A note on some generalized curvature tensor

Ryszard Deszcz, Małgorzata Głogowska, Marian Hotloś,
Miroslava Petrović-Torgašev, and Georges Zafindratafa

Dedicated to the memory of Professor Krishan Lal Duggal (1929–2022)

For any semi-Riemannian manifold $(M,g)$ we define
some generalized curvature tensor $E$ as a linear combination
of Kulkarni-Nomizu products formed by the metric tensor,
the Ricci tensor and its square of given manifold.
That tensor is closely related to quasi-Einstein spaces,
Roter spaces and some Roter type spaces.[2020
Mathematics Subject Classification.
Primary 53B20, 53B25; Secondary 53C25.
Keywords and phrases: warped product manifold,
Einstein manifold, quasi-Einstein manifold,
partially Einstein manifold,
generalized Einstein metric condition,
pseudosymmetry type curvature condition, Roter space,
generalized Roter space, hypersurface, principal curvature,
quasi-umbilical hypersurface.]

§ INTRODUCTION

Let $(M,g)$
be a semi-Riemannian manifold.
We denote by
$g$, $R$, $S$, $\kappa$ and $C$,
the metric tensor, the Riemann-Christoffel curvature tensor,
the Ricci tensor,
the scalar curvature and the Weyl conformal curvature tensor
of $(M,g)$, respectively. Further, let $A \wedge B$
be the Kulkarni-Nomizu product of symmetric
$(0,2)$-tensors $A$ and $B$.
Now we can define the $(0,2)$-tensors
$S^{2}$ and $S^{3}$,
the $(0,4)$-tensors
$R \cdot S$, $C \cdot S$ and
the $(0,6)$-tensors
$R \cdot R$,
$R \cdot C$,
$C \cdot R$,
$C \cdot C$
where $T$ is a generalized curvature tensor.
For precise definitions of the symbols used,
we refer to Section 2
of this paper, as well as to
<cit.> and

A semi-Riemannian manifold $(M,g)$, $\dim M = n \geq 2$, is said to be
an Einstein manifold [2],
or an Einstein space, if at every point of $M$
its Ricci tensor $S$ is proportional to $g$,

\begin{eqnarray}
S &=& \frac{\kappa}{n}\, g
\label{2020.10.3.c}
\end{eqnarray}

on $M$,
assuming that the scalar curvature $\kappa$ is constant when $n = 2$.
According to <cit.>
this condition is called the Einstein metric condition.

Let $(M,g)$ be a semi-Riemannian manifold of dimension $\dim M = n \geq 3$.
We set

\begin{eqnarray}
E &=& g \wedge S^{2} + \frac{n-2}{2} \, S \wedge S - \kappa \, g \wedge S
+ \frac{\kappa ^{2} - \mathrm{tr}_{g} (S^{2})}{2(n-1)}
\, g \wedge g .
\label{2022.11.10.aaa}
\end{eqnarray}

It is easy to check
that the tensor $E$ is a generalized curvature tensor.
we define the subsets ${\mathcal{U}}_{R}$ and ${\mathcal U}_{S}$ of $M$ by
${\mathcal{U}}_{R}  =
\{x \in M\, | \, R - \frac{\kappa }{(n-1) n}\, G \neq 0\ \mbox {at}\ x \}$
${\mathcal U}_{S} =  \{x \in M\, | \,
S - \frac{\kappa }{n}\, g \neq 0\ \mbox {at}\ x \}$, respectively,
where $G = \frac{1}{2}\,  g \wedge g$.
If $n \geq 4$ then
we define the set ${\mathcal U}_{C} \subset M$ as the set of all points
of $(M,g)$ at which which $C \neq 0$.
We note that if $n \geq 4$ then
${\mathcal{U}}_{S} \cup {\mathcal{U}}_{C} = {\mathcal{U}}_{R}$
(see, e.g., [24]).

An extension of the class of Einstein manifolds form
quasi-Einstein, $2$-quasi-Einstein and partially Einstein manifolds.
A semi-Riemannian manifold $(M,g)$,
$\dim M = n \geq 3$,
is said to be a quasi-Einstein manifold, or a quasi-Einstein space,

\begin{eqnarray}
\mathrm{rank}\, (S - \alpha\, g) &=& 1
\label{quasi02}
\end{eqnarray}

on ${\mathcal U}_{S} \subset M$, where $\alpha $ is some function
on ${\mathcal U}_{S}$.
It is known that every non-Einstein warped product manifold
$\overline{M} \times _{F} \widetilde{N}$
with a $1$-dimensional $(\overline{M}, \overline{g})$ base manifold and
a $2$-dimensional manifold $(\widetilde{N}, \widetilde{g})$
or an $(n-1)$-dimensional Einstein manifold
$(\widetilde{N}, \widetilde{g})$, $\dim M = n \geq 4$,
and a warping function $F$,
is a  quasi-Einstein manifold (see, e.g., <cit.>).
A Riemannian manifold
$(M,g)$, $\dim M = n \geq 3$, whose Ricci tensor has an eigenvalue
of multiplicity $n-1$ is a non-Einstein quasi-Einstein manifold
(cf. <cit.>).
We mention that quasi-Einstein manifolds arose during the study
of exact solutions
of the Einstein field equations and the investigation
on quasi-umbilical hypersurfaces
of conformally flat spaces
(see, e.g., <cit.> and references therein).
Quasi-Einstein hypersurfaces
in semi-Riemannian spaces of constant curvature
were studied among others in
<cit.> (see also [27]
and references therein).
Quasi-Einstein manifolds satisfying some pseudosymmetry
type curvature conditions
were investigated recently in
Quasi-Einstein hypersurfaces in conformally flat semi-Riemannian manifolds
were studied in <cit.>.
In those papers quasi-Einstein hypersurfaces were called
pseudo-Einstein hypersurfaces (see also <cit.>).
Similarly, in <cit.>
quasi-Einstein semi-Riemannian manifolds (hypersurfaces)
were called pseudo-Einstein manifolds (hypersurfaces).

Let $(M,g)$, $\dim M = n \geq 3$, be a semi-Riemannian manifold.
We note that (<ref>)
holds at a point $x \in {\mathcal U}_{S} \subset M$
if and only if
$(S - \alpha\, g) \wedge (S - \alpha\, g) = 0$
at $x$, i.e.,

\begin{eqnarray}
\frac{1}{2}\, S \wedge S   - \alpha\, g \wedge S
+ \frac{\alpha ^{2}}{2}\, g \wedge g &=& 0
\label{quasi03}
\end{eqnarray}

at $x$  (cf. <cit.>).
From (<ref>), by a suitable contraction,
we get immediately

\begin{eqnarray}
&=& (\kappa - (n-2) \alpha )\, S + \alpha ( (n-1) \alpha - \kappa )\, g.
\label{quasi03quasi03}
\end{eqnarray}

Using (<ref>) we can easily check that the following equation
is satisfied on any Einstein manifold $(M,g)$

\begin{eqnarray}
g \wedge S^{2} + \frac{n-2}{2} \, S \wedge S - \kappa \, g \wedge S
+ \frac{\kappa ^{2} - \mathrm{tr}_{g} (S^{2})}{2(n-1)} \, g \wedge g
&=& 0 ,
\label{2022.11.20.aa}
\end{eqnarray}

i.e., $E = 0$ on $M$, where the tensor $E$
is defined by  (<ref>).
as it was stated in <cit.>,
(<ref>)  is satisfied on every quasi-Einstein manifold
$(M,g)$, $n \geq 3$. The converse statement also is true. Precisely,
from Proposition 2.1 it follows that
if  $(M,g)$, $n \geq 4$, is a semi-Riemannian manifold satisfying
(<ref>) on ${\mathcal U}_{S} \subset M$ then
a condition of the form (<ref>) holds on ${\mathcal U}_{S}$,
where $\alpha $ is some function on this set.
In Section 2 we also present another result
related to the tensor $E$ (see Proposition 2.2).
Namely, we prove that if a generalized curvature tensor $T$
is a linear combination of the tensors
$R$, $S\wedge S$, $g \wedge S$, $g \wedge S^{2}$,
and $g \wedge g$ then the Weyl tensor of $T$ is a linear combination
of the tensors $C$ and $E$.
The tensor $E$ is determined by some Kulkarni-Nomizu products formed
by $g$, $S$ and $S^{2}$,
i.e.,  $E$ is defined by (<ref>).
In the same way, we can define the $(0,4)$-tensor $E(A)$ corresponding
to a symmetric  $(0,2)$-tensor $A$

\begin{eqnarray}
E(A) &=& g \wedge A^{2} + \frac{n-2}{2} \, A \wedge A
- \mathrm{tr}_{g} ( A)  \, g \wedge A
+ \frac{(\mathrm{tr}_{g} ( A) )^{2} - \mathrm{tr}_{g} (A^{2})}{2(n-1)}
\, g \wedge g .
\label{2022.12.33.aaa}
\end{eqnarray}

The semi-Riemannian manifold $(M,g)$, $\dim M = n \geq 3$, will be called
a partially Einstein manifold, or
a partially Einstein space
(cf. <cit.>, <cit.>),
if at every point $x \in {\mathcal{U}}_{S} \subset M$
its Ricci operator ${\mathcal{S}}$ satisfies
${\mathcal{S}}^{2} = \lambda {\mathcal{S}} + \mu  I\!d_{x}$,
or equivalently,

\begin{eqnarray}
S^{2} &=&  \lambda \, S + \mu \, g ,
\label{partiallyEinstein}
\end{eqnarray}

where $\lambda , \mu \in {\mathbb{R}}$
and $I\!d_{x}$ is the identity transformation of $T_{x} M$.
is a special case of (<ref>).
Thus every quasi-Einstein manifold is a partially Einstein manifold.
The converse statement is not true.
Contracting (<ref>) we get
$ \mathrm{tr}_{g} (S^{2}) = \lambda \, \kappa + n\, \mu$.
This together with
(<ref>) yields (cf. <cit.>)

\begin{eqnarray*}
S^{2} - \frac{ \mathrm{tr}_{g} (S^{2})}{n} \, g
&=&  \lambda \left( S - \frac{\kappa }{n} \, g \right) .
\end{eqnarray*}

In particular, a Riemannian manifold $(M,g)$, $\dim M = n \geq 3$,
is a partially Einstein space if at every point
$x \in {\mathcal{U}}_{S} \subset M$
its Ricci operator ${\mathcal{S}}$
has exactly two distinct eigenvalues
$\kappa _{1}$ and $\kappa _{2}$
with multiplicities $p$ and $n-p$, respectively, where
$1 \leq p \leq n-1$.
Evidently, if $p = 1$, or $p = n-1$,
then $(M,g)$ is a quasi-Einstein manifold.

