Pair of Associated η-Ricci–Bourguignon Almost Solitons with Vertical Torse-Forming Potential on Almost Contact Complex Riemannian Manifolds

Abstract

Each of the studied manifolds has a pair of B-metrics, interrelated by an almost contact structure. The case where each of these metrics gives rise to an $\eta$-Ricci–Bourguignon almost soliton, where $\eta$ is the contact form, is studied. In addition, the geometry-rich case where the soliton potential is torse-forming and is pointwise collinear on the Reeb vector field with respect to each of the two metrics is considered. Ricci tensors and scalar curvatures are expressed as functions of the parameters of the pair of almost solitons. Particular attention is paid to the special case when the manifold belongs to the only possible basic class of the corresponding classification. A necessary and sufficient condition has been found for these almost solitons to be $\eta$-Einstein for both metrics.

1. Introduction

The study of the Ricci–Bourguignon flow was proposed by Jean-Pierre Bourguignon in [1], developing some unpublished work of André Lichnerowicz in the 1960s and a paper of Thierry Aubin [2]. Suppose that $g\left(t\right)$ is a time-dependent family of (pseudo-)Riemannian metrics on a smooth manifold $\mathcal{M}$. This family is said to evolve under the action of the Ricci–Bourguignon flow if $g\left(t\right)$, its Ricci tensor $\rho \left(t\right)$ and the scalar curvature $\tau \left(t\right)$ satisfy the following evolution equation for a real constant ℓ:

$${\displaystyle \frac{\partial }{\partial t}}g=-2\left(\rho -\mathsf{\ell }\tau g\right),\hspace{1em}g\left(0\right)=g_0.$$

(1)

Let us note that the Ricci–Bourguignon flow contains quite a few other geometric flows for special values of the constant ℓ in (1). The famous Ricci flow [3] occurs at $\mathsf{\ell }=0$, the Schouten flow [4] at $\mathsf{\ell }={\displaystyle \frac{1}{2(m-1)}}$, the traceless Ricci flow [5] at $\mathsf{\ell }={\displaystyle \frac{1}{m}}$, and the Einstein flow [6] at $\mathsf{\ell }={\displaystyle \frac{1}{2}}$, where m is the dimension of $\mathcal{M}$ [7,8]. For $m=2$, the last three tensors are zero; therefore, the flow is static, and in a higher dimension, the values of ℓ are strictly ordered as above, in ascending order.

Furthermore, this family can be considered as an interpolation between the Ricci flow and the Yamabe flow [9,10], obtained as a limit when $\mathsf{\ell }\to -\infty$.

Ricci–Bourguignon flow also interpolates between the Ricci flow and the Yamabe flow when ℓ is non-positive. The short time existence and uniqueness for solution to the Ricci–Bourguignon flow (1) as a system of partial differential equations have been established in [11] for sufficiently small t and $\mathsf{\ell }<{\displaystyle \frac{1}{2(m-1)}}$.

The Ricci–Bourguignon soliton (briefly RB soliton) is determined by the following equation [11,12]:

$$\rho +{\displaystyle \frac{1}{2}}{\mathcal{L}}_{\vartheta }g+(\lambda +\mathsf{\ell }\tau )g=0,$$

(2)

where ${\mathcal{L}}_{\vartheta }g$ stands for the Lie derivative of g along the vector field $\vartheta$ called the soliton potential, and $\lambda$ is the soliton constant. The solution is called a RB almost soliton if $\lambda$ is a differential function on $\mathcal{M}$ [12].

A slightly more general notion of an RB (almost) soliton is obtained by perturbing (2) using a multiple of a $(0,2)$-tensor field $\eta \otimes \eta$ for a certain 1-form $\eta$ on the manifold. Namely, this is an $\eta$-Ricci–Bourguignon (almost) soliton (e.g., [13]).

The $\eta$-Ricci–Bourguignon almost solitons naturally arise as self-similar solutions to the Ricci–Bourguignon flow and thus play a fundamental role in understanding the qualitative behavior of this generalized curvature evolution. In particular, they serve as local models for singularity formation and provide canonical limits under appropriate rescaling procedures. As in the classical Ricci soliton setting, the presence of an almost soliton structure often imposes strong restrictions on the geometry, leading to rigidity phenomena and classification results under curvature, volume growth, or completeness assumptions.

From a structural point of view, these solitons interpolate between Ricci solitons, Yamabe solitons, and Einstein metrics, thereby unifying several well-studied geometric frameworks. Explicit examples can be constructed on space forms, warped products, and Lie groups equipped with left-invariant metrics, highlighting the richness of the theory and demonstrating that nontrivial almost solitons exist beyond the Einstein case. Such examples also provide counterexamples to naive rigidity expectations and serve as test cases for sharp analytic estimates.

The study of $\eta$-Ricci–Bourguignon almost solitons is closely related to modified Einstein equations appearing in higher-order gravity models, where scalar curvature corrections are incorporated. This relation suggests potential applications to geometric aspects of theoretical physics, as well as to the variational analysis of Perelman-type entropy functionals adapted to the Ricci–Bourguignon flow.

Torse-forming vector fields are defined by a certain recurrent condition for their covariant derivative with respect to the Levi–Civita connection of the basic metric [14]. Such a vector field is a geometric generalization of some important types of vector fields—recurrent, conformal, and parallel—and it causes specific directional deformation properties of the geometry of the manifold. This natural choice of soliton potential on manifolds with different structures has been studied by several authors (e.g., [15,16,17,18,19]).

Furthermore, the almost contact structure contains the Reeb vector field $\xi$; therefore, it is natural to choose the soliton potential vector field to be in the $\xi$-direction, determined the so-called vertical distribution. According to [20], the only basic class of almost contact B-metric manifolds that allows a torse-forming Reeb vector field is ${\mathcal{F}}_5$ in the Ganchev–Mihova–Gribachev classification [21]. This class is the counterpart of the class of $\beta$-Kenmotsu manifolds among the classes of almost contact metric manifolds.

2. Almost Contact Complex Riemannian Manifolds

In the present work, we study an almost contact complex Riemannian manifold (abbreviated accR manifold), also known as an almost contact manifold with B-metric, which is introduced in [21]. This means that we have a $(2n+1)$-dimensional smooth manifold $\mathcal{M}$ equipped with an almost contact structure $(\phi ,\xi ,\eta )$ and a pseudo-Riemannian metric (called B-metric) g with signature $(n+1,n)$, such that [21]

$$\begin{array}{lll}\phi \xi =0,& {\phi }^2=-\iota +\eta \otimes \xi , \eta \circ \phi =0,& \eta \left(\xi \right)=1,\\ & g(\phi x,\phi y)=-g(x,y)+\eta \left(x\right)\eta \left(y\right),& \end{array}$$

(3)

where $\iota$ is the identity in $\mathfrak{X}\left(\mathcal{M}\right)$, the Lie algebra of tangent vector fields on $\mathcal{M}$ [21]. In the last equality and further on, by x, y, z, w, we denote arbitrary elements of $\mathfrak{X}\left(\mathcal{M}\right)$ or vectors in the tangent space $T_p\mathcal{M}$ of $\mathcal{M}$ at an arbitrary point p of $\mathcal{M}$.

