Geometry of Weak Metric f-Manifolds: A Survey

Abstract

The interest of geometers in f-structures is motivated by the study of the dynamics of contact foliations, as well as their applications in physics. A weak f-structure on a smooth manifold, introduced by the author and R. Wolak, generalizes K. Yano’s f-structure. This generalization allows us to revisit classical theory and discover applications of Killing vector fields, totally geodesic foliations, Ricci-type solitons, and Einstein-type metrics. This article reviews the results regarding weak metric f-manifolds and their distinguished classes.

1. Introduction

Contact geometry has garnered increasing interest due to its significant role in physics, e.g., ref. [1]. An important class of contact metric manifolds are K-contact manifolds (thus, the structural vector field is Killing) with the following two subclasses: Sasakian and cosymplectic manifolds. Every cosymplectic manifold is locally the product of $\mathbb{R}$ and a Kähler manifold. A Riemannian manifold $(M^{2n+1},g)$ with contact 1-form $\eta$ is Sasakian if its Riemannian cone $M\times {\mathbb{R}}^{>0}$ with metric $t^{2}g+d{t}^2$ is a Kähler manifold. Recent research has been driven by the intriguing question of how Ricci solitons—self-similar solutions of the Ricci flow equation—can be significant for the geometry of contact metric manifolds. Some studies have explored the conditions under which a contact metric manifold equipped with a Ricci-type soliton structure carries a canonical metric, such as an Einstein-type metric, e.g., refs. [2,3,4,5,6].

An f-structure introduced by K. Yano on a smooth manifold $M^{2n+s}$ serves as a higher-dimensional analog of almost complex structures ($s=0$) and almost contact structures ($s=1$). This structure is defined by a (1,1)-tensor f of rank $2n$, such that $f^3+f=0$, see [7,8,9]. The tangent bundle splits into two complementary subbundles as follows: $TM=f\left(TM\right)\oplus kerf$. The restriction of f to the $2n$-dimensional distribution $f\left(TM\right)$ defines a complex structure. The existence of the f-structure on $M^{2n+s}$ is equivalent to a reduction of the structure group to $U\left(n\right)\times O\left(s\right)$, see [10]. A submanifold M of an almost complex manifold $(\overline{M},J)$ that satisfies the condition $dim(T_{x}M\cap J\left(T_{x}M\right))=const>0$ naturally possesses an f-structure, see [11]. An f-structure is a special case of an almost product structure, defined by two complementary orthogonal distributions of a Riemannian manifold $(M,g)$, with Naveira’s 36 distinguished classes, see [12]. Foliations appear when one or both distributions are involutive. An interesting case occurs when the subbundle $kerf$ is parallelizable, leading to a framed f-structure for which the reduced structure group is $U\left(n\right)\times {\mathrm{id}}_s$. In this scenario, there exist vector fields ${\left\{{\xi }_i\right\}}_{1\le i\le s}$ spanning $kerf$ with dual 1-forms ${\left\{{\eta }^i\right\}}_{1\le i\le s}$, satisfying $f^2=-\mathrm{id}+{\sum }_i{\eta }^i\otimes {\xi }_i$. Compatible metrics, i.e., $g(fX,fY)=g(X,Y)-{\sum }_i{\eta }^i\left(X\right){\eta }^i\left(Y\right)$, exist on any framed f-manifold, and we obtain the metric f-structure, see [7,8,9,10,13,14,15,16].

To generalize concepts and results from almost contact geometry to metric f-manifolds, geometers have introduced and studied various broad classes of metric f-structures. A notable class is Kenmotsu f-manifolds, see [17] (Kenmotsu manifolds when $s=1$, see [18]), characterized in terms of warped products of ${\mathbb{R}}^s$ and a Kähler manifold. A metric f-manifold is termed a $\mathcal{K}$-manifold if it is normal and $d\Phi =0$, where $\Phi (X,Y):=g(X,fY)$. Two important subclasses of $\mathcal{K}$-manifolds are $\mathcal{C}$-manifolds if $d{\eta }^i=0$ and $\mathcal{S}$-manifolds if $d{\eta }^i=\Phi$ for any i, see [10]. Omitting the normality condition, we obtain almost $\mathcal{K}$-manifolds, almost $\mathcal{S}$-manifolds and almost $\mathcal{C}$-manifolds, e.g., refs. [19,20,21]. The distribution $kerf$ of a $\mathcal{K}$-manifold is tangent to a $\mathfrak{g}$-foliation with flat totally geodesic leaves. An f-K-contact manifold is an almost $\mathcal{S}$-manifold, whose characteristic vector fields are Killing vector fields; the structure is an intermediate between an almost $\mathcal{S}$-structure and S-structure, see [15,22]. Note that there are no Einstein metrics on f-K-contact manifolds. The interest of geometers in f-structures is also motivated by the study of the dynamics of contact foliations. Contact foliations generalize, to higher dimensions, the flow of the Reeb vector field on contact manifolds, and $\mathcal{K}$-structures are a particular case of uniform contact structures, see [23,24].

Extrinsic geometry is concerned with the properties of submanifolds (as being totally geodesic) that depend on the second fundamental form, which, roughly speaking, describes how the submanifolds are located in the ambient Riemannian manifold. The extrinsic geometry of foliations (i.e., involutive distributions) is a field of Riemannian geometry which studies the properties expressed by the second fundamental tensor of the leaves. Although the Riemann tensor belongs to the field of intrinsic geometry, a special component called mixed sectional curvature is a part of the extrinsic geometry of foliations. Totally geodesic foliations have simple extrinsic geometry and appear on manifolds with degenerate tensor fields, see [25]. A key problem posed in this context in [7] is identifying suitable structures on manifolds that lead to totally geodesic foliations.

In [26], we initiated the study of weak f-structures on a smooth $(2n+s)$-dimensional manifold, that is, the linear complex structure on the subbundle $\mathcal{D}=f\left(TM\right)$ of a metric f-structure is replaced with a nonsingular skew-symmetric tensor. These generalize the metric f-structure (the weak almost contact metric structure for $s=1$, see [27]) and its satellites allow us to look at classical theory in a new way and find new applications of Killing, totally geodesic foliations, Einstein-type metrics, and Ricci-type solitons.

The article reviews the results of our works [25,26,28,29,30,31,32] regarding the geometry of weak metric f-manifolds and their distinguished classes. It is organized as follows. Section 2 (following the Introduction) presents the basics of weak metric f-manifolds and introduces their important subclasses. It also investigates the normality condition and derives the covariant derivative of f using a new tensor ${\mathcal{N}}^{\left(5\right)}$, showing that the distribution $kerf$ of a weak almost $\mathcal{S}$-manifold and a weak almost $\mathcal{C}$-manifold is tangent to a $\mathfrak{g}$-foliation with an abelian Lie algebra. Section 3 presents the basic characteristics of weak almost $\mathcal{S}$-manifolds and shows that these manifolds are endowed with totally geodesic foliations. Section 4 discusses weak $\mathcal{C}$-structures and shows that the weak metric f-structure is a weak $\mathcal{S}$-structure if and only if it is an $\mathcal{S}$-structure. Section 5 characterizes weak f-K-contact manifolds among all weak almost $\mathcal{S}$-manifolds by the property known for f-K-contact manifolds, and presents the sufficient conditions under which a Riemannian manifold endowed with a set of unit Killing vector fields is a weak f-K-contact manifold. We also find the Ricci curvature of a weak f-K-contact manifold in the $kerf$-directions, showing that there are no Einstein weak f-K-contact manifolds for $s>1$ and that the mixed sectional curvature is positive. Using this, the weak f-K-contact structure can be deformed to the f-K-contact structure, and we obtain a topological obstruction (including the Adams number) to the existence of weak f-K-contact manifolds. Section 6 presents the sufficient conditions for a weak f-K-contact manifold with a generalized gradient Ricci soliton to be a quasi-Einstein or Ricci flat manifold. In Section 7, we find that the sufficient conditions for a compact weak f-K-contact manifold with the $\eta$-Ricci structure of constant scalar curvature is $\eta$-Einstein. Section 8 and Section 9 show that a weak $\beta$-Kenmotsu f-manifold is locally a twisted product of ${\mathbb{R}}^s$ and a weak Kähler manifold, and in the case of an additional $\eta$-Ricci soliton structure, we explore their potential to be $\eta$-Einstein manifolds of a constant scalar curvature. The proofs (of some results) given in the article for the convenience of the reader use the properties of the new tensors, as well as the constructions needed in the classical case.

2. Preliminaries

In this section, we review the basics of the weak metric f-structure, see [26,28]. First, let us generalize the notion of a framed f-structure [8,9,13,14,16,33], called an f-structure with complemented frames in [10] or an f-structure with parallelizable kernel in [7].

Definition 1.

A framed weak f-structure on a smooth manifold $M^{2n+s}(n,s>0)$ is a set $(f,Q,{\xi }_i,{\eta }^i)$, where f is a $(1,1)$-tensor of rank $2n$, Q is a nonsingular $(1,1)$-tensor, ${\xi }_i(1\le i\le s)$ are structure vector fields, and ${\eta }^i(1\le i\le s)$ are 1-forms, satisfying

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

(1)

Then, theequality $f^3+fQ=0$ holds. If there exists a Riemannian metric g on $M^{2n+s}$ such that

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

(2)

then, $(f,Q,{\xi }_i,{\eta }^i,g)$ is a weak metric f-structure, and g is called a compatible metric.

Assume that a $2n$-dimensional contact distribution $\mathcal{D}={\bigcap }_{i}ker{\eta }^i$ is f-invariant. Note that for the framed weak f-structure, $\mathcal{D}=f\left(TM\right)$ is true, and

$$\begin{array}{c}\hfill f{\xi }_i=0,{\eta }^i\circ f=0,{\eta }^i\circ Q={\eta }^i,[Q,f]=0.\end{array}$$

(3)

Using the above, the distribution ${\mathcal{D}}^{\perp }=kerf$ is spanned by $\{{\xi }_1,\dots ,{\xi }_s\}$ and is invariant for Q.

Remark 1.

The concept of an almost paracontact structure is analogous to the concept of an almost contact structure and is closely related to an almost product structure. Similarly to (1), we define a framed weak para-f-structure, see details in [31], by

$$f^2=Q-{\sum }_i{\eta }^i\otimes {\xi }_i,{\eta }^i\left({\xi }_j\right)={\delta }_j^i,Q{\xi }_i={\xi }_i,$$

and we assume that a $2n$-dimensional contact distribution $\mathcal{D}={\bigcap }_{i}ker{\eta }^i$ is f-invariant.

The framed weak f-structure is called normal if the following tensor is zero:

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

(4)

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

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

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 ${\mathcal{N}}_i^{\left(2\right)},{\mathcal{N}}_i^{\left(3\right)}$, and ${\mathcal{N}}_{ij}^{\left(4\right)}$ on framed weak f-manifolds, see [28,30], are well known in the classical theory, see [10], and are expressed as follows:

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

Remark 2.

Let $M^{2n+s}(f,Q,{\xi }_i,{\eta }^i)$ be a weak framed 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 ${\partial }_1,\dots ,{\partial }_s$, and define tensors J and $\overline{Q}$ on $\overline{M}$ assuming $J(X,{\sum }_i{a}^i{\partial }_i)=(fX-{\sum }_i{a}^i{\xi }_i,{\sum }_j{\eta }^j\left(X\right){\partial }_j)$ and $\overline{Q}(X,{\sum }_i{a}^i{\partial }_i)=(QX,{\sum }_i{a}^i{\partial }_i)$ for $a_i\in C^{\infty }\left(M\right)$. It can be shown that $J^2=-\overline{Q}$. The tensors ${\mathcal{N}}_i^{\left(2\right)},{\mathcal{N}}_i^{\left(3\right)},{\mathcal{N}}_{ij}^{\left(4\right)}$ appear when we derive the integrability condition $[J,J]=0$ and express the normality condition ${\mathcal{N}}^{\left(1\right)}=0$ for $(f,Q,{\xi }_i,{\eta }^i)$.

