η-Ricci Solitons on Weak β-Kenmotsu f-Manifolds

Abstract

Recent interest among geometers in f-structures of K. Yano is due to the study of topology and dynamics of contact foliations, which generalize the flow of the Reeb vector field on contact manifolds to higher dimensions. Weak metric structures introduced by the author and R. Wolak as a generalization of Hermitian and Kähler structures, as well as f-structures, allow for a fresh perspective on the classical theory. In this paper, we study a new f-structure of this kind, called the weak $\beta$-Kenmotsu f-structure, as a generalization of K. Kenmotsu’s concept. We prove that a weak $\beta$-Kenmotsu f-manifold is a locally twisted product of the Euclidean space and a weak Kähler manifold. Our main results show that such manifolds with $\beta =const$ and equipped with an $\eta$-Ricci soliton structure whose potential vector field satisfies certain conditions are $\eta$-Einstein manifolds of constant scalar curvature.

1. Introduction

Contact Riemannian geometry is of growing interest to mathematicians because of its importance for physics, e.g., [1]. A metric f-structure on a smooth manifold $M^{2n+s}$, see [2,3], as a higher-dimensional analog of a contact metric structure ($s=1$), is defined by a skew-symmetric (1,1)-tensor f of rank $2n$, orthonormal vector fields ${\left\{{\xi }_i\right\}}_{1\le i\le s}$, and their dual 1-forms $\left\{{\eta }^i\right\}$ such that

$$\begin{array}{cc}\hfill & f^2=-\mathrm{id}+{\sum }_i{\eta }^i\otimes {\xi }_i,{\eta }^i\left({\xi }_j\right)={\delta }_j^i,\hfill \\ \hfill & g(fX,fY)=g(X,Y)-{\sum }_i{\eta }^i\left(X\right){\eta }^i\left(Y\right),X,Y\in {\mathfrak{X}}_M.\hfill \end{array}$$

A metric f-structure is a special case of an almost product structure (with Naveira’s 36 distinguished classes, see [4]), defined by complementary orthogonal distributions $\mathcal{D}=f\left(TM\right)$ and ${\mathcal{D}}^{\perp }=kerf$. For an f-contact manifold, the s-dimensional distribution ${\mathcal{D}}^{\perp }$ is tangent to a totally geodesic foliation, which is spanned by Killing vector fields $\left\{{\xi }_i\right\}$. Such so-called $\mathfrak{g}$-foliation is defined by a homomorphism of an s-dimensional Lie algebra $\mathfrak{g}$ to the Lie algebra of all vector fields on M, for example, [5]. The recent interest of geometers in f-structures is motivated by the study of the topology and dynamics of contact foliations, especially the existence of closed leaves. Contact foliations generalize to higher dimensions the flow of the Reeb vector field on contact manifolds, and K-structures are a particular case of contact structures, see [6]. A special class of f-manifolds, known as Kenmotsu f-manifolds, see [7,8,9,10] (Kenmotsu manifolds when $s=1$, see [11]), can be characterized in terms of warped products of ${\mathbb{R}}^s$ and Kähler manifolds.

In [5], we defined metric structures that generalize Hermitian—more specifically, Kähler—structures, as well as metric f-structures, in particular, $\mathcal{C}$-, $\mathcal{S}$-, K-, and f-K- contact structures. These so-called “weak” f-structures (i.e., the complex structure is replaced by a nonsingular skew-symmetric tensor) are useful for studying totally geodesic and totally umbilical foliations, Killing fields, and Einstein-type metrics, and they allow a fresh look at the classical theory. A weak metric f-structure is a special case of an almost product structure, defined by two complementary orthogonal distributions $\mathcal{D}=f\left(TM\right)$ and ${\mathcal{D}}^{\perp }=kerf$ of $(M^{2n+s},g)$. Foliations appear when one or both distributions are involutive. Weak metric f-manifolds form a broad class as follows: a warped product of a Euclidean space ${\mathbb{R}}^s$ and a weak Kähler manifold (Definition 4 and Example 1) is a weak $\beta$-Kenmotsu f-manifold (Definition 3), and the metric product of ${\mathbb{R}}^s$ and a weak Kähler manifold is a weak $\mathcal{C}$-manifold.

Solutions of some non-linear partial differential equations decompose at large time t into solitary waves running with constant speed—the so-called solitons. Ricci solitons (RS) ${\displaystyle \frac{1}{2}}{\pounds }_{V}g+\mathrm{Ric}=\lambda g$ (where £ is the Lie derivative, V a vector field, and $\lambda$ is a real constant), as self-similar solutions of the Ricci flow $\partial g/\partial t=-2{Ric}_g$, generalize (when ${\pounds }_{V}g=0$) the Einstein metrics $Ric=\lambda g$; see [12]. The Ricci flow and RS were studied for Hermitian and Kähler manifolds, as well as for almost contact metrics—in particular, Sasakian—manifolds. Since some compact manifolds (and f-K-contact manifolds) do not admit Einstein metrics. Cho-Kimura [13] generalized the notion of RS to $\eta$-RS as follows:

$$\begin{array}{c}\hfill (1/2){\pounds }_{V}g+\mathrm{Ric}=\lambda g+\mu \eta \otimes \eta \mathrm{for} \mathrm{some}\lambda ,\mu \in \mathbb{R},\end{array}$$

(1)

where $\eta$ is a 1-form on M. If V is a Killing vector field, then (1) reduces to an $\eta$-Einstein structure, which is defined by

$$\begin{array}{c}\hfill \mathrm{Ric}=ag+b\eta \otimes \eta \mathrm{for} \mathrm{some}a,b\in C^{\infty }\left(M\right).\end{array}$$

(2)

The following questions arise (see [1,13]): How do RS interact with weak f-structures? Does a weak metric f-manifold equipped with RS carry Einstein-type metrics? In this paper, we introduce weak $\beta$-Kenmotsu f-manifolds ($\beta f$-KM)—see Definition 3—as a generalization of K. Kenmotsu’s concept, and explore their properties and geometrical interpretations. We study when a weak $\beta f$-KM (it cannot be an Einstein manifold) equipped with an $\eta$-RS structure (16) carries an $\eta$-Einstein metric (15) of constant scalar curvature.

The paper consists of an introduction and four sections. In Section 2, we review the basics of the weak metric f-structure. In Section 3 and Section 4, we introduce weak $\beta f$-KM (weak f-KM when $\beta \equiv 1$, and weak $\mathcal{C}$-manifolds when $\beta \equiv 0$), derive their fundamental properties (Theorem 1); give their geometrical interpretation in terms of the twisted structure (Theorem 2); and prove that a weak $\beta f$-KM with a $\xi$-parallel Ricci tensor is an $\eta$-Einstein manifold (Theorem 3). In Section 5, we study the interaction of weak $\beta f$-KM with $\eta$-RS. We prove that an additional $\eta$-Einstein structure ensures the constancy of the scalar curvature (Theorem 4). We then show that if a weak $\beta f$-KM with $\beta =const$ is an $\eta$-RS whose non-zero potential vector field, either a contact vector field (Theorem 5) or collinear with ${\sum }_i{\xi }_i$ (Theorem 6), then it is an $\eta$-Einstein manifold. The results generalize some theorems in [14], where $s=1$ and can be extended to the case $\beta \in C^{\infty }\left(M\right)$.

2. Preliminaries: Weak Metric f-Manifolds

In this section, we review the basics of a weak metric f-structure as a higher-dimensional analog of the weak almost contact metric structure; see [5,15]. First, we generalize the notion of the framed f-structure, see [2,3,16], called the $f.pk$-structure in [9].

Definition 1 

(see [5]). A weak metric f-structure on a smooth manifold $M^{2n+s}(n>0,s>1)$ is a set $(f,Q,{\xi }_i,{\eta }^i,g)$, where f is a skew-symmetric $(1,1)$-tensor of rank $2n$; Q is a self-adjoint nonsingular $(1,1)$-tensor; ${\xi }_i(1\le i\le s)$ are orthonormal vector fields; ${\eta }^i$ are dual 1-forms; g is a Riemannian metric on M, satisfying

$$\begin{array}{c}\hfill f^2=-Q+{\sum }_i{\eta }^i\otimes {\xi }_i,{\eta }^i\left({\xi }_j\right)={\delta }_j^i,Q{\xi }_i={\xi }_i,\end{array}$$

(3)

$$\begin{array}{c}\hfill g(fX,fY)=g(X,QY)-{\sum }_i{\eta }^i\left(X\right){\eta }^i\left(Y\right),X,Y\in {\mathfrak{X}}_M;\end{array}$$

(4)

and $M^{2n+s}(f,Q,{\xi }_i,{\eta }^i,g)$ is called a weak metric f-manifold.

Assume that the distribution $\mathcal{D}:={\bigcap }_{i}ker{\eta }^i$ is f-invariant; thus, $\mathcal{D}=f\left(TM\right)$, $dim\mathcal{D}=2n$, and

$$f{\xi }_i=0,{\eta }^i\circ f=0,{\eta }^i\circ Q={\eta }^i,[Q,f]=0.$$

Using the above, the distribution ${\mathcal{D}}^{\perp }=kerf$ is spanned by $\{{\xi }_1,\dots ,{\xi }_s\}$ and is invariant for Q. We define a (1,1)-tensor $\tilde{Q}$ by $Q=\mathrm{id}+\tilde{Q}$, and we note that $[\tilde{Q},f]=0$ and ${\eta }^i\circ \tilde{Q}=0$. We also obtain $f^3+f=-\tilde{Q}f$. Putting $Y={\xi }_i$ in (4), we obtain ${\eta }^i\left(X\right)=g(X,{\xi }_i)$; thus, each ${\xi }_i$ is orthogonal to $\mathcal{D}$. Therefore, $TM$ splits as complementary orthogonal sum of its sub-bundles $\mathcal{D}$ and ${\mathcal{D}}^{\perp }$—an almost product structure.