The remarkable thing about any accR manifold is that every B-metric has its twin, another B-metric. That is, on $(M,\phi ,\xi ,\eta ,g)$, there exists a B-metric $\tilde{g}$ associated with the given B-metric g by means of the almost contact structure and is defined by

$$\tilde{g}(x,y)=g(x,\phi y)+\eta \left(x\right)\eta \left(y\right).$$

(4)

The studied manifolds are divided into eleven basic classes ${\mathcal{F}}_i$, $i\in \{1,2,\dots ,11\}$ in the Ganchev–Mihova–Gribachev classification [21]. It is made with respect to the conditions for the $(0,3)$-tensor F, defined by

$$F(x,y,z)=g(\left({\nabla }_x\phi )y,z)\right.$$

and having the following basic properties:

$$\begin{array}{c}\hfill \begin{array}{cc} & F(x,y,z)=F(x,z,y)=F(x,\phi y,\phi z)+\eta \left(y\right)F(x,\xi ,z)+\eta \left(z\right)F(x,y,\xi ),\\ & F(x,\phi y,\xi )=\left({\nabla }_x\eta \right)\left(y\right)=g({\nabla }_x\xi ,y).\end{array}\end{array}$$

(5)

The intersection of the basic classes is the special class ${\mathcal{F}}_0$, defined by the condition $F=0$, and is known as the class of cosymplectic accR manifolds.

The aforementioned classification uses the following 1-forms associated with F, also known as Lee forms of the considered manifold:

$$\theta =g^{ij}F(e_i,e_j,·),\hspace{1em}{\theta }^{*}=g^{ij}F(e_i,e_j,·),\hspace{1em}\omega =F(\xi ,\xi ,·),$$

where $\left(g^{ij}\right)$ denotes the inverse of the matrix $\left(g_{ij}\right)$ of g with respect to a basis $\left\{e_i;\xi \right\}$$(i=1, 2, \dots ,2n)$ of $T_p\mathcal{M}$ at $p\in \mathcal{M}$.

The research in the present work mainly concerns one of the basic classes, namely ${\mathcal{F}}_5$, which is the counterpart of the class of well-known $\beta$-Kenmotsu manifolds among almost contact metric manifolds.

This restriction is caused by the use of torse-forming potential vector fields and by the possibility of explicitly expressing the classification tensor field F and the resulting dependencies. The problem of investigating the present topic in the general case for this type of potentials, when the manifold belongs to the direct sum of this basic class with several other basic classes (see Proposition 1), remains open.

The condition for defining ${\mathcal{F}}_5$-manifolds is the following:

$$F(x,y,z)=-s\left\{g(x,\phi y)\eta \left(z\right)+g(x,\phi z)\eta \left(y\right)\right\},$$

where for brevity of notation, we denote the following function:

$$s={\displaystyle \frac{{\theta }^{*}\left(\xi \right)}{2n}}.$$

(6)

It is known that the Lee forms on every ${\mathcal{F}}_5$-manifold have the following properties:

$$\theta =0,\hspace{1em}{\theta }^{*}=2n s \eta ,\hspace{1em}\omega =0.$$

(7)

A subclass of the basic class ${\mathcal{F}}_5$ of accR manifolds with remarkable curvature properties is ${\mathcal{F}}_5^0$, defined by the additional condition that its only non-zero Lee form ${\theta }^{*}$ must be closed, i.e., $\mathrm{d}{\theta }^{*}=0$ holds [22].

Using the second equality in (7), we find that the closedness of ${\theta }^{*}$ is equivalent to the following:

$$\mathrm{d}s\left(x\right)\eta \left(y\right)=\mathrm{d}s\left(y\right)\eta \left(x\right),$$

which for an ${\mathcal{F}}_5$-manifold can be written in the following form:

$$\mathrm{d}s=\mathrm{d}s\left(\xi \right)\eta .$$

(8)

In [22], for each ${\mathcal{F}}_5^0$-manifold, a relation is given between the curvature tensors R and $\tilde{R}$ generated by the Levi–Civita connections ∇ and $\tilde{\nabla }$, corresponding to the B-metrics g and $\tilde{g}$, respectively. This relation can be written, using (6), in the following form:

$$\begin{array}{cc}\hfill \tilde{R}(x,y)z& =R(x,y)z-\left\{g(y,\phi z)+g(\phi y,\phi z)\right\}\left\{s^{2}x+\mathrm{d}s\left(\xi \right)\eta \left(x\right)\xi \right\} \hfill \\ \hfill \hspace{1em}& \hfill \phantom{=R(x,y)z}+\left\{g(x,\phi z)+g(\phi x,\phi z)\right\}\left\{s^{2}y+\mathrm{d}s\left(\xi \right)\eta \left(y\right)\xi \right\}.\end{array}$$

(9)

Obviously, the following property holds:

$$\tilde{R}(x,y)\xi =R(x,y)\xi .$$

By virtue of (4) and (9), we obtain

$$\begin{array}{cc}\hfill \tilde{R}(x,y,z,w)& =R(x,y,z,\phi w)+\eta \left(R\right(x,y\left)z\right)\eta \left(w\right)\hfill \\ \hfill & \phantom{=}-\left\{g(y,\phi z)+g(\phi y,\phi z)\right\}\left\{s^{2}g(x,\phi w)+\left[s^2+\mathrm{d}s\left(\xi \right)\right]\eta \left(x\right)\eta \left(w\right)\right\}\hfill \\ \hfill & \phantom{=}+\left\{g(x,\phi z)+g(\phi x,\phi z)\right\}\left\{s^{2}g(y,\phi w)+\left[s^2+\mathrm{d}s\left(\xi \right)\right]\eta \left(y\right)\eta \left(w\right)\right\}.\hfill \end{array}$$

(10)

Taking the trace of (10) for $x=e_i$ and $w=e_j$ by the following consequence of (4)

$${\tilde{g}}^{ij}=-{\phi }_m^j{g}^{im}+{\xi }^i{\xi }^j,$$

(11)

we get

$$\tilde{\rho }(y,z)=\rho (y,z)-\left\{2n s^2+\mathrm{d}s\left(\xi \right)\right\}\left\{g(y,\phi z)+g(\phi y,\phi z)\right\}.$$

(12)

As a corollary, we have the property $\xi ⌟ \tilde{\rho }=\xi ⌟ \rho$. Moreover, $\tilde{\rho }=\rho$ is true if and only if $\mathrm{d}s\left(\xi \right)=-2n s^2$ is fulfilled. The last differential equation has, for example, the solution $s={(2n t)}^{-1}$, where $\mathrm{d}t$ is the coordinate 1-form on ${\mathcal{H}}^{\perp }=span\left(\xi \right)$.

Applying (11) to (12) with $y=e_i$ and $z=e_j$, we express the scalar curvature for $\tilde{g}$ of an arbitrary ${\mathcal{F}}_5^0$-manifold. The expression uses the associated quantity ${\tau }^{*}$ of $\tau$ with respect to φ, which is defined by $\tau *=g^{\mathit{ij}}\rho \left(e_i,\phi e_j\right)$. The resulting relation has the following form:

$$\begin{array}{c}\hfill \tilde{\tau }=-{\tau }^{*}-2n\left\{2 \mathrm{d}s\left(\xi \right)+(2n+1)s^2\right\}.\end{array}$$

(13)

Analogously, we can consider the associated quantity ${\tilde{\tau }}^{*}$ of $\tilde{\tau }$ with respect to φ, which is defined by ${\tilde{\tau }}^{*}={\tilde{g}}^{\mathit{ij}}\tilde{\rho }\left(e_i,\phi e_j\right)$. Then, (11) and (12) imply the following relation on an arbitrary ${\mathcal{F}}_5^0$-manifold:

$$\begin{array}{c}\hfill {\tilde{\tau }}^{*}=\tau +2n\left\{2 \mathrm{d}s\left(\xi \right)+(2n+1)s^2\right\}.\end{array}$$

(14)

3. Pair of Associated η-Ricci–Bourguignon Almost Solitons on an accR Manifold

We consider the so-called $\eta$-Ricci–Bourguignon almost soliton (in short, $\eta$-RB almost soliton) induced by the metric g and generalizing the RB soliton (2) in the following way:

$$\rho +{\displaystyle \frac{1}{2}}{\mathcal{L}}_{\vartheta }g+(\lambda +\mathsf{\ell }\tau )g+\mu \eta \otimes \eta =0,$$

(15)

where $\mu$ is also a function on $\mathcal{M}$ [13]. We denote this almost soliton by $(g;\vartheta ;\lambda ,\mu ,\mathsf{\ell })$.