A framed weak f-manifold admits a compatible metric if f has a skew-symmetric representation, i.e., for any $x\in M$, there exist a frame $\left\{e_i\right\}$ on a neighborhood $U_x\subset M$, for which f has a skew-symmetric matrix, see [26]. For a weak metric f-manifold, the tensor f is skew-symmetric and Q is self-adjoint and positive definite. Putting $Y={\xi }_i$ in (2), and using $Q{\xi }_i={\xi }_i$, we obtain ${\eta }^i\left(X\right)=g(X,{\xi }_i)$. Hence, ${\xi }_i$ is orthogonal to $\mathcal{D}$ for any compatible metric. Thus, $TM=\mathcal{D}\oplus {\mathcal{D}}^{\perp }$—the sum of two complementary orthogonal subbundles.

A distribution $\tilde{\mathcal{D}}\subset TM$ is called totally geodesic if and only if its second fundamental form vanishes, i.e., ${\nabla }_{X}Y+{\nabla }_{Y}X\in \tilde{\mathcal{D}}$ for any vector fields $X,Y\in \tilde{\mathcal{D}}$—this is the case when any geodesic of M that is tangent to $\tilde{\mathcal{D}}$ at one point is tangent to $\tilde{\mathcal{D}}$ at all its points, e.g., Section 1.3.1 [25]. According to the Frobenius theorem, any involutive distribution is tangent to (the leaves of) a foliation. Any involutive and totally geodesic distribution is tangent to a totally geodesic foliation. A foliation whose orthogonal distribution is totally geodesic is called a Riemannian foliation.

A “small” (1,1)-tensor $\tilde{Q}=Q-{\mathrm{id}}_{TM}$ measures the difference between weak and classical f-structures. By (3), we obtain

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

Proposition 1

(see [28]). The normality condition 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)

Moreover, ${\mathcal{D}}^{\perp }$ is a totally geodesic distribution.

The coboundary formula for exterior derivative of a 2-form $\Phi$ is

$$\begin{array}{cc}\hfill d\Phi (X,Y,Z)& ={\displaystyle \frac{1}{3}}\{X\Phi (Y,Z)+Y\Phi (Z,X)+Z\Phi (X,Y)\hfill \\ \hfill & -\Phi \left(\right[X,Y],Z)-\Phi \left(\right[Z,X],Y)-\Phi \left(\right[Y,Z],X)\}.\hfill \end{array}$$

(8)

Note that $d\Phi (X,Y,Z)=-\Phi \left(\right[X,Y],Z)$ for $X,Y\in {\mathcal{D}}^{\perp }$. Therefore, for a weak metric f-structure, the distribution ${\mathcal{D}}^{\perp }$ is involutive if and only if $d\Phi =0$.

Only one new tensor ${\mathcal{N}}^{\left(5\right)}$ (vanishing at $\tilde{Q}=0$), which supplements the sequence of well-known tensors ${\mathcal{N}}^{\left(1\right)},{\mathcal{N}}_i^{\left(2\right)},{\mathcal{N}}_i^{\left(3\right)},{\mathcal{N}}_{ij}^{\left(4\right)}$, is needed to study the weak metric f-structure.

Proposition 2

(see [28]). 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 a skew-symmetric with respect to Y and Z tensor ${\mathcal{N}}^{\left(5\right)}(X,Y,Z)$ is defined by

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

For particular values of the tensor ${\mathcal{N}}^{\left(5\right)}$ we obtain

$$\begin{array}{cc}\hfill {\mathcal{N}}^{\left(5\right)}(X,{\xi }_i,Z)& =-{\mathcal{N}}^{\left(5\right)}(X,Z,{\xi }_i)=g({\mathcal{N}}_i^{\left(3\right)}\left(Z\right),\tilde{Q}X),\hfill \\ \hfill {\mathcal{N}}^{\left(5\right)}({\xi }_i,Y,Z)& =g([{\xi }_i,fZ],\tilde{Q}Y)-g([{\xi }_i,fY],\tilde{Q}Z),\hfill \\ \hfill {\mathcal{N}}^{\left(5\right)}({\xi }_i,{\xi }_j,Y)& ={\mathcal{N}}^{\left(5\right)}({\xi }_i,Y,{\xi }_j)=0.\hfill \end{array}$$

(10)

Similar to the classical case, we introduce broad classes of weak metric f-structures.

Definition 2.

A weak metric f-structure $(f,Q,{\xi }_i,{\eta }^i,g)$ is called

(i) A weak $\mathcal{K}$-structure if it is normal and $d\Phi =0$.

We define two subclasses of weak $\mathcal{K}$-manifolds as follows:

(ii) Weak $\mathcal{C}$-manifolds if $d{\eta }^i=0$ for any i.

(iii) Weak $\mathcal{S}$-manifolds if the following is valid:

$$\begin{array}{c}\hfill \Phi =d{\eta }^1=\dots =d{\eta }^s(\mathrm{hence},d\Phi =0).\end{array}$$

(11)

Omitting the normality condition, we obtain the following: a weak metric f-structure is called

(i) A weak almost $\mathcal{K}$-structure if $d\Phi =0$.

(ii) A weak almost $\mathcal{C}$-structure if Φ and ${\eta }^i(1\le i\le s)$ are closed forms.

(iii) A weak almost $\mathcal{S}$-structure (or, a weak f-contact structure), if (11) is valid.

A weak almost $\mathcal{S}$-structure, whose structure vector fields ${\xi }_i$ are Killing, i.e., the tensor

$$\begin{array}{c}\hfill \left({\pounds }_{{\xi }_i}g\right)(X,Y):={\xi }_i\left(g(X,Y)\right)-g([{\xi }_i,X],Y)-g(X,[{\xi }_i,Y])=g({\nabla }_Y{\xi }_i,X)+g({\nabla }_X{\xi }_i,Y),\end{array}$$

vanishes, is called a weak f-K-contact structure.

For a weak almost $\mathcal{K}$-structure (and its special cases, a weak almost $\mathcal{S}$-structure and a weak almost $\mathcal{C}$-structure), the distribution ${\mathcal{D}}^{\perp }$ is involutive. Moreover, for a weak almost $\mathcal{S}$-structure and a weak almost $\mathcal{C}$-structure, we obtain $[{\xi }_i,{\xi }_j]=0$; in other words, the distribution ${\mathcal{D}}^{\perp }$ of these manifolds is tangent to a $\mathfrak{g}$-foliation with an abelian Lie algebra.

Remark 3.

Let $\mathfrak{g}$ be a Lie algebra of dimension s. We can say that a foliation $\mathcal{F}$ of dimension s on a smooth connected manifold M is a $\mathfrak{g}$-foliation if there exist complete vector fields ${\xi }_1,\dots ,{\xi }_s$ on M, which, when restricted to each leaf of $\mathcal{F}$, form a parallelism of this submanifold with a Lie algebra isomorphic to $\mathfrak{g}$, see, for example, refs. [26,34].

The following diagram (well known for classical structures) summarizes the relationships between some classes of weak metric f-manifolds considered in this article:

$$\left|\begin{array}{c}\mathrm{weak}\mathrm{metric}\\ f\text{-manifold}\end{array}\right|\stackrel{\Phi =d{\eta }^i}{\supset }\left|\begin{array}{c}\mathrm{weak}\mathrm{almost}\\ \mathcal{S}\text{-manifold}\end{array}\right|\stackrel{{\xi }_i\text{-Killing}}{\supset }\left|\begin{array}{c}\mathrm{weak}\\ f\text{-K-contact}\end{array}\right|\stackrel{{\mathcal{N}}^{\left(1\right)}=0}{\supset }\left|\begin{array}{c}\mathrm{weak}\\ \mathcal{S}\text{-manifold}\end{array}\right|.$$

For $s=1$, we obtain the following diagram:

$$\left|\begin{array}{c}\mathrm{weak}\mathrm{almost}\\ \mathrm{contact}\mathrm{metric}\\ \mathrm{manifold}\end{array}\right|\stackrel{\Phi =d\eta }{\supset }\left|\begin{array}{c}\mathrm{weak}\mathrm{contact}\\ \mathrm{metric}\\ \mathrm{manifold}\end{array}\right|\stackrel{\xi -\mathrm{Killing}}{\supset }\left|\begin{array}{c}\mathrm{weak}\\ \text{K-contact}\\ \mathrm{manifold}\end{array}\right|\stackrel{{\mathcal{N}}^{\left(1\right)}=0}{\supset }\left|\begin{array}{c}\mathrm{weak}\\ \mathrm{Sasakian}\\ \mathrm{manifold}\end{array}\right|.$$

3. Geometry of Weak Almost $\mathcal{S}$-Manifolds

For a weak almost $\mathcal{S}$-structure, the distribution $\mathcal{D}$ is not involutive, since we have

$$g([X,fX],{\xi }_i)=2d{\eta }^i(fX,X)=g(fX,fX)>0(X\in \mathcal{D}\setminus \left\{0\right\}).$$

Proposition 3

(see Theorem 2.2 in [28]). For a weak almost $\mathcal{S}$-structure, the tensors ${\mathcal{N}}_i^{\left(2\right)}$ and ${\mathcal{N}}_{ij}^{\left(4\right)}$ vanish; moreover, ${\mathcal{N}}_i^{\left(3\right)}$ vanishes if and only if ${\xi }_i$ is a Killing vector field.

By ${\mathcal{N}}_{ij}^{\left(4\right)}=0$, we have $g(X,{\nabla }_{{\xi }_i}{\xi }_j)+g({\nabla }_X{\xi }_i,{\xi }_j)=0$ for all $X\in {\mathfrak{X}}_M$. Symmetrizing the above equality (with $i\leftrightarrow j$) and using $g({\xi }_i,{\xi }_j)={\delta }_{ij}$ yields ${\nabla }_{{\xi }_i}{\xi }_j+{\nabla }_{{\xi }_j}{\xi }_i=0$. From this and the equality $[{\xi }_i,{\xi }_j]=0$, it follows that weak almost $\mathcal{S}$-manifolds satisfy

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

Corollary 1.

For a weak almost $\mathcal{S}$-structure, the distribution ${\mathcal{D}}^{\perp }$ is tangent to a $\mathfrak{g}$-foliation with totally geodesic flat (that is $R_{{\xi }_i,{\xi }_j}{\xi }_k=0)$ leaves.

The following corollary of Propositions 2 and 3 generalizes well-known results with $Q={\mathrm{id}}_{TM}$, e.g., Proposition 1.4 [10] and Proposition 2.1 [35].

Proposition 4.

For a weak almost $\mathcal{S}$-structure we obtain

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

(12)

where $\overline{\eta }={\sum }_i{\eta }^i$. Taking $X={\xi }_i$ in (12), we obtain

$$\begin{array}{cc}\hfill 2g(\left({\nabla }_{{\xi }_i}f\right)Y,Z)& ={\mathcal{N}}^{\left(5\right)}({\xi }_i,Y,Z),1\le i\le s.\hfill \end{array}$$

(13)

The tensor ${\mathcal{N}}_i^{\left(3\right)}$ is important for weak almost $\mathcal{S}$-manifolds, see Proposition 3. Therefore, we define the tensor field $\mathbf{h}=(h_1,\dots ,h_s)$, where

$$\begin{array}{c}\hfill h_i={\displaystyle \frac{1}{2}}{\mathcal{N}}_i^{\left(3\right)}={\displaystyle \frac{1}{2}}{\pounds }_{{\xi }_i}f.\end{array}$$

Using $[{\xi }_i,{\xi }_j]=0$ and $f{\xi }_j=0$, we obtain $\left({\pounds }_{{\xi }_i}f\right){\xi }_j=[{\xi }_i,f{\xi }_j]-f[{\xi }_i,{\xi }_j]=0$; therefore, $h_i{\xi }_j=0$ is true. For $X\in \mathcal{D}$, using the equality $[{\xi }_i,{\xi }_j]=0$, we derive the following:

$$0=2\Phi ({\xi }_i,X)=2{\eta }^j({\xi }_i,X)=g({\nabla }_X{\xi }_j,{\xi }_i);$$

therefore, $g({\nabla }_X{\xi }_j,{\xi }_i)=0$ for all $X\in {\mathfrak{X}}_M$. Next, we calculate

$$\begin{array}{c}\hfill \left({\pounds }_{{\xi }_i}f\right)Y=\left({\nabla }_{{\xi }_i}f\right)Y-{\nabla }_{fY}{\xi }_i+f{\nabla }_Y{\xi }_i.\end{array}$$

Thus, using $g(\left({\nabla }_{{\xi }_i}f\right)Y,{\xi }_j)=0$, see (13) with $Z={\xi }_j$, we obtain ${\eta }^j\circ h_i=0$ for all $1\le i,j\le s$:

$$({\eta }^j\circ h_i)Y=g(\left({\pounds }_{{\xi }_i}f\right)Y,{\xi }_j)=g(\left({\nabla }_{{\xi }_i}f\right)Y,{\xi }_j)-g({\nabla }_{fY}{\xi }_i,{\xi }_j)+g(f{\nabla }_Y{\xi }_i,{\xi }_j)=0.$$

For an almost $\mathcal{S}$-structure, the tensor $h_i$ is self-adjoint, trace-free, and anti-commutes with f, i.e., $h_{i}f+f{h}_i=0$, see [35]. We generalize this result for a weak almost $\mathcal{S}$-structure.