In Section 3 we present definitions of some classes of semi-Riemannian manifolds
determined by curvature conditions of pseudosymmetry type.
Investigations of semi-Riemannian manifolds satisfying some particular
curvature conditions of pseudosymmetry type lead to
Roter spaces (see Propositon 4.1).
Roter spaces form an important subclass
of the class of non-conformally flat and non-quasi-Einstein
partially Einstein manifolds of dimension $\geq 4$.
Namely, a non-quasi-Einstein and non-conformally flat
semi-Riemannian manifold $(M,g)$, $\dim M = n \geq 4$,
satisfying on
${\mathcal U}_{S} \cap {\mathcal U}_{C} \subset M$
the following equation

\begin{eqnarray}
R &=& \frac{\phi}{2}\, S\wedge S + \mu\, g\wedge S
+ \frac{\eta}{2}\, g \wedge g ,
\label{eq:h7a}
\end{eqnarray}

$\phi$, $\mu $ and $\eta $ are some functions on this set,
is called a Roter type manifold, or a Roter manifold,
or a Roter space
(see, e.g., <cit.>,
Equation (<ref>) is called a Roter equation
(see, e.g., <cit.>).
In Section 4 we present results on such manifolds.
For instance,  every Roter space $(M,g)$, $\dim M = n \geq 4$,
satisfies among others the following pseudosymmetry type curvature condition
on ${\mathcal{U}}_{S} \cap {\mathcal{U}}_{C} \subset M$
(see Theorem 4.2)

\begin{eqnarray}
C \cdot R - R \cdot C = Q(S,C) - \frac{\kappa}{ n-1 }\, Q(g,C) .
\label{Roterformula}
\end{eqnarray}

Let $(M,g)$, $\dim M = n \geq 4$, be a non-partially-Einstein
and non-conformally flat semi-Riemannian manifold.
If its Riemann-Christoffel curvature $R$
is at every point of
${\mathcal U}_{S} \cap {\mathcal U}_{C} \subset M$
a linear combination of the Kulkarni-Nomizu products
formed by the tensors
$S^{0} = g$ and $S^{1} = S, \ldots , S^{p-1}, S^{p}$,
where $p$ is some natural
number $\geq 2$, then $(M,g)$
is called
a generalized Roter type manifold,
or a generalized Roter manifold,
or a generalized Roter type space, or
a generalized Roter space. For instance, when $p = 2$,
we have

\begin{eqnarray}
R &=&  \frac{\phi _{2} }{2}\, S^{2} \wedge S^{2}
+ \phi_{1}\, S \wedge S^{2}
+ \frac{\phi}{2}\, S \wedge S
+ \mu _{1}\, g \wedge S^{2}
+ \mu \, g \wedge S
+ \frac{ \eta }{2} \, g \wedge g ,
\label{B001simply}
\end{eqnarray}

$\phi $,
$\phi _{1}$,
$\phi _{2}$,
$\mu_{1}$, $\mu$ and
are functions on
${\mathcal U}_{S} \cap {\mathcal U}_{C}$.
Because $(M,g)$ is a non-partially Einstein manifold,
at least one of the functions
$\mu _{1}$, $\phi _{1}$ and $\phi _{2}$ is a non-zero function.
Equation (<ref>) is called a
Roter type equation
(see, e.g., <cit.>).
We refer to
for results on manifolds (hypersurfaces) satisfying (<ref>).

As it was stated in <cit.> (see Proposition 4.4),
if $(M,g)$, $\dim M = n \geq 4$, is a Roter space satisfying
on ${\mathcal U}_{S} \cap  {\mathcal U}_{C} \subset M$
then on this set

\begin{eqnarray}
C &=& \frac{\phi}{n-2}
\left(g \wedge S^{2} + \frac{n-2}{2} \, S \wedge S - \kappa \, g \wedge S
+ \frac{\kappa ^{2} - \mathrm{tr}_{g} (S^{2})}{2(n-1)}
\, g \wedge g \right) .
\label{2022.11.22.aa}
\end{eqnarray}

In Section 5 we recall results on some warped product manifolds
with $2$-dimensional base manifold obtained in [28].

In Section 6 we state that on every essentially conformally symmetric manifold
the following equation is satisfied

\begin{eqnarray}
F\, C
&=& \frac{n-2}{2 (n -2)} \, S \wedge S\nonumber\\
&=& \frac{1}{n-2}
\left(
g \wedge S^{2} + \frac{n-2}{2} \, S \wedge S - \kappa \, g \wedge S
+ \frac{\kappa ^{2} - \mathrm{tr}_{g} (S^{2})}{2(n-1)} \, g \wedge g
\right) .
\label{2023.03.03.a}
\end{eqnarray}

In Section 7 we recall some known results on hypersurfaces $M$,
$\dim M \geq 4$, isometrically immersed in a conformally flat spaces.
In particular,
we mention that
at every point of $M$ its Weyl conformal curvature tensor $C$
and the $(0,4)$-tensor $E(H)$, formed for the second fundamental tensor $H$
of $M$, are linearly dependent (see (<ref>)).

In the last section we investigate non-Einstein and non-conformally flat
hypersurfaces $M$, $\dim M \geq 4$, isometrically immersed
in semi-Riemannian spaces of constant curvature
satisfying some curvature conditions of pseudosymmetry type.
Under some additional assumptions imposed on the second fundamental tensor
$H$ of $M$ we obtain equations involved with the tensor $E$.

§ PRELIMINARIES.

Throughout this paper, all manifolds are assumed
to be connected paracompact
manifolds of class $C^{\infty }$. Let $(M,g)$, $\dim M = n \geq 3$,
be a semi-Riemannian manifold, and let $\nabla$
be its Levi-Civita connection and $\Xi (M)$ the Lie
algebra of vector fields on $M$. We define on $M$ the endomorphisms
$X \wedge _{A} Y$ and ${\mathcal{R}}(X,Y)$ of $\Xi (M)$ by
$(X \wedge _{A} Y)Z = A(Y,Z)X - A(X,Z)Y$ and

\begin{eqnarray*}
{\mathcal R}(X,Y)Z
\nabla _X \nabla _Y Z - \nabla _Y \nabla _X Z - \nabla _{[X,Y]}Z ,
\end{eqnarray*}

respectively, where
$X, Y, Z \in \Xi (M)$
$A$ is a symmetric $(0,2)$-tensor on $M$.
The Ricci tensor $S$, the Ricci operator ${\mathcal{S}}$
and the scalar curvature
$\kappa $ of $(M,g)$ are defined by

\begin{eqnarray*}
S(X,Y)\ =\ \mathrm{tr} \{ Z \rightarrow {\mathcal{R}}(Z,X)Y \} ,\ \
g({\mathcal S}X,Y)\ =\ S(X,Y) ,\ \
\kappa \ =\ \mathrm{tr}\, {\mathcal{S}},
\end{eqnarray*}

The endomorphism ${\mathcal{C}}(X,Y)$ is defined by

\begin{eqnarray*}
{\mathcal C}(X,Y)Z  &=& {\mathcal R}(X,Y)Z
- \frac{1}{n-2}(X \wedge _{g} {\mathcal S}Y + {\mathcal S}X \wedge _{g} Y
- \frac{\kappa}{n-1}X \wedge _{g} Y)Z .
\end{eqnarray*}

Now the $(0,4)$-tensor $G$,
the Riemann-Christoffel curvature tensor $R$ and
the Weyl conformal curvature tensor $C$ of $(M,g)$ are defined by
$G(X_1,X_2,X_3,X_4) = g((X_1 \wedge _{g} X_2)X_3,X_4)$ and

\begin{eqnarray*}
R(X_1,X_2,X_3,X_4) &=& g({\mathcal R}(X_1,X_2)X_3,X_4) ,\\
C(X_1,X_2,X_3,X_4) &=& g({\mathcal C}(X_1,X_2)X_3,X_4) ,
\end{eqnarray*}

respectively, where $X_1,X_2,\ldots \in \Xi (M)$.
For a symmetric $(0,2)$-tensor $A$ we denote by
${\mathcal{A}}$ the endomorphism related to $A$ by
$g({\mathcal{A}}X,Y) = A(X,Y)$.
The $(0,2)$-tensors
$A^{p}$, $p = 2, 3, \ldots $, are defined by
$A^{p}(X,Y) = A^{p-1} ({\mathcal{A}}X, Y)$,
assuming that $A^{1} = A$. In this way, for $A = S$ and
${\mathcal{A}} = {\mathcal S}$
we get the tensors $S^{p}$,
$p = 2, 3, \ldots $, assuming that $S^{1} = S$.

Let ${\mathcal B}$ be a tensor field sending any $X, Y \in \Xi (M)$
to a skew-symmetric endomorphism
${\mathcal B}(X,Y)$,
and let $B$ be
the $(0,4)$-tensor associated with ${\mathcal B}$ by

\begin{eqnarray}
B(X_1,X_2,X_3,X_4) &=&
g({\mathcal B}(X_1,X_2)X_3,X_4) .
\label{DS5}
\end{eqnarray}

The tensor $B$ is said to be a generalized curvature tensor  if the
following two conditions are fulfilled:
$B(X_1,X_2,X_3,X_4) = B(X_3,X_4,X_1,X_2)$ and

\begin{eqnarray*}
B(X_1,X_2,X_3,X_4) + B(X_2,X_3,X_1,X_4) + B(X_3,X_1,X_2,X_4) &=& 0 .
\end{eqnarray*}

For ${\mathcal B}$ as above, let $B$ be again defined by (<ref>).
We extend the endomorphism ${\mathcal B}(X,Y)$
to a derivation ${\mathcal B}(X,Y) \cdot \, $ of the algebra
of tensor fields on $M$,
assuming that it commutes with contractions and
${\mathcal B}(X,Y) \cdot \, f  = 0$ for any smooth function $f$ on $M$.
Now for a $(0,k)$-tensor field $T$,
$k \geq 1$, we can define the $(0,k+2)$-tensor $B \cdot T$ by

\begin{eqnarray*}
& & (B \cdot T)(X_1,\ldots ,X_k,X,Y) \ =\
({\mathcal B}(X,Y) \cdot T)(X_1,\ldots ,X_k)\\
&=& - T({\mathcal{B}}(X,Y)X_1,X_2,\ldots ,X_k)
- \cdots - T(X_1,\ldots ,X_{k-1},{\mathcal{B}}(X,Y)X_k) .
\end{eqnarray*}

If $A$ is a symmetric $(0,2)$-tensor then we define the
$(0,k+2)$-tensor $Q(A,T)$ by

\begin{eqnarray*}
& & Q(A,T)(X_1, \ldots , X_k, X,Y) \ =\
(X \wedge _{A} Y \cdot T)(X_1,\ldots ,X_k)\\
&=&- T((X \wedge _A Y)X_1,X_2,\ldots ,X_k)
- \cdots - T(X_1,\ldots ,X_{k-1},(X \wedge _A Y)X_k) .
\end{eqnarray*}

In this manner we obtain the $(0,6)$-tensors
$B \cdot B$ and $Q(A,B)$.

Substituting in the above formulas
${\mathcal{B}} = {\mathcal{R}}$ or ${\mathcal{B}} = {\mathcal{C}}$,
$T=R$ or $T=C$ or $T=S$, $A=g$ or $A=S$
we get the tensors $R\cdot R$, $R\cdot C$, $C\cdot R$,
$C\cdot C$, $R\cdot S$,
$Q(g,R)$, $Q(S,R)$, $Q(g,C)$,  $Q(S,C)$,
and $Q(g,S)$, $Q(g,S^{2})$,  $Q(S,S^{2})$.