Note that (almost) $\eta$-solitons of different types arise naturally when a manifold is endowed with a structural 1-form $\eta$, encoding preferred directions or geometric constraints. The additional $\eta \otimes \eta$ term in definitions of the type of (15) introduces anisotropic curvature effects compatible with this structure. Thus, (almost) $\eta$-solitons extend classical solitons in a geometrically principled way.

Analogously to (15), we also have an $\eta$-RB almost soliton induced by the other B-metric $\tilde{g}$ and defined for the corresponding Ricci tensor $\tilde{\rho }$ and scalar curvature $\tilde{\tau }$ as follows:

$$\tilde{\rho }+{\displaystyle \frac{1}{2}}{\mathcal{L}}_{\vartheta }\tilde{g}+(\tilde{\lambda }+\tilde{\mathsf{\ell }}\tilde{\tau })\tilde{g}+\tilde{\mu } \eta \otimes \eta =0,$$

(16)

where $\tilde{\lambda }$ and $\tilde{\mu }$ are differentiable functions on $\mathcal{M}$, and $\tilde{\mathsf{\ell }}$ is a real constant. We denote this almost soliton by $(\tilde{g};\vartheta ;\tilde{\lambda },\tilde{\mu },\tilde{\mathsf{\ell }})$.

The following concept was introduced in [23]. An accR manifold $(\mathcal{M}, \phi , \xi , \eta , g, \tilde{g})$ is said to be equipped with a pair of associated $\eta$-RB almost solitons with potential vector field $\vartheta$ if the corresponding Ricci tensors $\rho$ and $\tilde{\rho }$, as well as their scalar curvatures $\tau$ and $\tilde{\tau }$, satisfy (15) and (16), respectively.

The concept of an almost $\eta$-Einstein manifold equipped in the presence of some 1-form $\eta$, for example the contact form, is well known. The condition is that its Ricci tensor be expressed as a sum of the corresponding metric and $\eta \otimes \eta$, both multiplied by functions as coefficients, i.e.,

$$\rho =a g+b \eta \otimes \eta ,$$

(17)

where $(a,b)$ is some pair of functions on the manifold. This notion generalizes to almost Einstein manifolds for $b=0$, as well as $\eta$-Einstein manifolds and Einstein manifolds for constants a, b and a constant a and $b=0$, respectively.

In our case of an accR manifold, we also have the possibility of an almost $\eta$-Einstein manifold with respect to $\tilde{\rho }$ for $\tilde{g}$, i.e.,

$$\tilde{\rho }=\tilde{a} \tilde{g}+\tilde{b} \eta \otimes \eta$$

(18)

for a pair of functions $(\tilde{a},\tilde{b})$ on the manifold.

3.1. The Soliton Potential Is Double Torse-Forming

In the present study, we pay special attention to the case when the soliton potential $\vartheta$ is torse-forming with respect to the Levi–Civita connection ∇ of g. The definition of this type of vector field is as follows:

$${\nabla }_x\vartheta =f x+\gamma \left(x\right)\vartheta ,$$

(19)

where f is a differentiable function on $\mathcal{M}$ (called the conformal scalar of $\vartheta$) and $\gamma$ is a 1-form on $\mathcal{M}$ (called the generating form of $\vartheta$) [14,24].

Remark 1.

Some special types of torse-forming vector fields have been studied by various authors. Namely, a vector field ϑ determined by (19) is said to be of the following type if the corresponding specializing condition is satisfied: torqued, if $\gamma \left(\vartheta \right)=0$ [25]; concircular, if $\gamma =0$ [26]; concurrent, if $f-1=\gamma =0$ [27]; recurrent, if $f=0$ [28]; parallel, if $f=\gamma =0$ (e.g., [29]).

In (19), the Levi–Civita connection ∇ of the basic B-metric g is used. For a similar purpose, we can use the twin B-metric $\tilde{g}$ and its Levi–Civita connection $\tilde{\nabla }$ on the studied accR manifold. Furthermore, we require that the same vector field $\vartheta$ be torse-forming with respect to $\tilde{\nabla }$, i.e., the following condition be satisfied:

$${\tilde{\nabla }}_x\vartheta =\tilde{f} x+\tilde{\gamma }\left(x\right)\vartheta ,$$

(20)

where $\tilde{f}$ and $\tilde{\gamma }$ are also a differentiable function and a 1-form on $\mathcal{M}$, respectively. Then, $\tilde{f}$ and $\tilde{\gamma }$ are called the conformal scalar and the generating form of $\vartheta$ with respect to $\tilde{\nabla }$, respectively.

Further, when a vector field $\vartheta$ is torse-forming with respect to ∇ and $\tilde{\nabla }$, we briefly say that $\vartheta$ is double torse-forming.

Due to (19) and (20), we obtain the following expressions for the Lie derivatives of g and $\tilde{g}$ along $\vartheta$, which is a double torse-forming vector field:

$$\begin{array}{c}\left({\mathcal{L}}_{\vartheta }g\right)(x,y)=g({\nabla }_x\vartheta ,y)+g(x,{\nabla }_y\vartheta )=2fg(x,y)+\gamma \left(x\right)g(\vartheta ,y)+\gamma \left(y\right)g(\vartheta ,x),\end{array}$$

(21)

$$\begin{array}{c}\left({\mathcal{L}}_{\vartheta }\tilde{g}\right)(x,y)=\tilde{g}({\tilde{\nabla }}_x\vartheta ,y)+\tilde{g}(x,{\tilde{\nabla }}_y\vartheta )=2\tilde{f}\tilde{g}(x,y)+\tilde{\gamma }\left(x\right)\tilde{g}(\vartheta ,y)+\tilde{\gamma }\left(y\right)\tilde{g}(\vartheta ,x).\end{array}$$

(22)

Substituting (21) and (22) into (15) and (16), respectively, we obtain the following expressions:

$$\begin{array}{c}\rho (x,y)=-\left\{f+\lambda +\mathsf{\ell }\tau \right\}g(x,y)-\mu \eta \left(x\right)\eta \left(y\right)-{\displaystyle \frac{1}{2}}\left\{\gamma \left(x\right)g(\vartheta ,y)+\gamma \left(y\right)g(\vartheta ,x)\right\},\end{array}$$

(23)

$$\begin{array}{c}\tilde{\rho }(x,y)=-\left\{\tilde{f}+\tilde{\lambda }+\tilde{\mathsf{\ell }}\tilde{\tau }\right\}\tilde{g}(x,y)-\tilde{\mu } \eta \left(x\right)\eta \left(y\right)-{\displaystyle \frac{1}{2}}\left\{\tilde{\gamma }\left(x\right)\tilde{g}(\vartheta ,y)+\tilde{\gamma }\left(y\right)\tilde{g}(\vartheta ,x)\right\}.\end{array}$$

(24)

Theorem 1.

