1. Introduction

Ricci solitons play a crucial role in the study of Ricci flow, acting as self-similar solutions within this flow. Consequently, when investigating a geometric flow, it is essential to consider the solitons associated with that flow. Specifically, a family of metrics $g(t)$ on a Riemannian manifold $(M^n,g)$ is said to evolve according to the Ricci-Bourguignon flow (RB flow) if $g(t)$ satisfies the following evolution equation:

(1) \begin{equation} \frac {\partial g}{\partial t}=-2(\!\operatorname {Ric}-\rho r)g. \end{equation}

In this context, Ric represents the Ricci tensor corresponding to the metric, $r$ stands for the scalar curvature, and $\rho \in \mathbb{R}$ . The flow equation in (1) was first introduced by Jean-Pierre Bourguignon [Reference Bourguignon1], building on earlier work by Lichnerowicz and a publication by Aubin [Reference Aubin2]. Notably, when $\rho = 0$ , equation (1) reduces to the Ricci flow. The right-hand side of the evolution equation in (1) is significant for different values of $\rho$ . The parabolic theory of the RB flow was systematically investigated in [Reference Catino, Cremaschi, Djadli, Mantegazza and Mazzieri3], which marked a significant advancement in the field. Among their key findings, the authors demonstrated the short-term existence of the flow (1) on a closed n-dimensional manifold, starting with any initial metric $ g_0$ , for $ \rho \lt \frac {1}{2(n-1)}$ .

The concept of almost Ricci solitons was introduced in [Reference Pigola, Rigoli, Rimoldi and Setti4], where the traditional notion of a Ricci soliton was redefined by allowing the parameter $f$ , which was originally constant, to vary as a function. See [Reference Güler5–Reference Barros and Ribeiro7] for a more detailed understanding of almost Ricci solitons. These references offer comprehensive insights into the properties, applications, and recent advancements in the study of almost Ricci solitons, providing valuable resources for further research. Drawing inspiration from the Ricci flow framework, Dwivedi [Reference Dwivedi8] proceeded with the following considerations.

Definition 1. [Reference Dwivedi8] An almost Ricci-Bourguignon soliton (almost RB soliton for short) is a Riemannian manifold $(M^n, g)$ endowed with a vector field $V$ on $M$ and a soliton function $f\;:\; M\to \mathbb{R}$ that satisfies

(2) \begin{equation} \operatorname {Ric}+\frac {1}{2} \unicode{x00A3}_{V} g=(f +\rho \, r) g, \end{equation}

where $\unicode{x00A3}_{V} g$ denotes the Lie derivative of the metric $g$ with respect to the vector field $V$ , $r$ denotes the scalar curvature of the almost RB soliton and $\rho \in \mathbb{R}$ is a constant.

Again, if the vector field $V$ is a gradient of some smooth function $s$ on $M$ , then it is called a gradient almost RB soliton. In this case, the foregoing equation reduces to

(3) \begin{equation} \nabla ^{2} s + \operatorname {Ric}=(f+\rho r) g, \end{equation}

where $\nabla ^{2} s$ stands for the Hessian of $s$ . Note that if $s$ is constant, the soliton becomes trivial (Einstein). Perelman [Reference Perelman9] proved that the Ricci solitons on a compact manifold are always gradient Ricci solitons, and, hence, the potential vector field can be expressed as the sum of the gradient of a function and a Killing vector field. Such a result is also true for the almost RB solitons, if the scalar curvature is constant (see [Reference Aqib, Shah and Patra10]). In the following discussion, we will refer to $(M^n, g, V, f, \rho )$ as an almost RB-soliton, with $V$ being the potential vector field. If $f$ is constant, the almost RB soliton becomes an RB soliton. When the potential vector field $V$ is given by the gradient of a smooth function $s$ on $M$ , the almost RB soliton is termed a gradient almost RB soliton. Depending on whether $f$ is negative, zero, or positive, the soliton is classified as expanding, steady, or shrinking, respectively. If $V$ is a Killing vector field, the almost RB soliton reduces to an RB soliton with $f$ being constant, referred to as a trivial RB soliton. For further details on almost RB solitons; see [Reference Dwivedi8] and [Reference Ghosh11].