For a symmetric $(0,2)$-tensor $A$ and a $(0,k)$-tensor $T$, $k \geq 2$, we
define their Kulkarni-Nomizu tensor $A \wedge T$ by
(see, e.g., <cit.>)

\begin{eqnarray*}
& &(A \wedge T )(X_{1}, X_{2}, X_{3}, X_{4}; Y_{3}, \ldots , Y_{k})\\
A(X_{1},X_{4}) T(X_{2},X_{3}, Y_{3}, \ldots , Y_{k})
+ A(X_{2},X_{3}) T(X_{1},X_{4}, Y_{3}, \ldots , Y_{k} )\\
& &
- A(X_{1},X_{3}) T(X_{2},X_{4}, Y_{3}, \ldots , Y_{k})
- A(X_{2},X_{4}) T(X_{1},X_{3}, Y_{3}, \ldots , Y_{k}) .
\end{eqnarray*}

It is obvious that the following tensors
are generalized curvature tensors: $R$, $C$ and
$A \wedge B$, where $A$ and $B = T$ are symmetric $(0,2)$-tensors.
We have

\begin{eqnarray}
C &=& R - \frac{1}{n-2}\, g \wedge S + \frac{\kappa }{(n-2) (n-1)} \, G ,
\label{Weyl}\\
G &=& \frac{1}{2}\, g \wedge g ,
\label{2020.12.08.a}
\end{eqnarray}

and (see, e.g., <cit.>)

\begin{eqnarray}
\ \
Q(A, A \wedge B) \ =\ - \frac{1}{2}\, Q(B, A \wedge A ) ,\nonumber\\
\ \
A \wedge Q(A,B) \ =\ - \frac{1}{2}\, Q(B, A \wedge A ).
\label{DS7}
\end{eqnarray}

By an application of (<ref>)(a) we obtain on $M$ the identities

\begin{eqnarray}
Q(g, g \wedge S) &=& - Q(S,G) ,
\ \ Q(S, g \wedge S)\ = \ - \frac{1}{2}\, Q(g, S \wedge S) .
\label{dghz01}
\end{eqnarray}

Further, by making use of (<ref>), (<ref>)
and (<ref>), we get immediately

\begin{eqnarray}
Q(g,C) &=& Q(g,R) - \frac{1}{n-2}\, Q(g,g \wedge S)
\frac{\kappa }{(n-2) (n-1)} \, Q(g,G)  \nonumber\\
&=& Q(g,R) - \frac{1}{n-2}\, Q(g,g \wedge S)
\ = \ Q(g,R) + \frac{1}{n-2} \, Q(S, \frac{1}{2}\, g \wedge g) ,
\label{abcd07zz},\\
Q(S,C) &=& Q(S,R) - \frac{1}{n-2}\, Q(S,g \wedge S)
\frac{\kappa }{(n-2) (n-1)} \, Q(S,G) \nonumber\\
Q(S,R) + \frac{1}{2 (n-2)}\, Q(g,S \wedge S)
- \frac{\kappa }{(n-2) (n-1)} \, Q(g,g \wedge S) .
\label{abcd07}
\end{eqnarray}

From (<ref>) (a) it follows immmediately that $Q(g, g \wedge g) = 0$.
Thus we have

\begin{eqnarray}
Q(g, E) &=&
Q(g, g \wedge S^{2} + \frac{n-2}{2} \, S \wedge S - \kappa \, g \wedge S ) ,
\label{2022.11.10.bbb}
\end{eqnarray}

where the tensor $E$ is defined by  (<ref>).

Let $A_{1}$, $A_{2}$ and $B$ be
symmetric $(0,2)$-tensors.
We have (see, e.g.,
<cit.> and references therein)

\begin{eqnarray}
A_{1} \wedge Q(A_{2},B) + A_{2} \wedge Q(A_{1},B)
+ Q(B, A_{1} \wedge A_{2})
&=& 0 .
\label{2021.11.30.a4}
\end{eqnarray}

From (<ref>) we get easily
(see also <cit.>
and references therein)

\begin{eqnarray*}
Q(B, A_{1} \wedge A_{2})
+ Q(A_{1}, A_{2} \wedge B)
+ Q(A_{2}, B \wedge A_{1}) &=& 0 .
\end{eqnarray*}

Let $A$ be a symmetric $(0,2)$-tensor and
$T$ a $(0,k)$-tensor, $k = 2, 3, \ldots $.
The tensor $Q(A,T)$ is called the Tachibana tensor of $A$ and $T$,
or the Tachibana tensor for short (see, e.g., [36]).
Using the tensors $g$, $R$ and $S$ we can define the following
$(0,6)$-Tachibana tensors:
$Q(S,R)$, $Q(g,R)$, $Q(g,g \wedge S)$ and $Q(S, g \wedge S)$.
We can check, by making use of (<ref>)(a) and (<ref>),
that other $(0,6)$-Tachibana tensors
constructed from $g$, $R$ and $S$ may be expressed
by the four Tachibana tensors mentioned above
or vanish identically on $M$.

Let $T$ be a generalized curvature tensor on a semi-Riemannian manifold $(M,g)$,
$\dim M = n \geq 4$. We denote by
$\mathrm{Ric} (T)$, $\kappa (T)$ and $\mathrm{Weyl} (T)$ the Ricci tensor,
the scalar curvature and the Weyl tensor of the tensor $T$, respectively.
We refer to <cit.>,  <cit.>
or <cit.>
for definitions of the considered tensors.
In particular, we have

\begin{eqnarray}
\mathrm{Weyl} (T) &=& T - \frac{1}{n-2} \, g \wedge \mathrm{Ric} (T)
+  \frac{\kappa (T) }{2 (n-2)(n-1)}\, g \wedge g .
\label{2023.03.06a}
\end{eqnarray}

Let $A$ be a symmetric $(0,2)$-tensor on a semi-Riemannian manifold
$(M,g)$, $\dim M = n \geq 3$. Let $E(A)$ be the tensor
defined by (<ref>). It is easy to check that
$\mathrm{Ric} (E(A))$ is a zero tensor. Therefore, we also have
$\kappa (E(A)) = 0$. Any generalized curvature tensor $T$
defined on a $3$-dimensional semi-Riemannian manifold $(M,g)$ is expressed by
$T = g \wedge \mathrm{Ric} (T) - (\kappa (T)/4) g \wedge g$
<cit.> (see also <cit.>).
Thus we see that the tensor $T = E(A)$ on any $3$-dimensional
semi-Riemannian manifold $(M,g)$ is a zero tensor. In particular,
on any $3$-dimensional semi-Riemannian manifold $(M,g)$ we have
$E = 0$.

Let $A$ be a symmetric $(0,2)$-tensor on a semi-Riemannian manifold
$(M,g)$, $\dim M = n \geq 3$. We denote by ${\mathcal U}_{A}$ the set
of points of $M$ at which $A \neq \frac{\mathrm{tr}_{g} (A)}{n}\, g$.

Let $A$ be a symmetric $(0,2)$-tensor on a semi-Riemannian manifold
$(M,g)$, $\dim M = n \geq 4$.
(i) (cf. <cit.>)
If the following condition is satisfied on ${\mathcal U}_{A}  \subset M$

\begin{eqnarray}
\mathrm{rank} ( A - \alpha \, g ) &=& 1
\label{2022.12.22.aa}
\end{eqnarray}

\begin{eqnarray}
g \wedge A^{2} + \frac{n-2}{2}\, A \wedge A
- \mathrm{tr}_{g} (A) \, g \wedge A
+ \frac{(\mathrm{tr}_{g} (A))^{2}
- \mathrm{tr}_{g} (A^{2})}{2 (n-1)} \, g \wedge g
&=& 0
\label{2022.12.22.bb}
\end{eqnarray}

\begin{eqnarray}
A^{2} - \frac{ \mathrm{tr}_{g} (A^{2})}{n} &=&
( \mathrm{tr}_{g} (A) - (n - 2) \alpha )
\left( A - \frac{\mathrm{tr}_{g} (A) }{n} \, g \right)
\label{2022.12.20.kk}
\end{eqnarray}

on ${\mathcal U}_{A}$,
where $\alpha$ is some function on ${\mathcal U}_{A}$.
(ii) If (<ref>) is satisfied
on ${\mathcal U}_{A} \subset M$ then

\begin{eqnarray}
A^{2} - \frac{\mathrm{tr}_{g} (A^{2})}{n}\, g
\rho \left( A - \frac{\mathrm{tr}_{g} (A)}{n}\, g \right)
\label{2022.12.22.ii}
\end{eqnarray}

\begin{eqnarray}
\left(
A - \frac{ \mathrm{tr}_{g} (A) - \rho }{n-2}\, g
\right)
\wedge
\left(
A - \frac{ \mathrm{tr}_{g} (A) - \rho }{n-2}\, g
\right) &=& 0
\label{2022.12.22.jj}
\end{eqnarray}

on ${\mathcal U}_{A}$,
where $\rho$ is some function on ${\mathcal U}_{A}$.

(i) (cf. the proof of <cit.>)
Equation (<ref>) yields <cit.>

\begin{eqnarray}
\frac{1}{2}\, A \wedge A &=& \alpha \, g \wedge A
- \frac{\alpha ^{2} }{2}\, g \wedge g .
\label{2022.12.20.ll}
\end{eqnarray}

This, by suitable contractions yields

\begin{eqnarray}
A^{2} - \mathrm{tr}_{g} (A)\, A &=&
- (n-2) \alpha \, A - \alpha \mathrm{tr}_{g} (A)\, g
+ (n-1) \alpha ^{2}\, g ,
\label{2022.12.20.yy}\\
\mathrm{tr}_{g} (A^{2}) - (\mathrm{tr}_{g} (A))^{2}
&=& - 2(n-1) \alpha \mathrm{tr}_{g} (A) + n (n-1) \alpha ^{2} .
\label{2022.12.20.zz}
\end{eqnarray}

Now using (<ref>), (<ref>)
and (<ref>) we can easily check that
(<ref>) and (<ref>)
hold on ${\mathcal U}_{A}$.
(ii) (cf. the proof of <cit.>)
In the local coordinates (<ref>) reads

\begin{eqnarray}
& &
g_{hk}A^{2}_{ij} + g_{ij}A^{2}_{hk} - g_{hj}A^{2}_{ik} - g_{ik}A^{2}_{hj}
(n-2)\, (A_{hk}A_{ij} - A_{hj}A_{ik})
\nonumber\\
& &
- \mathrm{tr}_{g} (A) \,
(g_{hk}A_{ij} + g_{ij}A_{hk} - g_{hj}A_{ik} - g_{ik}A_{hj})\nonumber\\
& &
+ \frac{(\mathrm{tr}_{g} (A))^{2} - \mathrm{tr}_{g} (A^{2})}{n-1}
\, (g_{hk}g_{ij} - g_{hj}g_{ik}) \ = \ 0 .
\label{2022.12.22.cc}
\end{eqnarray}

Contracting (<ref>)  with
$A^{ij} = A_{rs}g^{ri}g^{sj}$ and $A^{k}_{\ l} = A_{rl}g^{rk}$ we find

\begin{eqnarray}
A^{3} &=& \frac{3 \mathrm{tr}_{g} (A)}{n}\, A^{2}
+ \left(
\frac{(n^{2} - 3n + 3) \mathrm{tr}_{g} (A^{2})}{ (n-1) n }
- \frac{(\mathrm{tr}_{g} (A) )^{2}}{n-1}
\right) A\nonumber\\
& &
+ \left(
\frac{  (\mathrm{tr}_{g} (A))^{3} }{(n-1) n}
- \frac{ \mathrm{tr}_{g} (A) \, \mathrm{tr}_{g} (A^{2})}{n-1}
+ \frac{ \mathrm{tr}_{g} (A^{3})}{n}
\right) g ,
\label{2022.12.22.dd}
\end{eqnarray}

\begin{eqnarray}
& &
A_{hl}A^{2}_{ij} - A_{il}A^{2}_{hj}
+ g_{ij}A^{3}_{hl} - g_{hj}A^{3}_{il}
+ (n-2)\, (A_{ij}A^{2}_{hl} - A_{hj}A^{2}_{il})
\nonumber\\
& &
- \mathrm{tr}_{g} (A) \,
(A_{hl}A_{ij} - A_{il}A_{hj}
+ g_{ij}A^{2}_{hl} - g_{hj}A^{2}_{il} )\nonumber\\
& &
+ \frac{(\mathrm{tr}_{g} (A))^{2} - \mathrm{tr}_{g} (A^{2})}{n-1}
\, (g_{ij}A_{hl} - g_{hj}A_{il}) \ = \ 0 ,
\label{2022.12.22.ee}
\end{eqnarray}

From (<ref>), by symmetrization in $l,j$, we obtain

\begin{eqnarray}
& &
Q(g, A^{3}) + (n-3) Q(A,A^{2}) - \mathrm{tr}_{g} (A) \, Q(g, A^{2})\nonumber\\
& &
\frac{(\mathrm{tr}_{g} (A))^{2}
- \mathrm{tr}_{g} (A^{2})}{n-1} \, Q(g,A) \ =\ 0 .
\label{2022.12.22.ff}
\end{eqnarray}

Applying (<ref>) into (<ref>) we get

\begin{eqnarray*}
(n-3) \left(
Q(A,A^{2}) - \frac{\mathrm{tr}_{g} (A)}{n}\, Q(g, A^{2})
+ \frac{\mathrm{tr}_{g} (A^{2})}{n}\, Q(g, A) \right)
&=& 0 ,
\end{eqnarray*}

which yields

\begin{eqnarray*}
Q\left( A - \frac{\mathrm{tr}_{g} (A)}{n}\, g ,
A^{2} - \frac{\mathrm{tr}_{g} (A^{2})}{n}\, g \right) &=& 0 .
\end{eqnarray*}

From this, in view of <cit.>,
it follows that (<ref>) holds on ${\mathcal U}_{A}$.
Now (<ref>) and (<ref>),
by an application of <cit.>,
lead to (<ref>), completing the proof of (ii).

