# Einstein-type metrics and generalized Ricci solitons
on weak $f$-K-contact manifolds

Vladimir Rovenski111Department of Mathematics, University of Haifa, Israel
e-mail<EMAIL_ADDRESS>
###### Abstract

A weak metric $f$-structure $(f,Q,\xi_{i},\eta^{i},g)\ (i=1,\ldots,s)$,
generalizes the metric $f$-structure on a smooth manifold, i.e., the complex
structure on the contact distribution is replaced with a nonsingular skew-
symmetric tensor $Q$. We study geometry of a weak $f$-K-contact structure,
which is a weak $f$-contact structure, whose characteristic vector fields are
Killing. We show that $\ker f$ of a weak $f$-contact manifold defines a
$\mathfrak{g}$-foliation with an abelian Lie algebra. Then we characterize
weak $f$-K-contact manifolds among all weak metric $f$-manifolds by the
property known for $f$-K-contact manifolds, and find when a Riemannian
manifold endowed with a set of orthonormal Killing vector fields is a weak
$f$-K-contact manifold. We show that for $s>1$, an Einstein weak $f$-K-contact
manifold is Ricci flat, then find sufficient conditions for a weak
$f$-K-contact manifold with parallel Ricci tensor or with a generalized
gradient Ricci soliton structure to be Ricci flat or a quasi Einstein
manifold. We prove positive definiteness of the Jacobi operators in the
characteristic directions and use this to deform a weak $f$-K-contact
structure to an $f$-K-contact structure. Finally, we define an $\eta$-Ricci
soliton structure on a weak metric $f$-manifold and find sufficient conditions
for a compact weak $f$-K-contact manifold with such a structure of constant
scalar curvature to be $\eta$-Einstein.

Keywords: Weak $f$-K-contact manifold, unit Killing vector field, totally
geodesic foliation, Einstein metric, $\eta$-Einstein manifold, generalized
Ricci soliton, $\eta$-Ricci soliton, $\xi$-curvature

Mathematics Subject Classifications (2010) 53C15, 53C25, 53D15

##  1 Introduction

Contact geometry is of growing interest due to its important role in mechanics
in explaining physical phenomena. In addition, many recent articles have been
motivated by the question: how interesting Ricci solitons – self-similar
solutions of the Ricci flow equation, $\partial g(t)/\partial
t=-2\,\operatorname{Ric}_{g(t)}$ – can be for geometry of contact metric
manifolds. Some of them consider conditions when a contact metric manifold
$M(g,f,\xi,\eta)$, equipped with a Ricci-type soliton structure carries a
canonical (e.g., constant curvature or Einstein-type) metric, e.g., [16, 17,
20, 21].

The Ricci soliton equation, where $V$ is a vector field and
$\lambda\in\mathbb{R}$,

$(1/2)\,\pounds_{V}\,g+\operatorname{Ric}=\lambda\,g,$

generalizes Einstein metrics, $\operatorname{Ric}=\lambda\,g$. In the case
that $V$ is the gradient of a potential function $\sigma$ one has a gradient
Ricci soliton equation ${\rm Hess}_{\,\sigma}+\operatorname{Ric}=\lambda\,g$.
Almost Ricci solitons (when $\lambda\in C^{\infty}(M)$, see [22]) are a
generalization of quasi-Einstein manifolds (see [9]) and Ricci solitons. The
following generalized Ricci soliton equation was studied in [21]:

$(1/2)\,\pounds_{V}\,g-c_{2}\operatorname{Ric}=\lambda\,g-c_{1}V^{\flat}\otimes
V^{\flat},$ (1)

where $c_{1},c_{2},\lambda\in\mathbb{R}$ and $V^{\flat}$ is the 1-form dual to
the vector field $V$. If $V=\nabla\sigma$ in (1), then using the definition
${\rm Hess}_{\,\sigma}=\frac{1}{2}\,\pounds_{\nabla\sigma}\,g$, we get the
generalized gradient Ricci soliton equation

${\rm Hess}_{\,\sigma}-c_{2}\operatorname{Ric}=\lambda\,g-c_{1}d\sigma\otimes
d\sigma\,,$ (2)

and the tensor product notation $(d\sigma\otimes
d\sigma)(Y,Z)=d\sigma(Y)\,d\sigma(Z)$ is used. Each equation above is a
generalization of the Einstein metric, $\operatorname{Ric}+\lambda\,g=0$. For
different values of $c_{1},c_{2}$ and $\lambda$, equation (1) generalizes
Killing equation ($c_{1}=c_{2}=\lambda=0$), equation for homotheties
($c_{1}=c_{2}=0$), Ricci soliton equation ($c_{1}=0,\,c_{2}=-1$), vacuum near-
horizon geometry equation ($c_{1}=1,\ c_{2}=1/2$), e.g., [16]. Catino et al.
[8] also introduced the notion of an Einstein-type structure on a manifold,

$\beta\,{\cal L}_{V}\,g+\alpha\operatorname{Ric}=(\rho\,{\rm
Scal}+\lambda)\,g+\mu\,V^{\flat}\otimes V^{\flat}$ (3)

(with $\alpha,\beta,\mu,\rho\in\mathbb{R}$) unifying various cases studied
recently, such as gradient Ricci solitons, and quasi-Einstein manifolds. There
are no Einstein metrics on some compact manifolds, which motivated the study
of generalizations of such metrics. A contact metric manifold is called
$\eta$-Einstein, if

$\operatorname{Ric}=\lambda\,g+\nu\,\eta\otimes\eta,\quad\lambda,\nu\in\mathbb{R}.$

Under certain conditions, a Sasaki-Ricci flow (an odd-dimensional counterpart
of the Kähler-Ricci flow) on a compact manifold converges to an
$\eta$-Einstein metric. The notion of $\eta$-Ricci soliton,

$(1/2)\,\pounds_{V}\,g+\operatorname{Ric}=\lambda\,g+\nu\,\eta\otimes\eta,\quad\nu\in\mathbb{R},$

see [11], generalizes Ricci solitons and $\eta$-Einstein metrics.

A metric $f$-structure on a $(2n+s)$-dimensional smooth manifold is a higher
dimensional analog of a contact structure, defined by a (1,1)-tensor $f$ of
constant rank $2n$, which satisfies $f^{3}+f=0$, and orthonormal vector fields
$\\{\xi_{i}\\}_{1\leq i\leq s}$ spanning $\ker f$ – the $2n$-dimensional
characteristic distribution, see [4, 5, 14, 31, 32, 33]. Foliations of simple
extrinsic geometry, i.e., vanishing second fundamental form of the leaves,
appear on manifolds with degenerate differential forms and curvature-like
tensors, e.g. [27]. The following problem was posed in [14]: find suitable
structures on manifolds, which lead to totally geodesic foliations. The
distribution $\ker f$ of an $f$-contact structure defines a totally geodesic
foliation. An interesting case occurs when $\ker f$ is defined by a
homomorphism of a $s$-dimensional Lie algebra $\mathfrak{g}$ to the Lie
algebra of all vector fields on $M$, i.e., $M$ admits a
$\mathfrak{g}$-foliation, see [1]. In the presence of a compatible metric,
such $\mathfrak{g}$-foliations are spanned by Killing vector fields, e.g.,
[7]. For a 1-dimensional Lie algebra, a ${\mathfrak{g}}$-foliation is
generated by a nonvanishing vector field, and we get contact metric manifolds
as well as $K$-contact and Sasakian ones, see [3].

An $f$-K-contact structure, i.e. an $f$-contact structure, whose
characteristic vector fields are unit Killing vector fields, see [18], can be
regarded as intermediate between a metric $f$-structure and $S$-structure (the
Sasaki structure when $s=1$). The influence of constant length Killing vector
fields on the geometry of Riemannian manifolds has been studied by several
authors from different points of view, e.g., [2, 12]. The curvature of
$f$-contact and $f$-K-contact manifolds was studied in [6, 29, 30]. Observe
that there are no Einstein metrics on $f$-K-contact manifolds.

In [24, 25, 28] the study of “weak” contact structures on a smooth
$(2n+s)$-dimensional manifold (i.e., the complex structure on the
characteristic distribution is replaced with a nonsingular skew-symmetric
tensor) was initiated. This generalizes a metric $f$-structure and its
satellites, and allows a new look at the classical theory and new
applications. In [24], we proved that the ${\cal S}$-structure is rigid, i.e.,
our weak ${\cal S}$-structure is the ${\cal S}$-structure, and a weak metric
$f$-structure with parallel tensor $f$ is a weak ${\cal C}$-structure. In
[25], we obtained a topological obstruction (including the Adams number) to
the existence of weak $f$-K-contact manifolds. In [28], we retracted weak
structures with positive partial Ricci curvature onto the subspace of
classical structures of the same type.

The article continues our study [24, 25, 28] of the geometry of weak
$f$-contact manifolds. Following the approach in [23] for $s=1$, we study the
sectional and Ricci curvature in the ${\xi}$-directions of a weak $f$-contact
structure and its particular case – a weak $f$-K-contact structure. The
characteristic distribution of a weak $f$-contact manifold defines a totally
geodesic foliation with flat leaves. Our goal is to show that the weak
structures can be a useful for studying unit Killing vector fields, totally
geodesic foliations and $\xi$-sectional curvature on Riemannian manifolds, and
that some results on $f$-K-contact manifolds can be extended to the case of
weak $f$-K-contact manifolds. For example, we answer the questions of when a
weak $f$-K-contact manifold (i) carries a generalized Ricci soliton structure
or just a gradient quasi Einstein (e.g., an Einstein) metric, and (ii) carries
an $\eta$-Ricci soliton structure or just an $\eta$-Einstein metric. The
proofs use the properties of new tensors, as well as the constructions
required in the classical case.

The article is organized as follows. Section 2, following the introductory
Section 1, presents basics of weak metric $f$-manifolds. Section 3 presents
basics of weak $f$-contact and weak $f$-K-contact manifolds and shows that
these manifolds are endowed with totally geodesic foliations. We show that
$\ker f$ of a weak $f$-contact manifold defines a $\mathfrak{g}$-foliation
with an abelian Lie algebra. In Section 4, we characterize (Theorem 4.1) weak
$f$-K-contact manifolds among all weak $f$-contact manifolds by the property
${f}=-\nabla{\xi_{i}}\ (i\leq s)$ known for $f$-K-contact manifolds, and find
sufficient conditions (Theorem 4.2) under which when a Riemannian manifold
endowed with a set of unit Killing vector fields is a weak $f$-K-contact
manifold. In Section 5, for a weak $f$-K-contact manifold, we calculate (in
Proposition 5.1) the Ricci curvature in the ${\xi}$-directions, prove that the
$\xi$-sectional curvature is positive (Corollary 5.2). This allows us to
deform a weak $f$-K-contact structure to the $f$-K-contact structure (Theorem
5.2). We show that for $s>1$, an Einstein weak $f$-K-contact manifold is Ricci
flat and find (Theorem 5.1) sufficient conditions for a weak $f$-K-contact
manifold with parallel Ricci tensor to be a Ricci flat manifold, or, for
$s=1$, an Einstein manifold. In Section 6, we find (Theorems 6.1 and 6.2)
sufficient conditions for a weak $f$-K-contact manifold with a generalized
gradient Ricci soliton to be a quasi Einstein or Ricci flat manifold, or, for
$s=1$, an Einstein manifold. In Section 7, we define an $\eta$-Ricci soliton
structure and find sufficient conditions for a compact weak $f$-K-contact
manifold with such a structure of constant scalar curvature to be
$\eta$-Einstein (Theorem 7.1).

##  2 Preliminaries

Here, we survey the basics of a weak metric $f$-structure (see [24, 28]) as a
higher dimensional analog of a weak almost contact structure (see [26]). We
define weak $f$-contact and weak $f$-K-contact structures and show that these
structures are endowed with totally geodesic foliations.

