Ricci-like Solitons with Arbitrary Potential and Gradient Almost Ricci-like Solitons on Sasaki-like Almost Contact B-metric Manifolds

Abstract

Ricci-like solitons with arbitrary potential are introduced and studied on Sasaki-like almost contact B-metric manifolds. A manifold of this type can be considered as an almost contact complex Riemannian manifold which complex cone is a holomorphic complex Riemannian manifold. The soliton under study is characterized and proved that its Ricci tensor is equal to the vertical component of both B-metrics multiplied by a constant. Thus, the scalar curvatures with respect to both B-metrics are equal and constant. In the 3-dimensional case, it is found that the special sectional curvatures with respect to the structure are constant. Gradient almost Ricci-like solitons on Sasaki-like almost contact B-metric manifolds have been proved to have constant soliton coefficients. Explicit examples are provided of Lie groups as manifolds of dimensions 3 and 5 equipped with the structures under study.

1 Introduction

Ricci soliton is a special self-similar solution of the Hamilton’s Ricci flow and it is a natural generalization of the notion of Einstein metric. According to [19], a (pseudo-)Riemannian manifold admits a Ricci soliton if the metric, its Ricci tensor and the Lie derivative of the metric along a vector field (called potential) are linear dependent. If the coefficients of this dependence are functions then the soliton is called an almost Ricci soliton [32]. If a function exists so that the potential is its gradient, then the (almost) Ricci soliton is called a gradient (almost) Ricci soliton (see, e.g., [14, 15, 35]).

The topic became more popular after Perelman’s proof of the Poincaré conjecture, following Hamilton’s program to use the Ricci flow (see [31]). Ricci solitons have been explored by a number of authors (see, e.g., [2, 3, 9,10,11, 17, 21, 30, 33]).

Ricci solitons are also of interest to physicists, and in physical literature are called quasi-Einstein (see, e.g., [12, 16]).

The presence of the structure 1-form $\eta$ on manifolds with almost contact or almost paracontact structure motivates the need to introduce so-called $\eta$-Ricci solitons. Then, $\eta \otimes \eta$ is the restriction of the metric on the orthogonal complement to the (para)contact distribution, determined by the structure vector field $\xi$. By adding a term proportional to $\eta \otimes \eta$ into the defining equality of a Ricci soliton, it is defined the notion of $\eta$-Ricci soliton, introduced in [13]. Later, it has been studied on almost contact and almost paracontact manifolds by many authors (e.g., [1, 4, 5, 8, 34]). For almost $\eta$-Ricci solitons, see, for example [6, 7].

Our global goal is to study the differential geometry of almost contact B-metric manifolds investigated since 1993 [18, 29].

Unlike almost contact metric manifolds, almost contact B-metric manifolds have two metrics that are mutually associated with structural endomorphism. The restrictions of both B-metrics on the orthogonal distribution to the contact distribution is $\eta \otimes \eta$. This is the reason for introducing in [27] a further generalization of the notions of a Ricci soliton and an $\eta$-Ricci soliton, the so-called Ricci-like soliton, using both B-metrics and $\eta \otimes \eta$. There, we have explored these objects with potential Reeb vector field on some important kinds of manifolds under consideration: Einstein-like, Sasaki-like and having a torse-forming Reeb vector field. In [28], we continue to study Ricci-like solitons, whose potential is the Reeb vector field or pointwise collinear to it.

In the present paper, our goal is to investigate Ricci-like solitons with arbitrary potential on almost contact B-metric manifolds of Sasaki-like type, as well as gradient almost Ricci-like solitons on these manifolds.

The paper is organized as follows. In Sect. 1, we recall basic definitions and properties of almost contact B-metric manifolds of Sasaki-like type and obtain several immediate consequences. Section 2 includes some necessary results and a 5-dimensional example for a Ricci-like soliton with a potential Reeb vector field. In Sect. 3, we study Ricci-like solitons with an arbitrary potential. Then, we prove an identity for the soliton constants and a property of the potential, as well as that the Ricci tensor is a constant multiple of $\eta \otimes \eta$. For the 3-dimensional case, we find the values of the sectional curvatures of the special 2-planes with respect to the structure and construct an explicit example. In Sect. 4, we introduce gradient almost Ricci-like solitons on Sasaki-like manifolds and prove that their Ricci tensor has the same form as in the previous section. For the example in Sect. 3, we find a potential function to illustrate the obtained results.

2 Sasaki-like Almost Contact B-metric Manifolds

A differentiable manifold M of dimension $(2n+1)$, equipped with an almost contact structure $(\varphi ,\xi ,\eta )$ and a B-metric g is called an almost contact B-metric manifold and it is denoted by $(M,{}\varphi ,{}\xi ,{}\eta ,{}{}g)$. More concretely, $\varphi$ is an endomorphism of the tangent bundle TM, $\xi$ is a Reeb vector field, $\eta$ is its dual contact 1-form and g is a pseudo-Riemannian metric of signature $(n+1,n)$ satisfying the following conditions [18]

$$\begin{aligned} \varphi \xi= & {} 0,\qquad \varphi ^2 = -\iota + \eta \otimes \xi ,\qquad \eta \circ \varphi =0,\qquad \eta (\xi )=1,\nonumber \\ g(\varphi x, \varphi y)= & {} - g(x,y) + \eta (x)\eta (y), \end{aligned}$$

(1.1)

where $\iota$ stands for the identity transformation on $\Gamma (TM)$.

In the latter equality and further, x, y, z, w will stand for arbitrary elements of $\Gamma (TM)$ or vectors in the tangent space $T_pM$ of M at an arbitrary point p in M.

The following equations are immediate consequences of (1.1)

$$\begin{aligned} \begin{array}{ll} g(\varphi x, y) = g(x,\varphi y),\qquad &{}g(x, \xi ) = \eta (x), \\ g(\xi , \xi ) = 1,\qquad &{}\eta (\nabla _x \xi ) = 0, \end{array} \end{aligned}$$

(1.2)

where $\nabla$ denotes the Levi-Civita connection of g.

The associated metric ${\tilde{g}}$ of g on M is also a B-metric and it is defined by

$$\begin{aligned} {\tilde{g}}(x,y)=g(x,\varphi y)+\eta (x)\eta (y). \end{aligned}$$

In [18], almost contact B-metric manifolds (also known as almost contact complex Riemannian manifolds) are classified with respect to the (0,3)-tensor F defined by

$$\begin{aligned} F(x,y,z)=g\bigl ( \left( \nabla _x \varphi \right) y,z\bigr ). \end{aligned}$$

It has the following basic properties:

$$\begin{aligned} F(x,y,z)= & {} F(x,z,y) =F(x,\varphi y,\varphi z)+\eta (y)F(x,\xi ,z) +\eta (z)F(x,y,\xi ),\\ F(x,\varphi y, \xi )= & {} (\nabla _x\eta )y=g(\nabla _x\xi ,y). \end{aligned}$$

This classification consists of eleven basic classes $\mathcal {F}_i$, $i\in \{1,2,\dots ,11\}$.

In [20], it is introduced the type of a Sasaki-like manifold among almost contact B-metric manifolds. The definition condition is its complex cone to be a Kähler-Norden manifold, i.e. with a parallel complex structure. A Sasaki-like manifold with almost contact B-metric structure is determined by the condition

$$\begin{aligned} \begin{array}{l} \left( \nabla _x\varphi \right) y=-g(x,y)\xi -\eta (y)x+2\eta (x)\eta (y)\xi , \end{array} \end{aligned}$$

(1.3)

which is equivalent to the following $\left( \nabla _x\varphi \right) y=g(\varphi x,\varphi y)\xi +\eta (y)\varphi ^2 x$.