Proposition 5

(see [28,29]). For a weak almost $\mathcal{S}$-structure $(f,Q,{\xi }_i,{\eta }^i,g)$, the tensor $h_i$ and its conjugate tensor $h_i^{*}$ satisfy

$$\begin{array}{ccc}\hfill g((h_i-h_i^{*})X,Y)& =& {\displaystyle \frac{1}{2}}{\mathcal{N}}^{\left(5\right)}({\xi }_i,X,Y),X,Y\in TM,\hfill \\ \hfill h_{i}f+f{h}_i& =& -{\displaystyle \frac{1}{2}}{\pounds }_{{\xi }_i}Q,\hfill \\ \hfill h_{i}Q-Q{h}_i& =& {\displaystyle \frac{1}{2}}[f,{\pounds }_{{\xi }_i}Q].\hfill \end{array}$$

Let us consider the following condition (which is trivially satisfied by metric f-manifolds):

$${\pounds }_{{\xi }_i}Q=0.$$

(14)

The following corollary generalizes the known property of almost $\mathcal{S}$-manifolds.

Corollary 2.

Let a weak almost $\mathcal{S}$-manifold satisfy (14), then, $h_i=h_i^{*}$ and $\mathrm{trace}{h}_i=0$ for all i.

Proof. 

Under the conditions and Proposition 5, $h_i$ commutes with Q. Since the self-adjoint tensor Q is positive definite, then, $h_i$ is also self-adjoint, that is, $h_i=h_i^{*}$. If $h_{i}X=\lambda X$, then using $h_{i}f=-f{h}_i$ (by assumptions and Proposition 5), we obtain $h_{i}fX=-\lambda fX$. Thus, if $\lambda$ is an eigenvalue of $h_i$, then $-\lambda$ is also an eigenvalue of $h_i$; hence, $\mathrm{trace}{h}_i=0$. □

Definition 3

(see [28]). Framed weak f-structures $(f,Q,{\xi }_i,{\eta }^i)$ and $(f^{\prime },Q^{\prime },{\xi }_i,{\eta }^i)$ on a smooth manifold are said to be homothetic if the following conditions:

$$\begin{array}{cc}\hfill & f=\sqrt{\lambda }{f}^{\prime },\hfill \end{array}$$

(15a)

$$\begin{array}{cc}\hfill & {Q|}_{\mathcal{D}}=\lambda Q^{\prime }{|}_{\mathcal{D}},\hfill \end{array}$$

(15b)

are valid for some real $\lambda >0$. Weak metric f-structures $(f,Q,{\xi }_i,{\eta }^i,g)$ and $(f^{\prime },Q^{\prime },{\xi }_i,{\eta }^i,g^{\prime })$ are said to be homothetic if they satisfy conditions (15a,b) and

$$\begin{array}{c}\hfill {g|}_{\mathcal{D}}={\lambda }^{-{\displaystyle \frac{1}{2}}}{g}^{\prime }{|}_{\mathcal{D}},g({\xi }_i,·)=g^{\prime }({\xi }_i,·).\end{array}$$

(16)

Proposition 6

(see [28]). Let a framed weak f-structure $(f,Q,{\xi }_i,{\eta }^i)$ satisfy ${Q|}_{\mathcal{D}}=\lambda {\mathrm{id}}_{\mathcal{D}}$ for some real $\lambda >0$. Then, $(f^{\prime },{\xi }_i,{\eta }^i)$ is a framed f-structure, where $f^{\prime }$ is given by (15a). Moreover, if $(f,Q,{\xi }_i,{\eta }^i,g)$ is a weak almost $\mathcal{S}$-structure and (15a) and (16) are valid, then $(f^{\prime },{\xi }_i,{\eta }^i,g^{\prime })$ is an almost $\mathcal{S}$-structure.

Denote by $\mathrm{Ric}(X,Y)=\mathrm{trace}(Z\to R_{Z,X}Y)$ the Ricci tensor, where $R_{X,Y}=({\nabla }_X{\nabla }_Y-{\nabla }_Y{\nabla }_X-{\nabla }_{[X,Y]})Z$ is the curvature tensor. The Ricci operator is given by $g({Ric}^{♯}X,Y)=\mathrm{Ric}(X,Y)$. The scalar curvature of g is given by $r={\mathrm{trace}}_{g}Ric$.

Remark 4.

For almost $\mathcal{S}$-manifolds, we have, see Proposition 2.6 [35],

$${Ric}^{♯}\left({\xi }_i\right)={\sum }_{i=1}^{2n}\left({\nabla }_{e_i}f\right)e_i=2n\overline{\xi }.$$

(17)

Can one generalize (17) for weak almost $\mathcal{S}$-manifolds ?

For a weak almost $\mathcal{S}$-manifold, the splitting tensor $C:{\mathcal{D}}^{\perp }\times \mathcal{D}\to \mathcal{D}$ is defined by

$$C_{\xi }\left(X\right)=-{\left({\nabla }_X\xi \right)}^{\top }(X\in \mathcal{D},\xi \in {\mathcal{D}}^{\perp },\parallel \xi \parallel =1),$$

(18)

where ^⊤: $TM\to \mathcal{D}$ is the orthoprojector, see [29]. The splitting tensor is decomposed as $C_{\xi }=A_{\xi }+T_{\xi }^{♯}$, where the skew-symmetric operator $T_{\xi }^{♯}$ and the self-adjoint operator $A_{\xi }$ are defined using the integrability tensor $T(X,Y)={\displaystyle \frac{1}{2}}{({\nabla }_{X}Y-{\nabla }_{Y}X)}^{\perp }$ and the second fundamental form $b(X,Y)={\displaystyle \frac{1}{2}}{({\nabla }_{X}Y+{\nabla }_{Y}X)}^{\perp }$ of $\mathcal{D}$ by

$$g(T_{\xi }^{♯}X,Y)=g(T(X,Y),\xi ),g(A_{\xi }X,Y)=g(b(X,Y),\xi ),X,Y\in \mathcal{D}.$$

(19)

Since ${\mathcal{D}}^{\perp }$ defines a totally geodesic foliation, see Corollary 1, then the distribution $\mathcal{D}$ is totally geodesic if and only if $C_{\xi }$ is skew-symmetric, and $\mathcal{D}$ is integrable if and only if the tensor $C_{\xi }$ is self-adjoint. Thus, $C_{\xi }\equiv 0$ if and only if $\mathcal{D}$ is integrable and defines a totally geodesic foliation; in this case, by de Rham Decomposition Theorem, the manifold splits (is locally the product of Riemannian manifolds, defined by distributions $\mathcal{D}$ and ${\mathcal{D}}^{\perp }$), e.g., ref. [25].

Theorem 1.

The splitting tensor of a weak almost $\mathcal{S}$-manifold has the following view:

$$C_{{\xi }_i}=f+Q^{-1}f{h}_i^{*}(i=1,\dots ,s).$$

The mixed scalar curvature of an almost product manifold $M({\mathcal{D}}_1,{\mathcal{D}}_2,g)$ is the function

$$r_{\mathrm{mix}}={\sum }_{a,i}g(R_{E_a,{\mathcal{E}}_i}{E}_a,{\mathcal{E}}_i),$$

where $\{{\mathcal{E}}_i,E_a\}$ is an adapted orthonormal frame, i.e., $\left\{E_a\right\}\subset {\mathcal{D}}_1$ and $\left\{{\mathcal{E}}_i\right\}\subset {\mathcal{D}}_2$. Let $b_i$ and $H_i$ be the second fundamental form and the mean curvature vector, and let $T_i$ be the integrability tensor of the distribution ${\mathcal{D}}_i$. The following formula:

$$r_{\mathrm{mix}}=\mathrm{div}(H_1+H_2)-\parallel b_1{\parallel }^2-\parallel b_2{\parallel }^2+\parallel H_1{\parallel }^2+\parallel H_2{\parallel }^2+\parallel T_1{\parallel }^2+{\parallel T_2\parallel }^2,$$

(20)

has many applications in Riemannian, Kähler and Sasakian geometries, see [25].

Theorem 2.

For the weak almost $\mathcal{S}$-structure on a closed manifold $M^{2n+s}$ satisfying condition (14), the following integral formula is true:

$$\begin{array}{c}\hfill {\int }_M\left\{{\sum }_i\left(\mathrm{Ric}({\xi }_i,{\xi }_i)+{\parallel Q^{-1}f{h}_i\parallel }^2-{\left(\mathrm{trace}{Q}^{-1}f{h}_i\right)}^2\right)-s{\parallel f\parallel }^2\right\}d\mathrm{vol}=0.\end{array}$$

(21)

Proof. 

According to (20), set ${\mathcal{D}}_1=\mathcal{D}$ and ${\mathcal{D}}_2={\mathcal{D}}^{\perp }$. By the assumptions, Proposition 5 and Corollary 2, $h_i^{*}=h_i$, $h_{i}f+f{h}_i=0$ and $[h_i,Q]=0$. Then, $b_2=H_2=T_2=0$ and

$$b_1(X,Y)={\sum }_{i}g(Q^{-1}f{h}_{i}X,Y){\xi }_i,H_1={\sum }_i\mathrm{trace}\left(Q^{-1}f{h}_i\right){\xi }_i,T_1(X,Y)=g(fX,Y)\overline{\xi },$$

where $\overline{\xi }={\sum }_i{\xi }_i$. For a weak almost $\mathcal{S}$-manifold, we have $r_{\mathrm{mix}}={\sum }_i\mathrm{Ric}({\xi }_i,{\xi }_i)$. Thus, (21) is the counterpart of (20) integrated on a closed manifold using the Divergence Theorem. □

Definition 4

(see [36]). An even-dimensional Riemannian manifold $(\tilde{M},\tilde{g})$ equipped with a skew-symmetric $(1,1)$-tensor J such that $J^2$ is negative-definite is called a weak Hermitian manifold. This manifold is called weak Kählerian if $\tilde{\nabla }J=0$, where $\tilde{\nabla }$ is the Levi–Civita connection of $\tilde{g}$.

An involutive distribution is regular if every point of the manifold has a neighborhood such that any integral submanifold passing through the neighborhood passes through only once, see, for example, ref. [21]. The next theorem states that a compact manifold with a regular weak almost $\mathcal{S}$-structure is a principal torus bundle over a weak Hermitian manifold, and we believe that its proof using Proposition 3 is similar to the proof of Theorem 4.2 [21].

Theorem 3.

Let $M^{2n+s}$ be a compact manifold equipped with a regular weak almost $\mathcal{S}$-structure $(f,Q,{\tilde{\xi }}_i,{\tilde{\eta }}^i,\tilde{g})$. Then, there exists a weak almost $\mathcal{S}$-structure $(f,Q,{\xi }_i,{\eta }^i,g)$ on M for which the structure vector fields ${\xi }_1,\dots ,{\xi }_s$ are the infinitesimal generators of a free and effective ${\mathbb{T}}^s$-action on M. Moreover, the quotient $N=M/{\mathbb{T}}^s$ is a smooth weak Hermitian manifold of dimension $2n$.