A framed weak $f$-structure on a smooth manifold $M^{2n+s}$ is a set
$({f},Q,{\xi_{i}},{\eta^{i}})$, where ${f}$ is a $(1,1)$-tensor, $Q$ is a
nonsingular $(1,1)$-tensor, ${\xi_{i}}\ (1\leq i\leq s)$ are characteristic
vector fields and ${\eta^{i}}$ are dual 1-forms, satisfying

${f}^{2}=-Q+\sum\nolimits_{\,i}{\eta^{i}}\otimes{\xi_{i}},\quad{\eta^{i}}({\xi_{j}})=\delta^{i}_{j},\quad
Q\,{\xi_{i}}={\xi_{i}}.$ (4)

The forms ${\eta^{i}}$ determine a $2n$-dimensional contact distribution
${\cal D}:=\bigcap_{\,i}\ker{\eta^{i}}$. Assume that ${\cal D}$ is
${f}$-invariant (that is true for framed $f$-structure [4, 33], where $Q={\rm
id}_{\,TM}$),

${f}X\in{\cal D}\quad(X\in{\cal D}).$ (5)

By (4) and (5), the distribution ${\cal D}$ is invariant for $Q$: $Q({\cal
D})={\cal D}$. Note that $f^{3}+fQ=0$. For a framed weak $f$-structure on a
manifold $M$, the tensor ${f}$ has rank $2n$, thus ${\cal D}=f(TM)$, and

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

A framed weak $f$-structure $({f},Q,{\xi_{i}},{\eta^{i}}^{i})$ is called
normal if the following tensor is zero:

$\displaystyle
N^{\,(1)}(X,Y)=[{f},{f}](X,Y)+2\sum\nolimits_{\,i}d{\eta^{i}}(X,Y)\,{\xi_{i}},\quad
X,Y\in\mathfrak{X}_{M},$

where the Nijenhuis torsion $[{f},{f}]$ of ${f}$ is given by

$\displaystyle[{f},{f}](X,Y)={f}^{2}[X,Y]+[{f}X,{f}Y]-{f}[{f}X,Y]-{f}[X,{f}Y],\quad
X,Y\in\mathfrak{X}_{M},$ (6)

and the exterior derivative of a differential form ${\eta^{i}}$ is given by

$d\eta^{i}(X,Y)=\frac{1}{2}\,\\{X({\eta^{i}}(Y))-Y({\eta^{i}}(X))-{\eta^{i}}([X,Y])\\},\quad
X,Y\in\mathfrak{X}_{M}.$ (7)

Recall the following formulas with the Lie derivative $\pounds_{Z}$ in the
$Z$-direction:

$\displaystyle(\pounds_{Z}{f})X$
$\displaystyle=[Z,{f}X]-{f}[Z,X]\quad(X,Y\in\mathfrak{X}_{M}),$
$\displaystyle(\pounds_{{\xi_{i}}}\,g)(X,Y)$
$\displaystyle={\xi_{i}}(g(X,Y))-g([{\xi_{i}},X],Y)-g(X,[{\xi_{i}},Y]).$ (8)

The following tensors $N^{\,(2)}_{i},N^{\,(3)}_{i}$ and $N^{\,(4)}_{ij}$ are
well known in the classical theory, see [4, 33]:

$\displaystyle N^{\,(2)}_{i}(X,Y)$
$\displaystyle=(\pounds_{{f}X}\,{\eta^{i}})(Y)-(\pounds_{{f}Y}\,{\eta^{i}})(X)\overset{\eqref{3.3A}}{=}2\,d{\eta^{i}}({f}X,Y)-2\,d{\eta^{i}}({f}Y,X),$
$\displaystyle N^{\,(3)}_{i}(X)$
$\displaystyle=(\pounds_{{\xi_{i}}}\,{f})X=[{\xi_{i}},{f}X]-{f}[{\xi_{i}},X],$
$\displaystyle N^{\,(4)}_{ij}(X)$
$\displaystyle=(\pounds_{{\xi_{i}}}\,{\eta^{j}})(X)={\xi_{i}}({\eta^{j}}(X))-{\eta^{j}}([{\xi_{i}},X])\overset{\eqref{3.3A}}{=}2\,d{\eta^{j}}({\xi_{i}},X),$

where $\pounds_{Z}$ is the Lie derivative in the ${Z}$-direction. For $s=1$,
the above tensors reduce to the following tensors on (weak) almost contact
manifolds:

$\displaystyle
N^{\,(2)}(X,Y)=(\pounds_{f\,X}\,\eta)Y-(\pounds_{f\,Y}\,\eta)X,\quad
N^{\,(3)}=\pounds_{\xi}\,f,\quad N^{\,(4)}=\pounds_{\xi}\,\eta.$

######  Remark 2.1 (see [23]).

Let $M^{2n+s}(f,Q,\xi_{i},\eta^{i})$ be a framed weak $f$-manifold. Consider
the product manifold $\bar{M}=M^{2n+s}\times\mathbb{R}^{s}$, where
$\mathbb{R}^{s}$ is a Euclidean space with a basis
$\partial_{1},\ldots,\partial_{s}$, and define tensors $\bar{f}$ and $\bar{Q}$
on $\bar{M}$ putting

$\displaystyle\bar{f}(X,\,\sum\nolimits_{\,i}a^{i}\partial_{i})$
$\displaystyle=(fX-\sum\nolimits_{\,i}a^{i}\xi_{i},\,\sum\nolimits_{\,j}\eta^{j}(X)\partial_{j}),$
$\displaystyle\bar{Q}(X,\,\sum\nolimits_{\,i}a^{i}\partial_{i})$
$\displaystyle=(QX,\,\sum\nolimits_{\,i}a^{i}\partial_{i}),$

where $a_{i}\in C^{\infty}(M)$. Hence, $\bar{f}(X,0)=(fX,0)$,
$\bar{Q}(X,0)=(QX,0)$ for $X\in{\cal D}$,
$\bar{f}(\xi_{i},0)=(0,\partial_{i})$, $\bar{Q}(\xi_{i},0)=(\xi_{i},0)$ and
$\bar{f}(0,\partial_{i})=(-\xi_{i},0)$,
$\bar{Q}(0,\partial_{i})=(0,\partial_{i})$. Then it is easy to verify that
$\bar{f}^{\,2}=-\bar{Q}$. The tensors
$N^{\,(1)},N^{\,(2)}_{i},N^{\,(3)}_{i},N^{\,(4)}_{ij}$ appear when we derive
the integrability condition $[\bar{f},\bar{f}]=0$ of $\bar{f}$ and express the
normality condition $N^{\,(1)}=0$ of a framed weak $f$-structure on $M$.

If there is a Riemannian metric $g$ on $M$ such that

$\displaystyle
g({f}X,{f}Y)=g(X,Q\,Y)-\sum\nolimits_{\,i}{\eta^{i}}(X)\,{\eta^{i}}(Y),\quad
X,Y\in\mathfrak{X}_{M},$ (9)

then $({f},Q,{\xi_{i}},{\eta^{i}},g)$ is called a weak metric $f$-structure on
$M$, and $g$ is called a compatible metric. Note that $X=X^{\top}+X^{\bot}$,
where $X^{\top}=\sum_{\,i}\eta^{i}(X)\,\xi_{i}$ is the projection of the
vector $X\in TM$ on $\ker f$. A framed weak $f$-manifold
$M({f},Q,{\xi_{i}},{\eta^{i}})$ endowed with a compatible Riemannian metric
$g$ is called a metric weak $f$-manifold and is denoted by
$M({f},Q,{\xi_{i}},{\eta^{i}},g)$. Putting $Y={\xi_{i}}$ in (9) and using
$Q\,{\xi_{i}}={\xi_{i}}$, we get, as in the classical theory,
${\eta^{i}}(X)=g(X,{\xi_{i}})$. In particular, ${\xi_{i}}$ is $g$-orthogonal
to ${\cal D}$ for any compatible metric $g$. Thus, the orthogonal complement
${\cal D}^{\bot}$ of ${\cal D}$ is the characteristic (or, Reeb) distribution
$\ker f$. Some sufficient conditions for the existence of a compatible metric
on a weak almost contact manifold are given in [28].

Using the Levi-Civita connection $\nabla$ of $g$, one can rewrite (6) as

$\displaystyle[{f},{f}](X,Y)=({f}\nabla_{Y}{f}-\nabla_{{f}Y}{f})X-({f}\nabla_{X}{f}-\nabla_{{f}X}{f})Y.$
(10)

For a weak metric $f$-structure, the distribution ${\cal D}$ is non-integrable
(has no integral $2n$-dimensional submanifolds) since for any nonzero
$X\in{\cal D}$ we get

$g([X,{f}X],{\xi_{i}})=2\,d{\eta^{i}}({f}X,X)=g({f}X,{f}X)>0.$

For a weak metric $f$-structure, the tensor ${f}$ is skew-symmetric and $Q$ is
self-adjoint, see [24, 28]:

$g({f}X,Y)=-g(X,{f}Y),\quad g(QX,Y)=g(X,QY),\quad X,Y\in\mathfrak{X}_{M}.$

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

$\displaystyle d\Phi(X,Y,Z)$
$\displaystyle=\frac{1}{3}\,\big{\\{}X\,\Phi(Y,Z)+Y\,\Phi(Z,X)+Z\,\Phi(X,Y)$
$\displaystyle-\Phi([X,Y],Z)-\Phi([Z,X],Y)-\Phi([Y,Z],X)\big{\\}}.$ (11)

######  Remark 2.2.

A differential $k$-form on a manifold $M$ is a skew-symmetric tensor field
$\omega$ of type $(0,k)$. According the conventions of e.g., [19], the formula

$\displaystyle
d\omega({X}_{1},\ldots,{X}_{k+1})=\frac{1}{k+1}\sum\nolimits_{\,i=1}^{k+1}(-1)^{i+1}{X}_{i}(\omega({X}_{1},\ldots,\hat{{X}}_{i}\ldots,{X}_{k+1}))$
$\displaystyle+\sum\nolimits_{\,i<j}(-1)^{i+j}\omega([{X}_{i},{X}_{j}],{X}_{1},\ldots,\hat{{X}}_{i},\ldots,\hat{{X}}_{j},\ldots,{X}_{k+1}),$
(12)

where ${X}_{i}\in\mathfrak{X}_{M}$ and $\,\hat{\cdot}\,$ denotes the operator
of omission, defines a $(k+1)$-form $d\omega$ called the exterior differential
of $\omega$. Note that (7) and (2) correspond to (2.2) with $k=1$ and $k=2$.

##  3 Weak $f$-contact and weak $f$-K-contact structures

Here, we define “weak” structures that generalize $f$-contact and
$f$-K-contact structures.

######  Definition 3.1.

A weak $f$-contact structure is a weak metric $f$-structure satisfying

$\displaystyle\Phi=d{\eta^{1}}=\ldots=d{\eta^{s}},$ (13)

thus, $d\Phi=0$. A weak $f$-contact structure is called a weak $f$-K-contact
structure if all characteristic vector fields ${\xi_{i}}$ are Killing, i.e.