Let an accR manifold $(\mathcal{M}, \phi , \xi , \eta , g, \tilde{g})$ be equipped with a pair of associated η-RB almost solitons $(g;\vartheta ;\lambda ,\mu ,\mathsf{\ell })$ and $(\tilde{g};\vartheta ;\tilde{\lambda },\tilde{\mu },\tilde{\mathsf{\ell }})$, where ϑ is double torse-forming. Then, we have:

(i) 

For $\mathsf{\ell }\ne -{\displaystyle \frac{1}{2n+1}}$, the Ricci tensor and the scalar curvature with respect to g of this manifold have the following form:

$$\begin{array}{cc}\hfill \rho (x,y)=& -{\displaystyle \frac{f+\lambda -\mathsf{\ell }\left[\mu +\gamma \left(\vartheta \right)\right]}{1+(2n+1)\mathsf{\ell }}}g(x,y)-\mu \eta \left(x\right)\eta \left(y\right)\hfill \\ & -{\displaystyle \frac{1}{2}}\left\{\gamma \left(x\right)g(\vartheta ,y)+\gamma \left(y\right)g(\vartheta ,x)\right\},\hfill \end{array}$$

(25)

$$\begin{array}{c}\tau =-{\displaystyle \frac{(2n+1)(f+\lambda )+\mu +\gamma \left(\vartheta \right)}{1+(2n+1)\mathsf{\ell }}}.\end{array}$$

(26)

(ii) 

For $\mathsf{\ell }=-{\displaystyle \frac{1}{2n+1}}$, the scalar curvature with respect to g cannot be expressed explicitly, and we have the following relation between the Ricci tensor and the scalar curvature with respect to g:

$$\begin{array}{cc}\hfill \rho (x,y)=& -\left\{f+\lambda -{\displaystyle \frac{1}{2n+1}}\tau \right\}g(x,y)-\mu \eta \left(x\right)\eta \left(y\right)\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}\left\{\gamma \left(x\right)g(\vartheta ,y)+\gamma \left(y\right)g(\vartheta ,x)\right\}.\hfill \end{array}$$

(27)

In addition, the following property is valid

$$\gamma \left(\vartheta \right)=-(2n+1)(f+\lambda )-\mu ,$$

(28)

and ϑ is geodesic with respect to ∇ if and only if $f=-{\displaystyle \frac{1}{2n}}\{(2n+1)\lambda +\mu \}$ holds.

(iii) 

For $\tilde{\mathsf{\ell }}\ne -{\displaystyle \frac{1}{2n+1}}$, the Ricci tensor and the scalar curvature with respect to $\tilde{g}$ of the considered manifold have the following form:

$$\begin{array}{cc}\hfill \tilde{\rho }(x,y)& =-{\displaystyle \frac{\tilde{f}+\tilde{\lambda }-\tilde{\mathsf{\ell }}\left[\tilde{\mu }+\tilde{\gamma }\left(\vartheta \right)\right]}{1+(2n+1)\tilde{\mathsf{\ell }}}}\tilde{g}(x,y)-\tilde{\mu } \eta \left(x\right)\eta \left(y\right)\hfill \\ \hfill & \phantom{= }-{\displaystyle \frac{1}{2}}\left\{\tilde{\gamma }\left(x\right)\tilde{g}(\vartheta ,y)+\tilde{\gamma }\left(y\right)\tilde{g}(\vartheta ,x)\right\},\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill \tilde{\tau }& =-{\displaystyle \frac{(2n+1)(\tilde{f}+\tilde{\lambda })+\tilde{\mu }+\tilde{\gamma }\left(\vartheta \right)}{1+(2n+1)\tilde{\mathsf{\ell }}}}.\hfill \end{array}$$

(30)

(iv) 

For $\tilde{\mathsf{\ell }}=-{\displaystyle \frac{1}{2n+1}}$, we have the following relation between $\tilde{\rho }$ and $\tilde{\tau }$, from which the scalar curvature cannot be expressed explicitly:

$$\begin{array}{cc}\hfill \tilde{\rho }(x,y)& =-\left\{\tilde{f}+\tilde{\lambda }-{\displaystyle \frac{1}{2n+1}}\tilde{\tau }\right\}\tilde{g}(x,y)-\tilde{\mu } \eta \left(x\right)\eta \left(y\right)\hfill \\ \hfill & \phantom{= }-{\displaystyle \frac{1}{2}}\left\{\tilde{\gamma }\left(x\right)\tilde{g}(\vartheta ,y)+\tilde{\gamma }\left(y\right)\tilde{g}(\vartheta ,x)\right\}.\hfill \end{array}$$

(31)

Furthermore, the following property is valid:

$$\tilde{\gamma }\left(\vartheta \right)=-(2n+1)(\tilde{f}+\tilde{\lambda })-\tilde{\mu },$$

(32)

and ϑ is geodesic with respect to $\tilde{\nabla }$ if and only if $\tilde{f}=-{\displaystyle \frac{1}{2n}}\{(2n+1)\tilde{\lambda }+\tilde{\mu }\}$ holds.

Proof. 

Contracting (23), we obtain the expression in (26) for the corresponding scalar curvature for g in the case $\mathsf{\ell }\ne -{\displaystyle \frac{1}{2n+1}}$. Substituting (26) into (23) gives us the result in (25), thus proving case (i).

In the case of $\mathsf{\ell }=-{\displaystyle \frac{1}{2n+1}}$, the equality in (23) implies (27), then $\tau$ cannot be expressed explicitly, but instead we get (28). As a consequence of the latter and (19), we obtain the property ${\nabla }_{\vartheta }\vartheta =-\{2n f+(2n+1)\lambda +\mu \}\vartheta$, from which the last part of the statement in case (ii) follows.

In a similar way, the contraction of (24) using $\tilde{g}$ gives the following equality:

$$\left\{1+(2n+1)\tilde{\mathsf{\ell }}\right\}\tilde{\tau }=-(2n+1)(\tilde{f}+\tilde{\lambda })-\tilde{\mu }-\tilde{\gamma }\left(\vartheta \right).$$

First, under the condition of case (iii), we obtain the expression in (30) for the scalar curvature with respect to $\tilde{g}$, and substituting it into (24), we obtain (29).

Otherwise, i.e., if $\tilde{\mathsf{\ell }}$ has the value $-{\displaystyle \frac{1}{2n+1}}$, then (32) is true and (24) takes the form in (31). The proof of case (iv) completes by establishing the truth of the fact at the end of this case, using (20) and (32). □

3.2. The Double Torse-Forming Potential Is Vertical

In addition to the condition that the soliton potential $\vartheta$ be torse-forming with respect to the pair of Levi–Civita connections, we require that it be vertical, i.e., the following holds:

$$\vartheta =k \xi ,$$

(33)

where k is a nowhere-vanishing function on $\mathcal{M}$, and obviously, $k=\eta \left(\vartheta \right)$ is true. This means that we exclude from consideration the trivial case in which $\vartheta$ is a zero soliton potential, or equivalently, $\vartheta$ is a torqued vector field according to Remark 1.

Using the expression of $\rho$ in (23), we calculate ${\tau }^{*}$ and obtain ${\tau }^{*}=-\gamma \left(\phi \vartheta \right)$, which due to (33) implies the following:

$${\tau }^{*}=0.$$

(34)

Similarly, based on (24), for $\tilde{\rho }$, we calculate ${\tilde{\tau }}^{*}$ and get ${\tilde{\tau }}^{*}=-\tilde{\gamma }$, which for vertical $\vartheta$ means that

$${\tilde{\tau }}^{*}=0.$$

(35)

Taking into account (19), (20), and (33), we get

$$\begin{array}{c}\mathrm{d}k\left(x\right)=f \eta \left(x\right)+k \gamma \left(x\right),\end{array}$$

(36)

$$\begin{array}{c}{\nabla }_x\xi =-{\displaystyle \frac{f}{k}} {\phi }^{2}x.\end{array}$$

(37)

Note that from the expression in (37), it follows that $\xi$ is geodesic with respect to ∇ and $\eta$ is closed.