4. Geometry of Weak $\mathcal{K}$-Manifolds and Their Two Subclasses

The following result generalizes Theorem 1.1 [10].

Theorem 4.

On a weak $\mathcal{K}$-manifold the structure vector fields ${\xi }_1,\dots ,{\xi }_s$ are Killing and

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

(22)

thus, the distribution ${\mathcal{D}}^{\perp }$ is tangent to a totally geodesic Riemannian foliation with flat leaves.

Proof. 

By Proposition 1, ${\mathcal{D}}^{\perp }$ is totally geodesic and ${\mathcal{N}}_i^{\left(3\right)}={\pounds }_{{\xi }_i}f=0$. Using ${\iota }_{{\xi }_i}\Phi =0$ and condition $d\Phi =0$ in the identity ${\pounds }_{{\xi }_i}={\iota }_{{\xi }_i}d+d{\iota }_{{\xi }_i}$, we obtain ${\pounds }_{{\xi }_i}\Phi =0$. By direct calculation we obtain the following:

$$\begin{array}{c}\hfill ({\pounds }_{{\xi }_i}\Phi )(X,Y)=\left({\pounds }_{{\xi }_i}g\right)(X,fY)+g(X,\left({\pounds }_{{\xi }_i}f\right)Y).\end{array}$$

(23)

Thus, from (23) we obtain $\left({\pounds }_{{\xi }_i}g\right)(X,fY)=0$. To show ${\pounds }_{{\xi }_i}g=0$, we will examine $\left({\pounds }_{{\xi }_i}g\right)(fX,{\xi }_j)$ and $\left({\pounds }_{{\xi }_i}g\right)({\xi }_k,{\xi }_j)$. Using ${\pounds }_{{\xi }_i}{\eta }^j=0$, we obtain $\left({\pounds }_{{\xi }_i}g\right)(fX,{\xi }_j)=\left({\pounds }_{{\xi }_i}{\eta }^j\right)fX-g(fX,[{\xi }_i,{\xi }_j])=-g(fX,[{\xi }_i,{\xi }_j])=0$. Next, using Proposition 1, we obtain

$$\left({\pounds }_{{\xi }_i}g\right)({\xi }_k,{\xi }_j)=-g({\xi }_i,{\nabla }_{{\xi }_k}{\xi }_j+{\nabla }_{{\xi }_j}{\xi }_k)=0.$$

Thus, ${\xi }_i$ is a Killing vector field, i.e., ${\pounds }_{{\xi }_i}g=0$. From $d\Phi (X,{\xi }_i,{\xi }_j)=0$ and (8) we obtain $g([{\xi }_i,{\xi }_j],fX)=0$, i.e., $kerf$ is integrable. Finally, from this and Proposition 1 we obtain (22); thus, the sectional curvature $K({\xi }_i,{\xi }_j)$ vanishes. □

By Proposition 4 with ${\mathcal{N}}^{\left(1\right)}=0$, we obtain the following.

Corollary 3.

For a weak $\mathcal{S}$-structure $(f,Q,{\xi }_i,{\eta }^i,g)$ we obtain

$$\begin{array}{c}\hfill g(\left({\nabla }_{X}f\right)Y,Z)=g(fX,fY)\overline{\eta }\left(Z\right)-g(fX,fZ)\overline{\eta }\left(Y\right)+{\displaystyle \frac{1}{2}}{\mathcal{N}}^{\left(5\right)}(X,Y,Z).\end{array}$$

(24)

Moreover, ${\xi }_i$ are Killing vector fields and ${\mathcal{D}}^{\perp }$ is tangent to a Riemannian totally geodesic foliation.

Using Corollary 3, we obtain a rigidity theorem for an $\mathcal{S}$-structure.

Theorem 5.

A weak metric f-structure is a weak $\mathcal{S}$-structure if and only if it is an $\mathcal{S}$-structure.

Proof. 

Let $(f,Q,{\xi }_i,{\eta }^i,g)$ be a weak $\mathcal{S}$-structure. Since ${\mathcal{N}}^{\left(1\right)}=0$, by Proposition 1, we obtain ${\mathcal{N}}_i^{\left(3\right)}=0$. By (10), we obtain ${\mathcal{N}}^{\left(5\right)}(·,{\xi }_i,·)={\mathcal{N}}^{\left(5\right)}(·,·,{\xi }_i)=0$. Since f is skew-symmetric, applying (24) with $Z={\xi }_i$ in (5) yields

$$\begin{array}{c}\hfill g([f,f](X,Y),{\xi }_i)=g(\left({\nabla }_{fX}f\right)Y,{\xi }_i)-g(\left({\nabla }_{fY}f\right)X,{\xi }_i)=-2g(QX,fY).\end{array}$$

From this and ${\mathcal{N}}^{\left(1\right)}=0$, we obtain $g(\tilde{Q}X,fY)=0$ for all $X,Y\in {\mathfrak{X}}_M$; therefore, $\tilde{Q}=0$. □

For $s=1$, from Theorem 5, we have the following

Corollary 4

(see [37]). A weak almost contact metric structure on $M^{2n+1}$ is weak Sasakian if and only if it is a Sasakian structure (i.e., a normal contact metric structure) on $M^{2n+1}$.

Next, we study a weak almost $\mathcal{C}$-structure.

Proposition 7.

For a weak $\mathcal{C}$-structure $(f,Q,{\xi }_i,{\eta }^i,g)$, we obtain

$$\begin{array}{cc}\hfill 2g(\left({\nabla }_{X}f\right)Y,Z)& ={\mathcal{N}}^{\left(5\right)}(X,Y,Z).\hfill \end{array}$$

(25)

A $\mathcal{K}$-structure is a $\mathcal{C}$-structure if and only if f is parallel, e.g., Theorem 1.5 [10]. The following theorem extends this result and characterizes weak $\mathcal{C}$-manifolds using the condition $\nabla f=0$.

Theorem 6.

A weak metric f-structure $(f,Q,{\xi }_i,{\eta }^i,g)$ with conditions $\nabla f=0$ and

$$\begin{array}{c}\hfill {[{\xi }_i,{\xi }_j]}^{\perp }=0,1\le i,j\le s,\end{array}$$

is a weak $\mathcal{C}$-structure with the property ${\mathcal{N}}^{\left(5\right)}=0$.

Proof. 

Using condition $\nabla f=0$, from (5), we obtain $[f,f]=0$. Hence, from (4), we obtain ${\mathcal{N}}^{\left(1\right)}(X,Y)=2{\sum }_{i}d{\eta }^i(X,Y){\xi }_i$, and from (5), with $S=f$ and $Y={\xi }_i$, we obtain

$$\begin{array}{c}\hfill {\nabla }_{fX}{\xi }_i-f{\nabla }_X{\xi }_i=0,X\in {\mathfrak{X}}_M.\end{array}$$

(26)

From (8), we calculate

$$3d\Phi (X,Y,Z)=g(\left({\nabla }_{X}f\right)Z,Y)+g(\left({\nabla }_{Y}f\right)X,Z)+g(\left({\nabla }_{Z}f\right)Y,X);$$

hence, using condition $\nabla f=0$ again, we obtain $d\Phi =0$. Next,

$$\begin{array}{c}\hfill {\mathcal{N}}_i^{\left(2\right)}(Y,{\xi }_j)=-{\eta }^i\left([fY,{\xi }_j]\right)=g({\xi }_j,f{\nabla }_{{\xi }_i}Y)=0.\end{array}$$

Thus, setting $Z={\xi }_j$ in Proposition 4 and using the condition $\nabla f=0$ and the properties $d\Phi =0$, ${\mathcal{N}}_i^{\left(2\right)}(Y,{\xi }_j)=0$, and ${\mathcal{N}}^{\left(1\right)}(X,Y)=2{\sum }_{i}d{\eta }^i(X,Y){\xi }_i$, we find $0=2d{\eta }^i(fY,X)-{\mathcal{N}}^{\left(5\right)}(X,{\xi }_i,Y)$. By (10) and (26), we obtain

$${\mathcal{N}}^{\left(5\right)}(X,{\xi }_i,Y)=g([{\xi }_i,fY]-f[{\xi }_i,Y],\tilde{Q}X)=g({\nabla }_{fY}{\xi }_i-f{\nabla }_Y{\xi }_i,\tilde{Q}X)=0;$$

hence, $d{\eta }^i(fY,X)=0$. From this and condition $g([{\xi }_i,{\xi }_j],{\xi }_k)=2d{\eta }^k({\xi }_j,{\xi }_i)=0$, we obtain $d{\eta }^i=0$. By the above, ${\mathcal{N}}^{\left(1\right)}=0$. Thus, $(f,Q,{\xi }_i,{\eta }^i,g)$ is a weak $\mathcal{C}$-structure. Finally, from (25) and condition $\nabla f=0$, we obtain ${\mathcal{N}}^{\left(5\right)}=0$. □

Example 1.

Let M be a $2n$-dimensional smooth manifold and $J:TM\to TM$ an endomorphism of rank $2n$ such that $\nabla J=0$. To construct a weak $\mathcal{C}$-structure on $M\times {\mathbb{R}}^s$ or $M\times T^s$, where $T^s$ is an s-dimensional torus, take any point $(x,t_1,\dots ,t_s)$ of either space and set ${\xi }_i=(0,\partial /\partial t_i)$, ${\eta }^i=(0,d{t}_i)$, and

$$f(X,Y)=(JX,0),Q(X,Y)=(-J^{2}X,Y).$$

where $X\in M_x$ and $Y\in \{{\mathbb{R}}_t^s,T_t^s\}$. Then, (1) holds and Theorem 6 can be used.

5. Geometry of Weak f-K-Contact Manifolds

Here, we characterize weak f-K-contact manifolds among all weak almost $\mathcal{S}$-manifolds and find conditions under which a Riemannian manifold endowed with a set of unit Killing vector fields becomes a weak f-K-contact manifold.

Lemma 1.

For a weak f-K-contact manifold we obtain

$$\begin{array}{ccc}\hfill {\mathcal{N}}^{\left(1\right)}({\xi }_i,·)& =& 0,\hfill \\ \hfill {\mathcal{N}}^{\left(5\right)}({\xi }_i,·,·)& =& {\mathcal{N}}^{\left(5\right)}(·,{\xi }_i,·)=0,\hfill \\ \hfill {\pounds }_{{\xi }_i}Q& =& {\nabla }_{{\xi }_i}Q=0,\hfill \\ \hfill {\nabla }_{{\xi }_i}f& =& 0.\hfill \end{array}$$

Recall the following property of f-K-contact manifolds, see [10,22]:

$$\nabla {\xi }_i=-f,1\le i\le s.$$

(27)

Using Proposition 3 and Theorem 1, we obtain the following.

Theorem 7.

A weak almost $\mathcal{S}$-structure is weak f-K-contact if and only if (27) is true.

From Proposition 5 and Theorem 1, using Lemma 1, we obtain $h_i=h_i^{*}=0(1\le i\le s)$.

The mapping $R_{{\xi }_i}:X\mapsto R_{X,{\xi }_i}{\xi }_i(\xi \in {\mathcal{D}}^{\perp },\parallel \xi \parallel =1)$ is called the Jacobi operator in the ${\xi }_i$-direction, e.g., ref. [25]. For a weak almost $\mathcal{S}$-manifold, by Proposition 4, we obtain $R_{{\xi }_i}\left(X\right)\in \mathcal{D}$.

Theorem 8.

Let $(M^{2n+s},g)$ be a Riemannian manifold with orthonormal Killing vector fields ${\xi }_i,\dots ,{\xi }_s$ such that $d{\eta }^1=\dots =d{\eta }^s$(where ${\eta }^i$ is the 1-form dual to ${\xi }_i)$ and the Jacobi operators $R_{{\xi }_i}(i\le s)$ are positive definite on the distribution $\mathcal{D}={\bigcap }_{i}ker{\eta }^i$. Then, the manifold is weak f-K-contact, and its structural tensors are as follows:

$$f=-\nabla {\xi }_i,QX=R_{{\xi }_i}\left(X\right)(X\in \mathcal{D},1\le i\le s).$$

Proof. 