$\displaystyle(\pounds_{{\xi_{i}}}\,g)(X,Y):={\xi_{i}}(g(X,Y))-g([{\xi_{i}},X],Y)-g(X,[{\xi_{i}},Y])$
$\displaystyle=g(\nabla_{X}{\xi_{i}},Y)+g(\nabla_{Y}{\xi_{i}},X)=0.$ (14)

For a weak $f$-contact structure and all $i$ and $j$, by (13), we get

$d\eta^{j}(\xi_{i},X)=\Phi(\xi_{i},X)=0.$

The relationships between the different classes of weak manifolds (considered
in this article) can be summarizes in the diagram well known for classical
structures:

$\left|\\!\\!\\!\begin{array}[]{c}\textrm{framed\ weak}\\\
f\textrm{-manifold}\\\
\end{array}\\!\\!\\!\right|\overset{\textrm{metric}}{\longrightarrow}\left|\\!\\!\\!\begin{array}[]{c}\textrm{metric\
weak}\\\ f\textrm{-manifold}\\\
\end{array}\\!\\!\\!\right|\overset{\Phi=d\eta^{i}}{\longrightarrow}\left|\\!\\!\\!\begin{array}[]{c}\textrm{weak}\\\
f\textrm{-contact}\\\
\end{array}\\!\\!\\!\right|\overset{\xi_{i}\,\textrm{-Killing}}{\longrightarrow}\left|\\!\\!\\!\begin{array}[]{c}\textrm{weak}\\\
f\textrm{-K-contact}\\\
\end{array}\\!\\!\\!\right|\overset{N^{\,(1)}=0}{\longrightarrow}\left|\\!\\!\\!\begin{array}[]{c}\textrm{weak}\\\
{\cal S}\textrm{-manifold}\\\ \end{array}\\!\\!\\!\right|.$

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

$\left|\\!\\!\\!\begin{array}[]{c}\textrm{weak\ almost}\\\ \textrm{contact}\\\
\end{array}\\!\\!\\!\right|\overset{\textrm{metric}}{\longrightarrow}\left|\\!\\!\begin{array}[]{c}\textrm{weak\
almost}\\\ \textrm{contact\ metric}\\\
\end{array}\\!\\!\right|\overset{\Phi=d\eta}{\longrightarrow}\left|\\!\\!\begin{array}[]{c}\textrm{weak\
contact}\\\ \textrm{metric}\\\
\end{array}\\!\\!\right|\overset{\xi\,\textrm{-Killing}}{\longrightarrow}\left|\\!\\!\\!\begin{array}[]{c}\textrm{weak}\\\
\textrm{K-contact}\\\
\end{array}\\!\\!\\!\right|\overset{N^{\,(1)}=0}{\longrightarrow}\left|\\!\\!\\!\begin{array}[]{c}\textrm{weak}\\\
\textrm{Sasakian}\\\ \end{array}\\!\\!\\!\right|.$

######  Remark 3.1.

In [13, 24, 29], the $f$-contact structure is called the almost ${\cal
S}$-structure, and the normal $f$-contact structure is called the ${\cal
S}$-structure. A weak metric $f$-structure with the properties $d\Phi=0$ and
$N^{\,(1)}=0$ is called a weak ${\cal K}$-structure; in this case,
$\xi_{1},\ldots,\xi_{s}$ are Killing vector fields, see [24]; thus, we get a
weak $f$-K-contact structure.

######  Proposition 3.1 (see Theorem 2.2 in [24]).

For a weak $f$-contact structure, the tensors $N^{\,(2)}_{i}$ and
$N^{\,(4)}_{ij}$ vanish; moreover, $N^{\,(3)}_{i}$ vanishes if and only if
$\,\xi_{i}$ is a Killing vector field. In particular, for a weak $f$-K-contact
structure, the tensors $N^{\,(2)}_{i}$, $N^{\,(3)}_{i}$ and $N^{\,(4)}_{ij}$
vanish.

A “small” (1,1)-tensor $\widetilde{Q}=Q-{\rm id}_{\,TM}$ measures the
difference between a weak $f$-contact structure and a contact one, and
$\widetilde{Q}=0$ means the classical $f$-contact geometry. We have

$[\widetilde{Q},{f}]=0,\quad\widetilde{Q}\,{\xi_{i}}=0.$

######  Proposition 3.2 (see Corollary 2.1 in [24]).

For a weak $f$-contact structure we get

$\displaystyle 2\,g((\nabla_{X}{f})Y,Z)$
$\displaystyle=g(N^{\,(1)}(Y,Z),{f}X)+2\,g(fX,fY)\,\bar{\eta}(Z)-2\,g(fX,fZ)\,\bar{\eta}(Y)$
$\displaystyle+N^{\,(5)}(X,Y,Z),$ (15)

where $\bar{\eta}=\sum\nolimits_{\,i}\eta^{i}$, and a skew-symmetric with
respect to $Y$ and $Z$ tensor $N^{\,(5)}(X,Y,Z)$ is given by

$\displaystyle N^{\,(5)}(X,Y,Z)$
$\displaystyle=({f}Z)\,(g(X,\widetilde{Q}Y))-({f}Y)\,(g(X,\widetilde{Q}Z))+g([X,{f}Z],\widetilde{Q}Y)-g([X,{f}Y],\widetilde{Q}Z)$
$\displaystyle+\,g([Y,{f}Z]-[Z,{f}Y]-{f}[Y,Z],\ \widetilde{Q}X).$

Taking $X=\xi_{i}$ in (3.2), we get

$\displaystyle 2\,g((\nabla_{\xi_{i}}{f})Y,Z)$
$\displaystyle=N^{\,(5)}(\xi_{i},Y,Z),\quad 1\leq i\leq s.$ (16)

Recall that a distribution $\widetilde{\cal D}\subset TM$ is totally geodesic
if and only if $\nabla_{X}Y+\nabla_{Y}X\in\widetilde{\cal D}$ for any vector
fields $X,Y\in\widetilde{\cal D}$ – this is the case when any geodesic of $M$
that is tangent to $\widetilde{\cal D}$ at one point is tangent to
$\widetilde{\cal D}$ at all its points, e.g., [27, Section 1.3.1]. An
integrable totally geodesic distribution determines a totally geodesic
foliation.

Let $\mathfrak{g}$ be a Lie algebra of dimension $s$. We 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
$\xi_{1},\ldots,\xi_{s}$ on $M$, which restricted to each leaf of
$\mathcal{F}$ form a parallelism of this submanifold which is Lie algebra
isomorphic to $\mathfrak{g}$, see, for example, [28]. A Lie algebra
$\mathfrak{g}$ is abelian if its bracket is identically zero. We show that
$\ker f$ of a weak $f$-contact manifold defines a $\mathfrak{g}$-foliation
with an abelian Lie algebra $\mathfrak{g}$.

######  Corollary 3.1.

For a weak $f$-contact structure, the distribution $\ker f$ is integrable and
defines a $\mathfrak{g}$-foliation with totally geodesic flat leaves.

###### Proof.

Recall that
$[\xi_{i},\xi_{j}]=\nabla_{\xi_{i}}\,\xi_{j}-\nabla_{\xi_{j}}\,\xi_{i}$.
Taking $Y=\xi_{j}$ in (16), we get

$\displaystyle\nabla_{\xi_{i}}\,\xi_{j}=0,\qquad 1\leq i,j\leq s.$ (17)

By (17), $\ker f$ of a weak $f$-contact structure defines a
$\mathfrak{g}$-foliation with an abelian Lie algebra (that is
$[\xi_{i},\xi_{j}]=0$) with totally geodesic (that is
$\nabla_{\xi_{i}}\,\xi_{j}=0$) flat leaves (that is
$R_{\xi_{i},\xi_{j}}\,\xi_{k}=0$). ∎

######  Remark 3.2.

Only one new tensor $N^{(5)}$ (vanishing at $\widetilde{Q}=0$), which
supplements the sequence of well-known tensors $N^{\,(i)},\ i=1,2,3,4$, is
needed to study the weak $f$-contact structure. For particular values of the
tensor $N^{\,(5)}$ we get

$\displaystyle N^{\,(5)}(X,\xi_{i},Z)$
$\displaystyle=-N^{\,(5)}(X,Z,\xi_{i})=g(N^{\,(3)}_{i}(Z),\,\widetilde{Q}X),$
$\displaystyle N^{\,(5)}(\xi_{i},Y,Z)$
$\displaystyle=g([\xi_{i},{f}Z],\widetilde{Q}Y)-g([\xi_{i},{f}Y],\widetilde{Q}Z),$
$\displaystyle N^{\,(5)}(\xi_{i},\xi_{j},Y)$
$\displaystyle=N^{\,(5)}(\xi_{i},Y,\xi_{j})=0.$ (18)

######  Definition 3.2 (see [24]).

Two framed weak $f$-structures $({f},Q,{\xi_{i}},{\eta^{i}})$ and
$({f}^{\prime},Q^{\prime},{\xi_{i}},{\eta^{i}})$ on a smooth manifold $M$ are
said to be homothetic if the following is valid for some real $\lambda>0$:

$\displaystyle{f}=\sqrt{\lambda}\ {f}^{\prime},\quad
Q\,|_{\,{\mathcal{D}}}=\lambda\,Q^{\prime}|_{\,\mathcal{D}}.$ (19)

Two weak metric $f$-structures $({f},Q,{\xi_{i}},{\eta^{i}},g)$ and
$({f}^{\prime},Q^{\prime},{\xi_{i}},{\eta^{i}},g^{\prime})$ on $M$ are said to
be homothetic if they satisfy conditions (19) and

$\displaystyle
g|_{\,{\mathcal{D}}}=\lambda^{-\frac{1}{2}}\,g^{\prime}|_{\,{\mathcal{D}}},\quad
g({\xi_{i}},\,\cdot)={g}^{\prime}({\xi_{i}},\,\cdot).$ (20)

######  Lemma 3.1 (see [24]).

Let $({f},Q,{\xi_{i}},{\eta^{i}})$ be a framed weak $f$-structure such that
$Q\,|_{\,{\mathcal{D}}}=\lambda\,{\rm id}_{\mathcal{D}}$ for some real
$\lambda>0$. Then the following is true:

$\bullet$ $({f}^{\prime},{\xi_{i}},{\eta^{i}})$ is a framed $f$-contact
structure, where ${f}^{\prime}$ is given by (19)1.

$\bullet$ If $({f},Q,{\xi_{i}},{\eta^{i}},g)$ is a weak $f$-contact structure
and (19)1 and (20) are valid, then
$({f}^{\prime},{\xi_{i}},{\eta^{i}},{g}^{\prime})$ is an $f$-contact
structure.

##  4 Killing vector fields of unit length

Here, we characterize weak $f$-K-contact manifolds among all weak $f$-contact
manifolds and find conditions under which a Riemannian manifold endowed with a
set of unit Killing vector fields is a weak $f$-K-contact manifold.

The tensor $N^{\,(3)}_{i}$ is important for $f$-contact manifolds. Therefore,
we define for a weak $f$-contact manifold the tensor field
$h=(h_{1},\ldots,h_{s})$, where

$\displaystyle
h_{i}=({1}/{2})\,N^{\,(3)}_{i}=({1}/{2})\,\pounds_{\xi_{i}}{f}.$

By Proposition 3.1, $h_{i}=0$ if and only if $\xi_{i}$ is a Killing vector
field. First, we calculate

$\displaystyle(\pounds_{\xi_{i}}{f})X$
$\displaystyle\overset{\eqref{3.3B}}{=}\nabla_{\xi_{i}}({f}X)-\nabla_{{f}X}\,\xi_{i}-{f}(\nabla_{\xi_{i}}X-\nabla_{X}\,\xi_{i})$
$\displaystyle=(\nabla_{\xi_{i}}{f})X-\nabla_{{f}X}\,\xi_{i}+{f}\nabla_{X}\,\xi_{i}.$
(21)

Taking $X=\xi_{i}$ in (4) and using $\nabla_{\xi_{i}}\,\xi_{j}=0$, see (17),
and $(\nabla_{\xi_{i}}{f})\,\xi_{j}=0$, see (16) with $Y=\xi_{j}$, we get the
equality $h_{i}\,\xi_{j}=0$.

For an $f$-contact structure, the linear operator $h_{i}$ is self-adjoint,
trace-free and anti-commutes with $f$: $h_{i}{f}+{f}\,h_{i}=0$, see [13]. We
generalize this for a weak $f$-contact structure.