Thus, while the direct relation of almost RB solitons to the long-time behavior of the Ricci-Bourguignon flow remains an open question, their study on almost $\alpha$ -cosymplectic manifolds is highly productive: it reveals classification patterns, clarifies the interplay between contact-like structures and curvature, and provides a systematic way to explore the geometric consequences of the parameter $\rho$ .

In this article, we investigate RB solitons on almost $\alpha$ -cosymplectic manifolds. RB solitons generalize quasi-Einstein metrics arising in string theory, while $\alpha$ -cosymplectic manifolds extend symplectic geometry to odd dimensions. Let $M$ be a $(2n+1)$ -dimensional Riemannian manifold. An almost contact structure on $M^{2n+1}$ is a triple $(\varphi, \xi, \eta )$ , where $\varphi$ is a $(1,1)$ -tensor, $\xi$ a global vector field, and $\eta$ a 1-form such that

(4) \begin{equation} \varphi ^{2}=-I+\eta \otimes \xi, \quad \eta (\xi )=1. \end{equation}

The vector field $\xi$ is called the Reeb vector field. Here we use the metric to identify the Reeb vector field $\xi$ with its dual $1$ -form $\eta$ that is $\eta (.)=g(.,\xi )$ . If there exists a Riemannian metric $g$ such that

(5) \begin{equation} g(\varphi X, \varphi Y)=g(X, Y)-\eta (X) \eta (Y), \end{equation}

for any $X, Y \in \mathfrak{X}(M)$ , the metric $g$ is said to be compatible with an almost contact structure, and $(\varphi, \xi, \eta, g)$ is said to be an almost contact metric structure. An almost contact metric structure is said to be normal, if the almost complex structure $J$ on $M^{2 n+1} \times \mathbb{R}$ is integrable, where $J$ is defined by $J\left (X, f \frac {d}{d t}\right )=\left (\varphi X-f \xi, \eta (X) \frac {d}{d t}\right )$ for $X \in \mathfrak{X}(M), t$ is the coordinate of $\mathbb{R}$ and $f$ a smooth function on $M \times \mathbb{R}$ . Or equivalently, an almost contact metric structure is said to be normal if the Nijenhuis tensor of $\varphi$ satisfies $[\varphi, \varphi ]=-2 d \eta \otimes \xi$ , where the Nijenhuis tensor of $\varphi$ is defined by

\begin{equation*} [\varphi, \varphi ]=\varphi ^{2}[X, Y]+[\varphi X, \varphi Y]-\varphi [\varphi X, Y]-\varphi [X, \varphi Y] \end{equation*}

for any $X, Y \in \mathfrak{X}(M)$ .

We denote a fundamental 2-form $\Phi$ on an almost contact metric manifold $M^{2 n+1}$ by $\Phi (X, Y)=g(X, \varphi Y)$ for any $X, Y \in \mathfrak{X}(M)$ . In general, the almost contact metric structure is called almost $\alpha$ -cosymplectic, if $d \eta =0$ and $d \Phi =2 \alpha \eta \wedge \xi$ for any constant $\alpha$ . Almost $\alpha$ -cosymplectic structure is said to be $\alpha$ - cosymplectic, if the almost contact metric structure is normal. In particular, if $\alpha$ is non-zero, the almost contact metric structure is said to be almost $\alpha$ -Kenmotsu. If $\alpha =0$ , the almost contact metric structure is called almost cosymplectic. If $\alpha =1$ , the almost contact metric structure is called almost Kenmotsu.

On an almost contact metric manifold, one defines a tensor field $h$ by $h=\frac {1}{2} \mathfrak{L}_{\xi } \varphi$ . On almost $\alpha$ -cosymplectic $(2 \textrm{n}+1)$ manifolds, we have the following properties from [Reference Perrone12] for the tensor $h$ and $\varphi$ :

(6) \begin{equation} \nabla _{\xi } \varphi =0, \quad \nabla \xi =h \varphi, \quad h \varphi =-\varphi h, \quad h \xi =0, \end{equation}

and

(7) \begin{equation} \nabla _{X} \xi =-\alpha \varphi ^2 X-\varphi h X=\alpha X-\alpha \eta (X) \xi -\varphi h X. \end{equation}

In particular, an almost $\alpha$ -cosymplectic three-manifold is $\alpha$ -cosymplectic if and only if $\mathfrak{L}_{\xi } h=0$ (see [Reference Perrone12]).