Obviously, Sasaki-like manifolds form a subclass of the class $\mathcal {F}_4$. Moreover, the following identities are valid for it [20]

$$\begin{aligned} \begin{array}{ll} \nabla _x \xi =-\varphi x, \qquad \qquad &{}\left( \nabla _x \eta \right) (y)=-g(x,\varphi y),\\ R(x,y)\xi =\eta (y)x-\eta (x)y, \qquad \qquad &{}\rho (x,\xi )=2n\, \eta (x), \\ R(\xi ,y)z=g(y,z)\xi -\eta (z)y,\qquad \qquad &{}\rho (\xi ,\xi )=2n, \end{array} \end{aligned}$$

(1.4)

where R and $\rho$ stand for the curvature tensor and the Ricci tensor of $\nabla$.

The corresponding curvature tensor of type (0, 4) is determined as usually by $R(x,y,z,w)=g(R(x,y)z,w)$.

Further, we use an arbitrary basis $\left\{ e_i\right\}$, $i\in \{1,2,\dots ,2n+1\}$ of $T_pM$, $p\in M$.

On an arbitrary almost contact B-metric manifold, there exists a (0, 2)-tensor $\rho ^*$, which is associated with $\rho$ regarding $\varphi$. It is defined by $\rho ^*(y,z)=g^{ij}R(e_i,y,z,\varphi e_j)$ and due to the first equality in (1.2) $\rho ^*$ is symmetric.

The following relation between $\rho ^*$ and $\rho$ is valid for a Sasaki-like manifold

$$\begin{aligned} \rho ^*(y,z)=\rho (y,\varphi z)+(2n-1)g(y,\varphi z). \end{aligned}$$

(1.5)

It follows by taking the trace for $x=e_i$ and $w=e_j$ the following property of a Sasaki-like manifold [20]

$$\begin{aligned}&R(x,y,\varphi z,w)-R(x,y,z,\varphi w)\nonumber \\&\quad =\left\{ g(y,z)-2\eta (y)\eta (z)\right\} g(x,\varphi w) +\left\{ g(y,w)-2\eta (y)\eta (w)\right\} g(x,\varphi z)\nonumber \\&\qquad -\left\{ g(x,z)-2\eta (x)\eta (z)\right\} g(y,\varphi w)-\left\{ g(x,w)-2\eta (x)\eta (w)\right\} g(y,\varphi z).\nonumber \\ \end{aligned}$$

(1.6)

As a corollary of (1.5) we have that $\rho (y,\varphi z)=\rho (\varphi y,z)$, i.e. $Q\circ \varphi =\varphi \circ Q$, where Q is the Ricci operator, i.e. $\rho (y,z)=g(Qy,z)$.

The scalar curvature ${{\tilde{\tau }}}$ of ${\tilde{g}}$ is defined by ${{\tilde{\tau }}}={\tilde{g}}^{ij}{\tilde{g}}^{kl}{\tilde{R}}(e_i,e_k,e_l,e_j)$, where ${\tilde{R}}$ is the curvature tensor of ${\tilde{g}}$ and ${\tilde{g}}^{ij}=-\varphi ^j_k g^{ik} + \xi ^i\xi ^j$ holds. In addition, the associated quantity $\tau ^*$ of $\tau$ with respect to $\varphi$ is determined by $\tau ^*=g^{ij}\rho (e_i,\varphi e_j)$. For them, using (1.6) for a Sasaki-like manifold, we infer the following relation

$$\begin{aligned} {{\tilde{\tau }}}=-\tau ^*+2n. \end{aligned}$$

(1.7)

In [26], it is given the following relations between $\tau$ and ${{\tilde{\tau }}}$

$$\begin{aligned} \mathrm {d}\tau \circ \varphi =\mathrm {d}{{\tilde{\tau }}}+2(\tau -2n)\eta ,\qquad \mathrm {d}{{\tilde{\tau }}}\circ \varphi =-\mathrm {d}\tau +2({{\tilde{\tau }}}-2n)\eta . \end{aligned}$$

As corollaries we have

$$\begin{aligned} \mathrm {d}\tau \circ \varphi ^2= & {} \mathrm {d}{{\tilde{\tau }}}\circ \varphi ,\qquad \mathrm {d}{{\tilde{\tau }}}\circ \varphi ^2=-\mathrm {d}\tau \circ \varphi , \end{aligned}$$

(1.8)

$$\begin{aligned} \mathrm {d}\tau (\xi )= & {} 2({{\tilde{\tau }}}-2n),\qquad \mathrm {d}{{\tilde{\tau }}}(\xi )=-2(\tau -2n). \end{aligned}$$

(1.9)

Proposition 1.1

On a Sasaki-like manifold $(M,{}\varphi ,{}\xi ,{}\eta ,{}{}g)$ of dimension $2n+1$, the following formulae for the Ricci operator Q are valid

$$\begin{aligned} (\nabla _x Q)\xi= & {} Q\varphi x-2n\,\varphi x, \end{aligned}$$

(1.10)

$$\begin{aligned} (\nabla _\xi Q)y= & {} 2Q\varphi y. \end{aligned}$$

(1.11)

Proof

For a Sasaki-like manifold, according to (1.4), the equalities $Q\xi =2n\,\xi$ and $\nabla _x \xi =-\varphi x$ holds. Using them, we obtain immediately the covariant derivative in (1.10).

Now, we apply $\nabla _z$ to the expression of $R(x,y)\xi$ in (1.4) and then, using the form of $\nabla \eta$ in (1.4), we get the following

$$\begin{aligned} \left( \nabla _z R\right) (x,y)\xi =R(x,y)\varphi z - g(y,\varphi z)x +g(x,\varphi z)y. \end{aligned}$$

We take the trace of the above equality for $z=e_i$ and $x=e_j$ and use (1.5) to obtain

$$\begin{aligned} g^{ij} (\nabla _{e_i} R)(e_j,y)\xi =-Q\varphi y - 2n \varphi y. \end{aligned}$$

By virtue of the following consequence the second Bianchi identity

$$\begin{aligned} g^{ij} (\nabla _{e_i} R)(\xi ,y)e_j=\left( \nabla _y Q\right) \xi -\left( \nabla _\xi Q\right) y, \end{aligned}$$

the symmetries of R and (1.10), we get (1.11). $\square$

As consequences of (1.10) and (1.11) we obtain respectively

$$\begin{aligned} \eta \bigl ((\nabla _x Q)\xi \bigr )=0,\qquad \eta \bigl ((\nabla _\xi Q)y\bigr )=0. \end{aligned}$$

(1.12)

Let us recall [27], an almost contact B-metric manifold $(M,{}\varphi ,{}\xi ,{}\eta ,{}{}g)$ is said to be Einstein-like if its Ricci tensor $\rho$ satisfies

$$\begin{aligned} \begin{array}{l} \rho =a\,g +b\,{\tilde{g}}+c\,\eta \otimes \eta \end{array} \end{aligned}$$

(1.13)

for some triplet of constants (a, b, c). In particular, when $b=0$ and $b=c=0$, the manifold is called an $\eta$-Einstein manifold and an Einstein manifold, respectively.

If a, b, c are functions on M, then the manifold is called almost Einstein-like, almost $\eta$-Einstein and almost Einstein, respectively.