Due to (5) and (37), we obtain

$$F(x,\phi y,\xi )=\left({\nabla }_x\eta \right)\left(y\right)=-s g(\phi x,\phi y),$$

$$s={\displaystyle \frac{f}{k}}={\displaystyle \frac{{\theta }^{*}\left(\xi \right)}{2n}}.$$

(38)

Similarly, we have

$$\begin{array}{c}\hfill \mathrm{d}k\left(x\right)=\tilde{f} \eta \left(x\right)+k \tilde{\gamma }\left(x\right),\end{array}$$

(39)

$$\begin{array}{c}\hfill {\tilde{\nabla }}_x\xi =-\tilde{s} {\phi }^{2}x,\hspace{1em}\tilde{s}={\displaystyle \frac{\tilde{f}}{k}}.\end{array}$$

(40)

It is clear that (40) shows that $\xi$ is also geodesic with respect to $\tilde{\nabla }$ in the considered case.

Let us recall a fact about the construction under consideration.

Lemma 1

([30]). Let us consider an accR manifold $(\mathcal{M}, \phi , \xi , \eta , g, \tilde{g})$ and a vertical vector field ϑ on it. If ϑ is torse-forming with respect to both ∇ and $\tilde{\nabla }$ with conformal scalars f, $\tilde{f}$ and generating forms γ, $\tilde{\gamma }$, respectively, then we have the following:

$$\tilde{f}=f,\hspace{1em}\tilde{\gamma }=\gamma ,\hspace{1em}\tilde{\nabla }\vartheta =\nabla \vartheta .$$

(41)

Combining (19), (33), and (36), we get

$${\nabla }_x\vartheta =-f {\phi }^{2}x+\mathrm{d}k\left(x\right)\xi$$

and due to (21) we have

$$\left({\mathcal{L}}_{\vartheta }g\right)(x,y)=-2fg(\phi x,\phi y)+h(x,y),$$

(42)

where h denotes the symmetric (0,2)-tensor determined by

$$h(x,y)=\mathrm{d}k\left(x\right)\eta \left(y\right)+\mathrm{d}k\left(y\right)\eta \left(x\right).$$

(43)

Similarly, (20), (33), (39), (41), and (22) imply

$$\left({\mathcal{L}}_{\vartheta }\tilde{g}\right)(x,y)=2fg(x,\phi y)+h(x,y).$$

(44)

Using (37), (38), and (40), we get the following for the curvature tensors R and $\tilde{R}$ of g and $\tilde{g}$, respectively:

$$\begin{array}{c}R(x,y)\xi =-\left\{\mathrm{d}s\left(x\right)+s^2\eta \left(x\right)\right\}{\phi }^{2}y+\left\{\mathrm{d}s\left(y\right)+s^2\eta \left(y\right)\right\}{\phi }^{2}x,\end{array}$$

(45)

$$\begin{array}{c}\tilde{R}(x,y)\xi =-\left\{\mathrm{d}\tilde{s}\left(x\right)+{\tilde{s}}^2\eta \left(x\right)\right\}{\phi }^{2}y+\left\{\mathrm{d}\tilde{s}\left(y\right)+{\tilde{s}}^2\eta \left(y\right)\right\}{\phi }^{2}x.\end{array}$$

(46)

Taking the trace of the maps $x\to R(x,\xi )\xi$ and $x\to \tilde{R}(x,\xi )\xi$, based on (45) and (46), we obtain the following:

$$\begin{array}{c}\rho (\xi ,\xi )=-2n\left\{\mathrm{d}s\left(\xi \right)+s^2\right\},\end{array}$$

(47)

$$\begin{array}{c}\tilde{\rho }(\xi ,\xi )=-2n\left\{\mathrm{d}\tilde{s}\left(\xi \right)+{\tilde{s}}^2\right\}.\end{array}$$

(48)

The expression in (47), due to (38), is equivalent to the following form:

$$\rho (\xi ,\xi )=-{\displaystyle \frac{2n}{k^2}}\left\{k\mathrm{d}f\left(\xi \right)-f\mathrm{d}k\left(\xi \right)+f^2\right\}.$$

(49)

Similarly, the expression in (48), using the second equality of (40), can be written in the following form:

$$\tilde{\rho }(\xi ,\xi )=-{\displaystyle \frac{2n}{k^2}}\left\{k\mathrm{d}\tilde{f}\left(\xi \right)-\tilde{f}\mathrm{d}k\left(\xi \right)+{\tilde{f}}^2\right\}.$$

From Theorem 1, assuming that $\vartheta$ is vertical, we have the following.

Theorem 2.

Let an accR manifold $(\mathcal{M}, \phi , \xi , \eta , g, \tilde{g})$ be equipped with a pair of associated η-RB almost solitons $(g;\vartheta ;\lambda ,\mu ,\mathsf{\ell })$ and $(\tilde{g};\vartheta ;\tilde{\lambda },\tilde{\mu },\tilde{\mathsf{\ell }})$, where ϑ is double torse-forming and vertical. Then, we have:

(i) 

For $\mathsf{\ell }\ne -{\displaystyle \frac{1}{2n+1}}$, the following expressions of ρ and τ are true:

$$\begin{array}{c}\rho =-{\displaystyle \frac{(1+\mathsf{\ell })f+\lambda -\mathsf{\ell }\left[\mu +\mathrm{d}k\left(\xi \right)\right]}{1+(2n+1)\mathsf{\ell }}}g+(f-\mu )\eta \otimes \eta -{\displaystyle \frac{1}{2}}h,\end{array}$$

(50)

$$\begin{array}{c}\tau =-{\displaystyle \frac{2n f+(2n+1)\lambda +\mu +\mathrm{d}k\left(\xi \right)}{1+(2n+1)\mathsf{\ell }}}.\end{array}$$

(51)

Furthermore, the following property for the functions used is valid:

$$\begin{array}{cc}\hfill & \left\{(1+2n \mathsf{\ell })k^2+2n\left[1+(2n+1)\mathsf{\ell }\right]f\right\}\mathrm{d}k\left(\xi \right)\hfill \\ \hfill & -2n\left[1+(2n+1)\mathsf{\ell }\right]\left\{k \mathrm{d}f\left(\xi \right)+f^2\right\}-\left\{2n \mathsf{\ell } f-\lambda -(1+2n \mathsf{\ell })\mu \right\}{k}^2=0.\hfill \end{array}$$

(52)

(ii) 

For $\mathsf{\ell }=-{\displaystyle \frac{1}{2n+1}}$, the following expressions of ρ and τ are true:

$$\begin{array}{cc}\hfill \rho =& -{\displaystyle \frac{1}{k^2}}\left\{2n\left[k \mathrm{d}f\left(\xi \right)+f\mu \right]+(2n+1)(\lambda +f)\left(k^2+2n f\right)\right\}g\hfill \\ \hfill & +(f-\mu )\eta \otimes \eta -{\displaystyle \frac{1}{2}}h,\hfill \\ \hfill \tau =& -{\displaystyle \frac{2n(2n+1)}{k^2}}\left\{k \mathrm{d}f\left(\xi \right)+f\mu +\left[(2n+1)f+k^2\right](\lambda +f)\right\}.\hfill \end{array}$$

Furthermore, the following property for the functions used is valid:

$$\mathrm{d}k\left(\xi \right)+2n f+(2n+1)\lambda +\mu =0$$

(53)

and k is a vertical constant if and only if $f=-{\displaystyle \frac{1}{2n}}\{(2n+1)\lambda +\mu \}$ holds.