Since ${\xi }_i$ are Killing vector fields, we obtain the following property (11):

$$d{\eta }^i(X,Y)=(1/2)(g({\nabla }_X{\xi }_i,Y)-g({\nabla }_Y{\xi }_i,X))=-g({\nabla }_Y{\xi }_i,X)=g(X,fY).$$

Set $QX=R_{{\xi }_i}\left(X\right)$ for some i and all $X\in \mathcal{D}$. Since ${\xi }_i$ is a unit Killing vector field, we obtain ${\nabla }_{{\xi }_i}{\xi }_i=0$ and ${\nabla }_X{\nabla }_Y{\xi }_i-{\nabla }_{{\nabla }_{X}Y}{\xi }_i=R_{X,{\xi }_i}Y$, see [9]. Thus, $f{\xi }_i=0$ is true, and

$$f^{2}Y={\nabla }_{{\nabla }_Y{\xi }_i}{\xi }_i=R_{{\xi }_i,Y}{\xi }_i=-R_{{\xi }_i}\left(Y\right)=-QY,Y\in \mathcal{D}.$$

By the conditions, the tensor Q is positive definite on the subbundle $\mathcal{D}$. Therefore, the rank of f restricted to $\mathcal{D}$ is $2n$. Set $Q{\xi }_i={\xi }_i(1\le i\le s)$. Thus, (1) and (2) are true. □

The sectional curvature of a plane containing unit vectors $\xi \in {\mathcal{D}}^{\perp }$ and $X\in \mathcal{D}$ is called mixed sectional curvature. The mixed sectional curvature of an almost $\mathcal{S}$-manifold is a spacial case of mixed sectional curvature of almost product manifolds, for example, ref. [25]. Note that the mixed sectional curvature of an f-K-contact manifold is constant and equal to 1.

Proposition 8.

A weak f-K-contact structure $(f,Q,{\xi }_i,{\eta }^i,g)$ of constant mixed sectional curvature, satisfying $K({\xi }_i,X)=\lambda >0$ for all $X\in \mathcal{D}$ and some $\lambda =const\in \mathbb{R}$, is homothetic to an f-K-contact structure $(f^{\prime },{\xi }_i,{\eta }^i,g^{\prime })$ after the transformation (15a,b)–(16).

Example 2.

According to Theorem 8, we can search for examples of weak f-K-contact (not f-K-contact) manifolds that can be found among Riemannian manifolds of positive sectional curvature admitting $s\ge 1$ mutually orthogonal unit Killing vector fields. Set $s=1$, and let $M^{n+1}$ be an ellipsoid with induced metric g of ${\mathbb{R}}^{2n+2}(n\ge 1)$,

$$M=\left\{(u_1,\dots ,u_{2n+2})\in {\mathbb{R}}^{2n+2}:{\sum }_{i=1}^{n+1}{u}_i^2+a{\sum }_{i=n+2}^{2n+1}{u}_i^2=1\right\},$$

where $0<a=const\ne 1$. The sectional curvature of $(M,g)$ is positive. It follows that

$$\xi =(-u_2,u_1,\dots ,-u_{n+1},u_n,-\sqrt{a}{u}_{n+3},\sqrt{a}{u}_{n+2},\dots ,-\sqrt{a}{u}_{2n+2},\sqrt{a}{u}_{2n+1})$$

is a Killing vector field on ${\mathbb{R}}^{2n+2}$, whose restriction to M has unit length. Since M is invariant under the flow of ξ, then ξ is a unit Killing vector field on $(M,g)$.

Since $K({\xi }_i,{\xi }_j)=0$ for a weak almost $\mathcal{S}$-manifold, the Ricci curvature in the ${\xi }_j$-direction is given by

$$\mathrm{Ric}({\xi }_j,{\xi }_j)={\sum }_{i=1}^{2n}g(R_{e_i,{\xi }_j}{\xi }_j,e_i),$$

where $\left(e_i\right)$ is a local orthonormal basis of $\mathcal{D}$. The next proposition generalizes some particular properties of f-K-contact manifolds for the case of weak f-K-contact manifolds.

Proposition 9.

Let $M^{2n+s}(f,Q,{\xi }_i,{\eta }^i,g)$ be a weak f-K-contact manifold, then, for all $i,j$ we obtain

$$\begin{array}{ccc}\hfill & & R_{{\xi }_i,Y}={\nabla }_{Y}f(Y\in {\mathfrak{X}}_M),\hfill \end{array}$$

(28)

$$\begin{array}{ccc}\hfill & & R_{{\xi }_i,Y}{\xi }_j=f^{2}Y(Y\in {\mathfrak{X}}_M),\hfill \end{array}$$

(29)

$$\begin{array}{ccc}\hfill & & {Ric}^{♯}{\xi }_i=divf,\hfill \end{array}$$

(30)

$$\begin{array}{ccc}\hfill & & \mathrm{Ric}({\xi }_i,{\xi }_j)=\mathrm{trace}Q>0.\hfill \end{array}$$

(31)

Proposition 10.

There are no Einstein weak f-K-contact manifolds with $s>1$.

Proof. 

A weak f-K-contact manifold with $s>1$ and ${\xi }^{\prime }={\displaystyle \frac{{\xi }_1+{\xi }_2}{\sqrt{2}}}$, satisfies the following:

$$\mathrm{Ric}({\xi }^{\prime },{\xi }^{\prime })={\displaystyle \frac{1}{2}}{\sum }_{i,j=1}^2\mathrm{Ric}({\xi }_i,{\xi }_j)\stackrel{\left(31\right)}{=}2\mathrm{trace}Q.$$

(32)

If the manifold is an Einstein manifold, then for the unit vector field ${\xi }^{\prime }$, we obtain $\mathrm{Ric}({\xi }^{\prime },{\xi }^{\prime })=\mathrm{Ric}({\xi }_i,{\xi }_i)=\mathrm{trace}Q$. Comparing this with (32) yields $\mathrm{trace}Q=0$—a contradiction. □

For a f-K-contact manifold, Equations (30) and (17) give ${Ric}^{♯}\left({\xi }_i\right)=2n\overline{\xi }(1\le i\le s)$ and $Ric({\xi }_i,{\xi }_i)=2n$.

Proposition 11.

For a weak f-K-contact manifold, the mixed sectional curvature is positive, as follows:

$$K({\xi }_i,X)=g(QX,X)>0(X\in \mathcal{D},\parallel X\parallel =1),$$

and the Ricci curvature satisfies the following: $\mathrm{Ric}({\xi }_i,{\xi }_j)>0$ for all $1\le i,j\le s$.

Proof. 

From (29), we obtain $K(\xi ,X)=g(fX,fX)>0$ for any unit vectors $\xi \in {\mathcal{D}}^{\perp },X\in \mathcal{D}$. Using (1) and non-singularity of f on $\mathcal{D}$, from (31) we obtain

$$Ric({\xi }_j,{\xi }_i)=\mathrm{trace}Q=-{\sum }_{p=1}^{2n}g(f^2{e}_p,e_p)={\sum }_{p=1}^{2n}g(f{e}_p,f{e}_p)>0,$$

where $\left(e_p\right)$ is a local orthonormal frame of $\mathcal{D}$, thus the second statement is valid. □

Theorem 9.

A weak f-K-contact manifold with conditions $(\nabla Ric)({\xi }_i,·)=0(1\le i\le s)$ and $\mathrm{trace}Q=const$ is an Einstein manifold and $s=1$.

Proof. 

Differentiating (31) and using (27) and the conditions, we have

$$0={\nabla }_Y\left(\mathrm{Ric}({\xi }_i,{\xi }_i)\right)=\left({\nabla }_Y\mathrm{Ric}\right)({\xi }_i,{\xi }_i)+2\mathrm{Ric}({\nabla }_Y{\xi }_i,{\xi }_i)=-2\mathrm{Ric}(fY,{\xi }_i),$$

hence $\mathrm{Ric}(Y,{\xi }_i)=\left(\mathrm{trace}Q\right){\eta }^i\left(Y\right)$. Differentiating this, then using

$$X\left({\eta }^i\left(Y\right)\right)=g({\nabla }_X{\xi }_i,Y)=-g(fX,Y)+g({\nabla }_{X}Y,{\xi }_i)$$

and assuming ${\nabla }_{X}Y=0$ at a point $x\in M$, gives

$$\begin{array}{cc}\hfill & \left(\mathrm{trace}Q\right)g(fX,Y)={\nabla }_Y\left(\mathrm{Ric}(X,{\xi }_i)\right)\hfill \\ \hfill & =\left({\nabla }_Y\mathrm{Ric}\right)(X,{\xi }_i)+2\mathrm{Ric}(X,{\nabla }_Y{\xi }_i)=-2\mathrm{Ric}(X,fY).\hfill \end{array}$$

Thus, $\mathrm{Ric}(X,fY)=\left(\mathrm{trace}Q\right)g(X,fY)$. Hence, $\mathrm{Ric}(X,Y)=\left(\mathrm{trace}Q\right)g(X,Y)$ for any vectors $X,Y\in T_{x}M$; therefore, $(M,g)$ is an Einstein manifold. By Proposition 10, $s=1$. □

The partial Ricci curvature tensor, see [26], is self-adjoint and is given by

$${Ric}^{\top }\left(X\right)={\sum }_{i=1}^s{\left(R_{X^{\top },{\xi }_i}{\xi }_i\right)}^{\top }.$$

Note that ${Ric}^{\top }=s{\mathrm{id}}_{\mathcal{D}}$ holds for f-K-contact manifolds. For weak f-K-contact manifolds, the tensor ${Ric}^{\top }$ is positive definite, see Proposition 11. In [26], applying the flow of metrics for a $\mathfrak{g}$-foliation, a deformation of a weak almost $\mathcal{S}$-structure with positive partial Ricci curvature onto the the classical structures of the same kind was constructed.

The next theorem, using Proposition 11 and the method of [26], shows that a weak f-K-contact manifold can be deformed into an f-K-contact manifold.

Theorem 10.

Let $M^{2n+s}(f_0,Q_0,{\xi }_i,{\eta }^i,g_0)$ be a weak f-K-contact manifold. Then, there exist metrics $g_{\tau }(\tau \in \mathbb{R})$, such that each $(f_t,Q_{\tau },{\xi }_i,{\eta }^i,g_{\tau })$ is a weak f-K-contact structure satisfying

$$Q_{\tau }=(1/s){Ric}_{\tau }^{\top },f_{\tau }{|}_{\mathcal{D}}=T_{{\xi }_i}^{♯}\left(\tau \right).$$

(33)

Moreover, $g_{\tau }$ converges exponentially fast, as $\tau \to -\infty$, to a metric $\widehat{g}$ with ${Ric}_{\widehat{g}}^{\top }=s{\mathrm{id}}_{\mathcal{D}}$ that gives an f-K-contact structure on M.

Proof. 