Tracing (1.13) and using (1.7), the scalar curvatures $\tau$ and ${{\tilde{\tau }}}$ of an Einstein-like almost contact B-metric manifold have the form

$$\begin{aligned} \tau =(2n+1)a+b+c,\qquad {{\tilde{\tau }}}=2n(b+1). \end{aligned}$$

(1.14)

For a Sasaki-like manifold $(M,{}\varphi ,{}\xi ,{}\eta ,{}{}g)$ with $\dim M = 2n+1$ and a scalar curvature $\tau$ regarding g, which is Einstein-like with a triplet of constants (a, b, c), the following equalities are given in [27]:

$$\begin{aligned} a + b + c = 2n, \qquad \tau = 2n(a + 1). \end{aligned}$$

(1.15)

Then, for ${{\tilde{\tau }}}$ on an Einstein-like Sasaki-like manifold we obtain

$$\begin{aligned} {{\tilde{\tau }}}=2n(b+1) \end{aligned}$$

(1.16)

and because (1.14)–(1.16), the expression (1.13) becomes

$$\begin{aligned} \begin{array}{l} \rho =\left( \dfrac{\tau }{2n}-1\right) g +\left( \dfrac{{{\tilde{\tau }}}}{2n}-1\right) {\tilde{g}}+\left( 2(n+1)-\dfrac{\tau +{{\tilde{\tau }}}}{2n}\right) \eta \otimes \eta . \end{array} \end{aligned}$$

Proposition 1.2

Let $(M,{}\varphi ,{}\xi ,{}\eta ,{}{}g)$ be a $(2n+1)$-dimensional Sasaki-like manifold. If it is almost Einstein-like with functions (a, b, c) then the scalar curvatures $\tau$ and ${{\tilde{\tau }}}$ of g and ${\tilde{g}}$, respectively, are constants

$$\begin{aligned} \tau = const, \qquad {{\tilde{\tau }}}=2n \end{aligned}$$

and $(M,{}\varphi ,{}\xi ,{}\eta ,{}{}g)$ is $\eta$-Einstein with constants

$$\begin{aligned} (a,b,c)=\left( \frac{\tau }{2n}-1,\,0,\,2n+1-\frac{\tau }{2n}\right) . \end{aligned}$$

Proof

If $(M,{}\varphi ,{}\xi ,{}\eta ,{}{}g)$ is almost Einstein-like then $\rho$ has the form in (1.13), where (a, b, c) are a triad of functions. Then, according to (1.7), (1.13), (1.14) and the expression for $\rho (\xi ,\xi )$ on a Sasaki-like manifold, given in (1.4), we have the following

$$\begin{aligned} a+b+c=2n,\qquad \tau =2n(a+1),\qquad {{\tilde{\tau }}}=2n(b+1). \end{aligned}$$

(1.17)

Using (1.4), we can express $R(x,y)\xi$ and $R(x,\xi )y$ as follows

$$\begin{aligned} R(x,y)\xi= & {} \dfrac{1}{4n^2}\bigl \{2n\left[ \eta (x) Qy - \eta (y) Qx + \rho (x,\xi )y - \rho (y,\xi )x \right] \\&+ (\tau -2n)\left[ \eta (y)x -\eta (x)y\right] + ({{\tilde{\tau }}}-2n)\left[ \eta (y)\varphi x -\eta (x)\varphi y\right] \bigr \},\\ R(x,\xi )y= & {} \dfrac{1}{4n^2}\bigl \{2n\left[ \rho (x,y)\xi + g(x,y)Q\xi - \rho (y,\xi )x - \eta (y)Qx \right] \\&+ (\tau -2n)\left[ \eta (y)x -g(x,y)\xi \right] \\&+ ({{\tilde{\tau }}}-2n)\left[ \eta (y)\varphi x -g(x,\varphi y)\xi \right] \bigr \}. \end{aligned}$$

Then, for $y=\xi$ in either of the last two equalities, we have

$$\begin{aligned} R(x,\xi )\xi = \eta (x)\xi -\dfrac{1}{2n} Qx -\dfrac{1}{4n^2}\bigl \{[\tau -2n(2n+1)]\varphi ^2 x - [{{\tilde{\tau }}}-2n]\varphi x \bigr \}. \end{aligned}$$

After that, we compute the covariant derivative of $R(x,\xi )\xi$ with respect to $\nabla _z$. Since (1.3) and (1.4), we obtain

$$\begin{aligned} \left( \nabla _z R\right) (x,\xi )\xi= & {} -\dfrac{1}{2n} \left\{ (\nabla _z Q)x-\eta (x)Q\varphi z\right\} -\dfrac{1}{4n^2}\bigl \{\mathrm {d}\tau (z)\varphi ^2 x - \mathrm {d}{{\tilde{\tau }}}(z)\varphi x \\&-[\tau -2n(2n+1)]g(x, \varphi z)\xi +[{{\tilde{\tau }}}-2n]g(\varphi x, \varphi z)\xi \bigr \} - \eta (x)\varphi z, \end{aligned}$$

which by taking the trace for $z=e_i$ and $x=e_j$ and (1.8) gives the following

$$\begin{aligned} g^{ij}g\bigl (\left( \nabla _{e_i} R\right) (e_j,\xi )\xi ,y\bigr ) = -\dfrac{1}{4n}\mathrm {d}\tau (y) -\left\{ \dfrac{{{\tilde{\tau }}}}{2n}-1\right\} \eta (y). \end{aligned}$$

(1.18)

By virtue of the following consequence of the second Bianchi identity

$$\begin{aligned} g^{ij} g\bigl ((\nabla _{e_i} R)(y,\xi )\xi ,e_j\bigr ) = \eta \bigl (\left( \nabla _y Q\right) \xi \bigr )-\eta \bigl (\left( \nabla _\xi Q\right) y\bigr ) \end{aligned}$$

(1.19)

and (1.12), we have that the trace in the left hand side of (1.19) vanishes. Then, (1.18) and (1.19) imply

$$\begin{aligned} \mathrm {d}\tau (y) = -2\{{{\tilde{\tau }}}-2n\}\eta (y), \end{aligned}$$

which comparing with (1.9) implies

$$\begin{aligned} \mathrm {d}\tau (\xi )=0,\qquad {{\tilde{\tau }}}=2n. \end{aligned}$$

The latter equalities together with (1.8) and (1.17) complete the proof. $\square$

3 Ricci-like Solitons with Potential Reeb Vector Field on Sasaki-like Manifolds

In [27], by a condition for Ricci tensor, it is introduced the notion of a Ricci-like soliton with potential $\xi$ on an almost contact B-metric manifold.

Now, we generalize this notion for a potential, which is an arbitrary vector field as follows. We say that $(M,{}\varphi ,{}\xi ,{}\eta ,{}{}g)$ admits a Ricci-like soliton with potential vector field v if the following condition is satisfied for a triplet of constants $(\lambda ,\mu ,\nu )$

$$\begin{aligned} \frac{1}{2} \mathcal {L}_{v} g + \rho + \lambda \, g + \mu \, {\tilde{g}}+ \nu \, \eta \otimes \eta =0, \end{aligned}$$

(2.1)

where $\mathcal {L}$ denotes the Lie derivative.

If $\mu =0$ (respectively, $\mu =\nu =0$), then (2.1) defines an $\eta$-Ricci soliton (respectively, a Ricci soliton) on $(M,{}\varphi ,{}\xi ,{}\eta ,{}{}g)$.

If $\lambda$, $\mu$, $\nu$ are functions on M, then the soliton is called almost Ricci-like soliton, almost $\eta$-Ricci soliton and almost Ricci soliton, respectively.

If $(M,{}\varphi ,{}\xi ,{}\eta ,{}{}g)$ is Sasaki-like, we have

$$\begin{aligned} \left( \mathcal {L}_{\xi } g\right) (x,y)=g(\nabla _x\xi ,y)+g(x,\nabla _y\xi )=-2g(x,\varphi y), \end{aligned}$$

i.e. $\frac{1}{2} \mathcal {L}_{\xi } g=-{\tilde{g}}+\eta \otimes \eta$. Then, because of (2.1), $\rho$ takes the form

$$\begin{aligned} \rho = -\lambda g + (1-\mu ) {\tilde{g}}- (1+\nu ) \eta \otimes \eta . \end{aligned}$$

Theorem 2.1

([27]). Let $(M,{}\varphi ,{}\xi ,{}\eta ,{}{}g)$ be a $(2n+1)$-dimensional Sasaki-like manifold and let a, b, c, $\lambda$, $\mu$, $\nu$ be constants that satisfy the following equalities:

$$\begin{aligned} a+\lambda =0,\qquad b+\mu -1=0,\qquad c+\nu +1=0. \end{aligned}$$

Then, the manifold admits a Ricci-like soliton with potential $\xi$ and constants $(\lambda ,{}\mu ,{}\nu )$, where $\lambda +\mu +\nu =-2n$, if and only if it is Einstein-like with constants (a, b, c), where $a+b+c=2n$.