(iii) 

For $\tilde{\mathsf{\ell }}\ne -{\displaystyle \frac{1}{2n+1}}$, the following expressions of $\tilde{\rho }$ and $\tilde{\tau }$ are true:

$$\begin{array}{c}\tilde{\rho }=-{\displaystyle \frac{\left(1+\tilde{\mathsf{\ell }}\right)\tilde{f}+\tilde{\lambda }-\tilde{\mathsf{\ell }}\left[\tilde{\mu }+\mathrm{d}k\left(\xi \right)\right]}{1+(2n+1)\tilde{\mathsf{\ell }}}}\tilde{g}+\left(\tilde{f}-\tilde{\mu }\right)\eta \otimes \eta -{\displaystyle \frac{1}{2}}h,\end{array}$$

(54)

$$\begin{array}{c}\tilde{\tau }=-{\displaystyle \frac{2n \tilde{f}+(2n+1)\tilde{\lambda }+\tilde{\mu }+\mathrm{d}k\left(\xi \right)}{1+(2n+1)\tilde{\mathsf{\ell }}}}.\end{array}$$

(55)

Furthermore, the following property for the functions used is valid:

$$\begin{array}{cc}\hfill & \left\{(1+2n \tilde{\mathsf{\ell }})k^2+2n\left[1+(2n+1)\tilde{\mathsf{\ell }}\right]\tilde{f}\right\}\mathrm{d}k\left(\xi \right)\hfill \\ \hfill & -2n\left[1+(2n+1)\tilde{\mathsf{\ell }}\right]\left\{k \mathrm{d}\tilde{f}\left(\xi \right)+{\tilde{f}}^{ 2}\right\}-\left\{2n \tilde{\mathsf{\ell }} \tilde{f}-\tilde{\lambda }-(1+2n \tilde{\mathsf{\ell }})\tilde{\mu }\right\}{k}^2=0.\hfill \end{array}$$

(iv) 

For $\tilde{\mathsf{\ell }}=-{\displaystyle \frac{1}{2n+1}}$, the following expressions of $\tilde{\rho }$ and $\tilde{\tau }$ are true:

$$\begin{array}{cc}\hfill \tilde{\rho }=& -{\displaystyle \frac{1}{k^2}}\left\{2n\left[k \mathrm{d}\tilde{f}\left(\xi \right)+\tilde{f}\tilde{\mu }\right]+(2n+1)\left(\tilde{\lambda }+\tilde{f}\right)\left(k^2+2n \tilde{f}\right)\right\}\tilde{g}\hfill \\ \hfill & +\left(\tilde{f}-\tilde{\mu }\right)\eta \otimes \eta -{\displaystyle \frac{1}{2}}h,\hfill \\ \hfill \tilde{\tau }=& -{\displaystyle \frac{2n(2n+1)}{k^2}}\left\{k \mathrm{d}\tilde{f}\left(\xi \right)+\tilde{f}\tilde{\mu }+\left[(2n+1)\tilde{f}+k^2\right]\left(\tilde{\lambda }+\tilde{f}\right)\right\}.\hfill \end{array}$$

Furthermore, the following property for the functions used is valid:

$$\mathrm{d}k\left(\xi \right)+2n \tilde{f}+(2n+1)\tilde{\lambda }+\tilde{\mu }=0,$$

(56)

and k is a horizontal constant if and only if $\tilde{f}=-{\displaystyle \frac{1}{2n}}\{(2n+1)\tilde{\lambda }+\tilde{\mu }\}$ holds.

Proof. 

Considering (33) and (36), we obtain the following expression:

$$\gamma \left(\vartheta \right)=\mathrm{d}k\left(\xi \right)-f.$$

(57)

Similarly, (33) and (39) imply

$$\tilde{\gamma }\left(\vartheta \right)=\mathrm{d}k\left(\xi \right)-\tilde{f}.$$

(58)

We use (33), (36) and (57) first in (25) and (26) to obtain (50) and (51) of case (i); and second in (27) and (28) to deduce the following relation between $\rho$ and $\tau$ for case (ii):

$$\begin{array}{cc}\hfill \rho & =-\left\{f+\lambda -{\displaystyle \frac{1}{2n+1}}\tau \right\}g+(f-\mu )\eta \otimes \eta -{\displaystyle \frac{1}{2}}h,\hfill \end{array}$$

(59)

which does not lead to an explicit expression of $\tau$, but the contraction of (59) implies (53) and through it the assertion in the last line of (ii).

The formula (50) in case (i) implies for $x=y=\xi$ the following equality

$$\begin{array}{cc}\hfill \rho (\xi ,\xi )& ={\displaystyle \frac{2n \mathsf{\ell } f-\lambda -(1+2n \mathsf{\ell })\left[\mu +\mathrm{d}k\left(\xi \right)\right]}{1+(2n+1)\mathsf{\ell }}},\hfill \end{array}$$

which together with (49) gives the condition in (52).

Taking into account (53) for case (ii), we obtain that (59) implies

$$\rho (\xi ,\xi )={\displaystyle \frac{1}{2n+1}}\tau +2n(f+\lambda ).$$

(60)

We compare (60) with (47) and obtain an expression of $\tau$ in this case as follows:

$$\tau =-2n(2n+1)\left\{\mathrm{d}s\left(\xi \right)+s^2+f+\lambda \right\}.$$

(61)

Based on (38) and (53), the formula in (61) can be written in terms of the functions $\lambda$, $\mu$, f, and k, as given in (ii).

Similarly, for cases (iii) and (iv), we substitute (33), (39), and (58) first into (29) and (30) to arrive at (54) and (55); and second into (31) and (32) to get (56) and the following relation between $\tilde{\rho }$ and $\tilde{\tau }$:

$$\begin{array}{cc}\hfill \tilde{\rho }& =-\left\{\tilde{f}+\tilde{\lambda }-{\displaystyle \frac{1}{2n+1}}\tilde{\tau }\right\}\tilde{g}+\left(\tilde{f}-\tilde{\mu }\right)\eta \otimes \eta -{\displaystyle \frac{1}{2}}h,\hfill \end{array}$$

which does not directly imply an explicit expression of $\tilde{\tau }$.

The rest of the proof for cases (iii) and (iv) is completed with similar reasoning as in the first two cases. □

3.3. The Double Torse-Forming Vertical Potential Is on an ${\mathcal{F}}_5^0$-Manifold

In this subsection, we recall two more well-known facts regarding the position of the considered manifolds in the most popular classification of accR manifolds from [21] and continue the reasoning in this direction in order to study the curvature properties of these manifolds.

Proposition 1

([30]). Let $(\mathcal{M}, \phi , \xi , \eta , g, \tilde{g})$ be an accR manifold with a vertical vector field ϑ that is torse-forming with respect to both ∇ and $\tilde{\nabla }$. Then, the manifold belongs to ${\mathcal{F}}_5$ or to a direct sum of ${\mathcal{F}}_5$ with ${\mathcal{F}}_1$, ${\mathcal{F}}_2$, ${\mathcal{F}}_3$, ${\mathcal{F}}_6$, and ${\mathcal{F}}_{10}$. Furthermore, ϑ is recurrent if and only if the component of F relating $(\mathcal{M}, \phi , \xi , \eta , g, \tilde{g})$ to ${\mathcal{F}}_5$ vanishes, i.e., ${\mathcal{F}}_5$ is restricted to ${\mathcal{F}}_0$.

Proposition 2

([30]). Let an accR manifold $(\mathcal{M}, \phi , \xi , \eta , g, \tilde{g})$ belong to ${\mathcal{F}}_5$, a vector field ϑ on $\mathcal{M}$ be vertical, and ϑ be torse-forming with respect to both ∇ and $\tilde{\nabla }$. Then, $(\mathcal{M}, \phi , \xi , \eta , g, \tilde{g})$ belongs to ${\mathcal{F}}_5^0$.

Proposition 3.