######  Proposition 4.1 (see [24, 25]).

For a weak $f$-contact structure $(f,Q,\xi_{i},\eta^{i},g)$, the tensor
$h_{i}$ and its conjugate $h_{i}^{*}$ satisfy

$\displaystyle g((h_{i}-h_{i}^{*})X,Y)$ $\displaystyle=$
$\displaystyle({1}/{2})\,N^{\,(5)}(\xi_{i},X,Y),\quad X,Y\in TM,$
$\displaystyle h_{i}{f}+{f}\,h_{i}$ $\displaystyle=$
$\displaystyle-(1/2)\,\pounds_{\xi_{i}}\widetilde{Q},$ $\displaystyle
h_{i}{Q}-{Q}\,h_{i}$ $\displaystyle=$
$\displaystyle(1/2)[\,f,\,\pounds_{\xi_{i}}\,\widetilde{Q}\,].$

The following corollary generalizes the known property of $f$-contact
manifolds.

######  Corollary 4.1.

Let a weak $f$-contact manifold satisfy $\pounds_{\xi_{i}}\,\widetilde{Q}=0$,
then

${\rm\,trace\,}h_{i}=0.$

###### Proof.

If $h_{i}X=\lambda X$, then using $h_{i}f=-fh_{i}$ (by the assumptions and
Proposition 4.1), we get $h_{i}fX=-\lambda fX$. Thus, if $\lambda$ is an
eigenvalue of $h_{i}$, then $-\lambda$ is also an eigenvalue of $h_{i}$;
hence, ${\rm\,trace\,}h_{i}=0$. ∎

######  Lemma 4.1.

On a weak $f$-K-contact manifold $M^{2n+s}(f,Q,\xi_{i},\eta^{i},g)$, we get
$N^{\,(1)}({\xi_{i}},\,\cdot)=0$ and

$\displaystyle N^{\,(5)}({\xi_{i}},\,\cdot\,,\,\cdot)$ $\displaystyle=$
$\displaystyle N^{\,(5)}(\,\cdot\,,\,{\xi_{i}},\,\cdot)=0,$ (22)
$\displaystyle\pounds_{{\xi_{i}}}\,\widetilde{Q}$ $\displaystyle=$
$\displaystyle\nabla_{\xi_{i}}\,\widetilde{Q}=0,$ (23)
$\displaystyle\nabla_{{\xi_{i}}}\,{f}$ $\displaystyle=$ $\displaystyle 0.$
(24)

###### Proof.

From (10) with $Y=\xi_{i}$, since ${f}\,\xi_{i}=0$, we get

$\displaystyle[{f},{f}](X,\xi_{i})={f}(\nabla_{\xi_{i}}{f})X+\nabla_{{f}X}\,\xi_{i}-{f}\,\nabla_{X}\,\xi_{i},\quad
X\in\mathfrak{X}_{M}.$ (25)

By (25) and $d{\eta^{j}}({\xi_{i}},\,\cdot)=\Phi(\xi_{i},\,\cdot)=0$ we have

$N^{\,(1)}({\xi_{i}},X)=[{f},{f}](X,{\xi_{i}})={f}^{2}[X,{\xi_{i}}]-{f}[{f}X,{\xi_{i}}]={f}N^{\,(3)}_{i}(X)=0.$

By (3.2)1, using $N^{\,(3)}_{i}=0$ we get
$N^{\,(5)}(\,\cdot\,,{\xi_{i}},\,\cdot)=N^{\,(5)}(\,\cdot\,,\cdot\,,\xi_{i})=0$.
Then, by Proposition 4.1 with $h_{i}:=\frac{1}{2}\,N^{\,(3)}_{i}=0$, we get
$N^{\,(5)}({\xi_{i}},\,\cdot\,,\,\cdot)=0$, $\pounds_{{\xi_{i}}}{Q}=0$ and

$g(Q\,\nabla_{X}\,{\xi_{i}},Z)=g({f}Z,QX)-(1/2)\,N^{\,(5)}(X,{\xi_{i}},{f}Z)=g({f}Z,QX).$

We use $[{f},{Q}]=0$ and $Q=\widetilde{Q}+{\rm id}_{\,TM}$ to obtain
$\nabla_{\xi_{i}}\,\widetilde{Q}=0$:

$(\pounds_{{\xi_{i}}}\,{Q})X=[{\xi_{i}},{Q}X]-{Q}[{\xi_{i}},X]=(\nabla_{\xi_{i}}{Q})X+[{f},{Q}]X=(\nabla_{\xi_{i}}\,{Q})X=(\nabla_{\xi_{i}}\,\widetilde{Q})X.$

This completes the proof of (22) and (23). Next, from (16) and (22) we get
(24). ∎

In the next theorem, we characterize weak $f$-K-contact manifolds among all
weak $f$-contact manifolds by the following well known property of
$f$-K-contact manifolds, see [4, 18]:

$\nabla\,{\xi_{i}}=-{f},\quad 1\leq i\leq s.$ (26)

######  Theorem 4.1.

A weak $f$-contact manifold is weak $f$-K-contact if and only if (26) holds.

###### Proof.

Let a weak $f$-contact manifold satisfy (26). By skew-symmetry of ${f}$, we
get

$(\pounds_{{\xi_{i}}}\,g)(X,Y)=g(\nabla_{X}\,{\xi_{i}},Y)+g(\nabla_{Y}\,{\xi_{i}},X)=-g({f}X,Y)-g({f}Y,X)=0,$

thus, ${\xi_{i}}$ are Killing vector fields.

Conversely, let our manifold be weak $f$-K-contact. By (3.2) with
$Y={\xi_{i}}$, using (22), we get

$g((\nabla_{X}\,{f})\,{\xi_{i}},Z)=g({f}^{2}X,Z).$

Hence, $(\nabla_{X}\,{f})\,{\xi_{i}}={f}^{2}X$. From this and

$0=\nabla_{X}\,({f}\,{\xi_{i}})=(\nabla_{X}\,{f})\,{\xi_{i}}+{f}\nabla_{X}\,{\xi_{i}},$

we obtain ${f}(\nabla_{X}\,{\xi_{i}}+{f}X)=0$. Since
$\nabla_{{X}}\,\xi_{i}+f{X}\in{\cal D}$ and $f$ is invertible when restricted
on ${\cal D}$, we get $\nabla_{{X}}\,\xi_{i}=-f{X}$ that proves (26). ∎

######  Corollary 4.2 (see [23]).

A weak contact manifold $M^{2n+1}(f,Q,\xi,\eta,g)$ is weak K-contact if and
only if $\nabla\,{\xi}=-{f}$ holds.

Denote by
$R_{{X},{Y}}Z=(\nabla_{X}\nabla_{Y}-\nabla_{Y}\nabla_{X}-\nabla_{[X,Y]})Z$ the
curvature tensor. The mapping $R_{\,{\xi_{i}}}:X\mapsto
R_{X,\,{{\xi_{i}}}}\,{{\xi_{i}}}\ (\xi\in\ker f,\ \|\xi\|=1)$ will be called
the Jacobi operator in the ${\xi_{i}}$-direction, e.g., [27]. For a weak
$f$-contact manifold, by Proposition 3.2, we get $R_{\,{\xi_{i}}}(X)\,\in{\cal
D}$. Recall that a Riemannian manifold with a unit Killing vector field and
the property $R_{\xi}(X)=X\ (X\bot\,{\xi})$ is a K-contact manifold, e.g.,
[33, p. 268].

We generalize this result and [23, Theorem 2] in the following.

######  Theorem 4.2.

A Riemannian manifold $(M^{2n+s},g)$ with orthonormal Killing vector fields
${\xi_{i}}\ (i=1,\ldots,s)$ such that $d\,\eta^{1}=\ldots=d\,\eta^{s}$
$($where $\eta^{i}$ is the 1-form dual to $\xi_{i})$ and the Jacobi operators
$R_{\,{\xi_{i}}}\ (i\leq s)$ are positive definite on the distribution ${\cal
D}=\bigcap_{\,i}\ker\eta^{i}$ is a weak $f$-K-contact manifold with the
following structural tensors:

${f}=-\nabla\,{\xi_{i}},\quad QX=R_{\xi_{i}}(X)\quad(X\in{\cal D},\ \
i=1,\ldots,s).$

###### Proof.

Put ${f}=-P\nabla\,{\xi_{i}}$ and $\Phi(X,Y)=g(X,{f}Y)$, where $P:TM\to{\cal
D}$ is the orthoprojector, that does no depend on $i$. Since ${\xi_{i}}$ are
Killing vector fields, we obtain the property (13):

$d{\eta^{i}}(X,Y)=(1/2)\,(g(\nabla_{X}\,{\xi_{i}},Y)-g(\nabla_{Y}\,{\xi_{i}},X))=-g(\nabla_{Y}\,{\xi_{i}},X)=g(X,{f}Y).$

Set $QX=R_{\,{\xi_{i}}}(X)$ for some $i$ and all $X\in{\cal D}$. Since
${\xi_{i}}$ is a unit Killing vector field, we get
$\nabla_{\xi_{i}}\,{\xi_{i}}=0$ and
$\nabla_{X}\nabla_{Y}\,{\xi_{i}}-\nabla_{\nabla_{X}\,Y}\,{\xi_{i}}=R_{\,{\xi_{i}},X}\,Y$.
Thus, ${f}\,{\xi_{i}}=0$ and

${f}^{2}X=\nabla_{\nabla_{X}\,{\xi_{i}}}\,{\xi_{i}}=R_{\,{\xi_{i}},X}\,{\xi_{i}}=-R_{\xi_{i}}(X)=-QX,\quad
X\in{\cal D}.$

By the conditions, the Jacobi operator $R_{\xi_{i}}$ does no depend on $i$ and
the tensor $Q$ is positive definite on ${\cal D}$. Thus, $f$ restricted to
$\cal D$ has constant rank $2\,n$. Put $Q\,{\xi_{i}}={\xi_{i}}\
(i=1,\ldots,s)$. Therefore, (4) and (9) are valid, that completes the proof. ∎

######  Corollary 4.3 (see [23]).

A Riemannian manifold $(M^{2n+1},g)$ with a unit Killing vector field $\xi$
such that $R_{\,\xi}$ is positive definite on ${\cal D}=\\{X\in
TM:g(X,\xi)=0\\}$ is a weak K-contact manifold with the structural tensors
$\eta(X)=g(X,\xi)$, $f=-\nabla\,\xi$ and $QX=R_{\,\xi}(X)$ for $X\in{\cal D}$.

If a plane contains unit vectors: $\xi\in\ker f$ and $X\in{\cal D}$, then its
sectional curvature is called $\xi$-sectional curvature. The $\xi$-sectional
curvature of an $f$-contact manifold is an example of mixed sectional
curvature of an almost product manifold, for example, [27]. It is well known
that the $\xi$-sectional curvature of an $f$-K-contact manifold is constant
equal to 1.

######  Proposition 4.2.

A weak $f$-K-contact structure $({f},Q,{\xi_{i}},{\eta^{i}},g)$ with constant
mixed sectional curvature, $K({\xi_{i}},X)=\lambda>0$ for some
$\lambda=const\in\mathbb{R}$ and all ${X}\in{\cal D}$, is homothetic to an
$f$-K-contact structure $({f}^{\prime},{\xi_{i}},{\eta^{i}},g^{\prime})$ after
the transformation (19)–(20) with $Q^{\prime}={\rm id}$.