A distribution ${\mathcal{D}}^{\perp }\subset TM$ (whether integrable or not) is said to be totally geodesic if its second fundamental form vanishes as follows: ${\nabla }_{X}Y+{\nabla }_{Y}X\in {\mathcal{D}}^{\perp }$ for any vector fields $X,Y\in {\mathcal{D}}^{\perp }$— this is the case when any geodesic of M that is tangent to ${\mathcal{D}}^{\perp }$ at one point is tangent to ${\mathcal{D}}^{\perp }$ at all its points. By Frobenius’ theorem, any involutive distribution is integrable, i.e., it is tangent to the leaves of the foliation. Any integrable and totally geodesic distribution determines a totally geodesic foliation.

A weak metric f-structure $(f,Q,{\xi }_i,{\eta }^i)$ is said to be normal if the following tensor is zero:

$$\begin{array}{c}\hfill {\mathcal{N}}^{\left(1\right)}(X,Y)=[f,f](X,Y)+2{\sum }_{i}d{\eta }^i(X,Y){\xi }_i,X,Y\in {\mathfrak{X}}_M.\end{array}$$

The Nijenhuis torsion of a (1,1)-tensor S and the derivative of a 1-form $\omega$ are given by

$$\begin{array}{cc}\hfill & [S,S](X,Y)=S^2[X,Y]+[SX,SY]-S[SX,Y]-S[X,SY],X,Y\in {\mathfrak{X}}_M,\hfill \\ \hfill & d\omega (X,Y)=(1/2)\{X\left(\omega \left(Y\right)\right)-Y\left(\omega \left(X\right)\right)-\omega \left([X,Y]\right)\},X,Y\in {\mathfrak{X}}_M.\hfill \end{array}$$

Using the Levi-Civita connection ∇ of g, one can rewrite $[S,S]$ as

$$\begin{array}{c}\hfill [S,S](X,Y)=(S{\nabla }_{Y}S-{\nabla }_{SY}S)X-(S{\nabla }_{X}S-{\nabla }_{SX}S)Y.\end{array}$$

(5)

The following tensors (see [15]): ${\mathcal{N}}_i^{\left(2\right)},{\mathcal{N}}_i^{\left(3\right)}$ and ${\mathcal{N}}_{ij}^{\left(4\right)}$, are well known for metric f-manifolds as follows [2]:

$$\begin{array}{cc}\hfill {\mathcal{N}}_i^{\left(2\right)}(X,Y)& =\left({\pounds }_{fX}{\eta }^i\right)\left(Y\right)-\left({\pounds }_{fY}{\eta }^i\right)\left(X\right)=2d{\eta }^i(fX,Y)-2d{\eta }^i(fY,X),\hfill \\ \hfill {\mathcal{N}}_i^{\left(3\right)}\left(X\right)& =\left({\pounds }_{{\xi }_i}f\right)X=[{\xi }_i,fX]-f[{\xi }_i,X],\hfill \\ \hfill {\mathcal{N}}_{ij}^{\left(4\right)}\left(X\right)& =\left({\pounds }_{{\xi }_i}{\eta }^j\right)\left(X\right)={\xi }_i\left({\eta }^j\left(X\right)\right)-{\eta }^j\left([{\xi }_i,X]\right)=2d{\eta }^j({\xi }_i,X).\hfill \end{array}$$

Remark 1. 

Let $M^{2n+s}(f,Q,{\xi }_i,{\eta }^i,g)$ be a weak metric f-manifold. Consider the product manifold $\overline{M}=M^{2n+s}\times {\mathbb{R}}^s$, where ${\mathbb{R}}^s$ is a Euclidean space with a basis ${\partial }_1,\dots ,{\partial }_s$. Define tensor J on $\overline{M}$ by $J(X,{\sum }_i{a}^i{\partial }_i)=(fX-{\sum }_i{a}^i{\xi }_i,{\sum }_j{\eta }^j\left(X\right){\partial }_j)$, where $a_i\in C^{\infty }\left(M\right)$. Tensors ${\mathcal{N}}^{\left(1\right)},{\mathcal{N}}_i^{\left(2\right)},{\mathcal{N}}_i^{\left(3\right)},$ and ${\mathcal{N}}_{ij}^{\left(4\right)}$ appear from the condition $[J,J]=0$ when we express the normality condition ${\mathcal{N}}^{\left(1\right)}=0$.

Proposition 1 

(see [15]). The condition ${\mathcal{N}}^{\left(1\right)}=0$ for a weak metric f-structure implies

$$\begin{array}{cc}\hfill & {\mathcal{N}}_i^{\left(3\right)}={\mathcal{N}}_{ij}^{\left(4\right)}=0,{\mathcal{N}}_i^{\left(2\right)}(X,Y)={\eta }^i\left([\tilde{Q}X,fY]\right),\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & {\nabla }_{{\xi }_i}{\xi }_j\in \mathcal{D},[X,{\xi }_i]\in \mathcal{D}(X\in \mathcal{D}).\hfill \end{array}$$

(7)

In this case, ${\mathcal{D}}^{\perp }$ is a totally geodesic distribution.

The fundamental 2-form $\Phi$ on $M(f,Q,{\xi }_i,{\eta }^i,g)$ is defined by $\Phi (X,Y)=g(X,fY)$ for all $X,Y\in {\mathfrak{X}}_M$. Recall the co-boundary formula for the exterior derivative of a 2-form $\Phi$,

$$\begin{array}{cc}\hfill 3d\Phi (X,Y,Z)& =X\Phi (Y,Z)+Y\Phi (Z,X)+Z\Phi (X,Y)\hfill \\ \hfill & -\Phi \left(\right[X,Y],Z)-\Phi \left(\right[Z,X],Y)-\Phi \left(\right[Y,Z],X).\hfill \end{array}$$

(8)

Proposition 2 

(see [15]). For a weak metric f-structure, we obtain

$$\begin{array}{cc}\hfill 2g(\left({\nabla }_{X}f\right)Y,Z)& =3d\Phi (X,fY,fZ)-3d\Phi (X,Y,Z)+g({\mathcal{N}}^{\left(1\right)}(Y,Z),fX)\hfill \\ \hfill & +{\sum }_i\left({\mathcal{N}}_i^{\left(2\right)}(Y,Z){\eta }^i\left(X\right)+2d{\eta }^i(fY,X){\eta }^i\left(Z\right)-2d{\eta }^i(fZ,X){\eta }^i\left(Y\right)\right)\hfill \\ \hfill & +{\mathcal{N}}^{\left(5\right)}(X,Y,Z),\hfill \end{array}$$

(9)

where ${\mathcal{N}}^{\left(5\right)}$ is the tensor field acting as

$$\begin{array}{cc}\hfill & {\mathcal{N}}^{\left(5\right)}(X,Y,Z)=fZ\left(g(X,\tilde{Q}Y)\right)-fY\left(g(X,\tilde{Q}Z)\right)\hfill \\ \hfill & +g([X,fZ],\tilde{Q}Y)-g([X,fY],\tilde{Q}Z)+g([Y,fZ]-[Z,fY]-f[Y,Z],\tilde{Q}X).\hfill \end{array}$$

Let ${Ric}^{♯}$ be a (1,1)-tensor adjoint to the Ricci tensor—the suitable trace of the curvature tensor—expressed as follows:

$$\mathrm{Ric}(X,Y)={\mathrm{trace}}_g(Z\to R_{Z,X}Y),R_{X,Y}=[{\nabla }_X,{\nabla }_Y]-{\nabla }_{[X,Y]}(X,Y,Z\in {\mathfrak{X}}_M).$$

The scalar curvature of a Riemannian manifold $(M,g)$ is defined as

$$r={\mathrm{trace}}_{g}Ric=\mathrm{trace}{Ric}^{♯}.$$

The following formulas are well known, e.g., ([17], Equations (6) and (7)):

$$\begin{array}{cc}\hfill & ({\pounds }_V\nabla )(X,Y)={\nabla }_X{\nabla }_{Y}V-{\nabla }_{{\nabla }_{X}Y}V+R_{V,X}Y,\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & ({\pounds }_V\nabla )(X,Y)={\pounds }_V\left({\nabla }_{X}Y\right)-{\nabla }_X\left({\pounds }_{V}Y\right)-{\nabla }_{{\pounds }_{V}X}Y,\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & \left({\nabla }_X{\pounds }_{V}g\right)(Y,Z)=g(({\pounds }_V\nabla )(X,Y),Z)+g(({\pounds }_V\nabla )(X,Z),Y),\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & ({\pounds }_V{\nabla }_{Z}g-{\nabla }_Z{\pounds }_{V}g-{\nabla }_{[V,Z]}g)(X,Y)=-g(({\pounds }_V\nabla )(Z,X),Y)-g(({\pounds }_V\nabla )(Z,Y),X),\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & {\left({\pounds }_{V}R\right)}_{X,Y}Z=({\nabla }_X{\pounds }_V\nabla )(Y,Z)-({\nabla }_Y{\pounds }_V\nabla )(X,Z).\hfill \end{array}$$

(14)

Definition 2. 

A weak metric f-manifold is said to be η-Einstein, if

$$\begin{array}{c}\hfill Ric=ag+b{\sum }_i{\eta }^i\otimes {\eta }^i+(a+b){\sum }_{i\ne j}{\eta }^i\otimes {\eta }^{j}for somea,b\in C^{\infty }\left(M\right).\end{array}$$

(15)

An η-RS on a weak metric f-manifold $M^{2n+s}(f,Q,{\xi }_i,{\eta }^i,g)$ is defined by

$$\begin{array}{c}\hfill (1/2){\pounds }_{V}g+Ric=\lambda g+\mu {\sum }_i{\eta }^i\otimes {\eta }^i+(\lambda +\mu ){\sum }_{i\ne j}{\eta }^i\otimes {\eta }^j,\end{array}$$

(16)

where V is a smooth vector field on M and λ and μ are real constants.