Let $T$ be a generalized curvature tensor on a semi-Riemannian manifold $(M,g)$,
$\dim M = n \geq 4$.
If the following condition is satisfied at a point $x \in M$

\begin{eqnarray}
T &=& \alpha _{1} \, R + \frac{\alpha _{2}}{2} \, S \wedge S
+ \alpha _{3}\, g \wedge S  +  \alpha _{4}\, g \wedge S^{2}
+  \frac{ \alpha _{5} }{2} \, g \wedge g
\label{2023.03.06b}
\end{eqnarray}

\begin{eqnarray}
\mathrm{Weyl} (T) &=&  \alpha _{1} \, C +   \frac{\alpha _{2}}{n-2} \, E
\label{2023.03.06c}
\end{eqnarray}

at this point, where the tensor $E$ is defined by (<ref>)
and $\alpha_{1}, \ldots , \alpha_{5} \in \mathbb{R}$.

Proof. From (<ref>), by a suitable contraction,
we get immediately

\begin{eqnarray}
\mathrm{Ric} (T)
&=& ( \alpha _{1} + \alpha _{2} \kappa + (n-2)\, \alpha _{3} )\, S
+  ( (n-2) \alpha _{4} - \alpha _{2} ) \, S^{2}
+ \alpha _{6} \, g  \, ,
\label{2023.03.06d}
\end{eqnarray}

where $\alpha _{6}$ is some real number. Now using
(<ref>),  (<ref>), (<ref>),
(<ref>), (<ref>) and (<ref>) we get

\begin{eqnarray*}
& &
\mathrm{Weyl} (T) \ =\ T - \frac{1}{n-2} \, g \wedge \mathrm{Ric} (T)
+  \frac{\kappa (T) }{2 (n-2)(n-1)}\, g \wedge g \nonumber\\
&=&   \alpha _{1} \, R + \frac{\alpha _{2}}{2} \, S \wedge S
+ \alpha _{3}\, g \wedge S  +  \alpha _{4}\, g \wedge S^{2}
+ \frac{\alpha _{7}}{2} \, g \wedge g  \nonumber\\
& &
- \frac{1}{n-2}  ( \alpha _{1} + \alpha _{2} \kappa
+ (n-2)\, \alpha _{3} )\, g \wedge S
-  \frac{1}{n-2}  ( (n-2) \alpha _{4} - \alpha _{2} )
\, g \wedge S^{2} \nonumber\\
\alpha _{1} \, R +  \frac{\alpha _{2}}{2} \, S \wedge S
- \frac{ \alpha _{1} + \alpha _{2} \kappa }{n-2} \, g \wedge S
+  \frac{ \alpha _{2}}{n-2}  \, g \wedge S^{2}
+ \frac{\alpha _{7}}{2} \, g \wedge g \nonumber\\
\alpha _{1} \, ( R - \frac{ 1 }{n-2} \, g \wedge S )
\frac{\alpha _{2}}{n-2}\, (  g \wedge S^{2} + \frac{n-2}{2}\, S \wedge S
- \kappa \, g \wedge S)
+ \frac{\alpha _{7}}{2} \, g \wedge g \nonumber\\
&=&  \alpha _{1} \, C  +
\frac{\alpha _{2}}{n-2}\, E + \frac{\alpha _{8}}{2} \, g \wedge g ,
\end{eqnarray*}

\begin{eqnarray}
\mathrm{Weyl} (T) &=& \alpha _{1} \, C  + \frac{\alpha _{2}}{n-2}\, E
+ \frac{\alpha _{8}}{2} \, g \wedge g ,
\label{2023.03.06e}
\end{eqnarray}

where $\alpha_{7}$ and $\alpha_{8}$ are some real numbers.
From (<ref>),
by suitable contraction, we get immediately
$\alpha _{8} = 0$, and in a consequence (<ref>),
completing the proof.

§ PSEUDOSYMMETRY TYPE CURVATURE CONDITIONS

It is well-known that if a semi-Riemannian manifold $(M,g)$,
$\dim M = n \geq 3$,
is locally symmetric then
$\nabla R = 0$
on $M$
(see, e.g., <cit.>).
implies  the following integrability condition
${\mathcal{R}}(X,Y ) \cdot R = 0$
in short $R \cdot R = 0$.
Semi-Riemannian manifold satisfying the last condition
is called semisymmetric (see, e.g.,
<cit.>, <cit.>,
<cit.>, <cit.>).
Semisymmetric manifolds form a subclass of the class
of pseudosymmetric manifolds.
A semi-Riemannian manifold $(M,g)$, $\dim M = n \geq 3$,
is said to be pseudosymmetric
if the tensors $R \cdot R$ and $Q(g,R)$
are linearly dependent at every point of $M$
(see, e.g.,
<cit.> and references therein).
This is equivalent to

\begin{eqnarray}
R \cdot R &=& L_{R}\, Q(g,R)
\label{pseudo}
\end{eqnarray}

on   ${\mathcal{U}}_{R}  \subset M$,
where $L_{R}$ is some function on ${\mathcal{U}}_{R}$.
Every semisymmetric manifold is pseudosymmetric.
The converse statement is not true (see, e.g., <cit.>).
We note that (<ref>) implies

\begin{eqnarray}
R \cdot S &=& L_{R}\, Q(g,S)
\label{Weyl-pseudo-bis}
\end{eqnarray}

\begin{eqnarray}
R \cdot C &=&  L_{R}\, Q(g,C) .
\label{Weyl-pseudo-ter}
\end{eqnarray}

(<ref>), (<ref>) and (<ref>)
are equivalent on the set ${\mathcal{U}}_{S} \cap {\mathcal{U}}_{C}$
of any warped product manifold  $\overline{M} \times _{F} \widetilde{N}$,
with $\dim \overline{M} = \dim \widetilde{N} = 2$
(see, e.g., <cit.> and references therein).

A semi-Riemannian manifold $(M,g)$, $\dim M = n \geq 3$,
is called Ricci-pseudosymmetric
if the tensors $R \cdot S$ and $Q(g,S)$
are linearly dependent at every point of $M$
(see, e.g., <cit.>,
<cit.>, [27]).
This is equivalent on ${\mathcal{U}}_{S}$ to

\begin{eqnarray}
R \cdot S &=& L_{S}\, Q(g,S) ,
\label{Riccipseudo07}
\end{eqnarray}

where $L_{S}$ is some function on ${\mathcal{U}}_{S}$.
Every warped product manifold $\overline{M} \times _{F} \widetilde{N}$
with a $1$-dimensional manifold  $(\overline{M}, \overline{g})$ and
an $(n-1)$-dimensional Einstein semi-Riemannian manifold
$(\widetilde{N}, \widetilde{g})$, $n \geq 3$,
and a warping function $F$,
is a Ricci-pseudosymmetric manifold,
see, e.g.,

A semi-Riemannian manifold $(M,g)$, $\dim M = n \geq 4$,
is said to be Weyl-pseudosymmetric
if the tensors $R \cdot C$ and $Q(g,C)$ are linearly dependent
at every point of $M$
This is equivalent on ${\mathcal{U}}_{C}$ to

\begin{eqnarray}
R \cdot C &=& L_{1}\, Q(g,C) ,
\label{Weyl-pseudo}
\end{eqnarray}

where $L_{1}$ is some function on ${\mathcal{U}}_{C}$.
We can easily check that
on every Einstein manifold $(M,g)$, $\dim M \geq 4$,
(<ref>) turns into

\begin{eqnarray*}
R \cdot R &=& L_{1}\, Q(g,R) .
\end{eqnarray*}

For a presentation of results on the problem
of the equivalence of pseudosymmetry,
Ricci-pseudosymmetry and Weyl-pseudosymmetry
we refer to <cit.>.

A semi-Riemannian manifold $(M,g)$, $\dim M = n \geq 4$, is said to be
a manifold with pseudosymmetric Weyl tensor
(to have a pseudosymmetric conformal Weyl tensor)
if the tensors $C \cdot C$ and $Q(g,C)$
are linearly dependent at every point of $M$
(see, e.g., <cit.>,
This is equivalent on ${\mathcal U}_{C}$ to

\begin{eqnarray}
C \cdot C &=& L_{C}\, Q(g,C) ,
\label{4.3.012}
\end{eqnarray}

where $L_{C}$ is some function on ${\mathcal{U}}_{C}$.
Every warped product manifold
$\overline{M} \times _{F} \widetilde{N}$,
with $\dim \overline{M} = \dim \widetilde{N} = 2$,
satisfies (<ref>)
(see, e.g., <cit.> and references therein).
Thus in particular,
the Schwarzschild spacetime, the Kottler spacetime
and the Reissner-Nordström spacetime satisfy (<ref>).
Semi-Riemannian manifolds with pseudosymmetric Weyl tensor
were studied among others in <cit.>.

Warped product manifolds $\overline{M} \times _{F} \widetilde{N}$,
of dimension $\geq 4$,
satisfying on
${\mathcal U}_{C} \subset \overline{M} \times _{F} \widetilde{N}$,
the condition

\begin{eqnarray}
R \cdot R - Q(S,R) &=& L\, Q(g,C) ,
\label{genpseudo01}
\end{eqnarray}

where $L$ is some function on ${\mathcal{U}}_{C}$,
were studied among others in [10]. In that paper
necessary and sufficient conditions for
$\overline{M} \times _{F} \widetilde{N}$
to be a manifold satisfying (<ref>) are given.
Moreover, in that paper it was proved that
any $4$-dimensional warped product manifold
$\overline{M} \times _{F} \widetilde{N}$,
with a $1$-dimensional base $(\overline{M},\overline{g})$,
satisfies (<ref>) <cit.>.

We refer to
for details on semi-Riemannian manifolds satisfying
(<ref>) and (<ref>)-(<ref>),
as well other conditions of this kind, named pseudosymmetry
type curvature conditions.
We also refer to <cit.>
for a recent survey on manifolds
satisfying such curvature conditions.
It seems that the condition (<ref>)
is the most important condition of that family of curvature conditions
(see, e.g., [34]).
The Schwarzschild spacetime, the Kottler spacetime,
the Reissner-Nordström spacetime,
as well as the Friedmann-Lemaître-Robertson-Walker spacetimes
are the “oldest” examples
of pseudosymmetric warped product manifolds
(see, e.g., <cit.>).
We finish this section with the following remarks.

(i) In view of <cit.>,
we can state that the following identity is satisfied
on every semi-Riemannian manifold
$(M,g)$, $\dim M = n \geq 3$,
with vanishing Weyl conformal curvature tensor $C$

\begin{eqnarray}
R \cdot R - Q(S,R) &=&
\frac{1}{(n-2)^{2}}\,
Q(g, g \wedge S^{2} + \frac{n-2}{2} \, S \wedge S - \kappa \, g \wedge S) .
\label{2022.11.20.zz}
\end{eqnarray}

From (<ref>), by (<ref>), we get

\begin{eqnarray*}
R \cdot R - Q(S,R) &=& \frac{1}{(n-2)^{2}}\, Q(g,E) ,
\end{eqnarray*}

where the tensor $E$ is defined by  (<ref>). In particular, if $n = 3$ then $E = 0$ on $M$.
(ii) As it was stated in <cit.>
on every $3$-dimensional semi-Riemannian manifold $(M,g)$
the identity $R \cdot R = Q(S,R)$ is satisfied.
(iii) From (i) it follows  that
on every semi-Riemannian conformally flat manifold
$(M,g)$, $\dim M = n \geq 4$, the conditions: $R \cdot R = Q(S,R)$
and (<ref>) are equivalent.