###### Proof.

Note that $K({\xi_{i}},X)=\lambda\ (X\in{\cal D})$ if and only if
$R_{\,{\xi_{i}}}(X)=\lambda\,X\ (X\in{\cal D})$. By $QX=R_{\,{\xi_{i}}}(X)\
(X\in{\cal D})$, see Theorem 4.2, we get $QX=\lambda X\ (X\in{\cal D})$. By
Lemma 3.1(ii), $({f}^{\prime},{\xi_{i}},{\eta^{i}},{g}^{\prime})$ is an
$f$-contact structure. Using (3.1), we get
$(\pounds_{{\xi_{i}}}\,g^{\prime})({\xi_{i}},\,\cdot)=0$ and

$(\pounds_{{\xi_{i}}}\,g^{\prime})(X,Y)=\lambda(\pounds_{{\xi_{i}}}\,g)(X,Y),\quad
X,Y\in{\cal D}.$

By $\pounds_{{\xi_{i}}}\,g=0$, we get $\pounds_{{\xi_{i}}}\,g^{\prime}=0$;
thus $({f}^{\prime},{\xi_{i}},{\eta^{i}},g^{\prime})$ is an $f$-K-contact
structure. ∎

######  Example 4.1.

By Theorem 4.2, we can search for examples of weak $f$-K-contact (not
$f$-K-contact) manifolds among Riemannian manifolds of positive sectional
curvature that admit a set of $s\geq 1$ orthonormal Killing vector fields.
Such example with $s=1$ is given in [23] using idea of [12, p. 5]. Indeed, let
$M$ be a convex hypersurface (ellipsoid) with induced metric $g$ of
$\mathbb{R}^{2n+2}$,

$M=\Big{\\{}(u_{1},\ldots,u_{2n+2})\in\mathbb{R}^{2n+2}:\sum\nolimits_{\,i=1}^{n+1}u_{i}^{2}+a\sum\nolimits_{\,i=n+2}^{2n+1}u_{i}^{2}=1\Big{\\}},$

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

$\xi=(-u_{2},u_{1},\ldots,-u_{n+1},u_{n},-\sqrt{a}\,u_{n+3},\sqrt{a}\,u_{n+2},\ldots,-\sqrt{a}\,u_{2n+2},\sqrt{a}\,u_{2n+1})$

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

##  5 The curvature in the characteristic direction

Denote by $\operatorname{Ric}^{\sharp}$ the Ricci operator of $g$ associated
with the Ricci tensor ${\rm Ric}$ and given by

${\rm
Ric}({X},{Y})=g(\operatorname{Ric}^{\sharp}{X},{Y}),\quad{X},{Y}\in\mathfrak{X}_{M}.$

Since $K(\xi_{i},\xi_{j})=0$ for a weak $f$-contact manifold, the Ricci
curvature in the ${\xi_{j}}$-direction is given by

${\rm
Ric}({{\xi_{j}}},{{\xi_{j}}})=\sum\nolimits_{\,i=1}^{\,2n}g(R_{e_{i},\,{{\xi_{j}}}}\,{{\xi_{j}}},e_{i}),$

where $(e_{i})$ is any local orthonormal basis of ${\cal D}$. In the next
proposition, we generalize three particular properties of $f$-K-contact
manifolds for weak $f$-K-contact manifolds.

######  Proposition 5.1.

For a weak $f$-K-contact manifold $M^{2n+s}(f,Q,\xi_{i},\eta^{i},g)$, we have
for all $i$:

$\displaystyle
R_{\,{\xi_{i}},\,X}=\nabla_{X}\,{f}\quad(X\in\mathfrak{X}_{M}),$ (28)
$\displaystyle
R_{X,\,{{\xi_{i}}}}\,{{\xi_{j}}}=-{f}^{2}X\quad(X\in\mathfrak{X}_{M}),$
$\displaystyle\operatorname{Ric}^{\sharp}{\xi_{i}}=\operatorname{div}f,$ (30)
$\displaystyle{\rm
Ric}({{\xi_{i}}},{{\xi_{j}}})={\rm\,trace\,}Q=2\,n+{\rm\,trace\,}\widetilde{Q}.$

###### Proof.

Using (26), we derive

$\displaystyle R_{Z,\,X}\,{{\xi_{i}}}$ $\displaystyle=$
$\displaystyle\nabla_{Z}(\nabla_{X}\,{{\xi_{i}}})-\nabla_{X}(\nabla_{Z}\,{{\xi_{i}}})-\nabla_{[Z,X]}\,{{\xi_{i}}})$
(31) $\displaystyle=$
$\displaystyle\nabla_{X}({f}Z)-\nabla_{Z}({f}X)+{f}([Z,X])=(\nabla_{X}\,{f})Z-(\nabla_{Z}\,{f})X.$

Note that

$(\nabla_{X}\Phi)(Y,Z)=g((\nabla_{X}\,{f})Z,Y)=-g((\nabla_{X}\,{f})Y,Z).$

Using condition $d\Phi=0$ and (2), we get

$(\nabla_{X}\Phi)(Y,Z)+(\nabla_{Y}\Phi)(Z,X)+(\nabla_{Z}\Phi)(X,Y)=0.$ (32)

From (31), using (32) and skew-symmetry of $\Phi$, we get (28):

$\displaystyle g(R_{{{\xi_{i}}},\,X}\,Y,Z)$ $\displaystyle=$ $\displaystyle
g(R_{Y,\,Z}\,{{\xi_{i}}},X)\overset{\eqref{Eq-R}}{=}(\nabla_{Z}\,\Phi)(X,Y)+(\nabla_{Y}\,\Phi)(Z,X)$
$\displaystyle\overset{\eqref{E-dPhi-three}}{=}$
$\displaystyle-(\nabla_{X}\,\Phi)(Y,Z)=g((\nabla_{X}\,{f})Y,Z).$

By (28) with $Y={\xi_{j}}$, using ${f}\,{\xi_{j}}=0$ and (26), we get (28):

$\displaystyle
R_{{{\xi_{i}}},X}\,{{\xi_{j}}}=(\nabla_{X}\,{f})\,{\xi_{j}}=-{f}\nabla_{X}\,{{\xi_{j}}}={f}^{2}X.$

For weak $f$-K-contact manifolds, by (28) and (24), we get the equality (33):

$\operatorname{Ric}(\xi_{j},X)=\sum\nolimits_{\,k=1}^{\,2n}g((\nabla_{e_{k}}\,f)\,X,e_{k})=g(\operatorname{div}f,X),$
(33)

where $(e_{k})$ is any local orthonormal basis of ${\cal D}$. By (33), we get
the equality (30):

$\operatorname{Ric}^{\sharp}(\xi_{j})=\sum\nolimits_{\,k=1}^{\,2n}(\nabla_{e_{k}}\,f)\,e_{k}=\operatorname{div}f.$

Using the definition ${\rm Ric}(\xi_{i},\xi_{j})={\rm\,trace\,}(X\to
R_{X,\xi_{i}}\,\xi_{j})$ and $f\,\xi_{i}=0$, we get for all $i,j$,

$\operatorname{Ric}({{\xi_{i}}},{{\xi_{j}}})\overset{\eqref{E-R1}}{=}-\sum\nolimits_{\,k=1}^{\,2n}g({f}^{2}e_{k},e_{k})\overset{\eqref{2.1}}{=}\sum\nolimits_{\,k=1}^{\,2n}g(Qe_{k},e_{k}).$

By the above and the equality
${\rm\,trace\,}Q=2\,n+{\rm\,trace\,}\widetilde{Q}$, (30) is true. ∎

######  Corollary 5.1.

There are no Einstein weak $f$-K-contact manifolds
$M^{2n+s}(f,Q,\xi_{i},\eta^{i},g)$ with $s>1$ and ${\rm\,trace\,}Q\neq 0$.

###### Proof.

For a weak $f$-K-contact manifold with $s>1$ and vector field
$\tilde{\xi}=(\xi_{1}+\xi_{2})/\sqrt{2}$, we get

${\rm Ric}(\tilde{\xi},\tilde{\xi})=(1/2)\sum\nolimits_{i,j=1}^{2}{\rm
Ric}(\xi_{i},\xi_{j})\overset{\eqref{E-R1b}}{=}2{\rm\,trace\,}Q.$ (34)

If the manifold is an Einstein manifold, then for the unit vector field
$\tilde{\xi}$ we get ${\rm Ric}(\tilde{\xi},\tilde{\xi})={\rm
Ric}(\xi_{i},\xi_{i})={\rm\,trace\,}Q$. Comparing with (34) gives
${\rm\,trace\,}Q=0$. ∎

######  Corollary 5.2.

For a weak $f$-K-contact manifold $M^{2n+s}(f,Q,\xi_{i},\eta^{i},g)$, the
$\xi$-sectional curvature is positive:

$K(\xi_{i},{X})=g(Q{X},{X})>0\quad({X}\in{\cal D},\ \|{X}\|=1),$

and for the Ricci curvature we get ${\rm Ric}({\xi}_{i},{\xi}_{j})>0$ for all
$1\leq i,j\leq s$.

###### Proof.

For any unit vectors $\xi\in\ker f,\ X\in{\cal D}$, from (28), we get
$K(\xi,X)=g(fX,fX)>0$. From (30), using (4) and non-singularity of $f$ on
${\cal D}$, we get the following:

$\operatorname{Ric}({{\xi}_{i}},{{\xi}_{j}})={\rm\,trace\,}Q=-\sum\nolimits_{k=1}^{2n}g(f^{2}e_{k},e_{k})=\sum\nolimits_{k=1}^{2n}g(fe_{k},fe_{k})>0,$

where $(e_{k})$ is a local orthonormal frame of ${\cal D}$, thus the second
statement is valid. ∎

The following theorem generalizes [33, Proposition 5.1] and [23, Theorem 3]
for $s=1$.

######  Theorem 5.1.

A weak $f$-K-contact manifold $M^{2n+s}(f,Q,\xi_{i},\eta^{i},g)$ satisfying
${\rm\,trace\,}Q=const$ and $(\nabla\operatorname{Ric})({\xi_{i}},\,\cdot)=0$,
e.g., the Ricci tensor is parallel, is a Ricci flat manifold $($an Einstein
manifold, for $s=1)$.

###### Proof.

Differentiating (30) and using (26) and the conditions, we have

$0=\nabla_{Y}\,({\rm Ric}({{\xi_{i}}},{{\xi_{i}}}))=(\nabla_{Y}\,{\rm
Ric})({{\xi_{i}}},{{\xi_{i}}})+2\,{\rm
Ric}(\nabla_{Y}\,{{\xi_{i}}},{{\xi_{i}}})=-2\,{\rm Ric}({f}Y,{{\xi_{i}}}),$

hence ${\rm Ric}(Y,{{\xi_{i}}})=({\rm\,trace\,}Q)\,{\eta^{i}}(Y)$.
Differentiating this, then using

$X({\eta^{i}}(Y))=g(\nabla_{X}{\xi_{i}},Y)=-g({f}X,Y)+g(\nabla_{X}Y,{\xi_{i}})$

and assuming $\nabla_{X}Y=0$ at $x\in M$, gives

$\displaystyle({\rm\,trace\,}Q)\,g({f}Y,X)=\nabla_{X}\,({\rm
Ric}(Y,{{\xi_{i}}}))$ $\displaystyle=(\nabla_{X}\,{\rm
Ric})(Y,{{\xi_{i}}})+2\,{\rm Ric}(Y,\nabla_{X}\,{{\xi_{i}}})=-2\,{\rm
Ric}(Y,{f}X).$

Thus, ${\rm Ric}(Y,{f}X)=({\rm\,trace\,}Q)\,g(Y,{f}X)$. Therefore, ${\rm
Ric}(X,Y)=({\rm\,trace\,}Q)\,g(X,Y)$ for any vector fields $X$ and $Y$ on $M$,
which means that $(M,g)$ is a Ricci flat manifold with ${\rm\,trace\,}Q=0$
$($an Einstein manifold, for $s=1)$. ∎

######  Corollary 5.3 (see [23]).

A weak K-contact manifold $M^{2n+1}(f,Q,\xi,\eta,g)$ with
$(\nabla\operatorname{Ric})(\xi,\,\cdot)=0$, e.g., the Ricci tensor is
parallel, and ${\rm\,trace\,}Q=const$ is an Einstein manifold of scalar
curvature $(2\,n+1){\rm\,trace\,}Q$.