If V is a Killing vector field, i.e., ${\pounds }_{V}g=0$, then (16) reduces to (15) with $a=\lambda$ and $b=\mu$. Taking the trace of (15) gives the scalar curvature $r=(2n+s)a+sb$. For $s=1$ and $Q=\mathrm{id}$, the definitions (15) and (16) are well known for almost contact metric manifolds as follows: (16) reduces to an $\eta$-RS (1), and (15) reduces to an $\eta$-Einstein metric (2).

3. Geometry of Weak $\mathit{\beta }\mathit{f}$-KM

In the following definition, we generalize the notions of $\beta$-KM ($s=1$), f-KM ($\beta =1$, $s>1$)— see [8,9,10,17]—and weak $\beta$-KM ($s=1$)—see [14].

Definition 3. 

A weak metric f-manifold $M^{2n+s}(f,Q,{\xi }_i,{\eta }^i,g)$ will be called a weak $\beta f$-KM (a weak f-KM) when $\beta \equiv 1)$, if

$$\begin{array}{c}\hfill \left({\nabla }_{X}f\right)Y=\beta \{g(fX,Y)\overline{\xi }-\overline{\eta }\left(Y\right)fX\}(X,Y\in {\mathfrak{X}}_M),\end{array}$$

(17)

where $\overline{\xi }={\sum }_i{\xi }_i$, $\overline{\eta }={\sum }_i{\eta }^i$, and $\beta \in C^{\infty }\left(M\right)$. For $\beta \equiv 0$, (17) defines a weak $\mathcal{C}$-manifold.

Note that $\overline{\eta }\left({\xi }_i\right)={\eta }^i\left(\overline{\xi }\right)=1$ and $\overline{\eta }\left(\overline{\xi }\right)=s$. Taking $X={\xi }_j$ in (17) and using $f{\xi }_j=0$, we obtain ${\nabla }_{{\xi }_j}f=0$, which implies ${\nabla }_{{\xi }_i}{\xi }_j\perp \mathcal{D}$. This and the first equality in (7) give

$$\begin{array}{c}\hfill {\nabla }_{{\xi }_i}{\xi }_j=0(1\le i,j\le s);\end{array}$$

(18)

thus, ${\mathcal{D}}^{\perp }$ of a weak $\beta f$-KM is tangent to a foliation with flat totally geodesic leaves.

Lemma 1. 

For a weak $\beta f$-KM, the following formulas are true:

$$\begin{array}{cc}\hfill & {\nabla }_X{\xi }_i=\beta \{X-{\sum }_j{\eta }^j\left(X\right){\xi }_j\}(1\le i\le s,X\in {\mathfrak{X}}_M),\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & \left({\nabla }_X{\eta }^i\right)\left(Y\right)=\beta \{g(X,Y)-{\sum }_j{\eta }^j\left(X\right){\eta }^j\left(Y\right)\}(1\le i\le s,X,Y\in {\mathfrak{X}}_M).\hfill \end{array}$$

(20)

Proof. 

Taking $Y={\xi }_i$ in (17) and using $g(fX,{\xi }_i)=0$ and $\overline{\eta }\left({\xi }_i\right)=1$, we obtain $f({\nabla }_X{\xi }_i-\beta X)=0$. Since f is non-degenerate on $\mathcal{D}$ and has rank $2n$, we obtain ${\nabla }_X{\xi }_i-\beta X={\sum }_p{c}^p{\xi }_p$. The inner product with ${\xi }_j$ gives $g({\nabla }_X{\xi }_i,{\xi }_j)=\beta g(X,{\xi }_j)-c^j$. Using (7) and (18), we find $g({\nabla }_X{\xi }_i,{\xi }_j)=g({\nabla }_{{\xi }_i}X,{\xi }_j)=0$; hence, $c^j=\beta {\eta }^j\left(X\right)$. This proves (19). Using $\left({\nabla }_X{\eta }^i\right)\left(Y\right)=g({\nabla }_X{\xi }_i,Y)$ and (19), we obtain (20).   □

The following result generalizes Theorem 3.4 in [10].

Theorem 1. 

A weak metric f-manifold $M^{2n+s}(f,Q,{\xi }_i,{\eta }^i,g)$ is a weak $\beta f$-KM if and only if the following conditions hold:

$$\begin{array}{c}\hfill {\mathcal{N}}^{\left(1\right)}=0,d{\eta }^i=0,d\Phi =2\beta \overline{\eta }\wedge \Phi ,{\mathcal{N}}^{\left(5\right)}(X,Y,Z)=2\beta \overline{\eta }\left(X\right)g(fY,\tilde{Q}Z).\end{array}$$

(21)

Proof. 

Let (17) be true. Using (19), we obtain

$$\begin{array}{c}\hfill \left({\nabla }_X{\eta }^i\right)Y=Xg({\xi }_i,Y)-g({\xi }_i,{\nabla }_{X}Y)=g({\nabla }_X{\xi }_i,Y)=\beta \{g(X,Y)-{\sum }_j{\eta }^j\left(X\right){\eta }^j\left(Y\right)\}\end{array}$$

(22)

for all $X,Y\in {\mathfrak{X}}_M$. By (22), $\left({\nabla }_X{\eta }^i\right)Y=\left({\nabla }_Y{\eta }^i\right)X$ is true. Thus, for $X,Y\in \mathcal{D}$, we obtain

$$0=\left({\nabla }_X{\eta }^i\right)Y-\left({\nabla }_Y{\eta }^i\right)X=-\beta g([X,Y],{\xi }_i),$$

which signifies the integrability of the distribution $\mathcal{D}$, or equivalently, $d{\eta }^i(X,Y)=0$, for all $i=1,\dots ,s$ and $X,Y\in \mathcal{D}$. By this and ${\mathcal{N}}_{ij}^{\left(4\right)}=0$, see (6), we find $d{\eta }^i=0$. Using (17) and (8), we obtain

$$3d\Phi (X,Y,Z)=2\beta \{\overline{\eta }\left(X\right)g(fZ,Y)+\overline{\eta }\left(Y\right)g(fX,Z)+\overline{\eta }\left(Z\right)g(fY,X)\}.$$

On the other hand, we have

$$3(\overline{\eta }\wedge \Phi )(X,Y,Z)=\overline{\eta }\left(X\right)g(fZ,Y)+\overline{\eta }\left(Y\right)g(fX,Z)+\overline{\eta }\left(Z\right)g(fY,X).$$

Thus, $d\Phi =2\beta \overline{\eta }\wedge \Phi$ is valid. By (5) with $S=f$ and (17), we obtain $[f,f]=0$; thus, using $d{\eta }^i=0$, yields ${\mathcal{N}}^{\left(1\right)}=0$. Finally, from (9), using (3) and (4), we obtain

$$\begin{array}{cc}\hfill & g(\left({\nabla }_{X}f\right)Y,Z)-{\displaystyle \frac{1}{2}}{\mathcal{N}}^{\left(5\right)}(X,Y,Z)=3\beta \left\{(\overline{\eta }\wedge \Phi )(X,fY,fZ)-(\overline{\eta }\wedge \Phi )(X,Y,Z)\right\}\hfill \\ \hfill & =\beta \left\{-\overline{\eta }\left(X\right)g(QZ,fY)+\overline{\eta }\left(X\right)g(Z,fY)-\overline{\eta }\left(Y\right)g(fX,Z)-\overline{\eta }\left(Z\right)g(X,fY)\right\}\hfill \\ \hfill & =\beta \left\{\overline{\eta }\left(Z\right)g(fX,Y)-\overline{\eta }\left(Y\right)g(fX,Z)-\overline{\eta }\left(X\right)g(fY,\tilde{Q}Z)\right\}.\hfill \end{array}$$

From this, using (17), we obtain ${\mathcal{N}}^{\left(5\right)}(X,Y,Z)=2\beta \overline{\eta }\left(X\right)g(fY,\tilde{Q}Z)$.

Conversely, using (3) and (21) in (9), we obtain

$$\begin{array}{cc}\hfill & 2g(\left({\nabla }_{X}f\right)Y,Z)=6\beta (\overline{\eta }\wedge \Phi )(X,fY,fZ)-6\beta (\overline{\eta }\wedge \Phi )(X,Y,Z)+2\beta \overline{\eta }\left(X\right)g(\tilde{Q}fY,Z)\hfill \\ \hfill & =2\beta \{\overline{\eta }\left(X\right)g(fY,\tilde{Q}Z)-\overline{\eta }\left(X\right)g(fY,QZ)-\overline{\eta }\left(X\right)g(fZ,Y)-\overline{\eta }\left(Y\right)g(fX,Z)\hfill \\ \hfill & -\overline{\eta }\left(Z\right)g(fY,X)\}=2\beta \{g(fX,Y)g(\overline{\xi },Z)-\overline{\eta }\left(Y\right)g(fX,Z)\};\hfill \end{array}$$

thus, (17) is true.   □

Definition 4 

([14]). An even-dimensional Riemannian manifold $(\overline{M},\overline{g})$ equipped with a skew-symmetric (1,1)-tensor J (other than a complex structure) is called a weak Hermitian manifold if $J^2$ is negative definite. If $\overline{\nabla }J=0$, where $\overline{\nabla }$ is the Levi-Civita connection of $\overline{g}$, then $(\overline{M},\overline{g},J)$ is called a weak Kähler manifold.

Remark 2. 

L. P. Eisenhart [18] proved that if a Riemannian manifold $(\overline{M},\overline{g})$ admits a parallel symmetric 2-tensor other than the constant multiple of $\overline{g}$, then it is reducible. Some authors studied and classified (skew-)symmetric parallel 2-tensors, e.g., [19,20].