Let $(M,g)$, $\dim M = n \geq 4$, be a semi-Riemannian manifold.
(i) <cit.>
The following identity is satisfied on ${\mathcal{U}}_{C} \subset M$

\begin{eqnarray}
C \cdot R + R \cdot C &=& R \cdot R + C \cdot C - \frac{1}{ (n-2)^{2}}
Q( g, g \wedge S^{2} - \frac{\kappa }{n-1}\, g \wedge S) .
\label{2022.12.29.aaa}
\end{eqnarray}

(ii) If
(<ref>) holds on ${\mathcal{U}}_{C} \subset M$
then (<ref>) turns into

\begin{eqnarray}
& &
C \cdot R + R \cdot C \ =\ C \cdot C + Q(S,R) + L  \, Q(g,C)\nonumber\\
& &
- \frac{1}{ (n-2)^{2}}\,
Q( g, g \wedge S^{2} + \frac{n-2}{2}\, S \wedge S
- \frac{\kappa }{n-1}\, g \wedge S) .
\label{2022.12.29.ddd}
\end{eqnarray}

Moreover, from (<ref>), by an application
of (<ref>) and (<ref>),
we get on ${\mathcal{U}}_{C} \subset M$

\begin{eqnarray}
C \cdot R + R \cdot C
&=& C \cdot C + Q(S,C) + L \, Q(g,C) - \frac{1}{ (n-2)^{2}} \, Q( g, E) ,
\label{2022.12.29.fff}
\end{eqnarray}

where the tensor $E$ is defined by  (<ref>).
(iii) (cf. <cit.>)
If (<ref>) and (<ref>)
hold on ${\mathcal{U}}_{C} \subset M$ then
(<ref>) turns into

\begin{eqnarray*}
C \cdot R + R \cdot C
&=& Q(S,C) + (L_{C} + L) \, Q(g,C) - \frac{1}{ (n-2)^{2}} \, Q( g, E) .
\end{eqnarray*}

§ ROTER SPACES

Some results of <cit.>
(cf. <cit.>)
we can present in the following proposition.

Let $(M,g)$, $\dim M = n \geq 4$,
be a non-conformally flat and non-Einstein semi-Riemannian manifold.
(i) <cit.>
If (<ref>) and (<ref>)
hold on ${\mathcal{U}}_{S} \cap {\mathcal{U}}_{C} \subset M$ then
at every point $x \in {\mathcal{U}}_{S} \cap {\mathcal{U}}_{C}$
(<ref>) or (<ref>) is satisfied.
(ii) <cit.>
If (<ref>) and (<ref>)
hold on ${\mathcal{U}}_{S} \cap {\mathcal{U}}_{C} \subset M$ then
at every point $x \in {\mathcal{U}}_{S} \cap {\mathcal{U}}_{C}$
(<ref>) or (<ref>) is satisfied.
(iii) (cf. <cit.>)
If (<ref>), (<ref>)
and $R \cdot S = Q(g,D)$,
for some symmetric $(0,2)$-tensor $D$,
hold on ${\mathcal{U}}_{S} \cap {\mathcal{U}}_{C} \subset M$ then
at every point $x \in {\mathcal{U}}_{S} \cap {\mathcal{U}}_{C}$
(<ref>) or (<ref>) is satisfied.

We recall that
a  non-quasi-Einstein and non-conformally flat
semi-Riemannian manifold $(M,g)$, $\dim M = n \geq 4$,
satisfying (<ref>) on
${\mathcal U}_{S} \cap {\mathcal U}_{C} \subset M$
is called a Roter type manifold, or a Roter manifold,
or a Roter space
(see, e.g., <cit.>,

Roter spaces and in particular Roter hypersurfaces
in semi-Riemannian spaces of constant curvature were studied in:
In particular,
(<ref>) and (<ref>)-(<ref>)
are satisfied on such manifolds. More precisely, we have

(see, e.g., <cit.>,
If $(M,g)$, $\dim M = n \geq 4$, is a semi-Riemannian
Roter space satisfying
(<ref>) on ${\mathcal U}_{S} \cap {\mathcal U}_{C} \subset M$
then on this set we have:

\begin{eqnarray*}
S^{2} &=& \alpha _{1}\, S + \alpha _{2} \, g ,
\ \ \
\alpha _{1}
\ =\
\kappa + \frac{(n-2)\mu -1 }{\phi} ,
\ \ \
\alpha _{2}
\ =\
\frac{\mu \kappa + (n-1) \eta }{\phi } ,\\
R \cdot C &=& L_{R}\, Q(g,C),
\ \ \
L_{R} \ =\ \frac{1}{\phi}
\left(  (n-2) (\mu ^{2} - \phi \eta) - \mu \right) ,\\
R \cdot R & = & L_{R}\, Q(g,R) ,\\
R \cdot S & = & L_{R}\, Q(g,S),
\end{eqnarray*}

\begin{eqnarray*}
R \cdot R &=& Q(S,R) + L \, Q(g,C) ,
\ \ \
L \ =\ L_{R} + \frac{\mu }{\phi }
\ =\
\frac{n-2}{\phi} (\mu ^{2} - \phi \eta),\\
C \cdot C &=& L_{C}\, Q(g,C) ,
\ \ \
L_{C} \ =\ L_{R} + \frac{1}{n-2} (\frac{\kappa }{n-1} - \alpha _{1} ) ,\\
C \cdot R & = & L_{C}\, Q(g,R),\\
C \cdot S & = & L_{C}\, Q(g,S) ,
\end{eqnarray*}

\begin{eqnarray*}
C \cdot R + R \cdot C
&=& Q(S,C) + \left( L + L_{C} - \frac{1}{ (n-2) \phi } \right) Q(g,C) ,\\
R \cdot C - C \cdot R &=&
\left( \frac{1}{\phi} ( \mu - \frac{1}{n-2} )
+ \frac{\kappa }{n-1} \right) Q(g,R)
\left( \frac{\mu}{\phi } ( \mu - \frac{1}{n-2}) - \eta \right) Q(S,G) ,
\end{eqnarray*}

(<ref>) and

\begin{eqnarray*}
R\cdot C - C \cdot R
Q \left( \left(\frac{\mu \kappa  }{n-1} + \eta \right) \, g
\left( \frac{1 }{n-2} - \mu - \frac{\phi \kappa }{n-1} \right)\, S,
g \wedge S \right) .
\end{eqnarray*}

(i) In the standard Schwarzschild coordinates
$(t; r; \theta; \phi)$,
and the physical units ($c = G = 1$),
the Reissner-Nordström-de Sitter ($\Lambda > 0$), and
Reissner-Nordström-anti-de Sitter ($\Lambda < 0$)
spacetimes are given by the line element (see, e.g., [79])

\begin{eqnarray}
ds^{2} &=& - h(r)\, dt^{2} + h(r)^{-1}\, dr^{2} + r^{2}\,
( d\theta^{2} + \sin ^{2}\theta \, d\phi^{2}) ,\label{rns01}\\
h(r) &=& 1 - \frac{2 M}{r} + \frac{Q^{2}}{r^{2}}
- \frac{\Lambda }{3} r^{3} ,\nonumber
\end{eqnarray}

$M$, $Q$ and $\Lambda$ are non-zero constants.
(see also
The metric (<ref>) satisfies (<ref>) with

\begin{eqnarray*}
\phi &=&
\frac{3}{2} ( Q^{2}  -  M r) r^{4} }{2 Q^{-4},\ \ \
\mu \ =\
\frac{1}{2} (Q^{4} + 3 Q^{2} \Lambda r^{4} - 3 \Lambda M r^{5}) Q^{-4}, \\
\eta
\frac{1}{12}
( 3 Q^{6}
+ 4 Q^{4} \Lambda r^{4}
- 3 Q^{4} M r
+ 9 Q^{2} \Lambda ^{2} r^{8}
- 9 \Lambda^{2} M r^{9})  r^{-4} Q^{-4} .
\end{eqnarray*}
If we set $\Lambda = 0$ in (\ref{rns01}) then
we obtain the line element of the Reissner-Nordstr\"{o}m
spacetime, see, e.g., {\cite[Section 9.2] {GrifPod}}
and references therein.
It seems that the Reissner-Nordstr\"{o}m spacetime is the oldest example
of the Roter warped product space.
\newline
In [41]
a particular class of Roter warped product spaces was determined
such that every manifold of that class
admits a non-trivial geodesic mapping onto some Roter warped product space.
Moreover, both geodesically related manifolds
are pseudosymmetric of constant type.
\newline
An algebraic classification of the Roter type $4$-dimensional
spacetimes is given in
\newline
Some comments on pseudosymmetric manifolds (also called
Deszcz symmetric spaces),
as well as Roter spaces, are given in {\cite[Section 1] {DecuP-TSVer}}
(see also {\cite[Remark 2 (iii)] {DDH-2021}},
{\cite[Remark 2.1 (iii)] {DH-Colloq}}):
"{\sl{From a geometric point of view, the Deszcz symmetric spaces may well
be considered to be the simplest Riemannian manifolds next
to the real space forms.}}"
"{\sl{From an algebraic point of view, Roter spaces may well
be considered to
be the simplest Riemannian manifolds next to the real space forms.}}"
For further comments we refer to [84].
\end{rem}

We finish this section with the following results.
\begin{prop} {\cite[Lemma 2.2] {DGHSaw-2022}}
If $(M,g)$, $\dim M = n \geq 4$, is a Roter space satisfying
on ${\mathcal U}_{S} \cap  {\mathcal U}_{C} \subset M$
then (\ref{2022.11.22.aa}) holds
on this set, i.e., $(n-2)\, C = \phi \, E$,
where the tensor $E$ is defined by  (\ref{2022.11.10.aaa}).
\end{prop}

Propositions 2.1, 4.1 and 4.4 lead to the following
\begin{prop}
Let $(M,g)$, $\dim M = n \geq 4$,
be a non-conformally flat and non-Einstein semi-Riemannian manifold.
(\ref{pseudo}) and (\ref{4.3.012}),
(\ref{pseudo}) and (\ref{genpseudo01}),
(\ref{4.3.012}), (\ref{genpseudo01})
and $R \cdot S = Q(g,D)$,
for some symmetric $(0,2)$-tensor $D$,
hold on ${\mathcal{U}}_{S} \cap {\mathcal{U}}_{C} \subset M$ then
$E = \lambda \, C$
on ${\mathcal{U}}_{S} \cap {\mathcal{U}}_{C}$,
where the tensor $E$ is defined by  (\ref{2022.11.10.aaa})
and $\lambda $ is some function on this set.
\end{prop}

\section{Warped product manifolds with 2-dimensional base manifold}

Proposition 2.1 and Proposition 4.4 imply
\begin{prop} {\cite[Proposition 2.3] {DGHSaw-2022}}
If $(M,g)$, $\dim M = n \geq 4$, is a semi-Riemannian manifold satisfying
(\ref{quasi02}) or (\ref{eq:h7a}) at every point
of ${\mathcal U}_{S} \cap  {\mathcal U}_{C} \subset M$
then the following equation is satisfied on this set
\begin{eqnarray}
\tau \, C &=&
g \wedge S^{2} + \frac{n-2}{2} \, S \wedge S - \kappa \, g \wedge S
+ \frac{\kappa ^{2} - \mathrm{tr}_{g} (S^{2})}{2(n-1)}
\, g \wedge g ,
\label{2022.11.22.jj}
\end{eqnarray}
where $\tau$ is some function
on ${\mathcal U}_{S} \cap {\mathcal U}_{C}$.
\end{prop}