For an almost $\alpha$ -cosymplectic three-manifold, according to the properties of tensor $h$ , there exists a $\varphi$ -basis $\{\xi, e, \varphi e\}$ such that

\begin{equation*}h \xi =0, \quad h e=\lambda e, \quad h \varphi e=-\lambda \varphi e.\end{equation*}

where $\lambda$ is a local smooth function. The Levi-Civita connection is given by (cf. [Reference Perrone12])

(8) \begin{equation} \begin{cases} \nabla _{e} e=-\alpha \xi -a \varphi e, \quad \nabla _{e} \varphi e=\lambda \xi +a e, \quad \nabla _{e} \xi =\alpha e-\lambda \varphi e \\ \nabla _{\varphi e} e=\lambda \xi -b \varphi e, \quad \nabla _{\varphi e} \varphi e=-\alpha \xi +b e, \quad \nabla _{\varphi e} \xi =-\lambda e+\alpha \varphi e \\ \nabla _{\xi } e=\mu \varphi e, \quad \nabla _{\xi } \varphi e=-\mu e, \quad \nabla _{\xi } \xi =0, \end{cases} \end{equation}

where $\lambda, \mu, a, b$ are local smooth functions. Then, the Lie brackets are given by

(9) \begin{equation} [\xi, e]=-\alpha e+(\lambda +\mu ) \varphi e, \quad [\xi, \varphi e]=(\lambda -\mu ) e-\alpha \varphi e, \quad [e, \varphi e]=a e+b \varphi e. \end{equation}

Then, using (9) in the Jacobi identity, we get

(10) \begin{equation} \begin{cases} a\alpha -e(\lambda )+e(\mu )+\xi (a)+b\lambda -b\mu =0,\\ b\alpha -\varphi e(\lambda )+\varphi e(\mu )+\xi (b)+a\lambda +a\mu =0. \end{cases} \end{equation}

According to (8) and (9), the Ricci operator $Q$ is given by (cf. [Reference Perrone12]):

(11) \begin{equation} \begin{cases} Q e=-\left (2 \alpha ^{2}+a^{2}+b^{2}+2 \lambda \mu +(\varphi e)(a)-e(b)\right ) e+(2 \alpha \lambda +\xi (\lambda )) \varphi e-(2 a \lambda +(\varphi e)(\lambda )) \xi, \\ Q \varphi e=(2 \alpha \lambda +\xi (\lambda )) e-\left (2 \alpha ^{2}+a^{2}+b^{2}-2 \lambda \mu +(\varphi e)(a)-e(b)\right ) \varphi e+(2 b \lambda -e(\lambda )) \xi, \\ Q \xi =-(2 a \lambda +(\varphi e)(\lambda )) e+(2 b \lambda -e(\lambda )) \varphi e-\left (2 \alpha ^{2}+\operatorname {tr }h^{2}\right ) \xi . \end{cases} \end{equation}

Then, the scalar curvature $r$ is expressed by

(12) \begin{equation} r=-6\alpha ^{2}-\operatorname {tr} h^{2}-2\left (a^{2}+b^{2}\right )-2(\varphi e)(a)+2 e(b). \end{equation}

In what follows, we will use the important Remark $3.3$ of [Reference Perrone12] for an almost $\alpha$ -cosymplectic three-manifold: If $\lambda$ is constant then $\nabla _\xi h=2\mu \varphi h$ and the condition $\nabla _\xi h=0$ is related to an interesting geometric property: it is equivalent to the property that the sectional curvatures of the planes $P_1 = span(\xi, e)$ and $P_2 = span(\xi, \varphi e)$ , at a given point, are equal.