######  Remark 5.1.

For $f$-contact manifolds, we have

$\sum\nolimits_{\,i=1}^{\,2n}(\nabla_{e_{i}}\,f)\,e_{i}=2\,n\,\bar{\xi},$ (35)

see [13, Proposition 2.6]. For $f$-K-contact manifolds, (30) and (35) give
$\operatorname{Ric}^{\sharp}(\xi_{i})=2\,n\,\bar{\xi}\ (i=1,\ldots,s)$ and
$\operatorname{Ric}(\xi_{i},\xi_{i})=2\,n$. Can one generalize this and (35)
for weak $f$-contact manifolds ?

Define the skew-symmetric operator ${T}^{\sharp}_{\xi}\ (\xi\in\ker f)$ by

$g({T}^{\sharp}_{\xi}X,Y)=g({T}(X,Y),\xi)\quad(X,Y\in{\cal D}),$

where $T(X,Y)=(1/2)(\nabla_{X}Y-\nabla_{Y}X)^{\bot}$ is the integrability
tensor of ${\cal D}$.

The self-adjoint partial Ricci curvature tensor, see [28], is given by

$\operatorname{Ric}^{\bot}(X)=\sum\nolimits_{\,i}({R}_{\,X^{\bot},\,\xi_{i}}\,\xi_{i})^{\bot},$

and for an $f$-K-contact manifold, we have $\operatorname{Ric}^{\bot}=s\,{\rm
id}^{\bot}$. By Corollary 5.2, the tensor $\operatorname{Ric}^{\bot}$ of a
weak $f$-K-contact manifold is positive definite. In [28], using the flow of
metrics on a $\mathfrak{g}$-foliation, a deformation retraction of a weak
almost ${\cal S}$-structures with positive partial Ricci curvature onto the
subspace of the aforementioned classical structures is constructed. In the
following theorem, using Corollary 5.2 and the method of [28], we show that a
weak $f$-K-contact structure can be transformed into an $f$-K-contact
structure.

######  Theorem 5.2.

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 $g_{t}\ (t\in\mathbb{R})$ such that
$(f(t),Q_{\,t},\xi_{i},\eta^{i},g_{t})$ is a weak $f$-K-contact structure on
$M$ with structural tensors defined on ${\cal D}$ as

$Q_{\,t}=(1/s)\operatorname{Ric}^{\bot}_{t},\quad f(t)\,|_{\,{\cal
D}}=T_{\xi_{i}}^{\sharp}(t).$ (36)

Moreover, $g_{t}$ converges exponentially fast, as $t\to-\infty$, to a limit
metric $\hat{g}$ with $\operatorname{Ric}^{\bot}_{\hat{g}}=s\,{\rm id}^{\bot}$
that gives an $f$-K-contact structure on $M$.

###### Proof.

The partial Ricci curvature tensor of a weak $f$-K-contact manifold is
positive definite. Thus, the result is similar to [28, Theorem 1]. Consider
the “partial Ricci flow” of metrics on $M$,

$\partial_{t}\,g_{t}=-2\,(\operatorname{Ric}^{\bot})^{\flat}_{g_{t}}+2\,s\,g^{\bot}_{t},$
(37)

where the $(0,2)$-tensor $g^{\bot}$ is defined by
$g^{\bot}(X,Y):=g(X^{\bot},Y^{\bot})$. For $t=0$, we get on ${\cal D}$:

$\operatorname{Ric}^{\bot}=-\sum\nolimits_{\,i}(T_{\xi_{i}}^{\sharp})^{2}=-\sum\nolimits_{\,i}\phi_{i}^{2}=s\,Q,$

and find
$T_{\xi_{i}}^{\sharp}\operatorname{Ric}^{\bot}=\operatorname{Ric}^{\bot}T_{\xi_{i}}^{\sharp}$,
see also [28]. Thus,
$\sum\nolimits_{i}T_{\xi_{i}}^{\sharp}\operatorname{Ric}^{\bot}T_{\xi_{i}}^{\sharp}=-(\operatorname{Ric}^{\bot})^{2}$.
By the above, we obtain the following ODE:

${\partial_{t}}\operatorname{Ric}^{\bot}=4\operatorname{Ric}^{\bot}(\operatorname{Ric}^{\bot}-\,s\,{\rm
id}^{\bot}).$

By the theory of ODE’s, there exists a unique solution
$\operatorname{Ric}^{\bot}_{t}$ for $t\in\mathbb{R}$; hence, a solution
$g_{t}$ of (37) exists for $t\in\mathbb{R}$ and it is unique. Observe that
$(f(t),\xi_{i},\eta^{i},Q_{t})$ with $f(t),Q_{t}$ given in (36) is a weak
$f$-K-contact structure on $(M,g_{t})$. By uniqueness of a solution, the flow
(37) preserves the directions of eigenvectors of $\operatorname{Ric}^{\bot}$,
and each eigenvalue $\mu_{i}>0$ of $\operatorname{Ric}^{\bot}$ satisfies ODE
(for $\mu=\mu_{i}$)

$\dot{\mu}_{i}=4\mu_{i}\,(\mu_{i}-s)$

with $\mu_{i}(0)>0$. This ODE has the following solution (function $\mu(t)$ on
$M$ for any $t\in\mathbb{R}$):

$\mu_{i}(t)=\frac{\mu_{i}(0)\,s}{\mu_{i}(0)+\exp(4\,s\,t)(s-\mu_{i}(0))}$

with $\lim\limits_{t\to-\infty}\mu_{i}(t)=s$. Thus,
$\lim\limits_{t\to-\infty}\operatorname{Ric}^{\bot}(t)=s\,{\rm id}^{\bot}$.
Let $\\{e_{i}(t)\\}$ be a $g_{t}$-orthonormal frame of ${\cal D}$ of
eigenvectors associated with $\mu_{i}(t)$, we then get
${\partial_{t}}e_{i}=(\mu_{i}-s)e_{i}$. Since $e_{i}(t)=z_{i}(t)\,e_{i}(0)$
with $z_{i}(0)=1$, then ${\partial_{t}}\log z_{i}(t)=\mu_{i}(t)-s$. By the
above, $z_{i}(t)=(\mu_{i}(t)/\mu_{i}(0))^{1/4}$. Hence,

$g_{t}(e_{i}(0),e_{j}(0))=z_{i}^{-1}(t)z_{j}^{-1}(t)\,g_{t}(e_{i}(t),e_{j}(t))=\delta_{ij}(\mu_{i}(0)\mu_{j}(0)/(\mu_{i}(t)\mu_{j}(t)))^{1/4}.$

As $t\to-\infty$, $g_{t}$ converges to the metric $\hat{g}$ determined by
$\hat{g}(e_{i}(0),e_{j}(0))=\delta_{ij}\sqrt{\mu_{i}(0)/s}$. ∎

######  Corollary 5.4.

Let $M^{2n+1}(f(0),Q_{\,0},\xi,\eta,g_{0})$ be a weak K-contact manifold. Then
there exist metrics $g_{t}\ (t\in\mathbb{R})$ such that
$(f(t),Q_{\,t},\xi,\eta,g_{t})$ is a weak K-contact structure on $M$ with
structural tensors defined on ${\cal D}$ as
$Q_{\,t}=\operatorname{Ric}^{\bot}_{t}$ and $f(t)\,|_{\,{\cal
D}}=T_{\xi}^{\sharp}(t)$. Moreover, $g_{t}$ converges exponentially fast, as
$t\to-\infty$, to a limit metric $\hat{g}$ with
$\operatorname{Ric}^{\bot}_{\hat{g}}={\rm id}^{\bot}$ that gives a K-contact
structure on $M$.

##  6 Weak $f$-K-contact structure as a generalized Ricci soliton

Corollary 5.1 motivates to consider quasi Einstein manifolds, introduced in
[10] by the condition

$\operatorname{Ric}=a\,g+b\,\mu\otimes\mu$

for all vector fields $X,Y$, where $a$ and $b\neq 0$ are real scalars, and
$\mu$ is a 1-form of unit norm, and later this structure has been studied by
several geometers. If $\mu$ is the differential of a function, then we get a
gradient quasi Einstein manifold.

The following generalization of (2) was given in [20]:

${\rm
Hess}_{\,\sigma_{1}}-c_{2}\operatorname{Ric}=\lambda\,g-c_{1}d\sigma_{2}\otimes
d\sigma_{2}$ (38)

for some $\sigma_{1},\sigma_{2}\in C^{\infty}(M)$ and real constants
$c_{1},c_{2}$ and $\lambda$. For $\sigma_{1}=\sigma_{2}$, (38) reduces to (2).

First, we formulate some lemmas.

######  Lemma 6.1 (see Lemma 3.1 in [16] or Lemma 4 in [23] for $s=1$).

For a weak $f$-K-contact manifold $M^{2n+s}(f,Q,\xi_{i},\eta^{i},g)$ the
following holds:

$(\pounds_{{\xi_{i}}}(\pounds_{X}\,g))(Y,{\xi_{i}})=g(X,Y)+g(\nabla_{\xi_{i}}\nabla_{\xi_{i}}\,X,Y)+Yg(\nabla_{\xi_{i}}\,X,{\xi_{i}})$

for any $i\leq s$ and all smooth vector fields $X,Y$ with $Y$ orthogonal to
$\ker f$.

###### Proof.

This uses the equalities $\nabla_{\xi_{i}}\,{\xi_{i}}=0$ and (28), and is the
same as for [23, Lemma 4]. ∎

######  Lemma 6.2 (see, for example, [16]).

Let $(M;g)$ be a Riemannian manifold and $\sigma$ be a smooth function on $M$.
Then the following holds for every vector fields $\xi,Y$ on $M:$

$\pounds_{{\xi}}(d\sigma\otimes
d\sigma)(Y,{\xi})=Y({\xi}(\sigma))\,{\xi}(\sigma)+Y(\sigma)\,{\xi}({\xi}(\sigma)).$

Recall that the Ricci curvature of any $f$-K-contact manifold satisfies the
following condition:

$\operatorname{Ric}(X,{\xi})=0\quad(X\in{\cal D},\ \xi\in\ker f).$ (39)

######  Lemma 6.3 (see Lemma 3.3 in [16] or Lemma 6 in [23] for $s=1$).

Let a weak $f$-K-contact manifold $M^{2n+s}(f,Q,\xi_{i},\eta^{i},g)$ satisfy
(39) and admit the generalized gradient Ricci soliton structure. Then

$\nabla_{\xi_{i}}\nabla\sigma=(\lambda+c_{2}{\rm\,trace\,}Q)\,{\xi_{i}}-c_{1}{\xi_{i}}(\sigma)\,\nabla\sigma.$

###### Proof.

By (30) and (39) we get

$\lambda\,{\eta^{i}}(Y)+c_{2}\operatorname{Ric}({\xi_{i}},Y)=(\lambda+c_{2}{\rm\,trace\,}Q)\,{\eta^{i}}(Y).$
(40)

Using (2) and (40), we get

${\rm
Hess}_{\,\sigma}({\xi_{i}},Y)=-c_{1}{\xi_{i}}(\sigma)\,g(\nabla\sigma,Y)+(\lambda+c_{2}{\rm\,trace\,}Q)\,{\eta^{i}}(Y).$
(41)

Thus, (41) and the condition (2) for the Hessian complete the proof. ∎

The following theorem generalizes [23, Theorem 4] and [16, Theorem 3.1] with
$s=1$.

######  Theorem 6.1.

Let a weak $f$-K-contact manifold $M^{2n+s}(f,Q,\xi_{i},\eta^{i},g)$ with
${\rm\,trace\,}Q=const$ satisfy the generalized gradient Ricci soliton
equation (2) with $c_{1}(\lambda+c_{2}{\rm\,trace\,}Q)\neq-1$:

${\rm Hess}_{\,\sigma}-c_{2}\operatorname{Ric}=\lambda\,g-c_{1}d\sigma\otimes
d\sigma\,.$

Suppose that condition (39) is true. Then $\sigma=const$. Furthermore, if
$c_{2}\neq 0$, then the manifold is an Einstein one for $s=1$ and is Ricci
flat $($with ${\rm\,trace\,}Q=0)$ for $s>1$.