Let $(\overline{M},\overline{g})$ be a Riemannian manifold. A twisted product ${\mathbb{R}}^s{\times }_{\sigma }\overline{M}$ is the product $M={\mathbb{R}}^s\times \overline{M}$ with the metric $g=d{t}^2\oplus {\sigma }^2\overline{g}$, where $\sigma >0$ is a smooth function on M. Set ${\xi }_i={\partial }_{t_i}$. The Levi-Civita connections, ∇ of g and $\overline{\nabla }$ of $\overline{g}$, are related as follows:

(i)

${\nabla }_{{\xi }_i}{\xi }_j=0$, ${\nabla }_X{\xi }_i={\nabla }_{{\xi }_i}X={\xi }_i(log\sigma )X$ for $X\perp Span\{{\xi }_1,\dots ,{\xi }_s\}$.

(ii)

${\pi }_{1\ast }\left({\nabla }_{X}Y\right)=-g(X,Y){\pi }_{1\ast }(\nabla log\sigma )$, where ${\pi }_1:M\to {\mathbb{R}}^s$ is the orthoprojector.

(iii)

${\pi }_{2\ast }\left({\nabla }_{X}Y\right)$ is the lift of ${\overline{\nabla }}_{X}Y$, where ${\pi }_2:M\to \overline{M}$ is the orthoprojector.

If $\sigma (t_1,\dots ,t_s)>0$ is a smooth function on ${\mathbb{R}}^s$, then we obtain a warped product ${\mathbb{R}}^s{\times }_{\sigma }\overline{M}$.

Theorem 2. 

A weak $\beta f$-KM is a locally twisted product ${\mathbb{R}}^s{\times }_{\sigma }\overline{M}$, where $\overline{M}(\overline{g},J)$ is a weak Hermitian manifold (a warped product if $\nabla \beta \perp \mathcal{D}$, and then $\overline{M}(\overline{g},J)$ is a weak Kähler manifold), $\overline{\xi }(log\sigma )=s\beta$, and $-\beta \overline{\xi }$ is the mean curvature vector of the distribution $\mathcal{D}$.

Proof. 

By (18), the distribution ${\mathcal{D}}^{\perp }$ is tangent to a foliation with flat totally geodesic leaves, and by the second equality of (7), the distribution $\mathcal{D}$ is tangent to a foliation. By (19), the Weingarten operator $A_{{\xi }_i}=-{(\nabla {\xi }_i)}^{\perp }(1\le i\le s)$ on $\mathcal{D}$ is conformal as follows: $A_{{\xi }_i}X=-\beta X(X\in \mathcal{D})$. Hence, $\mathcal{D}$ is tangent to a totally umbilical foliation with the mean curvature vector $H=-\beta \overline{\xi }$. By Theorem 1 [21], our manifold is a locally twisted product. By the above property (ii) of a twisted product, $\overline{\xi }(log\sigma )=s\beta$ is true. If $X\left(\beta \right)=0(X\in \mathcal{D})$, then we obtain a locally warped product; see [21] (Proposition 3). By (4), the (1,1)-tensor ${J=f|}_{\mathcal{D}}$ is skew-symmetric and $J^2$ is negative definite. To show $\overline{\nabla }J=0$, using (17), we find $\left({\overline{\nabla }}_{X}J\right)Y={\pi }_{2\ast }\left(\left({\nabla }_{X}f\right)Y\right)=0$ for $X,Y\in \mathcal{D}$. □

Example 1. 

(a) According to [18], a weak Kähler manifold with $J^2\ne c\overline{g}$, where $c\in \mathbb{R}$, is reducible. Take two (or even more) Hermitian manifolds $({\overline{M}}_i,{\overline{g}}_i,J_i)$, whence $J_i^2=-{\mathrm{id}}_i$. The product ${\prod }_i({\overline{M}}_i,{\overline{g}}_i,c_i{J}_i)$, where $c_i\ne 0$ are different constants, is a weak Hermitian manifold with $Q={⨁}_i{c}_i^2{\mathrm{id}}_i$. Moreover, if $({\overline{M}}_i,{\overline{g}}_i,J_i)$ are Kähler manifolds, then ${\prod }_i({\overline{M}}_i,{\overline{g}}_i,c_i{J}_i)$ is a weak Kähler manifold.

(b) Let $\overline{M}(\overline{g},J)$ be a weak Kähler manifold and $\sigma (t_1,\dots ,t_s)=c{e}^{\beta \sum t_i}$ a function on Euclidean space ${\mathbb{R}}^s$, where $c\ne 0$ and β are constants. Then, the warped product manifold $M={\mathbb{R}}^s{\times }_{\sigma }\overline{M}$ has a weak metric f-structure which satisfies (17). Using (5) with $S=J$ for a weak Kähler manifold, we obtain $[J,J]=0$; hence, ${\mathcal{N}}^{\left(1\right)}=0$ is true.

Corollary 1. 

A weak f-KM $M^{2n+s}(f,Q,{\xi }_i,{\eta }^i,g)$ is a locally warped product ${\mathbb{R}}^s(t_1,\dots ,t_s){\times }_{\sigma }\overline{M}$, where $\sigma =c{e}^{\sum t_i}(c=const\ne 0)$ and $\overline{M}(\overline{g},J)$ is a weak Kähler manifold.

To simplify the calculations in the rest of the paper, we assume that $\beta =const\ne 0$.

4. Curvature of Weak $\mathit{\beta }\mathit{f}$-KM

In this section, we study the curvature of weak $\beta f$-KM.

Proposition 3. 

For a weak $\beta f$-KM with $\beta =const$, we have (for all $X,Y\in {\mathfrak{X}}_M$)

$$\begin{array}{cc}\hfill & R_{X,Y}{\xi }_i={\beta }^2\left\{\overline{\eta }\left(X\right)Y-\overline{\eta }\left(Y\right)X+{\sum }_j\left(\overline{\eta }\left(Y\right){\eta }^j\left(X\right)-\overline{\eta }\left(X\right){\eta }^j\left(Y\right)\right){\xi }_j\right\},\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill & {Ric}^{♯}{\xi }_i=-2n{\beta }^2\overline{\xi },\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill & \left({\nabla }_{{\xi }_i}{Ric}^{♯}\right)X=-2\beta {Ric}^{♯}X-4n{\beta }^3\left\{sX-s{\sum }_j{\eta }^j\left(X\right){\xi }_j+\overline{\eta }\left(X\right)\overline{\xi }\right\},\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill & {\xi }_i\left(r\right)=-2\beta \{r+2sn(2n+1){\beta }^2\}(1\le i\le s),\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill & \left({\nabla }_X{Ric}^{♯}\right){\xi }_i=-\beta {Ric}^{♯}X-2n{\beta }^3\left\{sX-s{\sum }_j{\eta }^j\left(X\right){\xi }_j+\overline{\eta }\left(X\right)\overline{\xi }\right\}.\hfill \end{array}$$

(27)

Proof. 

Taking the covariant derivative of (19) along $Y\in {\mathfrak{X}}_M$, we obtain

$$\begin{array}{c}\hfill {\nabla }_Y{\nabla }_X{\xi }_i=-{\beta }^2\left\{\left(g(X,Y)-{\sum }_q{\eta }^q\left(Y\right){\eta }^q\left(X\right)\right)\overline{\xi }+\overline{\eta }\left(X\right)\left(Y-{\sum }_p{\eta }^p\left(Y\right){\xi }_p\right)\right\}.\end{array}$$

With the repeated application of (19) and the foregoing equation in the curvature tensor R of the Riemannian manifold, we obtain (23). Using a local orthonormal basis $\left(e_q\right)$ of the manifold, and the equality ${\sum }_{p,q}\left(\overline{\eta }\left(Y\right){\eta }^p\left(e_q\right)-\overline{\eta }\left(e_q\right){\eta }^p\left(Y\right)\right){\eta }^p\left(e_q\right)=(s-1)\overline{\eta }\left(Y\right)$, we derive from (23)

$$\begin{array}{cc}\hfill & g({Ric}^{♯}{\xi }_i,Y)={\sum }_{q}g(R_{e_q,Y}{\xi }_i,e_q)\hfill \\ \hfill & ={\beta }^2{\sum }_q\left\{\overline{\eta }\left(e_q\right)g(Y,e_q)-\overline{\eta }\left(Y\right)g(e_q,e_q)+\left(\overline{\eta }\left(Y\right){\eta }^p\left(e_q\right)-\overline{\eta }\left(e_q\right){\eta }^p\left(Y\right)\right){\eta }^p\left(e_q\right)\right\}\hfill \\ \hfill & ={\beta }^2\left\{g(Y,\overline{\xi })-(2n+s)\overline{\eta }\left(Y\right)+(s-1)\overline{\eta }\left(Y\right)\right\}=-2n{\beta }^{2}g(\overline{\xi },Y),\hfill \end{array}$$

from which we obtain (24). Next, using (19), we obtain

$$\begin{array}{c}\hfill \left({\pounds }_{{\xi }_i}g\right)(Y,Z)=g({\nabla }_Y{\xi }_i,Z)+g({\nabla }_Z{\xi }_i,Y)=2\beta \left(g(Y,Z)-{\sum }_j{\eta }^j\left(Y\right){\eta }^j\left(Z\right)\right).\end{array}$$

(28)

Taking the covariant derivative of (28) along X and using (19) gives

$$\begin{array}{c}\hfill \left({\nabla }_X{\pounds }_{{\xi }_i}g\right)(Y,Z)=2{\beta }^2\{{\sum }_j{\eta }^j\left(X\right)\left({\eta }^j\left(Y\right)\overline{\eta }\left(Z\right)+\overline{\eta }\left(Y\right){\eta }^j\left(Z\right)\right)-g(X,Y)\overline{\eta }\left(Z\right)-g(X,Z)\overline{\eta }\left(Y\right)\}\end{array}$$

for all $X,Y,Z\in {\mathfrak{X}}_M$. Using this in (12) with $V={\xi }_i$, we obtain

$$\begin{array}{cc}\hfill & g(({\pounds }_{{\xi }_i}\nabla )(X,Y),Z)+g(({\pounds }_{{\xi }_i}\nabla )(X,Z),Y)\hfill \\ \hfill & =2{\beta }^2\left\{{\sum }_j{\eta }^j\left(X\right)\left({\eta }^j\left(Y\right)\overline{\eta }\left(Z\right)+\overline{\eta }\left(Y\right){\eta }^j\left(Z\right)\right)-g(X,Y)\overline{\eta }\left(Z\right)-g(X,Z)\overline{\eta }\left(Y\right)\right\}.\hfill \end{array}$$