In particular, we get:

• (i) The manifold admits an $\eta$-Ricci soliton with potential $\xi$ and constants $(\lambda ,0,-2n-\lambda )$ if and only if the manifold is Einstein-like with constants $(-\lambda ,1,\lambda +2n-1)$.

• (ii) The manifold admits a shrinking Ricci soliton with potential $\xi$ and constant $-2n$ if and only if the manifold is Einstein-like with constants $(2n,1,-1)$.

• (iii) The manifold is $\eta$-Einstein with constants $(a,0,2n-a)$ if and only if it admits a Ricci-like soliton with potential $\xi$ and constants $(-a,1,a-2n-1)$.

• (iv) The manifold is Einstein with constant 2n if and only if it admits a Ricci-like soliton with potential $\xi$ and constants $(-2n,1,-1)$.

3.1 Example 1

In Example 2 of [20], it is given a Lie group G of dimension 5 (i.e. $n=2$) with a basis of left-invariant vector fields $\{e_0,\dots , e_{4}\}$ and the corresponding Lie algebra is defined as follows

$$\begin{aligned} \begin{array}{ll} {[}e_0,e_1] = p e_2 + e_3 + q e_4,\quad &{}[e_0,e_2] = - p e_1 - q e_3 + e_4,\\ {[}e_0,e_3] = - e_1 - q e_2 + p e_4,\quad &{}[e_0,e_4] = q e_1 - e_2 - p e_3,\qquad p,q\in \mathbb R. \end{array} \end{aligned}$$

After that G is equipped with an almost contact B-metric structure defined by

$$\begin{aligned} g(e_0,e_0)= & {} g(e_1,e_1)=g(e_2,e_2)=-g(e_{3},e_{3})=-g(e_{4},e_{4})=1,\\ g(e_i,e_j)= & {} 0,\quad i,j\in \{0,1,\dots ,4\},\; i\ne j,\\ \xi= & {} e_0, \quad \varphi e_1=e_{3},\quad \varphi e_2=e_{4},\quad \varphi e_3=-e_{1},\quad \varphi e_4=-e_{2}. \end{aligned}$$

It is verified that the constructed almost contact B-metric manifold $(G,\varphi ,{}\xi ,{}\eta ,{}g)$ is Sasaki-like.

In [27], it is proved that $(G,\varphi ,{}\xi ,{}\eta ,{}g)$ is $\eta$-Einstein with constants

$$\begin{aligned} (a,b,c)=(0,0,4). \end{aligned}$$

(2.2)

Moreover, it is clear that $\tau ={{\tilde{\tau }}}=4$.

It is also found there that $(G,\varphi ,{}\xi ,{}\eta ,{}g)$ admits a Ricci-like soliton with potential $\xi$ and constants

$$\begin{aligned} (\lambda ,\mu ,\nu )=(0,1,-5). \end{aligned}$$

(2.3)

Therefore, this example is in unison with Theorem 2.1 (iii) for $a=0$.

4 Ricci-like Solitons with Arbitrary Potential on Sasaki-like Manifolds

Theorem 3.1

Let $(M,{}\varphi ,{}\xi ,{}\eta ,{}{}g)$ be a $(2n+1)$-dimensional Sasaki-like manifold. If it admits a Ricci-like soliton with arbitrary potential vector field v and constants $(\lambda ,\mu ,\nu )$ then it is valid the following identities

$$\begin{aligned} \lambda +\mu +\nu= & {} -2n,\nonumber \\ \nabla _\xi v= & {} -\varphi v. \end{aligned}$$

(3.1)

Proof

According to (2.1), a Ricci-like soliton with arbitrary potential vector field v is defined by $\bigl (\mathcal {L}_v g\bigr )(y,z) = -2 \rho (y,z) - 2 \lambda \,g(y,z) - 2 \mu \,{\tilde{g}}(y,z) -2 \nu \,\eta (y)\eta (z).$ Then, bearing in mind (1.4), the covariant derivative with respect to $\nabla _x$ has the form

$$\begin{aligned} \bigl (\nabla _x \mathcal {L}_v g\bigr )(y,z)= & {} -\,2 \bigl (\nabla _x \rho \bigr )(y,z) -2\mu \{g(\varphi x,\varphi y)\eta (z)+g(\varphi x,\varphi z)\eta (y)\}\nonumber \\&+2(\mu +\nu )\{g( x,\varphi y)\eta (z)+g( x,\varphi z)\eta (y)\}. \end{aligned}$$

(3.2)

We use of the following formula from [36] for a metric connection $\nabla$

$$\begin{aligned} \bigl (\nabla _x \mathcal {L}_v g\bigr )(y,z) = g\left( (\mathcal {L}_v \nabla )(x,y),z\right) + g\left( (\mathcal {L}_v \nabla )(x,z),y \right) , \end{aligned}$$

which due to symmetry of $\mathcal {L}_v \nabla$ can read as

$$\begin{aligned} 2g\bigl ((\mathcal {L}_v \nabla )(x,y),z\bigr ) = \bigl (\nabla _x \mathcal {L}_v g\bigr )(y,z) + \bigl (\nabla _y \mathcal {L}_v g\bigr )(z,x) - \bigl (\nabla _z \mathcal {L}_v g\bigr )(x,y). \end{aligned}$$

(3.3)

Applying (3.3) to (3.2), we obtain

$$\begin{aligned} g\bigl ((\mathcal {L}_v \nabla )(x,y),z\bigr )= & {} - \bigl (\nabla _x \rho \bigr )(y,z)- \bigl (\nabla _y \rho \bigr )(z,x) + \bigl (\nabla _z \rho \bigr )(x,y) \nonumber \\&-2\mu \,g(\varphi x,\varphi y)\eta (z)+2(\mu +\nu )g( x,\varphi y)\eta (z). \end{aligned}$$

(3.4)

Setting $y=\xi$ in the equality above and using (1.10) and (1.11), we get

$$\begin{aligned} (\mathcal {L}_v \nabla )(x,\xi ) = -2 Q\varphi x. \end{aligned}$$

(3.5)

The covariant derivative of the above equation by using of (1.4) has the form

$$\begin{aligned} \bigl (\nabla _y \mathcal {L}_v \nabla \bigr )(x,\xi )= & {} (\mathcal {L}_v \nabla )(x,\varphi y) -2(\nabla _y Q)\varphi x +2\eta (x) Qy\nonumber \\&-4n\,g(x,y)-2(2n+1)\eta (x)\eta (y)\xi . \end{aligned}$$

(3.6)

We apply the latter equality to the following formula from [36]

$$\begin{aligned} (\mathcal {L}_v R)(x,y)z = \bigl (\nabla _x \mathcal {L}_v \nabla \bigr )(y,z) - \bigl (\nabla _y \mathcal {L}_v \nabla \bigr )(x,z) \end{aligned}$$

(3.7)

and owing to symmetry of $\mathcal {L}_v \nabla$, we obtain the following consequence of (3.5)–(3.7)

$$\begin{aligned} g\bigl ((\mathcal {L}_v R)(x,y)\xi ,z\bigr )= & {} -\,(\nabla _x \rho )(\varphi y,z)+(\nabla _{\varphi y} \rho )(x,z) -(\nabla _z \rho )(x,\varphi y)\nonumber \\&+(\nabla _y \rho )(\varphi x,z)-(\nabla _{\varphi x} \rho )(y,z) +(\nabla _z \rho )(\varphi x, y)\nonumber \\&-2\,\eta (x)\rho (y,z)+2\,\eta (y)\rho (x,z). \end{aligned}$$

(3.8)

Plugging $y=z=\xi$ in (3.8) and using (3.5), we obtain

$$\begin{aligned} (\mathcal {L}_v R)(x,\xi )\xi = 0. \end{aligned}$$

(3.9)