###### Proof.

Let $Y\bot\ker f$. Then by Lemma 6.1 with $X=\nabla f$, we obtain

$2\,(\pounds_{{\xi_{i}}}({\rm
Hess}_{\,\sigma}))(Y,{\xi_{i}})=Y(\sigma)+g(\nabla_{\xi_{i}}\nabla_{\xi_{i}}\nabla\sigma,Y)+Yg(\nabla_{\xi_{i}}\nabla\sigma,{\xi_{i}}).$
(42)

Using Lemma 6.3 in (42) and the properties $\nabla_{\xi_{i}}\,{\xi_{i}}=0$ and
$g({\xi_{i}},{\xi_{i}})=1$, yields

$\displaystyle 2\,(\pounds_{{\xi_{i}}}({\rm
Hess}_{\,\sigma}))(Y,{\xi_{i}})=Y(\sigma)+a\,g(\nabla_{\xi_{i}}\,{\xi_{i}},Y)$
$\displaystyle-
c_{1}g(\nabla_{\xi_{i}}({\xi_{i}}(\sigma)\nabla\sigma),Y)+a\,Y(g({\xi_{i}},{\xi_{i}}))-c_{1}Y({\xi_{i}}(\sigma)^{2})$
$\displaystyle=Y(\sigma)-c_{1}g(\nabla_{\xi_{i}}({\xi_{i}}(\sigma)\nabla\sigma),Y)-c_{1}Y({\xi_{i}}(\sigma)^{2}),$
(43)

where $a:=\lambda+c_{2}{\rm\,trace\,}Q$. Using Lemma 6.3 with $Y\in{\cal D}$,
from (6) it follows that

$2\,(\pounds_{{\xi_{i}}}({\rm
Hess}_{\,\sigma}))(Y,{\xi_{i}})=Y(\sigma)-c_{1}{\xi_{i}}({\xi_{i}}(\sigma))\,Y(\sigma)+c_{1}^{2}{\xi_{i}}(\sigma)^{2}Y(\sigma)-c_{1}Y({\xi_{i}}(\sigma)^{2}).$
(44)

Since ${\xi_{i}}$ is a Killing vector field, thus $\pounds_{{\xi_{i}}}\,g=0$,
this implies $\pounds_{{\xi_{i}}}\operatorname{Ric}=0$. Using the above fact
and applying the Lie derivative to equation (2), gives

$2\,(\pounds_{{\xi_{i}}}({\rm
Hess}_{\,\sigma}))(Y,{\xi_{i}})=-2\,c_{1}(\pounds_{{\xi_{i}}}(d\sigma\otimes
d\sigma))(Y,{\xi_{i}}).$ (45)

Using (44), (45) and Lemma 6.2, we obtain

$Y(\sigma)\big{(}1+c_{1}{\xi_{i}}({\xi_{i}}(\sigma))+c_{1}^{2}\,{\xi_{i}}(\sigma)^{2}\big{)}=0.$
(46)

By Lemma 6.3, we get

$c_{1}{\xi_{i}}({\xi_{i}}(\sigma))=c_{1}\,{\xi_{i}}(g({\xi_{i}},\nabla\sigma))=c_{1}g({\xi_{i}},\nabla_{\xi_{i}}\nabla\sigma)=c_{1}a-c_{1}^{2}\,{\xi_{i}}(\sigma)^{2}.$
(47)

Using (46) in (47), we get $Y(\sigma)(c_{1}a+1)=0$. This implies $Y(\sigma)=0$
provided by $c_{1}a+1\neq 0$. Hence, $\nabla\sigma$ is parallel to $\ker f$.
Taking the covariant derivative of
$\nabla\sigma=\sum_{\,i}{\xi_{i}}(\sigma)\,{\xi_{i}}$ and using (26) and
$\bar{\xi}=\sum_{\,i}\xi_{i}$, we obtain

$g(\nabla_{X}\,\nabla\sigma,Y)=\sum\nolimits_{\,i}X({\xi_{i}}(\sigma))\,{\eta^{i}}(Y)-{\bar{\xi}}(\sigma)\,g({\sigma}X,Y),\quad
X,Y\in\mathfrak{X}_{M}.$

From this, by symmetry of ${\rm Hess}_{\,\sigma}$, i.e.
$g(\nabla_{X}\,\nabla\sigma,Y)=g(\nabla_{Y}\,\nabla\sigma,X)$, we get
${\bar{\xi}}(\sigma)\,g({\sigma}X,Y)=0$. For $Y={\sigma}X$ with some $X\neq
0$, since $g({\sigma}X,{\sigma}X)>0$, we get ${\bar{\xi}}(\sigma)=0$.
Replacing $(\xi_{i})$ with another orthonormal frame from $\ker f$ preserves
the weak $f$-K-contact structure and allows reaching any direction
${\bar{\xi}}$ in $\ker f$. So $\nabla\sigma=0$, i.e. $\sigma=const$. Thus,
from (2) and $c_{2}\neq 0$ it follows the claim. ∎

######  Corollary 6.1 (see [23]).

Let a weak K-contact manifold $M(f,Q,\xi,\eta,g)$ with ${\rm\,trace\,}Q=const$
satisfy the generalized gradient Ricci soliton equation (2) with
$c_{1}(\lambda+c_{2}{\rm\,trace\,}Q)\neq-1$. Suppose that condition
$\operatorname{Ric}({\xi},X)=0\ (X\,\perp\,\xi)$ is true. Then $\sigma=const$.
Furthermore, if $c_{2}\neq 0$, then the manifold is an Einstein one.

The following theorem contains 3 cases and generalizes (and uses) Theorem 6.1.

######  Theorem 6.2.

Let a weak $f$-K-contact manifold $M^{2n+s}(f,Q,\xi_{i},\eta^{i},g)$ satisfy
the generalized gradient Ricci soliton equation (38) with $c_{1}a\neq-1$,
where $a=\lambda+c_{2}{\rm\,trace\,}Q$. Suppose that conditions
${\rm\,trace\,}Q=const$ and (39) are true. Then
$\tilde{\sigma}=\sigma_{1}+c_{1}a\sigma_{2}$ is constant and

$-c_{1}a\,{\rm Hess}_{\,\sigma_{2}}=-c_{1}\,d\sigma_{2}\otimes
d\sigma_{2}+c_{2}\operatorname{Ric}+\lambda\,g.$ (48)

Furthermore,

1\. if $c_{1}a\neq 0$, then (48) reduces to ${\rm
Hess}_{\,\sigma_{2}}=\frac{1}{a}\,d\sigma_{2}\otimes
d\sigma_{2}-\frac{c_{2}}{c_{1}a}\,\operatorname{Ric}-\frac{\lambda}{c_{1}a}\,g$.
By Theorem 6.1, if $c_{1}a\neq-1$, then $\sigma_{2}=const$; moreover, if
$c_{2}\neq 0$, then $(M,g)$ is an Einstein manifold for $s=1$ and is Ricci
flat $($with ${\rm\,trace\,}Q=0)$ for $s>1$.

2\. if $a=0$ and $c_{1}\neq 0$, then (48) reduces to
$0=c_{2}\operatorname{Ric}-c_{1}\,d\sigma_{2}\otimes d\sigma_{2}+\lambda\,g$.
If $c_{2}\neq 0$ and $\sigma_{2}\neq const$, then we get a gradient quasi
Einstein manifold.

3\. if $c_{1}=0$, then (48) reduces to $0=c_{2}\operatorname{Ric}+\lambda\,g$,
and for $c_{2}\neq 0$ we get an Einstein manifold for $s=1$ and Ricci flat
manifold $($with ${\rm\,trace\,}Q=0)$ for $s>1$.

###### Proof.

Similarly to Lemma 6.3, we get

$\nabla_{\xi_{i}}\nabla\sigma_{1}=a\,\xi_{i}-c_{1}\xi_{i}(\sigma_{2})\,\nabla\sigma_{2}.$
(49)

Using (49) and Lemmas 6.1 and 6.2, and slightly modifying the proof of Theorem
6.1, we find that the vector field $\nabla\tilde{\sigma}$ is parallel to $\ker
f$, where $\tilde{\sigma}=\sigma_{1}+c_{1}a\sigma_{2}$. As in the proof of
Theorem 6.1, we get $d\tilde{\sigma}=0$, i.e.,
$d\sigma_{1}=-c_{1}a\,d\sigma_{2}$. Using this in (38), we get (48). Analyzing
(48), we obtain the required 3 cases of the theorem. ∎

##  7 $\eta$-Ricci solitons on compact weak $f$-K-contact manifolds

A weak metric $f$-manifold $M^{2n+s}({f},Q,{\xi_{i}},{\eta^{i}},g)$ is said to
be $\eta$-Einstein, if

$\operatorname{Ric}=\alpha\,g+\beta\sum\nolimits_{\,i}\eta^{i}\otimes\eta^{i}+(\alpha+\beta)\sum\nolimits_{\,i<j}\eta^{i}\otimes\eta^{j}$
(50)

for some functions $\alpha,\beta\in C^{\infty}(M)$. For $s=1$ (and $Q={\rm
id}$), we get an $\eta$-Einstein structure
$\operatorname{Ric}=\alpha\,g+\beta\,\eta\otimes\eta$ on an almost contact
metric manifold, e.g., [17].

By an $\eta$-Ricci soliton we mean here a weak $f$-contact manifold, whose
Ricci tensor satisfies

$(1/2)\,\pounds_{V}\,g+\operatorname{Ric}=\lambda\,g+\nu\sum\nolimits_{\,i}\eta^{i}\otimes\eta^{i}+(\lambda+\nu)\sum\nolimits_{\,i<j}\eta^{i}\otimes\eta^{j}$
(51)

for some functions $\lambda,\nu\in C^{\infty}(M)$ and some smooth vector field
$V$ on $M$. For $s=1$, we get an $\eta$-Ricci soliton
$\frac{1}{2}\,\pounds_{V}\,g+\operatorname{Ric}=\lambda\,g+\beta\,\eta\otimes\eta$
on an almost contact metric manifold, e.g., [17].

If $\nu=0$, then (51) gives a Ricci almost soliton studied by several authors;
moreover, if $\lambda=const$, then it is R. Hamilton’s Ricci soliton. For
$s=1$ (and $Q={\rm id}$), (51) gives a generalized $\eta$-Ricci soliton
$\frac{1}{2}\,\pounds_{V}\,g+\operatorname{Ric}=\lambda\,g+\nu\,\eta\otimes\eta$
on an almost contact metric manifold, e.g., [17].

For weak $f$-K-manifolds and $V=\bar{\xi}$, the notion of $\eta$-Ricci soliton
reduces to $\eta$-Einstein structure. For this reason, in this section we
study an $\eta$-Ricci soliton on a weak $f$-K-contact manifold. We recall the
generalized Pohozaev-Schoen identity due to Gover-Orsted [15].

######  Lemma 7.1.

Let $(M^{n},g)$ be a compact Riemannian manifold without boundary. If $E$ is a
divergence free symmetric (0,2)-tensor and $V$ is a vector field on $M$, then

$\int_{\,M}V({\rm\,trace\,}_{\\!g}\,E)\,{\rm
d}\operatorname{vol}=\frac{n}{2}\int_{\,M}g(E^{0},\pounds_{V}\,g)\,{\rm
d}\operatorname{vol},$ (52)

where $E^{0}=E-\frac{1}{n}\,({\rm\,trace\,}_{\\!g}\,E)\,g$ and ${\rm
d}\operatorname{vol}$ is the volume form of $g$.

Denote by $r$ the scalar curvature of a Riemannian manifold $(M,g)$. The
constancy of $r$ plays an important role for the triviality of a compact
generalized $\eta$-Ricci soliton, see [17, Theorem 3.1]. We extend this result
for compact weak $f$-K-contact manifolds.