Proposition 5.1, {\cite[Theorem 7.1 (ii)] {2016_DGJZ}} and
{\cite[Theorem 4.1] {DeKow}} imply
\begin{thm} {\cite[Theorem 2.4] {DGHSaw-2022}}
Let $\overline{M} \times _{F} \widetilde{N}$
be the warped product manifold
with a $2$-dimensional semi-Riemannian manifold
an $(n-2)$-dimensional semi-Riemannian manifold
$n \geq 4$, a warping function $F$,
and let $(\widetilde{N},\widetilde{g})$
be a space of constant curvature when $n \geq 5$.
Then (\ref{2022.11.22.jj}) holds on
${\mathcal U}_{S} \cap {\mathcal U}_{C}
\subset \overline{M} \times _{F} \widetilde{N}$.
\end{thm}

\begin{example} {\cite[Example 2.1] {DGHSaw-2022}}
(i) Let $\mathbb{S}^{k}(r)$ be a $k$-dimensional standard sphere
of radius $r$ in $\mathbb{E}^{k+1}$, $k \geq 1$.
It is well-known that
the Cartesian product
$\mathbb{S}^{1}(r_{1}) \times \mathbb{S}^{n-1}(r_{2})$
of spheres $\mathbb{S}^{1}(r_{1})$ and $\mathbb{S}^{n-1}(r_{2})$,
$n \geq 4$,
and more generally,
the warped product manifold
$\mathbb{S}^{1}(r_{1}) \times _{F} \mathbb{S}^{n-1}(r_{2})$
of spheres $\mathbb{S}^{1}(r_{1})$ and $\mathbb{S}^{n-1}(r_{2})$,
$n \geq 4$, with a warping function $F$,
is a conformally flat manifold.
\newline
As it was stated in {\cite[Example 3.2] {G5}},
the Cartesian product
$\mathbb{S}^{p}(r_{1}) \times \mathbb{S}^{n-p}(r_{2})$
of spheres
$\mathbb{S}^{p}(r_{1})$ and $\mathbb{S}^{k}(r_{2})$
such that
$2 \leq p \leq n-2$ and
$(n-p-1) r_{1}^{2} \neq (p-1) r_{2}^{2}$
is a non-conformally flat and non-Einstein manifold
satisfying the Roter equation (\ref{eq:h7a}) on
${\mathcal U}_{S} \cap  {\mathcal U}_{C}
= \mathbb{S}^{p}(r_{1}) \times \mathbb{S}^{n-p}(r_{2})$.
\newline
{\cite[Example 4.1] {DeKow}}
The warped product manifold
$\mathbb{S}^{p}(r_{1}) \times _{F} \mathbb{S}^{n-p}(r_{2})$,
$2 \leq p \leq n-2$, with some special warping function $F$,
satisfies on ${\mathcal U}_{S} \cap  {\mathcal U}_{C}
\subset \mathbb{S}^{p}(r_{1}) \times \mathbb{S}^{n-p}(r_{2})$
the Roter equation (\ref{eq:h7a}). Thus some
warped product manifolds
$\mathbb{S}^{2}(r_{1}) \times _{F} \mathbb{S}^{n-2}(r_{2})$
are Roter spaces.
\newline
(iv)  Properties of pseudosymmetry type of warped products
with $2$-dimensional base manifold,
a warping function $F$,
and an $(n-2)$-dimensional fibre,
$n \geq 4$,
assumed to be of constant curvature when $n \geq 5$,
were determined in
{\cite[Sections 6 and 7] {2016_DGJZ}}.
warped product manifolds
$\mathbb{S}^{2}(r_{1}) \times _{F} \mathbb{S}^{n-2}(r_{2})$,
$n \geq 4$, are such manifolds.
Let $g$, $R$, $S$, $\kappa$ and $C$ denote the metric tensor,
the Riemann-Christoffel curvature tensor, the Ricci tensor,
the scalar curvature and
the Weyl conformal curvature tensor of
$\mathbb{S}^{2}(r_{1}) \times _{F} \mathbb{S}^{n-2}(r_{2})$, respectively.
From {\cite[Theorem 7.1] {2016_DGJZ}}
it follows that on set $V$ of all points of
${\mathcal U}_{S} \cap  {\mathcal U}_{C}
\subset \mathbb{S}^{2}(r_{1}) \times _{F} \mathbb{S}^{n-2}(r_{2})$
at which the tensor $S^{2}$ is not a linear combination
of the tensors $S$ and $g$,
the Weyl tensor $C$ is expressed by
\begin{eqnarray}
C &=& \lambda \left(
g \wedge S^{2} + \frac{n-2}{2} \, S \wedge S - \kappa \, g \wedge S
+ \frac{\kappa ^{2} - \mathrm{tr}_{g} (S^{2}) }{2(n-1)}
\, g \wedge g \right) ,
\label{2022.11.10.zz}
\end{eqnarray}
where $\lambda$ is some function on $V$.
This,  by (\ref{Weyl}), turns into
\begin{eqnarray*}
R &=& \lambda \,
g \wedge S^{2} + \frac{n-2}{2} \lambda \, S \wedge S
+ \left( \frac{1}{n-2} - \kappa \lambda \right) g \wedge S\\
& &
+ \frac{1}{2(n-1)} \left( (\kappa ^{2} - \mathrm{tr}_{g} (S^{2}) ) \lambda
- \frac{\kappa }{n-2} \right) g \wedge g .
\end{eqnarray*}
Thus  (\ref{B001simply}) is satisfied on $V$.
Moreover,  (\ref{eq:h7a}) holds
at all points of $({\mathcal U}_{S} \cap  {\mathcal U}_{C}) \setminus V$,
at which (\ref{quasi02}) is not satisfied.
From Lemma 2.2 it follows that (\ref{2022.11.10.zz})
holds at all points of ${\mathcal U}_{S} \cap  {\mathcal U}_{C} \subset
\mathbb{S}^{2}(r_{1}) \times _{F} \mathbb{S}^{n-2}(r_{2})$, $n \geq 4$,
at which (\ref{quasi02}) is not satisfied. Finally, in view of Theorem 5.2,
we can state that (\ref{2022.11.22.jj}) holds on
${\mathcal U}_{S} \cap  {\mathcal U}_{C}$.
\end{example}

\section{Essentially conformally symmetric manifolds}

Let $(M,g)$, $\dim M = n \geq 4$, be a semi-Riemannian manifold
with parallel Weyl conformal curvature tensor,
i.e. $\nabla C  = 0$ on $M$.
It is obvious that the last condition implies $R \cdot C  = 0$.
Moreover, let the manifold $(M,g)$ be neither conformally flat nor
locally symmetric.
Such manifolds are called {\sl{essentially conformally
symmetric manifolds}}, e.c.s. manifolds/metrics, or ECS manifolds/metrics,
in short (see, e.g.,
\cite{{DerRot01}, {DerRot02}, {DerRot2007PADGE},
{DerTer2210a}, {DerTer2210b}, {Hotlos}}).
E.c.s. manifolds
are semisymmetric manifolds
($R \cdot R  = 0$, {\cite[Theorem 9] {DerRot01}})
$\kappa  = 0$ and $Q(S,C) \, =\, 0$
({\cite[Theorems 7 and 8] {DerRot01}}).
In addition,
\begin{eqnarray}
F\, C &=& \frac{1}{2}\, S \wedge S
\label{roter77}
\end{eqnarray}
holds on $M$, where $F$ is some function on $M$, called the
{\sl{fundamental function}} [12].
At every point of $M$ we also have $\mbox{rank}\, S \leq 2$
{\cite[Theorem 5] {DerRot02}}.
We mention that the local structure
of e.c.s. manifolds is already determined.
We refer to \cite{{DerRot2007}, {DerRot2009}} for the final results
related to this subject.
We also mention that certain e.c.s. metrics are realized
on compact manifolds
(\cite{{DerRot2008}, {DerRot2010}, {DerTer2210a}, {DerTer2210b}}).

Equation (\ref{roter77}), by suitable contraction, leads immediately to
$S^{2} = \kappa \, S$, which by $\kappa = 0$, reduces to $S^{2} = 0$.
Evidently, $\mathrm{tr}_{g} (S^{2}) = 0$.
Now using (\ref{roter77}) we get (\ref{2023.03.03.a}).
Thus we have
\begin{thm}
(\ref{2022.11.22.jj}), with $\tau = (n-2) F$,
is satisfied on every essentially conformally symmetric manifold $(M,g)$.
\end{thm}

\section{Hypersurfaces in semi-Riemannian conformally flat spaces}

Let $M$, $\dim M = n \geq 4$, be a hypersurface
isometrically immersed in a semi-Riemannian conformally flat manifold
$N$, $\dim N = n + 1$.
Let $g_{ad}$, $H_{ad}$,
$G_{abcd} = g_{ad}g_{bc} - g_{ac}g_{bd}$ and $C_{abcd}$
be the local components of the metric tensor $g$,
the second fundamental tensor $H$, the $(0,4)$-tensor $G$
and the Weyl conformal curvature tensor $C$ of $M$,
As it was stated in {\cite[eq. (20)] {DV-1991}}
(see also {\cite[eq. (11)] {P38}}) we have
\begin{eqnarray}
\varepsilon \, (H_{ad}H_{bc} - H_{ac}H_{bd})
\frac{\varepsilon \, \mathrm{tr} (H)}{ n-2}\,
( g_{ad}H_{bc} + g_{bc}H_{bd} - g_{ac}H_{bd} - g_{bd}H_{ac} )
\nonumber\\
& &
\frac{\varepsilon }{ n-2}\,
( g_{ad}H^{2}_{bc} + g_{bc}H^{2}_{bd} - g_{ac}H^{2}_{bd} - g_{bd}H^{2}_{ac} )
\mu \, G_{abcd} \, ,
\label{2022.08.03.bb}
\end{eqnarray}
$\varepsilon = \pm 1$,
$\mathrm{tr} (H) = g^{ad}H_{ad}$,
$H^{2}_{ad} = g^{bc}H_{ab}H_{cd}$
and $\mu$ is some function on $M$.
From (\ref{2022.08.03.bb}), by contraction we get easily
\begin{eqnarray}
\mu
&=& \frac{ \varepsilon }{ (n-2)(n-1)}\,
( (\mathrm{tr} (H))^{2} - \mathrm{tr} (H^{2}) ) \, ,
\label{2022.08.03.cc}
\end{eqnarray}
$\mathrm{tr} (H^{2})
\, =\, g^{ad}H^{2}_{ad}$.
Now (\ref{2022.08.03.bb}) and (\ref{2022.08.03.cc}) yield
\begin{eqnarray}
C &=& \frac{\varepsilon}{n-2}
\left( g \wedge H^{2} + \frac{n-2}{2}\, H \wedge H
- \mathrm{tr} (H) \, g \wedge H
+ \frac{(\mathrm{tr} (H))^{2}
- \mathrm{tr}(H^{2})}{2 (n-1)} \, g \wedge g \right).
\label{2022.08.03.dd}
\end{eqnarray}
If $H = \frac{\mathrm{tr} (H)}{n} g$ at a point $x \in M$,
i.e., $M$ is umbilical at $x$, then from (\ref{2022.08.03.dd}) it follows
immediately that the tensor $C$ vanishes at $x$.
If at a non-umbilical point $x \in M$, we have
$\mathrm{rank} (H - \alpha g) = 1$,
for some $\alpha \in \mathbb{R}$,
i.e., $M$ is quasi-umbilical at $x$,
then in view of Proposition 2.1 (i), the tensor $C$ vanishes at $x$.
Conversely, if at a non-umbilical point $x \in M$ the tensor $C$ vanishes
then in view of Proposition 2.1 (ii) we have $\mathrm{rank} (H - \alpha g) = 1$,
for some $\alpha \in \mathbb{R}$.
Thus we can present {\cite[Theorem 4.1] {DV-1991}} in the folowing form.
\begin{thm}
Let $M$, $\dim M = n \geq 4$, be a hypersurface
isometrically immersed in a semi-Riemannian conformally flat manifold
$N$, $\dim N = n + 1$.
At every non-umbilical point $x \in M$ the tensor $C$ vanishes at $x$
if and only if
at $x$ we have $\mathrm{rank} (H - \alpha g) = 1$,
for some $\alpha \in \mathbb{R}$.
\end{thm}