On the other hand, applying $\mathcal {L}_v$ to the expression of $R(x,\xi )\xi$ from (1.4) and using (2.1), as well as the formulae for $R(x,y)\xi$ and $R(\xi ,y)z$ from the same referent equalities, we get

$$\begin{aligned} (\mathcal {L}_v R)(x,\xi )\xi = (\mathcal {L}_v \eta )(x)\xi +g(x,\mathcal {L}_v \xi )\xi -2 \eta (\mathcal {L}_v \xi )x \end{aligned}$$

or in an equivalent form

$$\begin{aligned} (\mathcal {L}_v R)(x,\xi )\xi = \{(\mathcal {L}_v \eta )(x)+g(x,\mathcal {L}_v \xi )-2 \eta (\mathcal {L}_v \xi )\eta (x)\}\xi +2 \eta (\mathcal {L}_v \xi )\varphi ^2x. \end{aligned}$$

(3.10)

Comparing (3.9) and (3.10), we obtain the following system of equations

$$\begin{aligned} (\mathcal {L}_v \eta )(x)+g(x,\mathcal {L}_v \xi )-2 \eta (\mathcal {L}_v \xi )\eta (x)=0,\qquad \eta (\mathcal {L}_v \xi )=0, \end{aligned}$$

i.e.

$$\begin{aligned} (\mathcal {L}_v \eta )(x)+g(x,\mathcal {L}_v \xi )=0,\qquad \eta (\mathcal {L}_v \xi )=0. \end{aligned}$$

(3.11)

According to (2.1) and $\rho (x,\xi )=2n\eta (x)$ from (1.4), we have for a Sasaki-like manifold

$$\begin{aligned} (\mathcal {L}_v g)(x,\xi )=-2 (\lambda +\mu +\nu +2n)\eta (x) \end{aligned}$$

(3.12)

and as a consequence for $x=\xi$ the following

$$\begin{aligned} (\mathcal {L}_v g)(\xi ,\xi )=-2 (\lambda +\mu +\nu +2n). \end{aligned}$$

(3.13)

The Lie derivative of $g(x,\xi )=\eta (x)$ with respect to v gives

$$\begin{aligned} (\mathcal {L}_v g)(x,\xi )=(\mathcal {L}_v \eta )(x)-g\left( x,\mathcal {L}_v \xi \right) , \end{aligned}$$

(3.14)

which for $x=\xi$ leads to

$$\begin{aligned} (\mathcal {L}_v g)(\xi ,\xi )=-2\eta \left( \mathcal {L}_v \xi \right) . \end{aligned}$$

(3.15)

From (3.13) and (3.15) we obtain

$$\begin{aligned} \eta (\mathcal {L}_v \xi )=\lambda +\mu +\nu +2n. \end{aligned}$$

The latter equality implies (3.1), by virtue of the second equality in (3.11).

Substituting (3.1) in (3.12) gives the vanishing of $(\mathcal {L}_v g)(x,\xi )$ and because of (3.14) we have $(\mathcal {L}_v \eta )(x)=g\left( x,\mathcal {L}_v \xi \right)$. Hence, bearing in mind the first equality in (3.11), we get

$$\begin{aligned} \mathcal {L}_v \xi = 0, \end{aligned}$$

which together with $\nabla \xi =-\varphi$ from (1.4) completes the proof. $\square$

Proposition 3.2

Let $(M,{}\varphi ,{}\xi ,{}\eta ,{}{}g)$ be a $(2n+1)$-dimensional Sasaki-like manifold. If it admits a Ricci-like soliton with arbitrary potential v then the Ricci tensor $\rho$ of g and the scalar curvatures $\tau$ and ${{\tilde{\tau }}}$ of g and ${\tilde{g}}$, respectively, satisfy the following equalities

$$\begin{aligned} \left( \mathcal {L}_v \rho \right) (x,\xi ) = 0, \qquad \tau =2n, \qquad {{\tilde{\tau }}}=const. \end{aligned}$$

Proof

By (3.7) we find the following

$$\begin{aligned} g\bigl (\left( \mathcal {L}_v R\right) (x,y)\xi ,z\bigr )= & {} -\, g\bigl (\left( \mathcal {L}_v \nabla \right) (x,\varphi y),z\bigr ) + g\bigl (\left( \mathcal {L}_v \nabla \right) (\varphi x, y),z\bigr )\\&- 2 \left( \nabla _x \rho \right) (\varphi y, z) + 2 \left( \nabla _y \rho \right) (\varphi x, z)\\&- 2 \eta (x)\rho (y,z) + 2 \eta (y)\rho (x,z). \end{aligned}$$

Taking the trace of the last equality for $x=e_i$ and $z=e_j$ and using (3.4) and (1.7), $\mathrm {d}\tau =2{{\,\mathrm{div}\,}}\rho$, we obtain successively

$$\begin{aligned} g^{ij}g\bigl (\left( \mathcal {L}_v \nabla \right) (e_i,\varphi y),e_j\bigr )= & {} - \mathrm {d}\tau (\varphi y),\nonumber \\ g^{ij}g\bigl (\left( \mathcal {L}_v \nabla \right) (\varphi e_i, y),e_j\bigr )= & {} \mathrm {d}{{\tilde{\tau }}}(y),\nonumber \\ g^{ij}g\bigl (\left( \mathcal {L}_v R\right) (e_i,y)\xi ,e_j\bigr )= & {} \left( \mathcal {L}_v \rho \right) (y,\xi ) \end{aligned}$$

(3.16)

and therefore the following formula is valid

$$\begin{aligned} \left( \mathcal {L}_v \rho \right) (y,\xi ) = -\, \mathrm {d}{{\tilde{\tau }}}(y)+2(\tau -2n)\eta (y), \end{aligned}$$

(3.17)

which for $y=\xi$ implies

$$\begin{aligned} \left( \mathcal {L}_v \rho \right) (\xi ,\xi ) = -\, \mathrm {d}{{\tilde{\tau }}}(\xi )+2(\tau -2n). \end{aligned}$$

(3.18)

On the other hand, according to (3.9) and (3.16), $\left( \mathcal {L}_v \rho \right) (\xi ,\xi )$ vanishes and therefore (3.17) and (3.18) imply

$$\begin{aligned} \left( \mathcal {L}_v \rho \right) (x,\xi ) = \mathrm {d}{{\tilde{\tau }}}(\varphi ^2x), \qquad \mathrm {d}{{\tilde{\tau }}}(\xi ) = 2\tau - 4n. \end{aligned}$$

(3.19)

The latter equalities, due to (1.8) and (1.9), imply consequently $\mathrm {d}{{\tilde{\tau }}}(\xi )=0$ and

$$\begin{aligned} \tau =2n, \qquad {{\tilde{\tau }}}=const. \end{aligned}$$

In conclusion, because of (3.19), we infer the assertion. $\square$

Theorem 3.3

Let $(M,{}\varphi ,{}\xi ,{}\eta ,{}{}g)$ be a $(2n+1)$-dimensional Einstein-like Sasaki-like manifold. If it admits a Ricci-like soliton with potential v then the Ricci tensor is $\rho =2n\, \eta \otimes \eta$ and the scalar curvatures are $\tau ={{\tilde{\tau }}}=2n$.

Proof

The assertion follows from Theorem 3.1, Propositions 1.2 and 3.2. $\square$

Corollary 3.4

Let $(M,{}\varphi ,{}\xi ,{}\eta ,{}{}g)$, $\dim M = 2n+1$, be an Einstein-like Sasaki-like manifold. Then it is $\eta$-Einstein with constants (0, 0, 2n), which is equivalent to the existence on M of a Ricci-like soliton with potential $\xi$ and constants $(0,1,-2n-1)$.