On an almost $\alpha$ -cosymplectic three-manifold $(M, \varphi, \xi, \eta, g)$ , the canonical foliation $\mathcal{F}$ of codimension one on $M$ is defined by the contact distribution $\operatorname {ker} \eta$ . We note that the extrinsic curvature $K_{e}$ and Gaussian curvature $K_{G}$ of the canonical foliation $\mathcal{F}$ are given ([Reference Perrone12], p.73) by

\begin{equation*} K_{e}=\alpha ^{2}-\frac {\operatorname {tr} h^{2}}{2}, \quad K_{G}=\frac {r}{2}+\frac {\operatorname {tr} h^{2}}{2}+3 \alpha ^{2}. \end{equation*}

Now we construct an $\alpha$ -cosymplectic manifold which admits an almost RB soliton as follows.

Example 1. We consider the $5$ -dimensional manifold $M=\mathbb{R}^{5}$ . Coordinatize $\mathbb{R}^{5}$ by $\{\left (x_{1}, x_{2}, y_{1}, y_{2}, z\right) \in \mathbb{R}^{5}\}$ , where $\left (x_{1}, x_{2}, y_{1}, y_{2}, z\right)$ are the standard coordinates in $\mathbb{R}^{5}$ . Let $e_{1}, e_{2}, e_{3}, e_{4}$ and $e_{5}$ be the vector fields on $M$ given by

\begin{equation*}e_{1}=e^{\alpha z}\frac{\partial }{\partial x_{1}}, \quad e_{2}=e^{\alpha z} \frac{\partial}{\partial x_{2}},\quad e_{3}=e^{\alpha z} \frac{\partial}{\partial y_{1}}, \quad e_{4}=e^{\alpha z} \frac{\partial}{\partial y_{2}}, \quad e_{5}=-\frac {\partial }{\partial z}=\xi, \end{equation*}

where $\alpha$ is some constant. Let $g$ be the Riemannian metric defined by

\begin{equation*}g\left (e_{i}, e_{j}\right )=\delta _{ij}, \quad i \neq j, \quad i, j=1,2,3,4,5.\end{equation*}

Let $\eta$ be the 1-form on $M$ defined by $\eta (X)=g\left (X, e_{5}\right )=g(X, \xi )$ for all $X \in \chi (M)$ . Let $\varphi$ be the (1, 1) tensor field on $M$ defined by

\begin{equation*}\varphi e_{1}=-e_{2},\quad \varphi e_{2}=e_{1},\quad \varphi e_{3}=-e_{4},\quad \varphi e_{4}=e_{3},\quad \varphi e_{5}=0.\end{equation*}

By applying the linearity of $\varphi$ and $g$ , we have

\begin{equation*} \begin{gathered} \eta (\xi )=1,\quad \varphi ^{2} X=-X+\eta (X) \xi, \quad \eta (\varphi X)=0, \\ g(X, \xi )=\eta (X),\quad g(\varphi X, \varphi Y)=g(X, Y)-\eta (X) \eta (Y), \end{gathered} \end{equation*}

for all $X, Y \in \chi (M)$ . Thus, $(\mathbb{R}^5,g,\varphi, \xi, \eta )$ defines an almost contact structure. Then, we have

\begin{equation*} \begin{gathered} {\left [e_{1}, e_{2}\right ]=\left [e_{1}, e_{3}\right ]=\left [e_{1}, e_{4}\right ]=\left [e_{2}, e_{3}\right ]=\left [e_{2}, e_{4}\right ]=\left [e_{3}, e_{4}\right ]=0}, \\ {\left [e_{1}, e_{5}\right ]=\alpha e_{1},\quad \left [e_{2}, e_{5}\right ]=\alpha e_{2},\quad \left [e_{3}, e_{5}\right ]=\alpha e_{3},\quad \left [e_{4}, e_{5}\right ]=\alpha e_{4}}. \end{gathered} \end{equation*}

We can easily calculate,

\begin{equation*} \begin{gathered} \nabla _{e_{1}} e_{1}=-\alpha e_{5}, \quad \nabla _{e_{1}} e_{5}=\alpha e_{1},\quad \nabla _{e_{4}} e_{4}=-\alpha e_{5}, \\ \nabla _{e_{2}} e_{2}=-\alpha e_{5}, \quad \quad \nabla _{e_{2}} e_{5}=\alpha e_{2},\quad \nabla _{e_{4}} e_{5}=\alpha e_{4}, \\ \nabla _{e_{3}} e_{3}=-\alpha e_{5}, \quad \nabla _{e_{3}} e_{5}=\alpha e_{3}. \end{gathered}\end{equation*}