(29)

By a combinatorial computation, we find

$$\begin{array}{cc}\hfill & g(({\pounds }_{{\xi }_i}\nabla )(Y,Z),X)+g(({\pounds }_{{\xi }_i}\nabla )(Y,X),Z)\hfill \\ \hfill & =2{\beta }^2\left\{{\sum }_j{\eta }^j\left(Y\right)\left({\eta }^j\left(Z\right)\overline{\eta }\left(X\right)+\overline{\eta }\left(Z\right){\eta }^j\left(X\right)\right)-g(Y,Z)\overline{\eta }\left(X\right)-g(Y,X)\overline{\eta }\left(Z\right)\right\},\hfill \\ \hfill & g(({\pounds }_{{\xi }_i}\nabla )(Z,X),Y)+g(({\pounds }_{{\xi }_i}\nabla )(Z,Y),X)\hfill \\ \hfill & =2{\beta }^2\left\{{\sum }_j{\eta }^j\left(Z\right)\left({\eta }^j\left(X\right)\overline{\eta }\left(Y\right)+\overline{\eta }\left(X\right){\eta }^j\left(Y\right)\right)-g(Z,X)\overline{\eta }\left(Y\right)-g(Z,Y)\overline{\eta }\left(X\right)\right\}.\hfill \end{array}$$

Subtracting (29) from the sum of the last two equations gives

$$\begin{array}{c}\hfill ({\pounds }_{{\xi }_i}\nabla )(Y,Z)=2{\beta }^2\left\{{\sum }_j{\eta }^j\left(Y\right){\eta }^j\left(Z\right)-g(Y,Z)\right\}\overline{\xi }(Y,Z\in {\mathfrak{X}}_M).\end{array}$$

(30)

Taking the covariant derivative of (30) along X and using (19) gives

$$\begin{array}{cc}\hfill & ({\nabla }_X{\pounds }_{{\xi }_i}\nabla )(Y,Z)=2{\beta }^3\left\{\right[(g(X,Y)-{\sum }_j{\eta }^j\left(X\right){\eta }^j\left(Y\right))\overline{\eta }\left(Z\right)+(g(X,Z)\hfill \\ \hfill & -{\sum }_j{\eta }^j\left(X\right){\eta }^j\left(Z\right)\left)\overline{\eta }\left(Y\right)\right]\overline{\xi }+s({\sum }_j{\eta }^j\left(Y\right){\eta }^j\left(Z\right)-g(Y,Z))(X-{\sum }_p{\eta }^p\left(X\right){\xi }_p)\}.\hfill \end{array}$$

Using this in (14) with $V={\xi }_i$, we obtain

$$\begin{array}{cc}\hfill & {\left({\pounds }_{{\xi }_i}R\right)}_{X,Y}Z=2{\beta }^3\{\left(g(X,Z)-{\sum }_j{\eta }^j\left(X\right){\eta }^j\left(Z\right)\right)\left(\overline{\eta }\left(Y\right)\overline{\xi }+s(Y-{\sum }_q{\eta }^q\left(Y\right){\xi }_q)\right)\hfill \\ \hfill & -\left(g(Y,Z)-{\sum }_j{\eta }^j\left(Y\right){\eta }^j\left(Z\right)\right)\left(\overline{\eta }\left(X\right)\overline{\xi }+sX-s{\sum }_q{\eta }^q\left(X\right){\xi }_q\right)\}.\hfill \end{array}$$

(31)

Contracting (31) over X, we deduce

$$\begin{array}{c}\hfill ({\pounds }_{{\xi }_i}Ric)(Y,Z)={\sum }_{a}g({\left({\pounds }_{{\xi }_i}R\right)}_{e_a,Y}Z,e_a)=-4sn{\beta }^3\left(g(Y,Z)-{\sum }_j{\eta }^j\left(Y\right){\eta }^j\left(Z\right)\right).\end{array}$$

(32)

Taking the Lie derivative of equality $Ric(Y,Z)=g({Ric}^{♯}Y,Z)$, we obtain

$$\begin{array}{c}\hfill ({\pounds }_{{\xi }_i}Ric)(Y,Z)=\left({\pounds }_{{\xi }_i}g\right)({Ric}^{♯}Y,Z)+g(\left({\pounds }_{{\xi }_i}{Ric}^{♯}\right)Y,Z).\end{array}$$

(33)

On the other hand, replacing Y by ${Ric}^{♯}Y$ in (28) and using (24), we obtain

$$\begin{array}{cc}\hfill \left({\pounds }_{{\xi }_i}g\right)({Ric}^{♯}Y,Z)& =2\beta \left(g({Ric}^{♯}Y,Z)-g({Ric}^{♯}{\xi }_j,Y){\eta }^j\left(Z\right)\right)\hfill \\ \hfill & =2\beta \left(g({Ric}^{♯}Y,Z)+2n{\beta }^2\overline{\eta }\left(Y\right)\overline{\eta }\left(Z\right)\right).\hfill \end{array}$$

(34)

Applying (33) and (34) in (32), we obtain

$$\begin{array}{c}\hfill \left({\pounds }_{{\xi }_i}{Ric}^{♯}\right)Y=-2\beta {Ric}^{♯}Y-4n{\beta }^3\left\{sY-s{\sum }_j{\eta }^j\left(Y\right){\xi }_j+\overline{\eta }\left(Y\right)\overline{\xi }\right\}.\end{array}$$

(35)

Using (19), we calculate

$$\begin{array}{cc}\hfill & \left({\pounds }_{{\xi }_i}{Ric}^{♯}\right)Y={\pounds }_{{\xi }_i}\left({Ric}^{♯}Y\right)-{Ric}^{♯}{\pounds }_{{\xi }_i}Y\hfill \\ \hfill & ={\nabla }_{{\xi }_i}\left({Ric}^{♯}Y\right)-{\nabla }_{{Ric}^{♯}Y}{\xi }_i-{Ric}^{♯}{\nabla }_{{\xi }_i}Y+{Ric}^{♯}{\nabla }_Y{\xi }_i=\left({\nabla }_{{\xi }_i}{Ric}^{♯}\right)Y.\hfill \end{array}$$

Using this in (35), gives (25). Contracting (25), we obtain (26). Taking the covariant derivative of (24) along X and using (19) gives (27).   □

According to (24), weak $\beta f$-KM with $s>1$ cannot be Einstein manifolds.

Proposition 4. 

For an η-Einstein (15) weak $\beta f$-KM with $\beta =const$, we obtain

$$\begin{array}{cc}\hfill & \mathrm{Ric}=\left(s{\beta }^2+{\displaystyle \frac{r}{2n}}\right)g-\left((2n+s){\beta }^2+{\displaystyle \frac{r}{2n}}\right){\sum }_j{\eta }^j\otimes {\eta }^j-2n{\beta }^2{\sum }_{i\ne j}{\eta }^i\otimes {\eta }^j.\hfill \end{array}$$

(36)

Proof. 

Tracing (15) gives $r=(2n+s)a+sb$. Putting $X=Y={\xi }_i$ in (15) and using (24), yields $a+b=-2n{\beta }^2$. Thus, $a=s{\beta }^2+{\displaystyle \frac{r}{2n}}$ and $b=-(2n+s){\beta }^2-{\displaystyle \frac{r}{2n}}$, and (15) yields (36).   □

The following theorem generalizes ([17], Theorem 1) with $\beta \equiv 1$ and $Q=\mathrm{id}$.

Theorem 3. 

Let a weak $\beta f$-KM $M^{2n+s}(f,Q,{\xi }_i,{\eta }^i,g)$ with $\beta =const$ satisfy ${\nabla }_{\overline{\xi }}{Ric}^{♯}=0$. Then, $(M,g)$ is an η-Einstein manifold (15) of scalar curvature $r=-2sn(2n+1){\beta }^2$.

Proof. 

By (25) and conditions,

$${Ric}^{♯}X=-2n{\beta }^2\left\{sX-s{\sum }_j{\eta }^j\left(X\right){\xi }_j+\overline{\eta }\left(X\right)\overline{\xi }\right\},$$

is valid. Since (15) with $a=-2sn{\beta }^2$ and $b=2(s-1)n{\beta }^2$ is true, $(M,g)$ is an $\eta$-Einstein manifold of constant scalar curvature $r=-2sn(2n+1){\beta }^2$. For $s=1$, $(M,g)$ is an Einstein manifold.   □

5. $\mathit{\eta }$-RS on Weak $\mathit{\beta }\mathit{f}$-KM

Here, we study the interaction of weak $\beta f$-KM with $\eta$-RS and generalize some results in [14].

First, we derive the following two lemmas.

The following result generalizes Lemma 4 in [14].

Lemma 2. 