######  Theorem 7.1.

Let a compact weak $f$-K-contact manifold $M^{2n+s}(f,\xi_{i},\eta^{i},Q,g)$
satisfy $r=const$ and ${\rm\,trace\,}Q=const$. If $(g,V,\lambda,\nu)$
represents an $\eta$-Ricci soliton, then $M$ is $\eta$-Einstein.

###### Proof.

We take the trace of (50) to get the scalar curvature
$r=(2n+s)\,\alpha+s\,\beta$. On the other hand, using (30) in (50), we obtain
$\alpha+\beta={\rm\,trace\,}Q$. From these it follows that

$\alpha=\frac{r-s{\rm\,trace\,}Q}{2\,n},\quad\beta=\frac{(2\,n+s){\rm\,trace\,}Q-r}{2\,n}.$
(53)

Define a traceless symmetric (0,2)-tensor $E$ as

$E=\operatorname{Ric}-\frac{r-s{\rm\,trace\,}Q}{2\,n}\,g+\frac{r-(2\,n+s){\rm\,trace\,}Q}{2\,n}\sum\nolimits_{\,i}\eta^{i}\otimes\eta^{i}-({\rm\,trace\,}Q)\sum\nolimits_{\,i<j}\eta^{i}\otimes\eta^{j}.$
(54)

We express (51), using (54), as

$\displaystyle(1/2)\,\pounds_{V}\,g+E$
$\displaystyle=\big{(}\lambda-\frac{r-s{\rm\,trace\,}Q}{2\,n}\big{)}\,g+\big{(}\nu+\frac{r-(2\,n+s){\rm\,trace\,}Q}{2\,n}\big{)}\sum\nolimits_{\,i}\eta^{i}\otimes\eta^{i}$
$\displaystyle+(\lambda+\nu-{\rm\,trace\,}Q)\sum\nolimits_{\,i<j}\eta^{i}\otimes\eta^{j}.$
(55)

Taking covariant derivative of (54) along $X$ and using (26) and
${\rm\,trace\,}Q=const$, gives

$\displaystyle(\nabla_{X}\,E)(Y,Z)=(\nabla_{X}\operatorname{Ric})(Y,Z)-\frac{X(r)}{2\,n}\,\big{(}g(Y,Z)-\sum\nolimits_{\,i}\eta^{i}(Y)\,\eta^{i}(Z)\big{)}$
$\displaystyle+\Big{(}(s-1){\rm\,trace\,}Q+\frac{(2\,n+s){\rm\,trace\,}Q-r}{2\,n}\Big{)}\,\big{(}\bar{\eta}(Z)\,g(fX,Y)+\bar{\eta}(Y)\,g(fX,Z)\big{)}.$
(56)

Taking the trace of (7) over $X$ and $Z$, and since $r=const$,
$\operatorname{div}_{g}\operatorname{Ric}=\frac{1}{2}\,d\,r$ (twice-contracted
2nd Bianchi identity, [19, p. 135]), and ${\rm\,trace\,}f=0$, we get the
equality $\operatorname{div}_{g}E=0$:

$(\operatorname{div}_{g}E)(Y)=(\operatorname{div}_{g}\operatorname{Ric})(Y)+\frac{s\,(2\,n+1){\rm\,trace\,}Q-r}{2\,n}\,\big{(}\bar{\eta}(e_{i})\,g(fe_{i},Y)+\bar{\eta}(Y)\,g(fe_{i},e_{i})\big{)}=0.$

Since the tensor $E$ is traceless, we get $E^{0}=E$ and
$V({\rm\,trace\,}_{\\!g}E)=0$. Thus, applying (52), we find
$\int_{\,M}g(E,\pounds_{V}\,g)\,{\rm d}\operatorname{vol}=0$. Applying (7) and
(30) in the above integral formula, and using equalities
$g(E,g)={\rm\,trace\,}_{\\!g}E=0$ and
$g(E,\eta^{i}\otimes\eta^{j})=E(\xi_{i},\xi_{j})=0$, we obtain

$\int_{\,M}g(E,E)\,{\rm d}\operatorname{vol}=0.$ (57)

From (57) we get $E=0$. Thus, our manifold is $\eta$-Einstein, see (50), where
$\alpha,\beta$ satisfy (53). ∎

The following consequence of Theorem 7.1 generalizes [17, Theorem 3.1(i)].

######  Corollary 7.1.

Let a compact weak K-contact manifold $M^{2n+1}(f,\xi,\eta,Q,g)$ satisfy
$r=const$ and ${\rm\,trace\,}Q=const$. If $(g,V,\lambda,\nu)$ represents a
generalized $\eta$-Ricci soliton on $M$, then $M$ is $\eta$-Einstein.

##  8 Conclusion

It is shown that a weak $f$-contact structure is a useful tool for studying
unit Killing vector fields, totally geodesic foliations and mixed sectional
curvature on Riemannian manifolds. Some results for $f$-contact and
$f$-K-contact structures [4, 33] have been extended to certain weak
structures. Conditions are found under which (i) a weak $f$-K-contact manifold
with parallel Ricci tensor or with a generalized gradient Ricci soliton
structure is a quasi Einstein or an Einstein manifold; (ii) a compact weak
$f$-K-contact manifold with constant scalar curvature and an $\eta$-Ricci
soliton structure is $\eta$-Einstein. Based on applications of the weak
$f$-contact structure in contact geometry considered in the article, we expect
that this structure will also be fruitful in theoretical physics.

In conclusion, we pose several questions. Is the condition “the
$\xi$-sectional curvature is positive” sufficient for a weak $f$-contact
manifold to be weak $f$-K-contact? Does a weak $f$-contact manifold of
dimension greater than $3$ have some positive $\xi$-sectional curvature? Is a
compact weak $f$-K-contact Einstein manifold an ${\cal S}$-manifold? When a
given weak $f$-K-contact manifold is a mapping torus (see [18]) of a manifold
of lower dimension? When a weak $f$-contact manifold equipped with a Ricci-
type soliton structure, carries a canonical (for example, with constant
sectional curvature or Einstein-type) metric? One could answer these questions
by generalizing some deep results on $f$-contact manifolds for weak
$f$-contact manifolds.

## References

  * [1] Alekseevsky, D. and Michor, P. Differential geometry of $\mathfrak{g}$-manifolds, _Differential Geom. Appl_. 1995, 5, 371–403
  * [2] Berestovskij, V. N. and Nikonorov, Yu. G. Killing vector fields of constant length on Riemannian manifolds. _Sib. Math. J._ 49, No. 3 (2008), 395–407
  * [3] Blair, D. E. _Riemannian geometry of contact and symplectic manifolds_ , 2nd edition, Springer-Verlag, New York, 2010
  * [4] Blair, D. E. Geometry of manifolds with structural group $U(n)\times O(s)$, _J. Diff. Geom._ 4 (1970), 155–167
  * [5] Brunetti, L. and Pastore, A. M. $S$-manifolds versus indefinite $S$-manifolds and local decomposition theorems. _International Electronic J. of Geometry_ , 2016, 9:1, 1–8
  * [6] Cabrerizo, J. L., Fernández, L. M. and Fernández, M. The curvature tensor fields on $f$-manifolds with complemented frames. _An. Stiint. Univ. Al. I. Cuza Iasi_ , 36 (1990), 151–161
  * [7] Cairns, G. A general description of totally geodesic foliations, _Tohoku Math. J._ 1986, 38, 37–55
  * [8] Catino, G., Mastrolia, P., Monticelli, D. and Rigoli, M. On the geometry of gradient Einstein-type manifolds, _Pac. J. Math._ 286 (1) (2017), 39–67
  * [9] Cao, H. D. Recent progress on Ricci soliton, _Adv. Lect. Math._ 11 (2009), 1–38
  * [10] Chaki, M. C. and Maity, R. K. On quasi Einstein manifolds, _Publ. Math. Debrecen_ , 57 (2000), 297–306
  * [11] Cho, J. T. and Kimura, M. Ricci solitons and real hypersurfaces in complex space form, _Tohoku Math. J._ 61 (2) (2009), 205–212
  * [12] Deshmukh, S. and Belova, O. On Killing vector fields on Riemannian manifolds, _Mathematics_ , 9, 259 (2021), 1–17
  * [13] Duggal, K. L., Ianus, S. and Pastore, A. M. Maps interchanging $f$-structures and their harmonicity. _Acta Appl. Math._ 67 (2001), 91–115
  * [14] Falcitelli, M., Ianus, S. and Pastore, A. _Riemannian Submersions and Related Topics_ , World Scientific, 2004.
  * [15] Gover, A. R. and Orsted, B. Universal principles for Kazdan-Warner and Pohozaev-Schoen type identities, _Commun. Contemp. Math._ 15 (4) (2013), Art. ID 1350002
  * [16] Ghosh, G. and De, U. C. Generalized Ricci soliton on K-contact manifolds, _Math. Sci. Appl. E-Notes_ , 8 (2020), 165–169
  * [17] Ghosh, A. K-contact and $(k,\mu)$-contact metric as a generalized $\eta$-Ricci soliton. _Math. Slovaca 73_ , No. 1, 185–194 (2023)
  * [18] Goertsches, O. and Loiudice, E. On the topology of metric $f$-K-contact manifolds, _Monatshefte für Mathematik_ , 192 (2020), 355–370
  * [19] Kobayashi, S. and Nomizu, K. Foundations of differential geometry, Vols. I, II, USA, Interscience Publishers, New York–London–Sydney, 1963, 1969
  * [20] Mohammed Cherif, A., Zegga, K. and Beldjilali, G. On the generalised Ricci solitons and Sasakian manifolds. _Communications in Mathematics_ , 30, Issue 1 (2022) 119–123
  * [21] Nurowski, P. and Randall, M. Generalised Ricci solitons, _J. Geom. Anal._ , 26 (2016), 1280–1345
  * [22] Pigola S., Rigoli M., Rimoldi M. and Setti A. G. Ricci almost solitons. _Ann. Scuola Norm. Sup. Pisa Cl. Sci_. 10 (4) (2011), 757–799
  * [23] Rovenski, V. Generalized Ricci solitons and Einstein metrics on weak K-contact manifolds. _Communications in Analysis and Mechanics_ , 2023, Vol. 15 (2), 177–188
  * [24] Rovenski, V. On the geometry of a weakened $f$-structure. 2022, 14 pages. arXiv.2205.02158
  * [25] Rovenski, V. On the splitting tensor of the weak $f$-contact structure. _Symmetry_ 2023, 15(6), 1215. https://doi.org/10.3390/sym15061215
  * [26] Rovenski, V. and Patra, D. S. On the rigidity of the Sasakian structure and characterization of cosymplectic manifolds, preprint, 2022, arXiv:2203.04597
  * [27] Rovenski, V. and Walczak, P. G. _Extrinsic geometry of foliations_ , Progress in Mathematics, vol. 339, Birkhäuser, Cham, 2021
  * [28] Rovenski, V. and Wolak, R. New metric structures on $\mathfrak{g}$-foliations, _Indagationes Mathematicae_ , 33 (2022), 518–532
  * [29] Terlizzi, L. Di On the curvature of a generalization of contact metric manifolds, _Acta Math. Hung._ 110, No. 3 (2006), 225–239
  * [30] Terlizzi, L. Di Correction to the paper “On the curvature of a generalization of a contact metric manifolds”, _Acta Math. Hung._ 124, No. 4 (2009), 399–401
  * [31] Di Terlizzi, L., Pastore, A.M. and Wolak, R. Harmonic and holomorphic vector fields on an $f$-manifold with parallelizable kernel. _An. Stiint. Univ. Al. I. Cuza Iausi, Ser. Noua, Mat_. 2014, 60, No. 1, 125–144
  * [32] Yano, K. On a structure $f$ satisfying $f^{3}+f=0$, _Technical Report No. 12_ , University of Washington, 1961
  * [33] Yano, K. and Kon, M. _Structures on Manifolds_ , Vol. 3 of Series in Pure Math. World Scientific Publ. Co., Singapore, 1985