For a weak f-K-contact manifold, the tensor ${Ric}^{\top }$ is positive definite. Thus, we can apply the method of Theorem 1 [26]. Consider the partial Ricci flow, see [25],

$${\partial }_{\tau }{g}_{\tau }=-2{\left({Ric}^{\top }\right)}_{g_{\tau }}^{♭}+2s{g}_{\tau }^{\top },$$

(34)

where the tensor $g^{\top }$ is given by $g^{\top }(X,Y)=g(X^{\top },Y^{\top })$. We obtain the following, based on $\mathcal{D}$ for $\tau =0$:

$${Ric}^{\top }=-{\sum }_i{\left(T_{{\xi }_i}^{♯}\right)}^2=-s{f}^2=sQ,$$

and find $T_{{\xi }_i}^{♯}{Ric}^{\top }={Ric}^{\top }{T}_{{\xi }_i}^{♯}$, see also [26]. Thus, ${\sum }_i{T}_{{\xi }_i}^{♯}{Ric}^{\top }{T}_{{\xi }_i}^{♯}=-{\left({Ric}^{\top }\right)}^2$. By the above, we obtain the following ordinary differential equation:

$${\partial }_{\tau }{Ric}^{\top }=4{Ric}^{\top }({Ric}^{\top }-s{\mathrm{id}}_{\mathcal{D}}).$$

According to ODE theory, there exists a unique solution ${Ric}_{\tau }^{\top }$ for $\tau \in \mathbb{R}$; thus, a solution $g_{\tau }$ of (34) exists for $\tau \in \mathbb{R}$ and it is unique. Observe that $(f\left(\tau \right),{\xi }_i,{\eta }^i,Q_{\tau })$ with $f\left(\tau \right),Q_{\tau }$, given in (33), is a weak f-K-contact structure on $(M,g_{\tau })$. By the uniqueness of the solution, the flow (34) preserves the directions of the eigenvectors of ${Ric}^{\top }$, and each eigenvalue ${\mu }_i>0$ satisfies the ODE ${\dot{\mu }}_i=4{\mu }_i({\mu }_i-s)$ with ${\mu }_i\left(0\right)>0$. This ODE has the following solution:

$${\mu }_i\left(\tau \right)={\displaystyle \frac{{\mu }_i\left(0\right)s}{{\mu }_i\left(0\right)+exp\left(4s\tau \right)(s-{\mu }_i\left(0\right))}}$$

(a function ${\mu }_i\left(\tau \right)$ on M for $\tau \in \mathbb{R}$) with $\underset{\tau \to -\infty }{lim}{\mu }_i\left(\tau \right)=s$. Thus, $\underset{\tau \to -\infty }{lim}{Ric}^{\top }\left(\tau \right)=s{\mathrm{id}}_{\mathcal{D}}$. Let $\left\{e_i\left(\tau \right)\right\}$ be a $g_{\tau }$-orthonormal frame of $\mathcal{D}$ of eigenvectors associated with ${\mu }_i\left(\tau \right)$, we then obtain ${\partial }_{\tau }{e}_i=({\mu }_i-s)e_i$. Since $e_i\left(\tau \right)=z_i\left(\tau \right)e_i\left(0\right)$ with $z_i\left(0\right)=1$, then ${\partial }_{\tau }log{z}_i\left(\tau \right)={\mu }_i\left(\tau \right)-s$. By the above, we obtain $z_i\left(\tau \right)={({\mu }_i\left(\tau \right)/{\mu }_i\left(0\right))}^{1/4}$. Hence,

$$g_t(e_i\left(0\right),e_j\left(0\right))=z_i^{-1}\left(\tau \right)z_j^{-1}\left(\tau \right)g_t(e_i\left(\tau \right),e_j\left(\tau \right))={\delta }_{ij}{({\mu }_i\left(0\right){\mu }_j\left(0\right)/\left({\mu }_i\left(\tau \right){\mu }_j\left(\tau \right)\right))}^{1/4}.$$

$\widehat{g}(e_i\left(0\right),e_j\left(0\right))={\delta }_{ij}\sqrt{{\mu }_i\left(0\right)/s}$. □

Denote by $\rho \left(n\right)-1$ the maximal number of point-wise linearly independent vector fields on a sphere $S^{n-1}$. The topological invariant $\rho \left(n\right)$, called the Adams number, is

$$\rho \left(\left(\mathrm{odd}\right)2^{4b+c}\right)=8b+2^c\mathrm{for}\mathrm{any}\mathrm{integers}b\ge 0,0\le c\le 3,$$

see

Table 1. The number of vector fields on the $(2n-1)$ sphere.

$2n-1$	1	3	5	7	9	11	13	15	17	19	21	23	25	27	29
$\rho \left(2n\right)-1$	1	3	1	7	1	3	1	8	1	3	1	7	1	3	1

, and the inequality $\rho \left(n\right)\le 2{log}_{2}n+2$ is valid, for example, Section 1.4.4 [25].

There are not many theorems in differential geometry that use $\rho \left(n\right)$. Applying the Adams number, we obtain a topological obstruction to the existence of weak f-K-contact manifolds.

Theorem 11.

For a weak f-K-contact manifold $M^{2n+s}(f,Q,{\xi }_i,{\eta }^i,g)$ we have $s<\rho \left(2n\right)$.

Proof. 

For a weak almost $\mathcal{S}$-structure, the following Riccati equation is true, e.g., ref. [25]:

$$\begin{array}{c}\hfill {\nabla }_{\xi }{C}_{\xi }+{\left(C_{\xi }\right)}^2+R_{\xi }=0(\xi \in {\mathcal{D}}^{\perp }).\end{array}$$

(35)

Since the splitting tensor $C_{\xi }$ is skew-symmetric for a weak f-K-contact manifold, i.e., $C_{\xi }=T_{\xi }^{♯}$ and $A_{\xi }=0$, see (19), and the Jacobi operator $R_{\xi }$ is self-adjoined, (35) reduces to two equations on $\mathcal{D}$, as follows:

$${\nabla }_{\xi }{T}_{\xi }^{♯}=0\left(\text{the skew-symmetric part}\right),{\left(T_{\xi }^{♯}\right)}^2=-R_{\xi }\left(\text{the self-adjoint part}\right).$$

By this and Proposition 11, we obtain $C_{\xi }\left(Y\right)\ne 0$ for any $\xi \ne 0$ and $Y\ne 0$. Note that a skew-symmetric linear operator $T_{\xi }^{♯}$ can only have zero real eigenvalues. Thus, for any point $x\in M$, the continuous vector fields $C_{{\xi }_i}\left(Y\right)(\parallel Y\parallel =1,1\le i\le s)$, are tangent to the unit sphere ${\mathbb{S}}_x^{2n-1}\subset {\mathcal{D}}_x^{\perp }$. If $s\ge \rho \left(2n\right)$. Then, these vector fields are linearly dependent at point $\tilde{Y}\in {\mathbb{S}}_x^{2n-1}$ with weights ${\lambda }_i$, i.e., ${\sum }_i{\lambda }_i{C}_{{\xi }_i}\left(\tilde{Y}\right)=0$. Then, the splitting tensor has a real eigenvector as follows: $C_{\xi }\left(\tilde{Y}\right)=\lambda \tilde{Y}$, where $\xi ={\sum }_i{\lambda }_i{\xi }_i\ne 0$ and $\lambda =g(C_{\xi }\left(\tilde{Y}\right),\tilde{Y})=0$, a contradiction. Thus, the inequality $s<\rho \left(2n\right)$ holds. □

6. Weak f-K-Contact Structure Equipped with a Generalized Ricci Soliton

The following three lemmas are used in the proof of Theorem 12 given below.

Lemma 2

(see Lemma 3.1 in [4]). For a weak f-K-contact manifold the following holds:

$$\left({\pounds }_{{\xi }_i}\left({\pounds }_{X}g\right)\right)(Z,{\xi }_i)=g(X,Z)+g({\nabla }_{{\xi }_i}{\nabla }_{{\xi }_i}X,Z)+Zg({\nabla }_{{\xi }_i}X,{\xi }_i)$$

for any $1\le i\le s$ and all vector fields $X,Z$ such that Z is orthogonal to ${\mathcal{D}}^{\perp }$.

Proof. 

This uses ${\nabla }_{{\xi }_i}{\xi }_i=0$ and (29). □

Lemma 3

(e.g., ref. [4]). Let σ be a smooth function on a Riemannian manifold $(M;g)$. Then for any vector fields ξ and Z on M we have the following:

$${\pounds }_{\xi }(d\sigma \otimes d\sigma )(Z,\xi )=Z\left(\xi \left(\sigma \right)\right)\xi \left(\sigma \right)+Z\left(\sigma \right)\xi \left(\xi \left(\sigma \right)\right).$$

Recall the following property of f-K-contact manifolds:

$$Ric(Y,\xi )=0(Y\in \mathcal{D},\xi \in {\mathcal{D}}^{\perp }).$$

(36)

Lemma 4

(see [4]). Let a weak f-K-contact manifold satisfy (36) and admit the generalized gradient Ricci soliton structure. Then,

$${\nabla }_{{\xi }_i}\nabla \sigma =(\lambda +c_2\mathrm{trace}Q){\xi }_i-c_1{\xi }_i\left(\sigma \right)\nabla \sigma .$$

The following theorem generalizes Theorem 3.1 [4].

Theorem 12.

Let a weak f-K-contact manifold with the properties $\mathrm{trace}Q=const$ and (36) satisfy the following generalized gradient Ricci soliton equation with $c_1(\lambda +c_2\mathrm{trace}Q)\ne -1$:

$${\mathrm{Hess}}_{\sigma }-c_{2}Ric=\lambda g-c_{1}d\sigma \otimes d\sigma$$

(37)

for some $\sigma \in C^{\infty }\left(M\right)$ and $c_1,c_2,\lambda \in \mathbb{R}$. Then, $\sigma =const$, and if $c_2\ne 0$, then, our manifold is an Einstein manifold and $s=1$.

Proof. 

Set $Z\in \mathcal{D}$. Using Lemma 2 with $X=\nabla f$, we obtain

$$2\left({\pounds }_{{\xi }_i}\left({\mathrm{Hess}}_{\sigma }\right)\right)(Z,{\xi }_i)=Z\left(\sigma \right)+g({\nabla }_{{\xi }_i}{\nabla }_{{\xi }_i}\nabla \sigma ,Z)+Zg({\nabla }_{{\xi }_i}\nabla \sigma ,{\xi }_i).$$

(38)

Using Lemma 4 in (38) and the properties ${\nabla }_{{\xi }_i}{\xi }_i=0$ and $g({\xi }_i,{\xi }_i)=1$, yields

$$\begin{array}{c}2\left({\pounds }_{{\xi }_i}\left({\mathrm{Hess}}_{\sigma }\right)\right)(Z,{\xi }_i)=Z\left(\sigma \right)+ag({\nabla }_{{\xi }_i}{\xi }_i,Z)\hfill \\ -c_{1}g({\nabla }_{{\xi }_i}({\xi }_i\left(\sigma \right)\nabla \sigma ),Z)+aZ\left(g({\xi }_i,{\xi }_i)\right)-c_{1}Z\left({\xi }_i{\left(\sigma \right)}^2\right)\hfill \\ =Z\left(\sigma \right)-c_{1}g({\nabla }_{{\xi }_i}({\xi }_i\left(\sigma \right)\nabla \sigma ),Z)-c_{1}Z\left({\xi }_i{\left(\sigma \right)}^2\right),\hfill \end{array}$$

(39)

where $a=\lambda +c_2\mathrm{trace}Q$. Using Lemma 4 with $Z\in \mathcal{D}$, from (39) we deduce

$$2\left({\pounds }_{{\xi }_i}\left({\mathrm{Hess}}_{\sigma }\right)\right)(Z,{\xi }_i)=Z\left(\sigma \right)-c_1{\xi }_i\left({\xi }_i\left(\sigma \right)\right)Z\left(\sigma \right)+c_1^2{\xi }_i{\left(\sigma \right)}^{2}Z\left(\sigma \right)-c_{1}Z\left({\xi }_i{\left(\sigma \right)}^2\right).$$

(40)

Since ${\xi }_i$ is a Killing vector field, we obtain ${\pounds }_{{\xi }_i}g=0$. This implies ${\pounds }_{{\xi }_i}Ric=0$. Using the above fact and applying the Lie derivative to Equation (37), gives

$$2\left({\pounds }_{{\xi }_i}\left({\mathrm{Hess}}_{\sigma }\right)\right)(Z,{\xi }_i)=-2c_1\left({\pounds }_{{\xi }_i}(d\sigma \otimes d\sigma )\right)(Z,{\xi }_i).$$

(41)

Using (40), (41), and Lemma 3, we obtain

$$Z\left(\sigma \right)\left(1+c_1{\xi }_i\left({\xi }_i\left(\sigma \right)\right)+c_1^2{\xi }_i{\left(\sigma \right)}^2\right)=0.$$

(42)

Using Lemma 4, we obtain

$$c_1{\xi }_i\left({\xi }_i\left(\sigma \right)\right)=c_1{\xi }_i\left(g({\xi }_i,\nabla \sigma )\right)=c_{1}g({\xi }_i,{\nabla }_{{\xi }_i}\nabla \sigma )=c_{1}a-c_1^2{\xi }_i{\left(\sigma \right)}^2.$$

(43)