Let $(g,V)$ represent an η-RS (16) on a weak $\beta f$-KM $M^{2n+s}(f,Q,{\xi }_i,{\eta }^i,g)$ with $\beta =const$. Then,

$$\begin{array}{cc}\hfill & ({\pounds }_V\nabla )(X,{\xi }_i)=2\beta {Ric}^{♯}X+4n{\beta }^3\left\{sX-s{\sum }_j{\eta }^j\left(X\right){\xi }_j+\overline{\eta }\left(X\right)\overline{\xi }\right\},\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill & {\left({\pounds }_{V}R\right)}_{X,Y}{\xi }_i=2\beta \{\left({\nabla }_X{Ric}^{♯}\right)Y-\left({\nabla }_Y{Ric}^{♯}\right)X\}+2{\beta }^2\{\overline{\eta }\left(X\right){Ric}^{♯}Y-\overline{\eta }\left(Y\right){Ric}^{♯}X\}\hfill \\ \hfill & +4n{\beta }^4\left\{\left[sY-s{\sum }_j{\eta }^j\left(Y\right){\xi }_j+\overline{\eta }\left(Y\right)\overline{\xi }\right]\overline{\eta }\left(X\right)-[sX-s{\sum }_j{\eta }^j\left(X\right){\xi }_j+\overline{\eta }\left(X\right)\overline{\xi }]\overline{\eta }\left(Y\right)\right\},\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill & {\left({\pounds }_{V}R\right)}_{X,{\xi }_j}{\xi }_i=0(X\in {\mathfrak{X}}_M,1\le i,j\le s).\hfill \end{array}$$

(39)

Proof. 

Taking the covariant derivative of (16) along $Z\in {\mathfrak{X}}_M$ and using (20), we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2}}\left({\nabla }_Z{\pounds }_{V}g\right)(X,Y)=-\left({\nabla }_Z\mathrm{Ric}\right)(X,Y)+\beta [\mu +(\lambda +\mu )]\times \hfill \\ \hfill & \times \left\{\left(g(X,Z)-{\sum }_j{\eta }^j\left(X\right){\eta }^j\left(Z\right)\right)\overline{\eta }\left(Y\right)+\left(g(Y,Z)-{\sum }_j{\eta }^j\left(Y\right){\eta }^j\left(Z\right)\right)\overline{\eta }\left(X\right)\right\}\hfill \end{array}$$

(40)

for all $X,Y\in {\mathfrak{X}}_M$. Since the Riemannian metric tensor is parallel, $\nabla g=0$, it follows from (13) that

$$\begin{array}{c}\hfill \left({\nabla }_Z{\pounds }_{V}g\right)(X,Y)=g(({\pounds }_V\nabla )(Z,X),Y)+g(({\pounds }_V\nabla )(Z,Y),X).\end{array}$$

(41)

Plugging (41) into (40), we obtain

$$\begin{array}{cc}\hfill & g(({\pounds }_V\nabla )(Z,X),Y)+g(({\pounds }_V\nabla )(Z,Y),X)=-2\left({\nabla }_Z\mathrm{Ric}\right)(X,Y)+\beta [\mu +(s-1)(\lambda +\mu )]\times \hfill \\ \hfill & \times \left\{\left(g(X,Z)-{\sum }_j{\eta }^j\left(X\right){\eta }^j\left(Z\right)\right)\overline{\eta }\left(Y\right)+\left(g(Y,Z)-{\sum }_j{\eta }^j\left(Y\right){\eta }^j\left(Z\right)\right)\overline{\eta }\left(X\right)\right\}\hfill \end{array}$$

(42)

for all $X,Y,Z\in {\mathfrak{X}}_M$. Cyclically rearranging $X,Y$, and Z in (42), we obtain

$$\begin{array}{cc}\hfill g(({\pounds }_V\nabla )(X,Y),Z)& =\left({\nabla }_Z\mathrm{Ric}\right)(X,Y)-\left({\nabla }_X\mathrm{Ric}\right)(Y,Z)-\left({\nabla }_Y\mathrm{Ric}\right)(Z,X)\hfill \\ \hfill & +2\beta [\mu +(s-1)(\lambda +\mu )]\left(g(X,Y)-{\sum }_j{\eta }^j\left(X\right){\eta }^j\left(Y\right)\right)\overline{\eta }\left(Z\right).\hfill \end{array}$$

(43)

Substituting $Y={\xi }_i$ in (43) yields the following:

$$\begin{array}{cc}\hfill g(({\pounds }_V\nabla )(X,{\xi }_i),Z)& =\left({\nabla }_Z\mathrm{Ric}\right)(X,{\xi }_i)-\left({\nabla }_X\mathrm{Ric}\right)({\xi }_i,Z)-\left({\nabla }_{{\xi }_i}\mathrm{Ric}\right)(Z,X).\hfill \end{array}$$

Applying (25) and (27) to this, we obtain (37). Next, using (19) in the covariant derivative of (37) along $Y\in {\mathfrak{X}}_M$, and calculating ${\nabla }_Y\left(s{\sum }_j{\eta }^j\left(X\right){\xi }_j-\overline{\eta }\left(X\right)\overline{\xi }\right)=0$, yields

$$\begin{array}{cc}\hfill & \left({\nabla }_Y({\pounds }_V\nabla )\right)(X,{\xi }_i)+\beta ({\pounds }_V\nabla )(X,Y)=2\beta \left({\nabla }_Y{Ric}^{♯}\right)X+2{\beta }^2\overline{\eta }\left(Y\right){Ric}^{♯}X\hfill \\ \hfill & +4n\overline{\eta }\left(Y\right){\beta }^4\left\{sX-s{\sum }_j{\eta }^j\left(X\right){\xi }_j+\overline{\eta }\left(X\right)\overline{\xi }\right\}\hfill \end{array}$$

for any $X\in {\mathfrak{X}}_M$. Plugging this in (14) with $Z={\xi }_i$, we obtain (38) for all $X,Y\in {\mathfrak{X}}_M$. Substituting $Y={\xi }_j$ in (38) gives

$$\begin{array}{cc}\hfill {\left({\pounds }_{V}R\right)}_{X,{\xi }_j}{\xi }_i& =2\beta \{\left({\nabla }_X{Ric}^{♯}\right){\xi }_j-\left({\nabla }_{{\xi }_j}{Ric}^{♯}\right)X\}+2{\beta }^2\{\overline{\eta }\left(X\right){Ric}^{♯}{\xi }_j-{Ric}^{♯}X\}\hfill \\ \hfill & -4sn{\beta }^4\left(X-{\sum }_j{\eta }^j\left(X\right){\xi }_j\right).\hfill \end{array}$$

(44)

Then, using (24), (25), and (27) in (44) yields (39).   □

The following lemma generalizes Lemma 6 in [14].

Lemma 3. 

Let $(g,V)$ represent an η-RS (16), a weak $\beta f$-KM $M^{2n+s}(f,Q,{\xi }_i,{\eta }^i,g)$ with $\beta =const$. Then, $\lambda +\mu =-2n{\beta }^2$ is true.

Proof. 

Using (23) and $g(R_{X,{\xi }_i}Z,W)=-g(R_{Z,W}{\xi }_i,X)$, we derive

$$\begin{array}{c}\hfill R_{X,{\xi }_i}Z={\beta }^2\left\{g(X,Z)\overline{\xi }-\overline{\eta }\left(Z\right)X+{\sum }_j{\eta }^j\left(X\right)\left(\overline{\eta }\left(Z\right){\xi }_j-{\eta }^j\left(Z\right)\overline{\xi }\right)\right\}.\end{array}$$

(45)

Taking the Lie derivative along V of

$$R_{X,{\xi }_j}{\xi }_i={\beta }^2\left\{{\sum }_k{\eta }^k\left(X\right){\xi }_k-X\right\},$$

see (45), with $Z={\xi }_j$ (or (23) with $Y={\xi }_j$), and using (23) and (45) gives

$$\begin{array}{cc}\hfill & {\left({\pounds }_{V}R\right)}_{X,{\xi }_j}{\xi }_i=-{\beta }^2\left\{\overline{\eta }\left(X\right){\pounds }_V{\xi }_j-\overline{\eta }\left({\pounds }_V{\xi }_j\right)X+{\sum }_k\{\overline{\eta }\left({\pounds }_V{\xi }_j\right){\eta }^k\left(X\right)-\overline{\eta }\left(X\right){\eta }^k\left({\pounds }_V{\xi }_j\right)\}{\xi }_k\right\}\hfill \\ \hfill & -{\beta }^2\left\{g(X,{\pounds }_V{\xi }_i)\overline{\xi }-\overline{\eta }\left({\pounds }_V{\xi }_i\right)X+{\sum }_k{\eta }^k\left(X\right)\{\overline{\eta }\left({\pounds }_V{\xi }_i\right){\xi }_k-{\eta }^k\left({\pounds }_V{\xi }_i\right)\overline{\xi }\}\right\}\hfill \\ \hfill & +{\beta }^2{\sum }_k\left\{\left({\pounds }_V{\eta }^k\right)\left(X\right){\xi }_k+{\eta }^k\left(X\right){\pounds }_V{\xi }_k\right\}.\hfill \end{array}$$

(46)

Here we used ${\pounds }_V\left(R_{X,{\xi }_j}{\xi }_i\right)={\left({\pounds }_{V}R\right)}_{X,{\xi }_j}{\xi }_i+R_{X,{\pounds }_V{\xi }_j}{\xi }_i+R_{X,{\xi }_j}{\pounds }_V{\xi }_i$ with

$$\begin{array}{cc}\hfill & R_{X,{\pounds }_V{\xi }_j}{\xi }_i={\beta }^2\left\{\overline{\eta }\left(X\right){\pounds }_V{\xi }_j-\overline{\eta }\left({\pounds }_V{\xi }_j\right)X+{\sum }_k\{\overline{\eta }\left({\pounds }_V{\xi }_j\right){\eta }^k\left(X\right)-\overline{\eta }\left(X\right){\eta }^k\left({\pounds }_V{\xi }_j\right)\}{\xi }_k\right\},\hfill \\ \hfill & R_{X,{\xi }_j}{\pounds }_V{\xi }_i={\beta }^2\left\{g(X,{\pounds }_V{\xi }_i)\overline{\xi }-\overline{\eta }\left({\pounds }_V{\xi }_i\right)X+{\sum }_k{\eta }^k\left(X\right)\{\overline{\eta }\left({\pounds }_V{\xi }_i\right){\xi }_k-{\eta }^k\left({\pounds }_V{\xi }_i\right)\overline{\xi }\}\right\}.\hfill \end{array}$$