Proof

Using Theorem 3.3, we obtain the following expression $\mathcal {L}_v g =-2\lambda \,g-2\mu \,{\tilde{g}}$ $+2(\lambda +\mu )\eta \otimes \eta ,$ which holds for $\lambda =0$, $\mu =1$ in the case $v=\xi$. Therefore Theorem 2.1 is restricted to its case (iii) and $a=0$. $\square$

Let us recall, every non-degenerate 2-plane (or section) $\beta$ with a basis $\{x,y\}$ with respect to g in $T_pM$, $p \in M$, has the following sectional curvature

$$\begin{aligned} k(\beta ;p)=\frac{R(x,y,y,x)}{g(x,x)g(y,y)-[g(x,y)]^2}. \end{aligned}$$

(3.20)

A section $\beta$ is said to be $\varphi$-holomorphic if the condition $\beta = \varphi \beta$ holds. Every $\varphi$-holomorphic section has a basis of the form $\{\varphi x,\varphi ^2 x\}$. A section $\beta$ is called a $\xi$-section if it has a basis of the form $\{x,\xi \}$.

Theorem 3.5

Let $(M,{}\varphi ,{}\xi ,{}\eta ,{}{}g)$ be a 3-dimensional Sasaki-like manifold. If it admits a Ricci-like soliton with potential v then:

• (i) the sectional curvatures of its $\varphi$-holomorphic sections are equal to $-1$;

• (ii) the sectional curvatures of its $\xi$-sections are equal to 1.

Proof

It is well known that the curvature tensor of a 3-dimensional manifold has the form

$$\begin{aligned} R(x,y)z= & {} g(y,z)Qx - g(x,z)Qy +\rho (y,z)x - \rho (x,z)y \nonumber \\&-\dfrac{\tau }{2}\{g(y,z)x - g(x,z)y\}. \end{aligned}$$

(3.21)

Then, substituting $y=z=\xi$ and recalling (1.4), we have

$$\begin{aligned} \rho = \frac{1}{2}\{(\tau -2)g-(\tau -6)\eta \otimes \eta \}, \end{aligned}$$

which means that the manifold is $\eta$-Einstein. Therefore, because of Theorem 3.3, we have

$$\begin{aligned} \rho =2\, \eta \otimes \eta ,\qquad \tau ={{\tilde{\tau }}}=2. \end{aligned}$$

Substituting the latter two equalities for $\tau$ and $\rho$ in (3.21), we get

$$\begin{aligned} R(x,y)z= & {} -\,[g(y,z)-2\,\eta (y)\eta (z)]x +2\,g(y,z)\eta (x)\xi \\&+[g(x,z) -2\,\eta (x)\eta (z)]y - 2\,g(x,z)\eta (y)\xi . \end{aligned}$$

Using a basis $\{\varphi x,\varphi ^2 x\}$ of an arbitrary $\varphi$-holomorphic section, we calculate its sectional curvature by (3.20), replacing x and y by $\varphi x$ and $\varphi ^2 x$, respectively. Then, bearing in mind (1.1) and (1.2), we obtain $k(\varphi x,\varphi ^2 x)=-1$.

Similarly, for a $\xi$-section with a basis $\{x,\xi \}$, we get $k(x,\xi )=1$, which completes the proof. $\square$

Remark 3.6

Examples of 3-dimensional Sasaki-like manifolds as a Lie group from type $Bia(VII_0)(1)$, a matrix Lie group, an $S^1$-solvable extension on a Kähler-Norden 2-manifold, and their geometrical properties are studied in [22,23,24,25], respectively.

Remark 3.7

The constructed 5-dimensional example in Sect. 3.1 of a Sasaki-like manifold with the results in (2.2) and (2.3) supports also Theorem 3.1, Proposition 3.2, Theorem 3.3 and Corollary 3.4 for the case of $v=\xi$ and $n=2$.

4.1 Example 2

Let us consider M as a set of points in $\mathbb R^3$ with coordinates $(x^1,x^2,x^3)$ and let M be equipped with an almost contact B-metric structure defined by

$$\begin{aligned} \begin{array}{c} g\left( \partial _1,\partial _1\right) =-g\left( \partial _2,\partial _2\right) =\cos 2x^3,\qquad g\left( \partial _1,\partial _2\right) =\sin 2x^3, \\ g(\partial _1,\partial _3)=g(\partial _2,\partial _3)=0,\qquad g(\partial _3,\partial _3)=1, \\ \varphi \partial _1=\partial _2,\qquad \varphi \partial _2=-\partial _1,\qquad \xi =\partial _3, \end{array} \end{aligned}$$

where $\partial _1$, $\partial _2$, $\partial _3$ denote briefly $\frac{\partial }{\partial {x^1}}$, $\frac{\partial }{\partial {x^2}}$, $\frac{\partial }{\partial {x^3}}$, respectively. Then, the vectors determined by

$$\begin{aligned} e_1=\cos x^3 \partial _1+\sin x^3 \partial _2,\qquad e_2=-\sin x^3 \partial _1+\cos x^3 \partial _2,\qquad e_3=\partial _3 \end{aligned}$$

(3.22)

form an orthonormal $\varphi$-basis of $T_pM$, $p\in M$, i.e.

$$\begin{aligned} \begin{array}{c} g(e_1,e_1)=-g(e_2,e_2)=g(e_3,e_3)=1\\ g(e_i,e_j)=0,\quad i,j\in \{1,2,3\},\; i\ne j, \\ \varphi e_1=e_2,\qquad \varphi e_2=-e_1,\qquad \xi =e_3. \end{array} \end{aligned}$$

(3.23)

Immediately from (3.22) we obtain the commutators of $e_i$ as follows

$$\begin{aligned} {[}e_0,e_1]=e_2,\qquad [e_0,e_2]=-e_1,\qquad [e_1,e_2]=0. \end{aligned}$$

(3.24)

Then, according to Example 1 in [20] for $n=1$, the solvable Lie group of dimension 3 with a basis of left-invariant vector fields $\{e_1,e_2, e_3\}$ defined by (3.24) and equipped with the $(\varphi ,\xi ,\eta ,g)$-structure from (3.23) is a Sasaki-like almost contact B-metric manifold.

In the well-known way, we calculate the components of the Levi-Civita connection $\nabla$ for g and from there the corresponding components $R_{ijkl}=R(e_i,e_j,e_k,e_l)$ and $\rho _{ij}=\rho (e_i,e_j)$ of the curvature tensor R and the Ricci tensor $\rho$, respectively. The non-zero ones of them are the following (keep in mind the symmetries of R)

$$\begin{aligned} \nabla _{e_1} e_2= & {} \nabla _{e_2} e_1= -e_3,\qquad \nabla _{e_1} e_3= -e_2,\qquad \nabla _{e_2} e_3= e_1; \end{aligned}$$

(3.25)

$$\begin{aligned} R_{1221}= & {} R_{1331}=-R_{2332}=1,\qquad \rho _{33}=2. \end{aligned}$$

(3.26)

The latter equality means that the Ricci tensor has the following form

$$\begin{aligned} \rho =2\eta \otimes \eta , \end{aligned}$$

(3.27)

i.e. the manifold is Einstein-like with constants $(a,b,c)=(0,0,2)$. Therefore, the scalar curvatures with respect to g and ${\tilde{g}}$ are $\tau ={{\tilde{\tau }}}=2$.

The values of $R_{ijkl}$ in (3.26) imply the sectional curvatures

$$\begin{aligned} k_{12}=-k_{13}=-k_{23}=-1, \end{aligned}$$

which supports Theorem 3.5.

Let us consider a vector field, determined by the following

$$\begin{aligned} v= & {} v^1e_1 + v^2e_2 + v^3e_3, \nonumber \\ v^1= & {} -\{c_1\cos x^3 +c_2\sin x^3\}x^1 +\{c_2\cos x^3 -c_1\sin x^3\}x^2 + \sin x^3, \nonumber \\ v^2= & {} -\{c_2\cos x^3 -c_1\sin x^3\}x^1 -\{c_1\cos x^3 +c_2\sin x^3\}x^2 + \cos x^3, \nonumber \\ v^3= & {} c_3, \end{aligned}$$

(3.28)

and $c_1$, $c_2$, $c_3$ are arbitrary constants.