\begin{rem}
Let $M$, $\dim M = n \geq 4$, be a hypersurface
isometrically immersed in a semi-Riemannian conformally flat manifold
$N$, $\dim N = n + 1$.
\newline
We assume that at all points of ${\mathcal U}_{C} \subset M$ the tensor $H^{2}$
is a linear combination of $H$ and $g$, i.e.,
\begin{eqnarray}
H^{2} &=& \alpha_{1}\, H + \alpha_{2}\, g
\label{2022.12.12.mm}
\end{eqnarray}
on ${\mathcal U}_{C}$,
where $\alpha_{1}$ and $\alpha_{2}$ are some functions on this set.
Now (\ref{2022.08.03.dd}) turns into
\begin{eqnarray}
\frac{\varepsilon }{2}\, H \wedge H
+ \frac{\varepsilon (\alpha_{1} - \mathrm{tr} (H) ) }{n-2}
\, g \wedge H
+ \frac{\varepsilon}{n-2}
\left( \alpha_{2} +  \frac{(\mathrm{tr} (H))^{2}
- \mathrm{tr}(H^{2})}{2 (n-1)} \right)  g \wedge g \nonumber\\
\frac{\alpha }{2}\, H \wedge H
+ \beta \, g \wedge H
+ \frac{\gamma }{2} \, g \wedge g ,
\label{2022.08.03.ff}
\end{eqnarray}
\begin{eqnarray}
\alpha \ =\ \varepsilon,\ \ \
\beta \ =\  \frac{\varepsilon (\alpha_{1} - \mathrm{tr} (H) ) }{n-2} ,\ \ \
\gamma \ =\ \frac{\varepsilon}{n-2}
\left( 2 \alpha_{2} +  \frac{(\mathrm{tr} (H))^{2}
- \mathrm{tr}(H^{2})}{n-1} \right) .
\label{2022.08.03.gg}
\end{eqnarray}
From (\ref{2022.08.03.ff}) and (\ref{2022.08.03.gg}),
in view of {\cite[Theorem 3.1 (i)] {Kow02}}, we get
\begin{eqnarray}
C \cdot C
&=& (n-2) \left( \frac{\beta ^{2}}{\alpha } - \gamma \right)  Q(g,C)
\ =\  (n-2) ( \varepsilon \beta ^{2} - \gamma )\,  Q(g,C)
\label{2022.08.03.hh}
\end{eqnarray}
on ${\mathcal U}_{C}$, with $\alpha$, $\beta$ and $\gamma$
defined by (\ref{2022.08.03.gg}).
Thus $M$ is a hypersurface with pseudosymmetric Weyl tensor.
We also note that from (\ref{2022.12.12.mm}) we get immediately
$\alpha _{2} = \frac{1}{n} ( \mathrm{tr}(H^{2}) - \alpha _{1} \mathrm{tr} (H) )$
\begin{eqnarray*}
H^{2} - \frac{\mathrm{tr}(H^{2})}{n}\, g
&=& \alpha_{1} \left( H - \frac{\mathrm{tr} (H)}{n}\, g \right) .
\end{eqnarray*}
(ii) The above presented result, i.e.,
if (\ref{2022.12.12.mm}) is satisfied at every point
of ${\mathcal U}_{C} \subset M$
then (\ref{4.3.012}) holds on this set, was already obtained in
{\cite[Proposition 3.1] {DVY-1999}}.
We mention that Proposition 3.1 of [53]
was proved without application of {\cite[Theorem 3.1 (i)] {Kow02}}.
\newline
We assume that the tensor $H$ satisfies on ${\mathcal U}_{C} \subset M$
\begin{eqnarray}
H^{3} &=& \mathrm{tr} (H) \,  H^{2} + \psi \,   H,
\label{2022.10.01.kk}
\end{eqnarray}
where $\psi $ is some function on this set
and the $(0,2)$-tensor $H^{3}$ is defined by
$H^{3}_{ad} = g^{bc}H^{2}_{ab}H_{cd}$.
\begin{eqnarray}
C \cdot C &=&
\left(
\frac{\varepsilon }{(n-2)(n-1)}
(\mathrm{tr} (H))^{2} - \mathrm{tr}(H^{2})
+ \frac{\varepsilon \psi}{n-2}
\right)
& &
- \frac{n-3 }{n-2} \, Q(H^{2}, \frac{1}{2}\, H \wedge H)
\label{2022.10.01.ll}
\end{eqnarray}
on ${\mathcal U}_{C}$ {\cite[eq. (10)] {P37}}, see also
{\cite[the proof of Lemma 4.1] {DVY-1999}}.
We refer to \cite{{P37}, {DVY-1999}} for further results on hypersurfaces
$M$ in conformally flat manifold $N$ satisfying (\ref{2022.10.01.kk}).
\newline
(iv) Recently curvature properties of pseudosymmetry type of hypersurfaces
isometrically immersed in a semi-Riemannian conformally flat manifold
were investigated in [57] and [69].
\end{rem}

\section{Hypersurfaces in semi-Riemannian space forms}

Let now $N_{s}^{n+1}(c)$, $n  \geq 4$,
be a semi-Riemannian space of constant curvature
with signature $(s,n+1-s)$, where
$c = \frac{\widetilde{\kappa}}{n (n+1)}$ and
$\widetilde{\kappa}$ is its scalar curvature.
Let $M$, $\dim M = n \geq 4$, be a connected hypersurface
isometrically immersed in $N_{s}^{n+1}(c)$.
We denote by
$g$, $R$, $S$, $\kappa$ and $C$,
the metric tensor, the Riemann-Christoffel curvature tensor,
the Ricci tensor,
the scalar curvature and the Weyl conformal curvature tensor
of the hypersurface $M$, respectively.
The Gauss equation of $M$ in $N_{s}^{n+1}(c)$ reads
(see, e.g., \cite{{2016_DGHZhyper}, {DGP-TV02},
{DGPSS}, {2020_DGZ}, {Saw114}})
\begin{eqnarray}
R - \frac{\widetilde{\kappa}}{2 n (n+1)}\, g \wedge g
\frac{\varepsilon }{2}  \,  H \wedge H ,\  \  \  \varepsilon \ =\ \pm 1 .
\label{realC5}
\end{eqnarray}
From (\ref{realC5}), by suitable contractions, we obtain
\begin{eqnarray}
S -   \frac{ (n-1) \widetilde{\kappa}}{ n (n+1)}\, g
&=&  \varepsilon \, ( \mathrm{tr} (H)\, H - H^{2}) ,
\label{realC6}\\
\frac{ \kappa }{n-1}  - \frac{  \widetilde{\kappa} }{n+1}
&=& \frac{ \varepsilon}{n-1}
\, ( ( \mathrm{tr} (H))^{2} - \mathrm{tr} (H^{2}) ) .
\label{realC6bb}
\end{eqnarray}
Now using (\ref{realC5}),  (\ref{realC6}) and (\ref{realC6bb})
we get immediately
\begin{eqnarray}
Q(H^{2}, \frac{1}{2}\, H \wedge H)
- Q(  \mathrm{tr} (H)\, H  - H^{2} , \frac{1}{2}\, H \wedge H)  \nonumber\\
- Q( \varepsilon ( \mathrm{tr} (H)\, H  -  H^{2} ),
\frac{\varepsilon}{2}\, H \wedge H) \nonumber\\
- Q\left( S - \frac{ (n-1) \widetilde{\kappa}}{ n (n+1)}\, g ,
R - \frac{\widetilde{\kappa}}{2 n (n+1)}\, g \wedge g \right ) .
\label{realRa}
\end{eqnarray}

We also recall that the curvatere condition of pseudosymmetry type
(\ref{genpseudo01}) is satisfied on $M$. Precisely, we have on $M$
{\cite[Proposition 3.1] {DV-1991}}
(see also {\cite[eqs. (3.3) and (3.4)] {2020_DGZ}})
\begin{eqnarray}
R \cdot R - Q(S,R) &=& - \frac{ (n-2) \widetilde{\kappa} }{n (n+1)}\, Q(g,C) .
\label{2023.03.21.aa}
\end{eqnarray}
Now (\ref{2022.12.29.fff}), by (\ref{2023.03.21.aa}), turns into
({\cite[Propopsition 4.7, eq. (4.36)] {2020_DGZ}})
\begin{eqnarray}
C \cdot R + R \cdot C
&=& C \cdot C + Q(S,C)
- \frac{ (n-2) \widetilde{\kappa} }{n (n+1)}\,  Q(g,C)
- \frac{1}{ (n-2)^{2}} \, Q( g, E) .
\label{2022.12.29.rrr}
\end{eqnarray}

Let ${\mathcal U}_{H} \subset M$ be the set of all points
at which the tensor
$H^{2}$ is not a linear combination of the metric tensor $g$
and the second fundamental tensor $H$ of $M$. We have
${\mathcal U}_{H} \subset {\mathcal U}_{S} \cap {\mathcal U}_{C} \subset M$
(see, e.g., \cite{{R99}, {DGPSS}, {2020_DGZ}} or {\cite[p. 137] {G108}}).

We assume that
the following conditions are satisfied on ${\mathcal U}_{H} \subset M$
\begin{eqnarray}
H^{3} &=& \mathrm{tr} (H) \,  H^{2} + \psi \, H + \rho \, g ,
\label{2022.10.01.lla}\\
C \cdot C &=& Q(g, T) ,
\label{2022.10.01.llb}
\end{eqnarray}
where $T$ is a generalized curvature tensor
and $\psi$ and $\rho$ some functions on ${\mathcal U}_{H}$.
Now, in view of {\cite[Theorem 4.5] {2020_DGZ}}, we obtain
\begin{eqnarray}
T &=& \left(
\frac{\kappa + 2 \varepsilon \psi }{n-1}
- \frac{\widetilde{\kappa}}{n+1} \right) C
+ \frac{\lambda_{1} }{2}\, g \wedge g \nonumber\\
& &
- \frac{n-3}{ (n-2)^{2} (n-1)}
\left( g \wedge S^{2} + \frac{n-2}{2}\, S \wedge S - \kappa \, g \wedge S
\right)
\label{2022.10.01.llc}
\end{eqnarray}
on ${\mathcal U}_{H}$, where $\lambda _{1} $ is some function on this set.
(\ref{2022.11.10.aaa}), (\ref{2022.10.01.llb}) and (\ref{2022.10.01.llc})
we get immediately
\begin{eqnarray*}
T &=& \left(
\frac{\kappa + 2 \varepsilon \psi }{n-1}
- \frac{\widetilde{\kappa}}{n+1} \right) C
- \frac{n-3}{ (n-2)^{2} (n-1)}\, E
+ \frac{\lambda }{2}\, g \wedge g
\end{eqnarray*}
\begin{eqnarray}
C \cdot C &=&
\left(
\frac{\kappa + 2 \varepsilon \psi }{n-1}
- \frac{\widetilde{\kappa}}{n+1} \right) Q(g,C)
- \frac{n-3}{ (n-2)^{2} (n-1)}\,Q(g,E)
\label{2022.11.01.ll}
\end{eqnarray}
on ${\mathcal U}_{H}$, where $\lambda $ is some function on this set.
In addition, if we assume that (\ref{4.3.012}) holds on ${\mathcal U}_{H}$
then from  (\ref{2022.10.01.ll}) it follows that
\begin{eqnarray}
\left(
\frac{\kappa + 2 \varepsilon \psi }{n-1}
- \frac{\widetilde{\kappa}}{n+1} -  L_{C} \right)  C
&=&  \frac{n-3}{ (n-2)^{2} (n-1)}\,E
+ \frac{\lambda _{2} }{2}\, g \wedge g
\label{2023.03.23.aa}
\end{eqnarray}
on ${\mathcal U}_{H}$,
where $\lambda _{2}$ is some function on this set.
We note that (\ref{2023.03.23.aa}),
by a suitable contraction, yields $\lambda _{2} = 0$,
and in a consequence we obtain
\begin{eqnarray}
\left(
\frac{\kappa + 2 \varepsilon \psi }{n-1}
- \frac{\widetilde{\kappa}}{n+1} -  L_{C} \right)  C
&=&  \frac{n-3}{ (n-2)^{2} (n-1)}\, E .
\label{2023.03.23.bb}
\end{eqnarray}
From the above presented considerations it follows immediately
the following result.
\begin{thm}
Let $M$ be a non-Einstein and non-conformally flat hypersurface
in $N_{s}^{n+1}(c)$, $n \geq 4$.
If (\ref{4.3.012}) and (\ref{2022.10.01.lla}) are satisfied on
${\mathcal U}_{H} \subset M$ then (\ref{2023.03.23.bb}) holds on
${\mathcal U}_{H}$.
\end{thm}