Let an ${\mathcal{F}}_5^0$-manifold $(\mathcal{M}, \phi , \xi , \eta , g, \tilde{g})$ be equipped with a pair of associated η-RB almost solitons $(g;\vartheta ;\lambda ,\mu ,\mathsf{\ell })$ and $(\tilde{g};\vartheta ;\tilde{\lambda },\tilde{\mu },\tilde{\mathsf{\ell }})$, where ϑ is vertical and double torse-forming. Then, the scalar curvatures for g and $\tilde{g}$ are equal and have the following expression:

$$\tau =\tilde{\tau }=-2n\left\{2 \mathrm{d}s\left(\xi \right)+(2n+1)s^2\right\}.$$

(62)

Proof. 

Since the soliton potential $\vartheta$ is vertical (i.e., (33) is valid), then we have $\phi \vartheta =0$, and the vanishing of ${\tau }^{*}$ and ${\tilde{\tau }}^{*}$ holds because of (34) and (35). Thereafter, due to (13) and (14) for an ${\mathcal{F}}_5^0$-manifold, we obtain that $\tau$ and $\tilde{\tau }$ are equal and (62) is valid. □

An example of a 3-dimensional ${\mathcal{F}}_5^0$-manifold satisfying (62) is given in Section 4 of [30] for $n=1$ and $s=t^{-1}$, where $\eta =\mathrm{d}t$. This accR manifold is constructed on a cone over a 2-dimensional flat manifold with complex structure and Norden metric. In this case, (62) takes the following form: $\tau =\tilde{\tau }=-2t^{-2}$.

We continue by considering of the following symmetric $(0,2)$-tensors

$$\alpha =\rho +\left\{\mathrm{d}s\left(\xi \right)+2n s^2\right\}g,\hspace{1em}\tilde{\alpha }=\tilde{\rho }+\left\{\mathrm{d}\tilde{s}\left(\xi \right)+2n {\tilde{s}}^2\right\}\tilde{g}.$$

(63)

Then, we have the following:

Proposition 4.

Let an ${\mathcal{F}}_5^0$-manifold $(\mathcal{M}, \phi , \xi , \eta , g, \tilde{g})$ be equipped with a pair of associated η-RB almost solitons with the same potential ϑ, which is vertical and double torse-forming. Then, α is invariant under the change of ∇ by $\tilde{\nabla }$ and vice versa, i.e., $\alpha =\tilde{\alpha }$ holds.

Proof. 

Considering (38), the second equality of (40), and the first equality of (41), we conclude that $\tilde{s}=s$. Then, (47) and (48) imply the following equality:

$$\tilde{\rho }(\xi ,\xi )=\rho (\xi ,\xi ).$$

(64)

Moreover, taking into account (3) and (4), we obtain from (12) that

$$\tilde{\rho }+\left\{\mathrm{d}\tilde{s}\left(\xi \right)+2n {\tilde{s}}^2\right\}\tilde{g}=\rho +\left\{\mathrm{d}s\left(\xi \right)+2n s^2\right\}g.$$

The last equality means that on every ${\mathcal{F}}_5^0$-manifold with the conditions in the statement, the symmetric $(0,2)$-tensor $\alpha$, defined by (63), is invariant under the replacement of g by $\tilde{g}$, and vice versa. □

Corollary 1.

Let the requirements of Proposition 4 be fulfilled for an ${\mathcal{F}}_5^0$-manifold $(\mathcal{M}, \phi , \xi , \eta , g, \tilde{g})$. Then, the invariant tensor α has the following expression:

$$\alpha =\rho -{\displaystyle \frac{1}{2n}}\left\{\tau -\rho (\xi ,\xi )\right\}g.$$

(65)

Moreover, α vanishes if and only if the manifold is Einstein with negative scalar curvatures with respect to each B-metric of the pair $(g,\tilde{g})$. Then, the Lee form ${\theta }^{*}$ is a constant multiple of the contact form η.

Proof. 

Using (47) and (62), we obtain the equality in (65). Taking into account it and $\alpha =0$, we get

$$\tau =(2n+1)\rho (\xi ,\xi )$$

(66)

and therefore $\rho ={\displaystyle \frac{\tau }{2n+1}}g$ is valid, i.e., the manifold is Einstein. Then, applying (47) and (62) to (66), we obtain that $\mathrm{d}s\left(\xi \right)=0$ is valid, which is equivalent to concluding that ${\theta }^{*}\left(\xi \right)$ is constant in the vertical direction. Then, from (8), we have that ${\theta }^{*}\left(\xi \right)$ (as well as s) is a constant in general and due to (7), and we obtain the last part of the statement. The proof is complete once we notice that (66) implies $\tau =\tilde{\tau }=-2n(2n+1)s^2<0$, where s is non-zero for every ${\mathcal{F}}_5$-manifold that is not in ${\mathcal{F}}_0$. □

Let us now consider $(g;\vartheta ;\lambda ,\mu ,\mathsf{\ell })$ and $(\tilde{g};\vartheta ;\tilde{\lambda },\tilde{\mu },\tilde{\mathsf{\ell }})$ without restrictions on the value of the constants ℓ and $\tilde{\mathsf{\ell }}$.

Proposition 5.

Let an ${\mathcal{F}}_5^0$-manifold $(\mathcal{M}, \phi , \xi , \eta , g, \tilde{g})$ be equipped with a pair of associated η-RB almost solitons $(g;\vartheta ;\lambda ,\mu ,\mathsf{\ell })$ and $(\tilde{g};\vartheta ;\tilde{\lambda },\tilde{\mu },\tilde{\mathsf{\ell }})$, where ϑ is double torse-forming and vertical, determined by the functions f, k, and the 1-form γ. Then, all the parameters that determine the pair of almost solitons and their Ricci tensors are expressed in terms of f and k as follows:

$$\begin{array}{c}\hfill \lambda =-f+{\displaystyle \frac{1}{k^2}}\left\{(1+4n \mathsf{\ell })\left[k \mathrm{d}f\left(\xi \right)-f \mathrm{d}k\left(\xi \right)\right]+2n\left[1+(2n+1)\mathsf{\ell }\right]f^2\right\}.\end{array}$$

(67)

$$\tilde{\lambda }=-f+{\displaystyle \frac{1}{k^2}}\left\{(1+4n \tilde{\mathsf{\ell }})\left[k \mathrm{d}f\left(\xi \right)-f \mathrm{d}k\left(\xi \right)\right]+2n\left[1+(2n+1)\tilde{\mathsf{\ell }}\right]f^2\right\}.$$

(68)

$$\begin{array}{c}\hfill \mu =\tilde{\mu }=f+(2n-1){\displaystyle \frac{1}{k}}\mathrm{d}f\left(\xi \right)-\left[1+(2n-1){\displaystyle \frac{f}{k^2}}\right]\mathrm{d}k\left(\xi \right).\end{array}$$

(69)

$$\rho =-\left\{\mathrm{d}s\left(\xi \right)+2n s^2\right\}g+\left\{\mathrm{d}k\left(\xi \right)-(2n-1)\mathrm{d}s\left(\xi \right)\right\}\eta \otimes \eta -{\displaystyle \frac{1}{2}}h.$$

(70)

$$\tilde{\rho }=-\left\{\mathrm{d}s\left(\xi \right)+2n s^2\right\}\tilde{g}+\left\{\mathrm{d}k\left(\xi \right)-(2n-1)\mathrm{d}s\left(\xi \right)\right\}\eta \otimes \eta -{\displaystyle \frac{1}{2}}h.$$

(71)

Moreover, the considered manifold is η-Einstein with respect to both B-metrics g and $\tilde{g}$ if and only if k is a horizontal constant, i.e., $\mathrm{d}k\circ \phi =0$ is valid.

Proof. 