Using (3.22), (3.23), (3.25) and (3.28), we obtain the following

$$\begin{aligned} \nabla _{e_1} v= & {} -c_1e_1 -(c_2+c_3)e_2 - v^2 e_3, \nonumber \\ \nabla _{e_2} v= & {} (c_2+c_3)e_1 -c_1e_2 - v^1 e_3, \nonumber \\ \nabla _{e_3} v= & {} v^2e_1 -v^1 e_2, \end{aligned}$$

(3.29)

that allow us to calculate the components $\left( \mathcal {L}_v g\right) _{ij}=\left( \mathcal {L}_v g\right) (e_i,e_j)$ of the Lie derivative $\mathcal {L}_v g$. Then, we get the following nonzero ones

$$\begin{aligned} \left( \mathcal {L}_v g\right) _{11}= -\left( \mathcal {L}_v g\right) _{22}=-2c_1,\qquad \left( \mathcal {L}_v g\right) _{12}=2(c_2+c_3), \end{aligned}$$

which implies that this tensor has the following expression

$$\begin{aligned} \mathcal {L}_v g = -2\,c_1\,g -2(c_2+c_3){\tilde{g}}+2(c_1+c_2+c_3)\eta \otimes \eta . \end{aligned}$$

(3.30)

Substituting the latter equality and (3.27) in (2.1), we obtain that $(M,\varphi ,\xi ,\eta ,g)$ admits a Ricci-like soliton with potential v determined by (3.28) and the potential constants are

$$\begin{aligned} \lambda =c_1,\qquad \mu =c_2+c_3,\qquad \nu =-c_1-c_2-c_3-2. \end{aligned}$$

These results are in accordance with Theorems 3.1 and 3.3. The conclusion in Proposition 3.2 follows from (3.27) and the subsequent formula $(\mathcal {L}_v\rho )(x,\xi )=-2g(\varphi x,v)+2\eta (\nabla _x v)$, together with the equalities in (3.28) and (3.29).

5 Gradient Almost Ricci-like Solitons

Let us consider a Ricci-like soliton, defined by (2.1) with the condition $\lambda$, $\mu$, $\nu$ to be functions on M. If its potential v is a gradient of a differentiable function f, i.e. $v={{\,\mathrm{grad}\,}}f$, then the soliton is called a gradient almost Ricci-like soliton of $(M,{}\varphi ,{}\xi ,{}\eta ,{}{}g)$. In this case (2.1) is reduced to the following condition

$$\begin{aligned} {{\,\mathrm{Hess}\,}}f+\rho +\lambda g+\mu {\tilde{g}}+\nu \eta \otimes \eta =0, \end{aligned}$$

(4.1)

where ${{\,\mathrm{Hess}\,}}$ denotes the Hessian operator with respect to g, i.e. ${{\,\mathrm{Hess}\,}}f$ is defined by

$$\begin{aligned} ({{\,\mathrm{Hess}\,}}f)(x,y) := (\nabla _x \mathrm {d}f)(y)=g(\nabla _x {{\,\mathrm{grad}\,}}f,y). \end{aligned}$$

(4.2)

Taking the trace of (4.1), we obtain

$$\begin{aligned} \Delta f+\tau +(2n+1)\lambda +\mu +\nu =0, \end{aligned}$$

where $\Delta :={{\,\mathrm{tr}\,}}\circ {{\,\mathrm{Hess}\,}}$ is the Laplacian operator of g. Also for the Laplacian of f, the formula $\Delta f = {{\,\mathrm{div}\,}}({{\,\mathrm{grad}\,}}f)$ is valid, where ${{\,\mathrm{div}\,}}$ stands for the divergence operator.

The gradient Ricci-like soliton is said to be trivial when f is constant. Further, we consider only non-trivial gradient Ricci-like solitons.

Equality (4.1) with the recall of (4.2) provides the following

$$\begin{aligned} \nabla _x v=-Qx-\lambda x-\mu \varphi x-(\mu +\nu )\eta (x)\xi , \end{aligned}$$

(4.3)

where Q is the Ricci operator and $v={{\,\mathrm{grad}\,}}f$.

Theorem 4.1

Let $(M,{}\varphi ,{}\xi ,{}\eta ,{}{}g)$ be a Sasaki-like almost contact B-metric manifold of dimension $2n+1$. If it admits a gradient almost Ricci-like soliton with functions $(\lambda ,\mu ,\nu )$ and a potential function f, then $(M,{}\varphi ,{}\xi ,{}\eta ,{}{}g)$ has constant scalar curvatures $\tau ={{\tilde{\tau }}}=2n$ for both B-metrics g and ${\tilde{g}}$, respectively, and its Ricci tensor is $\rho =2n\,\eta \otimes \eta$.

Proof

Using (4.3), we compute the following curvature tensor

$$\begin{aligned} R(x,y)v= & {} -\left( \nabla _x Q\right) y+\left( \nabla _y Q\right) x\nonumber \\&+\{\mathrm {d}\lambda (y)+\mu \eta (y)\}x-\{\mathrm {d}\lambda (x)+\mu \eta (x)\}y\nonumber \\&+\{\mathrm {d}\mu (y)+(\mu +\nu )\eta (y)\}\varphi x-\{\mathrm {d}\mu (x)+(\mu +\nu )\eta (x)\}\varphi y\nonumber \\&+\mathrm {d}(\mu +\nu )(y)\eta (x) \xi - \mathrm {d}(\mu +\nu )(x)\eta (y) \xi . \end{aligned}$$

(4.4)

The latter expression implies the following equality

$$\begin{aligned} R(\xi ,y)v= & {} -\left( \nabla _\xi Q\right) y+\left( \nabla _y Q\right) \xi +\{\mathrm {d}\lambda (\xi )+\mu \}\varphi ^2 y-\{\mathrm {d}\mu (\xi )+\mu +\nu \}\varphi y\\&+\mathrm {d}(\lambda +\mu +\nu )(y)\xi -\mathrm {d}(\lambda +\mu +\nu )(\xi )\eta (y) \xi , \end{aligned}$$

where we apply (1.10) and (1.11) and get

$$\begin{aligned} R(\xi ,y)v= & {} -Q\varphi y +\{\mathrm {d}\lambda (\xi )+\mu \}\varphi ^2y-\{\mathrm {d}\mu (\xi )+\mu +\nu +2n\}\varphi y\nonumber \\&+\mathrm {d}(\lambda +\mu +\nu )(y)\xi -\mathrm {d}(\lambda +\mu +\nu )(\xi )\eta (y) \xi . \end{aligned}$$

(4.5)

We put $z=v$ in the equality for $R(\xi ,y)z$ in (1.4) and obtain the following expression

$$\begin{aligned} R(\xi ,y)v=\mathrm {d}f(y)\xi -\mathrm {d}f(\xi )y. \end{aligned}$$

(4.6)

Combining (4.5) and (4.6), we find the following formula

$$\begin{aligned} Q\varphi y= & {} \{\mathrm {d}(\lambda -f)(\xi )+\mu \}\varphi ^2y-\{\mathrm {d}\mu (\xi )+\mu +\nu +2n\}\varphi y\nonumber \\&+\mathrm {d}(\lambda +\mu +\nu -f)(y)\xi -\mathrm {d}(\lambda +\mu +\nu -f)(\xi )\eta (y) \xi . \end{aligned}$$

(4.7)

We apply $\eta$ of equality (4.7) and since $Q\circ \varphi =\varphi \circ Q$ for a Sasaki-like manifold, we obtain the following

$$\begin{aligned} \mathrm {d}(\lambda +\mu +\nu -f)(y)=\mathrm {d}(\lambda +\mu +\nu -f)(\xi )\eta (y), \end{aligned}$$