All the others are zero. It can be easily verified that the manifold $M$ is an $\alpha$ -cosymplectic manifold. Ricci tensor components are given by

\begin{equation*}\operatorname {Ric}\!(e_i,e_j)=-4\alpha ^2\delta _{ij}, \;\;\forall i, j.\end{equation*}

Therefore, the scalar curvature $r=-20\alpha ^2$ . Let us choose a potential vector field.

\begin{equation*}V=f(z)e_1+g(z)e_2+h(z)e_3+k(z)e_4+m(z)e_5, \end{equation*}

where $f(z),g(z),h(z),k(z), m(z)$ are arbitrary smooth functions. By the soliton equation (2) , we get the soliton function $\lambda$ ,

\begin{equation*}\lambda =(1/2)\left [(f'(z)+g'(z)+h'(z)+k'(z))e^{\alpha z}+m'(z)\right ]+20\rho \alpha ^2-4\alpha ^2.\end{equation*}

Thus, $(\mathbb{R}^5,V,\lambda, \rho, g)$ defines almost RB soliton.

This article is divided into $3$ sections. In Section 2, we classify $3$ -dimensional almost $\alpha$ -cosymplectic manifold admitting an almost RB soliton. For this purpose, we use the classification of $3$ -dimensional almost $\alpha$ -cosymplectic Lie groups as in [Reference Perrone12]. In Section 3, we explore Ricci semisymmetric manifolds admitting RB solitons. In the sequel, $(\tilde M, g)$ denotes the simply connected covering space of $(M,g)$ . First, we describe the motivation behind our present exploration.

Motivation. The study of almost Ricci-Bourguignon solitons on almost $\alpha$ -cosymplectic manifolds is motivated by several interrelated geometric goals:

1. 1. Unification and generalization. The Ricci-Bourguignon soliton equation

   \begin{equation*} \textrm{Ric} + \frac 12\unicode{x00A3}_{V} g = (f + \rho r)g \end{equation*}
   provides a single framework that interpolates between Ricci, Einstein, and traceless Ricci solitons as $\rho$ varies. Investigating this general structure on almost $\alpha$ -cosymplectic manifolds (includes almost $\alpha$ -Kenmotsu manifolds) yields classification results applicable to a whole family (depending on each $\alpha$ ) of geometric flows simultaneously.

2. 2. Geometric rigidity. Almost contact metric structures are rich but complex. Almost contact metric manifolds are analogues of almost Hermitian manifolds in odd dimensions. The presence of a soliton imposes a strong differential constraint that often forces the underlying geometry to become more regular. In our present research, we show that an almost RB soliton can, under some appropriate natural conditions, force an almost $\alpha$ -cosymplectic manifold to become $\alpha$ -cosymplectic, or even reduce to an almost cosymplectic manifold ( $\alpha =0$ ). This soliton thus acts as an instrument of geometric reduction in the family.

3. 3. Link to curvature and symmetry. In dimension three, the soliton equation simplifies the complicated curvature expressions of almost $\alpha$ -cosymplectic manifolds, allowing a complete classification in terms of Lie group structures (Theorem2.1). This reveals precisely how the parameters $(\lambda, a, b, \alpha, \mu )$ interact with the soliton function and the scalar curvature.

4. 4. Geometry behind the condition being $\boldsymbol{\eta}$ -parallel in contact geometry: The contact and complex geometries interact very extensively. In almost Hermitian manifolds, the complex structure $J$ being parallel, viz., $\nabla J = 0$ , corresponds to the vanishing of the Nijenhuis tensor being zero. This implies that $J$ is integrable. Analogously, in an almost contact manifold, an associated CR-structure is integrable if and only if the structure tensor $\phi$ is $\eta$ parallel. In classical contact geometry, $\eta$ -parallel conditions on tensors such as $h$ or $\tau$ have been shown to impose strong curvature restrictions (e.g., forcing $(k,\mu )$ -structures or constant curvature). By studying almost RB solitons on almost $\alpha$ -cosymplectic manifolds with $\eta$ -parallel $\varphi h$ , we extend this notion to a broader setting and obtain new rigidity theorems (Section 3).