Applying (42) in (43), we obtain $Z\left(\sigma \right)(c_{1}a+1)=0$. This implies $Z\left(\sigma \right)=0$ provided by $c_{1}a+1\ne 0$. Hence, $\nabla \sigma \in {\mathcal{D}}^{\perp }$. Taking the covariant derivative of $\nabla \sigma ={\sum }_i{\xi }_i\left(\sigma \right){\xi }_i$ and using (27) and $\overline{\xi }={\sum }_i{\xi }_i$, yields

$$g({\nabla }_X\nabla \sigma ,Z)={\sum }_{i}X\left({\xi }_i\left(\sigma \right)\right){\eta }^i\left(Z\right)-\overline{\xi }\left(\sigma \right)g(\sigma X,Z),X,Z\in {\mathfrak{X}}_M.$$

From this, by symmetry of ${\mathrm{Hess}}_{\sigma }$, i.e., $g({\nabla }_X\nabla \sigma ,Z)=g({\nabla }_Z\nabla \sigma ,X)$, we obtain the equality $\overline{\xi }\left(\sigma \right)g(\sigma X,Z)=0$. For $Z=\sigma X$ with some $X\ne 0$, since $g(\sigma X,\sigma X)>0$, we obtain $\overline{\xi }\left(\sigma \right)=0$. Replacing $\left({\xi }_i\right)$ with another orthonormal frame from ${\mathcal{D}}^{\perp }$ preserves the weak f-K-contact structure and allows reaching any direction $\tilde{\xi }$ in ${\mathcal{D}}^{\perp }$. So $\nabla \sigma =0$, i.e., $\sigma =const$. Thus, from (37) and $c_2\ne 0$, the claim holds. □

Motivated by Proposition 10, we consider quasi-Einstein manifolds, defined by

$$Ric(X,Y)=ag(X,Y)+b\mu \left(X\right)\mu \left(Y\right),$$

where a and b are nonzero real scalars, and $\mu$ is a 1-form of unit norm. If $\mu$ is the differential of a function, then we obtain a gradient quasi-Einstein manifold.

The following theorem generalizes (and uses) Theorem 12.

Theorem 13.

Let a weak f-K-contact manifold with the properties $\mathrm{trace}Q=const$ and (36) satisfy the following generalized gradient Ricci soliton equation:

$${\mathrm{Hess}}_{{\sigma }_1}-c_{2}Ric=\lambda g-c_{1}d{\sigma }_2\otimes d{\sigma }_2$$

(44)

with $c_{1}a\ne -1$, where $a=\lambda +c_2\mathrm{trace}Q$. Then, $\tilde{\sigma }={\sigma }_1+c_{1}a{\sigma }_2$ is constant and

$$-c_{1}a{\mathrm{Hess}}_{{\sigma }_2}=-c_{1}d{\sigma }_2\otimes d{\sigma }_2+c_{2}Ric+\lambda g.$$

(45)

Furthermore,

1. 

If $c_{1}a\ne 0$, then (45) reduces to ${\mathrm{Hess}}_{{\sigma }_2}={\displaystyle \frac{1}{a}}d{\sigma }_2\otimes d{\sigma }_2-{\displaystyle \frac{c_2}{c_{1}a}}Ric-{\displaystyle \frac{\lambda }{c_{1}a}}g$. By Theorem 12, if $c_{1}a\ne -1$, then ${\sigma }_2=const$; moreover, if $c_2\ne 0$, then $(M,g)$ is an Einstein manifold and $s=1$.

2. 

If $a=0$ and $c_1\ne 0$, then (45) reduces to $0=c_{2}Ric-c_{1}d{\sigma }_2\otimes d{\sigma }_2+\lambda g$. If $c_2\ne 0$ and ${\sigma }_2\ne const$, then we obtain a gradient quasi-Einstein manifold.

3. 

If $c_1=0$, then (45) reduces to $0=c_{2}Ric+\lambda g$, and for $c_2\ne 0$ we obtain an Einstein manifold and $s=1$.

Proof. 

Similarly to Lemma 4, we obtain

$${\nabla }_{{\xi }_i}\nabla {\sigma }_1=a{\xi }_i-c_1{\xi }_i\left({\sigma }_2\right)\nabla {\sigma }_2.$$

(46)

Using (46) and Lemmas 2 and 3, and slightly modifying the proof of Theorem 12, we find that the vector field $\nabla \tilde{\sigma }$ belongs to ${\mathcal{D}}^{\perp }$, where $\tilde{\sigma }={\sigma }_1+c_{1}a{\sigma }_2$. As in the proof of Theorem 12, we obtain $d\tilde{\sigma }=0$, i.e., $d{\sigma }_1=-c_{1}ad{\sigma }_2$. Applying this in (44), we obtain (45). Finally, from (45), we obtain the required three cases specified in the theorem. □

7. Compact Weak f-K-Contact Manifold Equipped with an η-Ricci Soliton

The following concepts were introduced in [30,32].

Definition 5.

An η-Ricci soliton is a weak metric f-manifold $M(f,Q,{\xi }_i,{\eta }^i,g)$ satisfying

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

(47)

for some smooth vector field V on M and functions $\lambda ,\mu \in C^{\infty }\left(M\right)$. A weak metric f-manifold $M(f,Q,{\xi }_i,{\eta }^i,g)$ is said to be η-Einstein, if

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

(48)

Remark 5.

For a Killing vector field V, e.g., $V={\xi }_i$ or $V=\overline{\xi }$, Equation (47) reduces to (48). Taking the trace of (48), gives the scalar curvature $r=(2n+s)a+sb$. For $s=1$, (48) and (47) give the following well-known definitions: from (48) we obtain an η-Einstein structure $Ric=ag+b\eta \otimes \eta$, and (47) gives an η-Ricci soliton ${\displaystyle \frac{1}{2}}{\pounds }_{V}g+Ric=\lambda g+\mu \eta \otimes \eta$ on an almost contact metric manifold.

We will use the generalized Pohozaev–Schoen identity.

Lemma 5

(e.g., ref. [38]). Let E be a divergence free symmetric (0,2)-tensor, $E^0=E-{\displaystyle \frac{1}{n}}\left({\mathrm{trace}}_{g}E\right)g$, and V a vector field on a compact Riemannian manifold $(M^n,g)$ without a boundary. Then,

$${\int }_{M}V\left({\mathrm{trace}}_{g}E\right)\mathrm{d}vol={\displaystyle \frac{n}{2}}{\int }_{M}g(E^0,{\pounds }_{V}g)\mathrm{d}vol.$$

(49)

The constancy of the scalar curvature r of a Riemannian manifold is important for proving the triviality of compact generalized $\eta$-Ricci solitons, see [6]. The next theorem extends this result for weak f-K-contact manifolds.

Theorem 14

(see [30]). Let $M^{2n+s}(f,{\xi }_i,{\eta }^i,Q,g)$ be a compact weak f-K-contact manifold with $r=const$ and $\mathrm{trace}Q=const$. Suppose that $(g,V,\lambda ,\nu )$ represents an η-Ricci soliton. Then, M is an η-Einstein manifold.

Proof. 

Taking the trace of (48), yields $r=(2n+s)\alpha +s\beta$. Using (31) in (48), we obtain $\alpha +\beta =\mathrm{trace}Q$. The solution of the linear system is $\alpha ={\displaystyle \frac{1}{2n}}(r-s\mathrm{trace}Q)$ and $\beta ={\displaystyle \frac{1}{2n}}((2n+s)\mathrm{trace}Q-r)$. Define a traceless (0,2)-tensor

$$E:=Ric-{\displaystyle \frac{r-s\mathrm{trace}Q}{2n}}g+{\displaystyle \frac{r-(2n+s)\mathrm{trace}Q}{2n}}{\sum }_i{\eta }^i\otimes {\eta }^i-\mathrm{trace}Q{\sum }_{i\ne j}{\eta }^i\otimes {\eta }^j.$$

Using this, we express $\eta$-Ricci soliton (47) as

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{\mathcal{L}}_{V}g+E& =\left(\lambda -{\displaystyle \frac{r-s\mathrm{trace}Q}{2n}}\right)g+\left(\nu +{\displaystyle \frac{r-(2n+s)\mathrm{trace}Q}{2n}}\right){\sum }_i{\eta }^i\otimes {\eta }^i\hfill \\ \hfill & +(\lambda +\nu -\mathrm{trace}Q){\sum }_{i\ne j}{\eta }^i\otimes {\eta }^j.\hfill \end{array}$$

(50)

We obtain the equality ${div}_{g}E=0$. Since E is traceless, we obtain $E^0=E$ and $V\left({\mathrm{trace}}_{g}E\right)=0$. Then, applying (49), gives ${\int }_{M}g(E,{\mathcal{L}}_{V}g)\mathrm{d}vol=0$. Using (50) and (31) in the above integral formula, and using the following equalities: $g(E,g)={\mathrm{trace}}_{g}E=0$ and $g(E,{\eta }^i\otimes {\eta }^j)=E({\xi }_i,{\xi }_j)=0$, we obtain ${\int }_{M}g(E,E)\mathrm{d}vol=0$. By this, $E=0$ is true. Therefore, our manifold is an $\eta$-Einstein manifold. □

8. Geometry of Weak $\mathit{\beta }$-Kenmotsu $\mathit{f}$-Manifolds

The next definition generalizes the notions of $\beta$-Kenmotsu manifolds and Kenmotsu f-manifolds ($\beta =1,s>1$), see [5,17], and weak $\beta$-Kenmotsu manifolds ($s=1$), see [36].

Definition 6

([32]). A normal (i.e., ${\mathcal{N}}^{\left(1\right)}=0$) weak metric f-manifold $M^{2n+s}(f,Q,{\xi }_i,{\eta }^i,g)$ is called a weak β-Kenmotsu f-manifold (a weak Kenmotsu f-manifold when $\beta \equiv 1$) if

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

(51)

where $\overline{\xi }={\sum }_i{\xi }_i$, $\overline{\eta }={\sum }_i{\eta }^i$, and β is a nonzero smooth function on M.

Remark 6.

Many of the results below on weak β-Kenmotsu f-manifolds hold if we replace (51) with a weaker condition involving non-zero smooth functions ${\beta }_i(1\le i\le s)$ on M, as follows:

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

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 (51) and using $f{\xi }_j=0$, we obtain ${\nabla }_{{\xi }_j}f=0$, which implies $f\left({\nabla }_{{\xi }_i}{\xi }_j\right)=0$, and so ${\nabla }_{{\xi }_i}{\xi }_j\in {\mathcal{D}}^{\perp }$. This and the 1st equality in (7) give

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

(52)

thus, ${\mathcal{D}}^{\perp }$ (of weak $\beta$-Kenmotsu f-manifolds) is tangent to a totally geodesic $\mathfrak{g}$-foliation with an abelian Lie algebra.

Proposition 12.

A weak metric f-manifold with condition (51) is a weak β-Kenmotsu f-manifold if and only if the following formula holds:

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

(53)

Proof. 

(⇒) Consider a weak $\beta$-Kenmotsu f-manifold. Taking $Y={\xi }_i$ in (51) 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 (52), 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 (53).

(⇐) Consider a weak metric f-manifold with conditions (51) and (53). It follows from (5) with $S=f$ and (51) that

$$[f,f](X,Y)=2\beta g(X,fY)f\overline{\xi }=0.$$

Using (53) in the formula for $d{\eta }^j$ gives

$$d{\eta }^j(X,Y)={\sum }_j\{g\left({\nabla }_X{\xi }_j\right),Y)-g\left({\nabla }_Y{\xi }_j\right),X\left)\right\}=0.$$

Therefore, the manifold is normal and hence a weak $\beta$-Kenmotsu f-manifold. □

It follows from (53) and $\beta \ne 0$ that every structure vector field ${\xi }_i$ (of a weak $\beta$-Kenmotsu f-manifold) is not a Killing vector field.

Theorem 15.

A weak metric f-manifold is a weak β-Kenmotsu f-manifold if and only if the following conditions are valid:

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

(54)

Proof. 