(4.8)

which changes (4.7) and (4.5) as follows

$$\begin{aligned} Q\varphi y= & {} \{\mathrm {d}(\lambda -f)(\xi )+\mu \}\varphi ^2y-\{\mathrm {d}\mu (\xi )+\mu +\nu +2n\}\varphi y,\\ R(\xi ,y)v= & {} -Q\varphi y +\{\mathrm {d}\lambda (\xi )+\mu \}\varphi ^2y-\{\mathrm {d}\mu (\xi )+\mu +\nu +2n\}\varphi y\\&+\{\mathrm {d}f(y)-\mathrm {d}f(\xi )\eta (y)\}\xi \end{aligned}$$

and therefore we have

$$\begin{aligned} R(\xi ,y,v,z)\,= & {} - \rho (y,\varphi z) +\{\mathrm {d}\lambda (\xi )+\mu \}g(\varphi y,\varphi z) \nonumber \\&- \{\mathrm {d}\mu (\xi )+\mu +\nu +2n\}g( y,\varphi z)\nonumber \\&+\{\mathrm {d}f(y)-\mathrm {d}f(\xi )\eta (y)\}\eta (z). \end{aligned}$$

(4.9)

On the other hand, the expression of $R(x,y)\xi$ from (1.4) and equality (4.4) imply respectively the following two equalities

$$\begin{aligned} R(x,y,\xi ,v)= & {} \mathrm {d}f(x)\eta (y)-\mathrm {d}f(y)\eta (x),\\ R(x,y,v,\xi )= & {} -\eta \left( (\nabla _x Q)y-(\nabla _y Q)x\right) +\mathrm {d}(\lambda +\mu +\nu )(y)\eta (x)\\&-\mathrm {d}(\lambda +\mu +\nu )(x)\eta (y). \end{aligned}$$

By summation of the latter two equalities, we find the following formula

$$\begin{aligned} \eta \bigl ((\nabla _x Q)y-(\nabla _y Q)x\bigr )= & {} -\,\mathrm {d}(\lambda +\mu +\nu -f)(x)\eta (y)\\&+\mathrm {d}(\lambda +\mu +\nu -f)(y)\eta (x), \end{aligned}$$

which because of (4.8) is simplified to the following form

$$\begin{aligned} \eta \bigl ((\nabla _x Q)y-(\nabla _y Q)x\bigr )=0. \end{aligned}$$

On the other hand, the expression of $R(\xi ,y)z$ from (1.4) yield

$$\begin{aligned} R(\xi ,y,z,v)\,{=}-\mathrm {d}f(\xi )g(\varphi y, \varphi z)-\{\mathrm {d}f(y)-\mathrm {d}f(\xi )\eta (y) \}\eta (z), \end{aligned}$$

which together with (4.9) and the form of $\rho (x,\xi )$ from (1.4) implies

$$\begin{aligned} \rho (y,z)= & {} \{\mathrm {d}\mu (\xi )+\mu +\nu +2n\}g(\varphi y, \varphi z)+\{\mathrm {d}(\lambda -f)(\xi )+\mu \}g(y,\varphi z)\\&+2n\eta (y)\eta (z). \end{aligned}$$

The latter equality can be rewritten in the form

$$\begin{aligned} \rho= & {} -\,\{\mathrm {d}\mu (\xi )+\mu +\nu +2n\}g+\{\mathrm {d}(\lambda -f)(\xi )+\mu \}{\tilde{g}}\\&+\{4n+\nu -\mathrm {d}(\lambda -\mu -f)(\xi )\}\eta \otimes \eta , \end{aligned}$$

which means that the manifold is almst Einstein-like with coefficient functions

$$\begin{aligned} a\,= & {} -\mathrm {d}\mu (\xi )-\mu -\nu -2n,\qquad b=\mathrm {d}(\lambda -f)(\xi )+\mu ,\nonumber \\ c\,= & {} -\mathrm {d}(\lambda -\mu -f)(\xi )+\nu +4n. \end{aligned}$$

(4.10)

Then, using (1.14), we obtain

$$\begin{aligned} \tau \,{=}-2n\{\mathrm {d}\mu (\xi )+\mu +\nu +2n-1\},\qquad {{\tilde{\tau }}}=2n\{\mathrm {d}(\lambda -f)(\xi )+\mu +1\}.\nonumber \\ \end{aligned}$$

(4.11)

Contracting (4.4) with respect to x, we obtain

$$\begin{aligned} \rho (y,v)=\frac{1}{2} \mathrm {d}\tau (y)+2n\, \mathrm {d}\lambda (y)+\mathrm {d}(\mu +\nu )(y)-\mathrm {d}\mu (\varphi y)-\{\mathrm {d}(\mu +\nu )(\xi )-2n\,\mu \}\eta (y) \end{aligned}$$

and consequently for $y=\xi$ we have

$$\begin{aligned} \rho (\xi ,v)=\frac{1}{2} \mathrm {d}\tau (\xi )+2n\, \mathrm {d}\lambda (\xi )+2n\,\mu . \end{aligned}$$

(4.12)

We compute the left side of (4.12) by the formula $\rho (x,\xi )=2n\, \eta (x)$ from (1.4) and then from (4.12) and (1.9), we obtain

$$\begin{aligned} {{\tilde{\tau }}}=-2n \{\mathrm {d}(\lambda -f)(\xi )+\mu -1\}. \end{aligned}$$

Comparing the latter equality with (4.11), we have

$$\begin{aligned} \mathrm {d}(\lambda -f)(\xi )= & {} -\mu ,\\ {{\tilde{\tau }}}= & {} 2n. \end{aligned}$$

The former equality implies $b=0$ in (4.10) and therefore the manifold is almost $\eta$-Einstein and the latter one means that $\mathrm {d}{{\tilde{\tau }}}=0$ and using (1.9), we obtain for $\tau$ the following

$$\begin{aligned} \tau =2n. \end{aligned}$$

Then, substituting the value of $\tau$ in (4.11), we obtain

$$\begin{aligned} \mathrm {d}\mu (\xi )=-\mu -\nu -2n, \end{aligned}$$

which implies $a=0$ in (4.10) and finally we get $(a,b,c)=(0,0,2n)$. $\square$

5.1 Example 3

Let $(M,{}\varphi ,{}\xi ,{}\eta ,{}{}g)$ be the 3-dimensional Sasaki-like manifold, given in Example 2 of sect. 3.1. Now, let f be a differentiable function on M, defined by

$$\begin{aligned} f=-\frac{1}{2}s\,\{(x^1)^2+(x^2)^2\} +x^2 + t\,x^3 \end{aligned}$$

for arbitrary constants s and t. Then, the gradient of f with respect to the B-metric g is the following

$$\begin{aligned} {{\,\mathrm{grad}\,}}f= & {} -\,\{s\,x^1\cos x^3 + (s\,x^2-1)\sin x^3\}e_1\nonumber \\&+\{s\,x^1\sin x^3 - (s\,x^2-1)\cos x^3\}e_2 + t\, e_3. \end{aligned}$$

(4.13)

Using (3.22), we compute the components of $\mathcal {L}_{{{\,\mathrm{grad}\,}}f}g$ as follows

$$\begin{aligned} \left( \mathcal {L}_{{{\,\mathrm{grad}\,}}f}g\right) _{11}=-\left( \mathcal {L}_{{{\,\mathrm{grad}\,}}f}g\right) _{22}=-2s,\qquad \left( \mathcal {L}_{{{\,\mathrm{grad}\,}}f}g\right) _{12}=2t, \end{aligned}$$

which give us the following expression

$$\begin{aligned} \left( \mathcal {L}_{{{\,\mathrm{grad}\,}}f}g\right) =-2s\, g -2t {\tilde{g}}+2(s+t)\,\eta \otimes \eta . \end{aligned}$$

The latter equality coincides with (3.30) for $s=c_1$, $t=c_2+c_3$. Therefore, $(M,\varphi ,\xi ,\eta ,g)$ admits a Ricci-like soliton with potential $v={{\,\mathrm{grad}\,}}f$ determined by (4.13) and the potential constants are

$$\begin{aligned} \lambda =s,\qquad \mu =t,\qquad \nu =-s-t-2. \end{aligned}$$

In conclusion, the constructed 3-dimensional example of a Sasaki-like manifold with $\tau ={{\tilde{\tau }}}=2$ and gradient Ricci-like soliton supports also Theorem 4.1.