In view of (39), Equation (46) divided by ${\beta }^2$ becomes

$$\begin{array}{cc}\hfill & {\sum }_k\left({\pounds }_V{\eta }^k\right)\left(X\right){\xi }_k+{\sum }_k{\eta }^k\left(X\right){\pounds }_V{\xi }_k-\overline{\eta }\left(X\right){\pounds }_V{\xi }_j+\overline{\eta }\left({\pounds }_V{\xi }_j\right)X\hfill \\ \hfill & -\overline{\eta }\left({\pounds }_V{\xi }_j\right){\sum }_k{\eta }^k\left(X\right){\xi }_k+\overline{\eta }\left(X\right){\sum }_k{\eta }^k\left({\pounds }_V{\xi }_j\right){\xi }_k-g(X,{\pounds }_V{\xi }_i)\overline{\xi }+\overline{\eta }\left({\pounds }_V{\xi }_i\right)X\hfill \\ \hfill & -\overline{\eta }\left({\pounds }_V{\xi }_i\right){\sum }_k{\eta }^k\left(X\right){\xi }_k+{\sum }_k{\eta }^k\left(X\right){\eta }^k\left({\pounds }_V{\xi }_i\right)\overline{\xi }=0.\hfill \end{array}$$

(47)

For $X\in \mathcal{D}$, Equation (47) reduces to the following:

$$\begin{array}{c}\hfill {\sum }_k\left({\pounds }_V{\eta }^k\right)\left(X\right){\xi }_k+\overline{\eta }\left({\pounds }_V{\xi }_i\right)X+\overline{\eta }\left({\pounds }_V{\xi }_j\right)X-g(X,{\pounds }_V{\xi }_i)\overline{\xi }=0.\end{array}$$

(48)

Taking the $\mathcal{D}$- and ${\mathcal{D}}^{\perp }$- components of (48) yields for all $i,k$,

$$\begin{array}{c}\hfill \overline{\eta }\left({\pounds }_V{\xi }_i\right)=0,\left({\pounds }_V{\eta }^k\right)\left(X\right)=g(X,{\pounds }_V{\xi }_i)(X\in {\mathfrak{X}}_M).\end{array}$$

(49)

Using (24), we write (16) with $Y={\xi }_k$ as

$$\begin{array}{c}\hfill \left({\pounds }_{V}g\right)(X,{\xi }_k)=2(2n{\beta }^2+\lambda +\mu )\overline{\eta }\left(X\right).\end{array}$$

(50)

Using the equality

$$\left({\pounds }_{V}g\right)({\xi }_i,{\xi }_k)=-g({\xi }_i,{\pounds }_V{\xi }_k)-g({\xi }_k,{\pounds }_V{\xi }_i)=-{\eta }^i\left({\pounds }_V{\xi }_k\right)-{\eta }^k\left({\pounds }_V{\xi }_i\right),$$

Equation (50) for $X={\xi }_i$ reduces to

$$\begin{array}{c}\hfill {\eta }^i\left({\pounds }_V{\xi }_k\right)+{\eta }^k\left({\pounds }_V{\xi }_i\right)=-2(2n{\beta }^2+\lambda +\mu )\Rightarrow \overline{\eta }\left({\pounds }_V\overline{\xi }\right)=-s^2(2n{\beta }^2+\lambda +\mu ).\end{array}$$

(51)

Comparing (51) with (49), we achieve the result $\lambda +\mu =-2n{\beta }^2$.   □

The following theorem generalizes Theorem 3 (where $s=1$) in [14].

Theorem 4. 

Let $(g,V)$ represent an η-RS (16) on a weak η-Einstein (15) $\beta f$-KM with $\beta =const$. If $s>1$, then we assume $V\in \mathcal{D}$. Then, $a=-2sn{\beta }^2$, $b=2(s-1)n{\beta }^2$, and the scalar curvature is equal to $r=-2sn(2n+1){\beta }^2$; moreover, if $s=1$, then the manifold is an Einstein manifold.

Proof. 

Taking the covariant derivative of (36) along Y and using (19), we obtain

$$\begin{array}{cc}\hfill & \left({\nabla }_Y{Ric}^{♯}\right)X={\displaystyle \frac{Y\left(r\right)}{2n}}\left\{X-{\sum }_j{\eta }^j\left(X\right){\xi }_j\right\}\hfill \\ \hfill & -\left((2n+s){\beta }^2+{\displaystyle \frac{r}{2n}}\right)\beta \left\{g(X,Y)\overline{\xi }+\overline{\eta }\left(X\right)\left(Y-{\sum }_j{\eta }^j\left(Y\right){\xi }_j\right)-{\sum }_i{\eta }^i\left(X\right){\eta }^i\left(Y\right)\overline{\xi }\right\}\hfill \\ \hfill & -2(s-1)n{\beta }^3\left\{\left(g(X,Y)-{\sum }_p{\eta }^p\left(X\right){\eta }^p\left(Y\right)\right)\overline{\xi }+\overline{\eta }\left(X\right)\left(Y-{\sum }_p{\eta }^p\left(Y\right){\xi }_p\right)\right\}.\hfill \end{array}$$

(52)

Contracting (52) over Y and using the well-known identity ${\mathrm{div}}_{g}Ric={\displaystyle \frac{1}{2}}dr$, we obtain

$$\begin{array}{c}\hfill (n-1)X\left(r\right)=-{\sum }_i{\eta }^i\left(X\right){\xi }_i\left(r\right)-2n\beta \left\{r+2sn(2n+1){\beta }^2\right\}\overline{\eta }\left(X\right).\end{array}$$

(53)

Using (26) in (53) yields

$$\begin{array}{c}\hfill X\left(r\right)=-2\beta \left\{r+2sn(2n+1){\beta }^2\right\}\overline{\eta }\left(X\right)(X\in {\mathfrak{X}}_M),\end{array}$$

(54)

hence r is constant along the leaves of $\mathcal{D}$. Using (36) and (52) in (38), and then applying (54) and Lemma 2, gives ${\left({\pounds }_{V}R\right)}_{X,Y}{\xi }_i=0$ for all $X,Y\in {\mathfrak{X}}_M$. Therefore, we obtain

$$\begin{array}{c}\hfill ({\pounds }_{V}Ric)(Y,{\xi }_i)=\mathrm{trace}\{X\to {\left({\pounds }_{V}R\right)}_{X,Y}{\xi }_i\}=0.\end{array}$$

(55)

Equation (24) gives $Ric(Y,{\xi }_i)=-2n{\beta }^2\overline{\eta }\left(Y\right)$. Taking its Lie derivative along V yields

$$({\pounds }_{V}Ric)(Y,{\xi }_i)+Ric(Y,{\pounds }_V{\xi }_i)=-2n{\beta }^2\left({\pounds }_V\overline{\eta }\right)\left(Y\right)$$

for all $Y\in {\mathfrak{X}}_M$. Inserting (55) in the preceding equation, we have

$$\begin{array}{c}\hfill Ric(Y,{\pounds }_V{\xi }_i)=-2n{\beta }^2\left\{\left({\pounds }_{V}g\right)(Y,\overline{\xi })+g(Y,{\pounds }_V\overline{\xi })\right\}=-2n{\beta }^{2}g(Y,{\pounds }_V\overline{\xi }).\end{array}$$

(56)

In view of (16), (24), and (36), Equation (56) becomes

$$\begin{array}{cc}\hfill & \left(s{\beta }^2+{\displaystyle \frac{r}{2n}}\right)g(Y,{\pounds }_V{\xi }_i)-\left((2n+s){\beta }^2+{\displaystyle \frac{r}{2n}}\right){\sum }_p{\eta }^p\left(Y\right){\eta }^p\left({\pounds }_V{\xi }_i\right)\hfill \\ \hfill & -2n{\beta }^2{\sum }_{p\ne q}{\eta }^p\left(Y\right){\eta }^q\left({\pounds }_V{\xi }_i\right)=-2n{\beta }^{2}g(Y,{\pounds }_V\overline{\xi }).\hfill \end{array}$$

(57)

For $Y\in \mathcal{D}$, (57) reduces to the following:

$$\begin{array}{c}\hfill \left(s{\beta }^2+{\displaystyle \frac{r}{2n}}\right)g(Y,{\pounds }_V{\xi }_i)=-2n{\beta }^{2}g(Y,{\pounds }_V\overline{\xi })(Y\in \mathcal{D}),\end{array}$$

(58)

from which we obtain

$$\begin{array}{c}\hfill \left(2sn(2n+1){\beta }^2+r\right)g(Y,{\pounds }_V\overline{\xi })=0(Y\in \mathcal{D}).\end{array}$$

(59)

Case I. Let us assume that $(M,g)$ has constant scalar curvature $r=-2sn(2n+1){\beta }^2$. Then, by (36), we obtain

$${\mathrm{Ric}}^{♯}X=-2sn{\beta }^{2}X+2(s-1)n{\beta }^2{\eta }^j\left(X\right){\xi }_j-2n{\beta }^2{\sum }_{i\ne j}{\eta }^i\left(X\right){\xi }_j.$$

Hence, $(M,g)$ is an $\eta$-Einstein manifold (15) with $a=-2sn{\beta }^2$ and $b=2(s-1)n{\beta }^2$.

