [q_3,q_4]& =0,\hfill \\ \hfill [q_1,q_3]& =0,\hspace{1em}\hspace{1em}\hfill & \hfill [q_1,q_5]& =\alpha q_1,\hfill \\ \hfill [q_1,q_4]& =0,\hspace{1em}\hspace{1em}\hfill & \hfill [q_2,q_5]& =\alpha q_2,\hfill \\ \hfill [q_2,q_4]& =0,\hspace{1em}\hspace{1em}\hfill & \hfill [q_3,q_5]& =\alpha q_3,\hfill \\ \hfill [q_2,q_3]& =0,\hspace{1em}\hspace{1em}\hfill & \hfill [q_4,q_5]& =\alpha q_4.\hfill \end{array}$$

Let ∇ be the Levi–Civita connection of $g^M$. From Koszul’s formula, for arbitrary $q_x,q_y,q_w\in \chi \left(M\right)$,

$$\begin{array}{cc}\hfill 2g^M({\nabla }_{q_x}{q}_y,q_w)& =q_x\left(g^M(q_y,q_w)\right)+q_y\left(g^M(q_w,q_x)\right)-q_w\left(g^M(q_x,q_y)\right)\hfill \\ \hfill & -g^M(q_x,[q_y,q_w])-g^M(q_y,[q_x,q_w])+g^M(q_w,[q_x,q_y]),\hfill \end{array}$$

we obtain

$$\left\{\begin{array}{c}{\nabla }_{q_1}{q}_1=-\alpha q_5,{\nabla }_{q_1}{q}_2=0,{\nabla }_{q_1}{q}_3=0,{\nabla }_{q_1}{q}_4=0,{\nabla }_{q_1}{q}_5=\alpha q_1,\hfill \\ {\nabla }_{q_2}{q}_1=0,{\nabla }_{q_2}{q}_2=-\alpha q_5,{\nabla }_{q_2}{q}_3=0,{\nabla }_{q_2}{q}_4=0,{\nabla }_{q_2}{q}_5=\alpha q_2,\hfill \\ {\nabla }_{q_3}{q}_1=0,{\nabla }_{q_3}{q}_2=0,{\nabla }_{q_3}{q}_3=-\alpha q_5,{\nabla }_{q_3}{q}_4=0,{\nabla }_{q_3}{q}_5=\alpha q_3,\hfill \\ {\nabla }_{q_4}{q}_1=0,{\nabla }_{q_4}{q}_2=0,{\nabla }_{q_4}{q}_3=-\alpha q_5,{\nabla }_{q_4}{q}_4=-\alpha q_5,{\nabla }_{q_4}{q}_5=\alpha q_4,\hfill \\ {\nabla }_{q_5}{q}_1=0,{\nabla }_{q_5}{q}_2=0,{\nabla }_{q_5}{q}_3=0,{\nabla }_{q_5}{q}_4=0,{\nabla }_{q_5}{q}_5=0.\hfill \end{array}\right.$$

Therefore,

$${\nabla }_{q_x}\xi =\alpha \left(q_x-\eta \left(q_x\right)\xi \right),\left({\nabla }_{q_x}\psi \right)q_y=\alpha \left(g^M(\psi q_x,q_y)\xi -\eta \left(q_y\right)\psi q_x\right),$$

for arbitrary $q_x,q_y\in \chi \left(M\right)$, where $\xi =q_5$. Hence, $(M,\psi ,\xi ,\eta ,g^M)$ becomes an $\alpha$-cosymplectic manifold.

The non-vanishing components of the curvature tensor are

$$\left\{\begin{array}{c}{R}^M(q_1,q_2)q_1={\alpha }^2{q}_2,R^M(q_1,q_3)q_1={\alpha }^2{q}_3,R^M(q_2,q_3)q_2={\alpha }^2{q}_3,R^M(q_5,q_3)q_5={\alpha }^2{q}_3,\hfill \\ R^M(q_1,q_2)q_2=-{\alpha }^2{q}_1,R^M(q_1,q_3)q_3=-{\alpha }^2{q}_1,R^M(q_1,q_4)q_4=-{\alpha }^2{q}_1,R^M(q_1,q_5)q_5=-{\alpha }^2{q}_1,\hfill \\ R^M(q_2,q_3)q_3=-{\alpha }^2{q}_2,R^M(q_2,q_4)q_4=-{\alpha }^2{q}_2,R^M(q_2,q_5)q_5=-{\alpha }^2{q}_2,R^M(q_3,q_4)q_4=-{\alpha }^2{q}_3,\hfill \\ R^M(q_1,q_4)q_1={\alpha }^2{q}_4,R^M(q_2,q_4)q_2={\alpha }^2{q}_4,R^M(q_3,q_5)q_3={\alpha }^2{q}_4,R^M(q_5,q_4)q_5={\alpha }^2{q}_4,\hfill \\ R^M(q_1,q_5)q_1={\alpha }^2{q}_5,R^M(q_1,q_5)q_2={\alpha }^2{q}_5,R^M(q_4,q_5)q_4={\alpha }^2{q}_5,R^M(q_3,q_5)q_3={\alpha }^2{q}_5.\hfill \end{array}\right.$$

From these results, the non-vanishing components of the Ricci tensor are obtained as

$$S^M(q_1,q_1)=S^M(q_2,q_2)=S^M(q_3,q_3)=S^M(q_4,q_4)=S^M(q_5,q_5)=-4{\alpha }^2.$$

(69)

In this case, Equation (23) reduces to

$$S^M(q_1,q_1)=S^M(q_2,q_2)=S^M(q_3,q_3)=S^M(q_4,q_4)=S^M(q_5,q_5)=-(\lambda +\beta ).$$

(70)

It is clear that from (69) and (70) that $\lambda =4{\alpha }^2-\beta .$ For $n=2$, the Theorem 1 is verified. Consequently, we obtain the following cases:

(i) Z-soliton is expanding if $\beta <4{\alpha }^2,$

(ii) Z-soliton is shrinking if $\beta >4{\alpha }^2,$

(iii) Z-soliton is steady if $\beta =4{\alpha }^2.$