According to (3) and (42), the expression (15) takes the following form:

$$\rho =-\left(f+\lambda +\mathsf{\ell }\tau \right)g+\left(f-\mu \right)\eta \otimes \eta -{\displaystyle \frac{1}{2}}h.$$

(72)

Therefore, from (72) and (43), we have

$$\rho (\xi ,\xi )=-\left\{\lambda +\mathsf{\ell }\tau +\mu +\mathrm{d}k\left(\xi \right)\right\}.$$

(73)

According to (62) and (47) for the considered ${\mathcal{F}}_5^0$-manifold, (73) implies the following differential equation for the functions used in the construction of $(g;\vartheta ;\lambda ,\mu ,\mathsf{\ell })$ without restrictions on the value of the constant ℓ:

$$\lambda +\mu +\mathrm{d}k\left(\xi \right)=2n\left\{\left(1+2\mathsf{\ell }\right)\mathrm{d}s\left(\xi \right)+\left[1+(2n+1)\mathsf{\ell }\right]s^2\right\}.$$

(74)

By virtue of (38), the relation in (74) takes the following form:

$$\begin{array}{cc}\hfill \lambda +\mu =-{\displaystyle \frac{1}{k^2}}& \left\{\left[k^2+2n\left(1+2\mathsf{\ell }\right)f\right]\mathrm{d}k\left(\xi \right)\right.\hfill \\ \hfill & \left.-2n\left[\left(1+2\mathsf{\ell }\right)k \mathrm{d}f\left(\xi \right)+\left[1+(2n+1)\mathsf{\ell }\right]f^2\right]\right\}.\hfill \end{array}$$

Contracting (72) and using (62) and (74), we get the following expression:

$$\begin{array}{c}\hfill \tau =-2n\left\{f+\lambda +\left(1-4n\mathsf{\ell }\right)\mathrm{d}s\left(\xi \right)+\left[1-2n(2n+1)\mathsf{\ell }\right]s^2\right\}.\end{array}$$

(75)

Then, from (75) and (62), we obtain the following expression of the sum of f and $\lambda$ as a function of s, which is the quotient of f and k:

$$\begin{array}{c}\hfill f+\lambda =\left(1+4n\mathsf{\ell }\right)\mathrm{d}s\left(\xi \right)+2n\left[1+(2n+1)\mathsf{\ell }\right]s^2.\end{array}$$

(76)

Taking into account (38) in (76), we can express for $\lambda$ in terms of f and k as in (67).

Excluding $\lambda$ from (74) and (76), we express the function $\mu$ as follows:

$$\begin{array}{c}\hfill \mu =f-\mathrm{d}k\left(\xi \right)+(2n-1)\mathrm{d}s\left(\xi \right).\end{array}$$

(77)

By virtue of (38), we can rewrite (77) as an expression for $\mu$ in terms of f and k, given in (69).

Substituting (76), (62), and (77) into (72), we obtain (70).

Taking into account (4), (44) and the first equality of (41), the expression (16) takes the following form:

$$\tilde{\rho }=-\left(f+\tilde{\lambda }+\tilde{\mathsf{\ell }}\tilde{\tau }\right)\tilde{g}+\left(f-\tilde{\mu }\right)\eta \otimes \eta -{\displaystyle \frac{1}{2}}h.$$

(78)

Applying the last equality to the arguments $(\xi ,\xi )$, we obtain from (78) and (43) that

$$\tilde{\rho }(\xi ,\xi )=-\left\{\tilde{\lambda }+\tilde{\mathsf{\ell }}\tilde{\tau }+\tilde{\mu }+\mathrm{d}k\left(\xi \right)\right\}.$$

(79)

According to (62) and (64) for the considered ${\mathcal{F}}_5^0$-manifold, (79) implies the following differential equation for the functions used in constructing $(\tilde{g};\vartheta ;\tilde{\lambda },\tilde{\mu },\tilde{\mathsf{\ell }})$:

$$\tilde{\lambda }+\tilde{\mu }+\mathrm{d}k\left(\xi \right)=2n\left\{\left(1+2\tilde{\mathsf{\ell }}\right)\mathrm{d}s\left(\xi \right)+\left[1+(2n+1)\tilde{\mathsf{\ell }}\right]s^2\right\}.$$

(80)

We substitute (38) into (80), and the last equality takes the following form:

$$\begin{array}{cc}\hfill \tilde{\lambda }+\tilde{\mu }=-{\displaystyle \frac{1}{k^2}}& \left\{\left[k^2+2n\left(1+2\tilde{\mathsf{\ell }}\right)f\right]\mathrm{d}k\left(\xi \right)\right.\hfill \\ \hfill & \left.-2n\left[\left(1+2\tilde{\mathsf{\ell }}\right)k \mathrm{d}f\left(\xi \right)+\left[1+(2n+1)\tilde{\mathsf{\ell }}\right]f^2\right]\right\}.\hfill \end{array}$$

Taking the trace of (78) by $\tilde{g}$ and then using (62) and (80), we get the following expression for $\tilde{\tau }$:

$$\begin{array}{c}\hfill \tilde{\tau }=-2n\left\{f+\tilde{\lambda }+\left(1-4n\tilde{\mathsf{\ell }}\right)\mathrm{d}s\left(\xi \right)+\left[1-2n(2n+1)\tilde{\mathsf{\ell }}\right]s^2\right\}.\end{array}$$

(81)

Then, comparing (81) and (62), we obtain the following condition for the sum of f and $\tilde{\lambda }$ in terms of s:

$$\begin{array}{c}\hfill f+\tilde{\lambda }=\left(1+4n\tilde{\mathsf{\ell }}\right)\mathrm{d}s\left(\xi \right)+2n\left[1+(2n+1)\tilde{\mathsf{\ell }}\right]s^2\end{array}$$

(82)

or replacing s by (38), we rewrite (82) in terms of f and k as in (68).

Excluding $\tilde{\lambda }$ from (80) and (82), we get that $\tilde{\mu }=\mu$ is true, since (77) holds, i.e., we have the expression for $\tilde{\mu }$ in (69).

Subtracting (76) and (82), we get $\tilde{\lambda }=\lambda +2n\left(\tilde{\mathsf{\ell }}-\mathsf{\ell }\right)\left\{2\mathrm{d}s\left(\xi \right)+(2n+1)s^2\right\}.$ Therefore, the equality $\tilde{\lambda }=\lambda$ is true if and only if either $\tilde{\mathsf{\ell }}=\mathsf{\ell }$ or $\mathrm{d}s\left(\xi \right)=-{\displaystyle \frac{1}{2}}(2n+1)s^2$ is valid. One solution to the last ODE is, for example, $s={\displaystyle \frac{2}{2n+1}}{t}^{-1}$, where t is the coordinate along $\xi$.

Using the analogue of (77) for $\tilde{\mu }$, (82) and (62) in (78), we obtain (71).

Due to (17), (18), (43), (72), and (78), the last statement in the proposition is true precisely when the condition $\mathrm{d}k=\mathrm{d}k\left(\xi \right)\eta$ holds, which is equivalent to the horizontal constancy of k. □

4. Conclusions

The accR manifolds equipped with a pair of associated $\eta$-RB almost solitons with a double torse-forming vertical potential are studied. Specifically, the Ricci tensors and scalar curvatures with respect to the two associated B-metrics of the considered accR manifolds are studied. These curvature characteristics are expressed as functions of the parameters of the pair of almost solitons. The case when the considered manifolds belong to the only possible basic class of the corresponding classification has been specially studied. Then, all the parameters that determine the pair of almost solitons, and their Ricci tensors are expressed in terms of the conformal scalar f of the potential $\vartheta$ and $k=\eta \left(\vartheta \right)$. It is proven that these almost solitons are $\eta$-Einstein for both metrics if and only if k is a horizontal constant.