Case II. Let us assume that $s(2n+1){\beta }^2+{\displaystyle \frac{r}{2n}}\ne 0$ on an open set $\mathcal{U}$ of M. Then, ${\pounds }_V\overline{\xi }=[V,\overline{\xi }]=0$ on $\mathcal{U}$, see (59) and (7). Let us show that this leads to a contradiction. If ${\pounds }_V{\xi }_i\ne 0$ for some i, then from (58) and ${\pounds }_V{\xi }_i\in \mathcal{D}$, see (7), we obtain $s{\beta }^2+{\displaystyle \frac{r}{2n}}=0$. Using the previous equality in (36), we obtain ${Ric}^{♯}X=0$ for all $X\in \mathcal{D}$. By this and (24), the following is true:

$${Ric}^{♯}X=-2n{\beta }^2\overline{\eta }\left(X\right)\overline{\xi }.$$

Therefore, using Lemma 1, we obtain

$$\left({\nabla }_{{\xi }_i}{Ric}^{♯}\right)X=-2n{\beta }^2{\nabla }_{{\xi }_i}\left(\overline{\eta }\left(X\right)\overline{\xi }\right)-{Ric}^{♯}\left({\nabla }_{{\xi }_i}X\right)=0(1\le i\le s).$$

By the previous equality, $\left({\nabla }_{\overline{\xi }}{Ric}^{♯}\right)X=0$ is true, hence by Theorem 3, we obtain $r=-2sn(2n+1){\beta }^2$—a contradiction. Therefore, ${\pounds }_V{\xi }_i=[V,{\xi }_i]=0$ for all i on some open set $\mathcal{V}\subset \mathcal{U}$. It follows that

$$\begin{array}{c}\hfill {\nabla }_{{\xi }_i}V={\nabla }_V{\xi }_i=\beta \{V-{\sum }_p{\eta }^p\left(V\right){\xi }_p\},\end{array}$$

(60)

where we have used (19). Replacing Y by ${\xi }_i$ in (10) and using (19), (23), and (60), we obtain

$$\begin{array}{c}\hfill ({\pounds }_V\nabla )(X,{\xi }_i)=-{\beta }^2\{g(X,V)-{\sum }_j{\eta }^j\left(X\right){\eta }^j\left(V\right)\}\overline{\xi }.\end{array}$$

(61)

Further, from (37) and (61), we obtain

$$\begin{array}{c}\hfill {Ric}^{♯}X=-2n{\beta }^2\left\{s\left(X-{\sum }_j{\eta }^j\left(X\right){\xi }_j\right)+\overline{\eta }\left(X\right)\overline{\xi }\right\}-(\beta /2)\{g(X,V)-{\sum }_j{\eta }^j\left(X\right){\eta }^j\left(V\right)\}\overline{\xi }.\end{array}$$

Comparing the $\mathcal{D}$-components of the above equation and (36), yields the following contradiction: $r=-2sn(2n+1){\beta }^2$ on $\mathcal{V}$.   □

Definition 5. 

A vector field X on a weak metric f-manifold $M(f,Q,{\xi }_i,{\eta }^i,g)$ is called a contact vector field, if there exists a function $\rho \in C^{\infty }\left(M\right)$ such that

$$\begin{array}{c}\hfill {\pounds }_X{\eta }^i=\rho {\eta }^i(1\le i\le s),\end{array}$$

(62)

and if $\rho =0$, i.e., the flow of X preserves ${\eta }^i$, then X is called a strictly contact vector field.

We study the interaction of a weak $\beta f$-KM with an $\eta$-RS whose potential vector field V is either a contact vector field or collinear to $\overline{\xi }$. The following theorem generalizes Theorem 4 ($s=1$) in [14].

Theorem 5. 

Let $M^{2n+s}(f,Q,{\xi }_i,{\eta }^i,g)$ be a weak $\beta f$-KM with $\beta =const$. If $(g,V)$ represents an η-RS (16), with a contact potential vector field V, then V is strictly contact and the manifold is η-Einstein (15) with $a=-2sn{\beta }^2,b=2(s-1)n{\beta }^2$ of scalar curvature $r=-2sn(2n+1){\beta }^2$.

Proof. 

Taking the Lie derivative of ${\eta }^i\left(X\right)=g(X,{\xi }_i)$ along V and using $\left({\pounds }_{V}g\right)(X,{\xi }_i)=0$, see (50), and (62), we obtain ${\pounds }_V{\xi }_i=\rho {\xi }_i$. Then, using ${\pounds }_V{\xi }_i\in \mathcal{D}$, see the second equality in (7), we obtain $\rho =0$. Thus, ${\pounds }_V{\xi }_i=0$ and V is a strictly contact vector field. Furthermore, (62) gives ${\pounds }_V{\eta }^j=0$. Setting $Y={\xi }_i$ in (11) and using (19) and the equality $\left({\pounds }_V{\eta }^j\right)\left(X\right)=V\left({\eta }^j\left(X\right)\right)-{\eta }^j\left([V,X]\right)$, we find

$$\begin{array}{cc}\hfill & ({\pounds }_V\nabla )(X,{\xi }_i)=\beta {\pounds }_V(X-{\sum }_p{\eta }^p\left(X\right){\xi }_p)-\beta ({\pounds }_{V}X-{\sum }_p{\eta }^p\left({\pounds }_{V}X\right){\xi }_p)\hfill \\ \hfill & =-\beta {\sum }_p\{\left({\pounds }_V{\eta }^p\right)\left(X\right){\xi }_p+{\eta }^p\left(X\right){\pounds }_V{\xi }_p\}+\beta {\sum }_p{\eta }^p\left({\pounds }_{V}X\right){\xi }_p.\hfill \end{array}$$

(63)

From (63), since ${\pounds }_V{\eta }^p={\pounds }_V{\xi }_p=0$ is true and the distribution $\mathcal{D}$ is involutive, i.e., ${\pounds }_{Y}X\in \mathcal{D}(X,Y\in \mathcal{D})$, we obtain $({\pounds }_V\nabla )(X,{\xi }_i)=0$. Using (37), we obtain

$$\begin{array}{cc}\hfill {Ric}^{♯}X& =-2n{\beta }^2\left\{sX-s{\sum }_j{\eta }^j\left(X\right){\xi }_j+\overline{\eta }\left(X\right)\overline{\xi }\right\}.\hfill \end{array}$$

Therefore, our $(M,g)$ is an $\eta$-Einstein manifold (15) with $a=-2sn{\beta }^2,b=2(s-1)n{\beta }^2$ and constant scalar curvature $r=-2sn(2n+1){\beta }^2$.   □

The following theorem generalizes Theorem 5 in [14] (where $s=1$).

Theorem 6. 

Let $M^{2n+s}(f,Q,{\xi }_i,{\eta }^i,g)$, be a weak $\beta f$-KM with $\beta =const$. If $(g,V)$ represents an η-RS (16) with a potential vector field V collinear to $\overline{\xi }$: $V=\delta \overline{\xi }$ for a smooth function $\delta \ne 0$ on M, then $\delta =const$ and the manifold is η-Einstein (15), with $a=-2sn{\beta }^2$ and $b=2(s-1)n{\beta }^2$ for constant scalar curvature $r=-2sn(2n+1){\beta }^2$.

Proof. 

Using (17) in the covariant derivative of $V=\delta \overline{\xi }$ with any $X\in {\mathfrak{X}}_M$ yields

$${\nabla }_{X}V=X\left(\delta \right)\overline{\xi }+\delta \beta (X-{\sum }_j{\eta }^j\left(X\right){\xi }_j)(X\in {\mathfrak{X}}_M).$$

Using this and calculations

$$\begin{array}{cc}\hfill & \left({\pounds }_{\delta \overline{\xi }}g\right)(X,Y)=\delta \left({\pounds }_{\overline{\xi }}g\right)(X,Y)+X\left(\delta \right)\overline{\eta }\left(Y\right)+Y\left(\delta \right)\overline{\eta }\left(X\right),\hfill \\ \hfill & \left({\pounds }_{\overline{\xi }}g\right)(X,Y)=2s\beta \{g(X,Y)-{\sum }_j{\eta }^j\left(X\right){\eta }^j\left(Y\right)\},\hfill \end{array}$$

we transform the $\eta$-RS Equation (16) into

$$\begin{array}{cc}\hfill & 2\mathrm{Ric}(X,Y)=-X\left(\delta \right)\overline{\eta }\left(Y\right)-Y\left(\delta \right)\overline{\eta }\left(X\right)+2(\lambda -\delta \beta )g(X,Y)\hfill \\ \hfill & +2(\delta \beta +\mu ){\sum }_j{\eta }^j\left(X\right){\eta }^j\left(Y\right)-4n{\beta }^2{\sum }_{i\ne j}{\eta }^i\left(X\right){\eta }^j\left(Y\right),X,Y\in {\mathfrak{X}}_M.\hfill \end{array}$$

(64)

Inserting $X=Y={\xi }_i$ in (64) and using (24) and $\lambda +\mu =-2n{\beta }^2$, see Lemma 3, we obtain ${\xi }_i\left(\delta \right)=0$. It follows from (64) and (24) that $X\left(\delta \right)=0(X\in \mathcal{D})$. Thus, $\delta$ is constant on M, and (64) reads as

$$\begin{array}{c}\hfill \mathrm{Ric}=(\lambda -\delta \beta )g+(\delta \beta +\mu ){\sum }_j{\eta }^j\otimes {\eta }^j-2n{\beta }^2{\sum }_{i\ne j}{\eta }^i\otimes {\eta }^j.\end{array}$$

This shows that $(M,g)$ is an $\eta$-Einstein manifold with $a=\lambda -\delta \beta$ and $b=\mu +\delta \beta$ in (15). Therefore, from Proposition 4, we conclude that $\lambda =\delta \beta -2sn{\beta }^2$, $\mu =-\delta \beta +2(s-1)n{\beta }^2$, and the scalar curvature of $(M,g)$ is $r=-2sn(2n+1){\beta }^2$.   □

6. Conclusions

The author poses the following question: How do RS interact with weak f-structures (recently introduced by the author and R. Wolak)? The paper defines weak $\beta f$-KM as a distinguished class of weak metric f-manifolds $M^{2n+s}$ (for $s>1$, they cannot be Einstein manifolds) and studies when a weak $\beta f$-KM equipped with an $\eta$-RS structure (16) carries an $\eta$-Einstein metric (15). Some results on the interaction of RS and KM $M^{2n+1}$ have been extended to $\beta f$-KM $M^{2n+s}$.