(⇒) For a weak $\beta$-Kenmotsu f-manifold, using (53), we obtain

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

(55)

for all $X,Y\in {\mathfrak{X}}_M$. By (55), $\left({\nabla }_X{\eta }^i\right)Y=\left({\nabla }_Y{\eta }^i\right)X$ is true. Thus, for $X,Y\in \mathcal{D}$ we obtain

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

that means $d{\eta }^i(X,Y)=0$ for all $i=1,\dots ,s$ and $X,Y\in \mathcal{D}$, that is, the distribution $\mathcal{D}$ is involutive. By this and ${\mathcal{N}}_{ij}^{\left(4\right)}=0$, see (6), we find $d{\eta }^i=0$. Using (51) and (8), we obtain

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

We also have the following:

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

Thus, $d\Phi =2\beta \overline{\eta }\wedge \Phi$ is valid. By (5) with $S=f$, and (51), we obtain $[f,f]=0$; thus, ${\mathcal{N}}^{\left(1\right)}=0$. Finally, from (9), using (1) and (2), 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 (51), we obtain ${\mathcal{N}}^{\left(5\right)}(X,Y,Z)=2\beta \overline{\eta }\left(X\right)g(fY,\tilde{Q}Z)$.

(⇐) Conversely, for a weak metric f-manifold, using (1) and (54) in (9), we obtain

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

thus (51) is true. □

Theorem 16.

A weak β-Kenmotsu f-manifold is locally a twisted product ${\mathbb{R}}^s{\times }_{\sigma }\overline{M}$ (a warped product when $X\left(\beta \right)=0$ for $X\in \mathcal{D})$, where $\overline{M}(\overline{g},J)$ is a weak Kähler manifold.

Proof. 

By (52), the distribution ${\mathcal{D}}^{\perp }$ is tangent to a totally geodesic foliation. By $d{\eta }^j=0$ (see the proof of Theorem 15), the distribution $\mathcal{D}$ is tangent to a foliation. By (53), the splitting tensor (18) is conformal as follows: $C_{{\xi }_i}X=-\beta X(X\in \mathcal{D})$. Hence, $\mathcal{D}$ is tangent to a totally umbilical foliation with the mean curvature vector $H=-\beta \overline{\xi }$. By Theorem 1 in [39], our manifold is locally a twisted product. If $X\left(\beta \right)=0(X\in \mathcal{D})$, then we locally obtain a warped product, see Proposition 3 in [39]. By (2), the (1,1)-tensor ${J=f|}_{\mathcal{D}}$ is skew-symmetric and $J^2$ is negative definite. Using (51), we find $\left({\overline{\nabla }}_{X}J\right)Y={\pi }_{2*}\left(\left({\nabla }_{X}f\right)Y\right)=0$ for $X,Y\in \mathcal{D}$, thus $\overline{\nabla }J=0$. □

Example 3.

Let $\overline{M}(\overline{g},J)$ be a weak Kähler manifold and $\sigma =c{e}^{\beta \sum t_i}$ a function on Euclidean space ${\mathbb{R}}^s(t_1,\dots ,t_s)$, where $\beta ,c$ are nonzero constants. Then, the warped product manifold $M={\mathbb{R}}^s{\times }_{\sigma }\overline{M}$ has a weak metric f-structure which satisfies (51). 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 5.

A weak Kenmotsu f-manifold $M^{2n+s}(f,Q,{\xi }_i,{\eta }^i,g)$ is locally a warped product ${\mathbb{R}}^s{\times }_{\sigma }\overline{M}$, 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 $\beta =const$.

Proposition 13.

The following formulas hold for weak β-Kenmotsu f-manifolds with $\beta =const$:

$$\begin{array}{cc}\hfill & R_{X,Y}{\xi }_i={\beta }^2\left\{\overline{\eta }\left(X\right)Y-\overline{\eta }\left(Y\right)X+{\sum }_j\left(\overline{\eta }\left(Y\right){\eta }^j\left(X\right)-\overline{\eta }\left(X\right){\eta }^j\left(Y\right)\right){\xi }_j\right\}(X,Y\in {\mathfrak{X}}_M),\hfill \\ \hfill & {Ric}^{♯}{\xi }_i=-2n{\beta }^2\overline{\xi },\hfill \\ \hfill & \left({\nabla }_{{\xi }_i}{Ric}^{♯}\right)X=-2\beta {Ric}^{♯}X-4n{\beta }^3\left\{s\left(X-{\sum }_j{\eta }^j\left(X\right){\xi }_j\right)+\overline{\eta }\left(X\right)\overline{\xi }\right\}(X\in {\mathfrak{X}}_M),\hfill \\ \hfill & {\xi }_i\left(r\right)=-2\beta \{r+2sn(2n+1){\beta }^2\}.\hfill \end{array}$$

The following theorem generalizes Theorem 1 [5] with $\beta \equiv 1$ and $Q=\mathrm{id}$.

Theorem 17.

Let $M^{2n+s}(f,Q,{\xi }_i,{\eta }^i,g)$ be a weak β-Kenmotsu f-manifold satisfying $\beta =const$. If ${\nabla }_{{\xi }_i}{Ric}^{♯}=0$, then $(M,g)$ is an η-Einstein manifold (48) of constant scalar curvature $r=-2sn(2n+1){\beta }^2$.

Proof. 

By conditions and Proposition 13, ${Ric}^{♯}Y=2n{\beta }^2\left\{s(Y-{\sum }_j{\eta }^j\left(Y\right){\xi }_j)+\overline{\eta }\left(Y\right)\overline{\xi }\right\}$; thus, $r=-2sn(2n+1){\beta }^2$. Since (48) with $a=2sn{\beta }^2$ and $b=2(1-s)n{\beta }^2$ holds, $(M,g)$ is $\eta$-Einstein. □

9. $\mathit{\eta }$-Ricci Solitons on Weak $\mathit{\beta }$-Kenmotsu $\mathit{f}$-Manifolds

The following lemmas are used in the proof of Theorem 18 given below.

Lemma 6.

Let $M^{2n+s}(f,Q,{\xi }_i,{\eta }^i,g)$ be a weak β-Kenmotsu f-manifold with $\beta =const$. If g represents an η-Ricci soliton (47), then $\lambda +\mu =-2n{\beta }^2$.

Proof. 

For a weak $\beta$-Kenmotsu f-manifold equipped with an $\eta$-Ricci soliton (47), using (7), we obtain

$$\begin{array}{c}\hfill \left({\pounds }_{V}g\right)({\xi }_i,{\xi }_j)=g({\xi }_i,[V,{\xi }_j])=0.\end{array}$$

Thus, using (47) in the Lie derivative of $g({\xi }_i,{\xi }_j)={\delta }_{ij}$, we obtain $Ric({\xi }_i,{\xi }_j)=\lambda +\mu$. Finally, using the equality $Ric({\xi }_i,{\xi }_j)=-2n{\beta }^2$, see Proposition 13, we achieve the result. □

Lemma 7

(see [32]). Let $M^{2n+s}(f,Q,{\xi }_i,{\eta }^i,g)$ be a weak β-Kenmotsu f-manifold with $\beta =const$. If g represents an η-Ricci soliton (47), then ${\left({\pounds }_{V}R\right)}_{X,{\xi }_j}{\xi }_i=0$ for all $i,j$.

Lemma 8.

On an η-Einstein (48) weak β-Kenmotsu f-manifold with $\beta =const$, we obtain

$$\begin{array}{c}\hfill {\mathrm{Ric}}^{♯}X=\left(s{\beta }^2+{\displaystyle \frac{r}{2n}}\right)X-\left((2n+s){\beta }^2+{\displaystyle \frac{r}{2n}}\right){\sum }_i{\eta }^j\left(X\right){\xi }_j-2n{\beta }^2{\sum }_{i\ne j}{\eta }^i\left(X\right){\xi }_j.\end{array}$$

(56)

Proof. 

Tracing (48) gives $r=(2n+s)a+sb$. Putting $X=Y={\xi }_i$ in (48) and using Proposition 13, 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 (48) gives (56). □

Next, we consider an $\eta$-Einstein weak $\beta$-Kenmotsu f-manifold as an $\eta$-Ricci soliton.

Theorem 18.

Let g represent an η-Ricci soliton (47) on a weak β-Kenmotsu f-manifold with $dimM>3$ and $\beta =const$. If the manifold is also η-Einstein (48), then $a=-2sn{\beta }^2$, $b=2(s-1)n{\beta }^2$, and the scalar curvature is $r=-2sn(2n+1){\beta }^2$.

Definition 7.

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

$$\begin{array}{c}\hfill {\pounds }_X{\eta }^i=\rho {\eta }^i,\end{array}$$

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

We consider the interaction of a weak $\beta$-Kenmotsu f-structure with an $\eta$-Ricci soliton whose potential vector field V is a contact vector field, or V is collinear to $\overline{\xi }$.

Theorem 19.

Let $M^{2n+s}(f,Q,{\xi }_i,{\eta }^i,g)$ be a weak β-Kenmotsu f-manifold with $dimM>3$ and $\beta =const$. If g represents an η-Ricci soliton (47) with a contact potential vector field V, then V is strict contact and the manifold is η-Einstein (48) with $a=-2sn{\beta }^2,b=2(s-1)n{\beta }^2$ of constant scalar curvature $r=-2sn(2n+1){\beta }^2$.

Theorem 20.

Let $M^{2n+s}(f,Q,{\xi }_i,{\eta }^i,g)$ be a weak β-Kenmotsu f-manifold with $dimM>3$ and $\beta =const$. If g represents an η-Ricci soliton (47) with a potential vector field V collinear to $\overline{\xi }$, as follows: $V=\delta \overline{\xi }$ for a smooth function $\delta \ne 0$ on M, then, $\delta =const$ and the manifold is η-Einstein (48) with $a=-2sn{\beta }^2$ and $b=2(s-1)n{\beta }^2$ of constant scalar curvature $r=-2sn(2n+1){\beta }^2$.

Proof. 

Using (51) in the derivative of $V=\delta \overline{\xi }$, 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 $\eta$-Ricci soliton Equation (47) into

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

(57)

Inserting $X=Y={\xi }_i$ in (57) and using Proposition 13 and $\lambda +\mu =-2n{\beta }^2$, see Lemma 6, we obtain ${\xi }_i\left(\delta \right)=0$. It follows from (57) and Proposition 13 that $X\left(\delta \right)=0(X\in \mathcal{D})$. Thus, $\delta$ is constant on M, and (57) 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 (48). Therefore, from Theorem 18, 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 $r=-2sn(2n+1){\beta }^2$. □

Remark 7.

For the case $dimM=3$ (thus, $s=1$) in Theorems 18–20, see [36].

10. Conclusions and Future Directions

This review paper demonstrates that the weak metric f-structure is a valuable tool for exploring various geometric properties on manifolds, including Killing vector fields, totally geodesic foliations, twisted products, Ricci-type solitons, and Einstein-type metrics. Several results for metric f-manifolds have been extended to manifolds with weak structures, providing new insights and applications.

In conclusion, we pose the following open questions: 1. Is the condition “the mixed sectional curvature is positive” sufficient for a weak metric f-manifold to be weak f-K-contact? 2. Does a weak metric f-manifold of a dimension greater than 3 have some positive mixed sectional curvature? 3. Is a compact weak f-K-contact Einstein manifold an $\mathcal{S}$-manifold? 4. When is a given weak f-K-contact manifold a mapping torus (see [22]) of a manifold of a lower dimension? 5. When does a weak metric f-manifold equipped with a Ricci-type soliton structure, carry a canonical (e.g., of constant sectional curvature or Einstein-type) metric? 6. Can Theorems 13–16 be extended to the case where $\beta$ is not constant? These questions highlight the potential for further exploration and development in the field of weak metric f-manifolds, encouraging continued research and discovery.

We delegate the following to the future:

• The study of integral formulas, variational problems, and extrinsic geometric flows for weak metric f-manifolds and their distinguished classes using the methodology of [25].
• The study of weak nearly $\mathcal{S}$- and weak nearly $\mathcal{C}$-manifolds as well as weak nearly Kenmotsu f-manifolds, and the generalization to the case $s>1$ of our results on weak nearly Sasakian/cosymplectic manifolds, see the survey in [27].
• The study of geometric inequalities with the mutual curvature invariants and with Chen-type invariants (and the case of equality in them) for submanifolds in weak metric f-manifolds and in their distinguished classes using the methodology of [40].