According to {\cite[Corollary 4.1] {R99}},
if on the subset ${\mathcal U}_{H}$
of a hypersurface $M$ in $N_{s}^{n+1}(c)$, $n \geq 4$,
one of the tensors $R \cdot C$, $C \cdot R$
or $R \cdot C - R \cdot C$
is a linear combination of $R \cdot R$ and of a finite sum
of tensors of the form $Q(A,T)$, where
$A$ is a symmetric $(0,2)$-tensor
and $T$ a generalized curvature tensor, then (\ref{2022.10.01.lla})
holds on ${\mathcal U}_{H}$.
In particular if one of  the following conditions is satisfied on
${\mathcal U}_{H} \subset M$: $R \cdot C = Q(g,T_{1})$,
$C \cdot R = Q(g,T_{2})$ or $R \cdot C - C \cdot R = Q(g,T_{3})$,
where $T_{1}$, $T_{2}$ and $T_{3}$ are generalized curvature tensors,
then (\ref{2022.10.01.lla}) holds on ${\mathcal U}_{H}$.
Now from Theorems 5.2, 5.3 and 5.4 of [36],
in view of Proposition 2.2, it follows that
\begin{eqnarray}
\mathrm{Weyl} (T_{1}) &=&
\left(
\frac{\kappa + \varepsilon \psi }{n-1}
- \frac{(n-1) \widetilde{\kappa}}{n (n+1)} \right)  C
- \frac{1}{(n-2)(n-1)}\, E ,\label{2023.03.23.cc1}\\
\mathrm{Weyl} (T_{2}) &=&
\left(
\frac{\kappa + 2 \varepsilon \psi }{n-1}
- \frac{ \widetilde{\kappa}}{n+1} \right)  C
- \frac{n-3}{(n-2)^{2} (n-1)}\, E ,\label{2023.03.23.cc2}\\
\mathrm{Weyl} (T_{3}) &=&
\left(
\frac{ \widetilde{\kappa}}{n (n+1)} - \frac{ \varepsilon \psi }{n-1} \right)  C
- \frac{1}{ (n-2)^{2} (n-1)}\, E .\label{2023.03.23.cc3}
\end{eqnarray}
Thus we have
\begin{thm}
Let $M$ be a non-Einstein and non-conformally flat hypersurface
in $N_{s}^{n+1}(c)$, $n \geq 4$, satisfying  (\ref{2022.10.01.lla})
on ${\mathcal U}_{H} \subset M$,
and let $T_{1}$, $T_{2}$ and $T_{3}$ be generalized curvature tensors
defined on ${\mathcal U}_{H}$.
If one of the following conditions:
$R \cdot C = Q(g,T_{1})$, respectively,
$C \cdot R = Q(g,T_{2})$ and $R \cdot C - C \cdot R = Q(g,T_{3})$,
is satisfied on ${\mathcal U}_{H}$
then (\ref{2023.03.23.cc1}), respectively (\ref{2023.03.23.cc2})
and (\ref{2023.03.23.cc3}), holds on ${\mathcal U}_{H}$.
\end{thm}

Finally, we assume that the tensor $H$ satisfies (\ref{2022.10.01.kk})
on ${\mathcal U}_{H} \subset M$.
Now (\ref{2022.10.01.ll}), by making use of (\ref{realC6bb})
and  (\ref{realRa}), turns into
\begin{eqnarray}
\frac{n-2}{n-3} \, C \cdot C
&=& \rho \, Q(g,C)
+  Q \left( S -   \frac{ (n-1) \widetilde{\kappa}}{ n (n+1)}\, g ,
R - \frac{\widetilde{\kappa}}{2 n (n+1)}\, g \wedge g \right) .
\label{2023.03.24.aa}\\
\rho &=&
\frac{1}{n-3}
\left( \frac{ \kappa }{n-1}  - \frac{  \widetilde{\kappa} }{n+1}
+ \varepsilon \psi \right) .
\label{2023.03.24.bb}
\end{eqnarray}
From  (\ref{2023.03.24.aa}), by an application of (\ref{abcd07zz}), we obtain
\begin{eqnarray*}
\frac{n-2}{n-3} \, C \cdot C
&=& \rho \, Q(g,R) + \frac{\rho}{2 (n-2)}\, Q(S, g \wedge g)
+  Q \left( S -   \frac{ (n-1) \widetilde{\kappa}}{ n (n+1)}\, g ,
R - \frac{\widetilde{\kappa}}{2 n (n+1)}\, g \wedge g \right) \nonumber\\
\frac{\rho}{2 (n-2)}\, Q(S,  g \wedge g)
+ \rho \,
Q\left(g,R - \frac{\widetilde{\kappa}}{2 n (n+1)}\,
g \wedge g \right) \nonumber\\
& & +  Q \left( S -   \frac{ (n-1) \widetilde{ \kappa}}{ n (n+1)}\, g ,
R - \frac{\widetilde{\kappa}}{2 n (n+1)}\, g \wedge g \right)  \nonumber\\
& = &
Q\left( S, \frac{\rho}{2 (n-2)}\, g \wedge g \right)
+  Q \left(  S -  \left(  \frac{ (n-1) \widetilde{ \kappa}}{ n (n+1)}
- \rho \right) g , R - \frac{\widetilde{\kappa}}{2 n (n+1)}\,
g \wedge g \right)
\nonumber\\
& = &
Q \left( S  -  \left(  \frac{ (n-1) \widetilde{ \kappa}}{ n (n+1)}
- \rho \right) g ,  \frac{\rho}{2 (n-2)}\, g \wedge g \right) \nonumber\\
& & +  Q \left(  S -  \left(  \frac{ (n-1) \widetilde{ \kappa}}{ n (n+1)}
- \rho \right) g , R - \frac{\widetilde{\kappa}}{2 n (n+1)}\, g \wedge g \right)
\nonumber\\
Q \left(  S -  \left(  \frac{ (n-1) \widetilde{ \kappa}}{ n (n+1)}
- \rho \right) g ,
R - \left( \frac{\widetilde{\kappa}}{ n (n+1)}
- \frac{\rho}{n-2} \right) \frac{1}{2}\, g \wedge g \right)  .
\end{eqnarray*}
Thus we see that if the tensor $H$ satisfies (\ref{2022.10.01.kk})
on ${\mathcal U}_{H} \subset M$ then
\begin{eqnarray}
\ \ \ \
C \cdot C &=&
\frac{n-3}{n-2} \,
Q \left(  S -  \left(  \frac{ (n-1) \widetilde{ \kappa}}{ n (n+1)}
- \rho \right) g ,
R - \left( \frac{\widetilde{\kappa}}{ n (n+1)}
-   \frac{\rho}{n-2} \right) \frac{1}{2}\, g \wedge g \right)
\label{2023.03.24.cc}
\end{eqnarray}
on  ${\mathcal U}_{H}$, where the function $\rho$
is defined by (\ref{2023.03.24.bb}).

In addition, we assume that (\ref{4.3.012}) holds
on  ${\mathcal U}_{H} \subset M$. Now
(\ref{2023.03.24.aa}) turns into
\begin{eqnarray}
& &
\tau \,  Q(g,C)
- Q\left( S -   \frac{ (n-1) \widetilde{\kappa}}{ n (n+1)}\, g ,
R - \frac{\widetilde{\kappa}}{2 n (n+1)}\, g \wedge g \right)  \ =\ 0 ,
\label{2023.03.24ee}\\
& &
\tau  \ =\  \rho  - \frac{n-2 }{n-3} \, L_{C} .
\label{2023.03.23.ff}
\end{eqnarray}
From the presented above calculations it follows that
(\ref{2023.03.24ee}) yields
\begin{eqnarray}
Q\left( S - \left(\frac{ (n-1) \widetilde{\kappa}}{ n (n+1)}
- \tau  \right) g, R
- \left( \frac{ \widetilde{\kappa}}{ n (n+1)}
- \frac{\tau}{n-2} \right) \frac{1}{2}\, g \wedge g \right) &=& 0 .
\label{2023.03.23.gg}
\end{eqnarray}
\begin{eqnarray}
\mathrm{rank} \left(
S - \left(\frac{ (n-1) \widetilde{\kappa}}{ n (n+1)}
- \tau  \right) g \right) &=& 1
\label{2023.03.23.ww}
\end{eqnarray}
at a point $x \in {\mathcal U}_{H}$ then in view of
Proposition 2.1
$E = 0$ at $x$, where the tensor $E$ is defined by  (\ref{2022.11.10.aaa}).
\begin{eqnarray}
\mathrm{rank} \left(
S - \left(\frac{ (n-1) \widetilde{\kappa}}{ n (n+1)}
- \tau  \right) g \right) & > & 1
\label{2023.03.23.zz}
\end{eqnarray}
at a point $x \in {\mathcal U}_{H}$ then by an application of
{\cite[Proposition 2.4] {DGHHY}}
(or, {\cite[Proposition 2.1] {DGHZ01}})
it follows that the following equation is satisfied at $x$
\begin{eqnarray}
& &
R - \left(
\frac{ \widetilde{\kappa}}{ n (n+1)}
- \frac{\tau}{n-2} \right) \frac{1}{2}\, g \wedge g\nonumber\\
&=& \phi
\left(
S - \left(\frac{ (n-1) \widetilde{\kappa}}{ n (n+1)}
- \tau  \right) g \right)
\wedge
\left(
S - \left(\frac{ (n-1) \widetilde{\kappa}}{ n (n+1)}
- \tau  \right) g \right), \ \ \ \phi \in \mathbb{R} .
\label{2023.03.23.yy}
\end{eqnarray}
This by Proposition 4.4 implies
$(n-2)\, C = \phi \, E$.
Thus we have

\begin{thm}
Let $M$ be a non-Einstein and non-conformally flat hypersurface
in $N_{s}^{n+1}(c)$, $n \geq 4$.
\newline
If (\ref{2022.10.01.kk})  is satisfied on ${\mathcal U}_{H} \subset M$ then
(\ref{2023.03.24.cc}) holds on ${\mathcal U}_{H}$, where
the function $\rho$ is defined by (\ref{2023.03.24.bb}) on this set.
\newline
If (\ref{4.3.012}) and (\ref{2022.10.01.kk}) are satisfied
on ${\mathcal U}_{H} \subset M$ then
holds on ${\mathcal U}_{H}$, where the function $\tau$
is defined by (\ref{2023.03.23.ff}) on this set.
Moreover, $\lambda \, C = E$ on  ${\mathcal U}_{H}$,
where $\lambda$ is some function on this set.
\end{thm}

\newpage

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\end{thebibliography}

\vspace{5mm}

\noindent
\footnotesize{Ryszard Deszcz\\
retired employee of the
Department of Applied Mathematics\\
Wroc\l aw University of Environmental and Life Sciences\\
Grunwaldzka 53, 50-357 Wroc\l aw, Poland}\\
{\sf E-mail<EMAIL_ADDRESS>
\vspace{1mm}

\noindent
\footnotesize{Ma\l gorzata G\l ogowska\\
Department of Applied Mathematics \\
Wroc\l aw University of Environmental and Life Sciences\\
Grunwaldzka 53, 50-357 Wroc\l aw, Poland}\\
{\sf E-mail<EMAIL_ADDRESS>
\vspace{1mm}

\noindent
\footnotesize{Marian Hotlo\'{s}\\
retired employee of the Department of Applied Mathematics\\
Wroc{\l}aw University of Science and Technology\\
Wybrze\.{z}e Wyspia\'{n}skiego 27\\
50-370 Wroc{\l}aw, Poland}\\
{\sf E-mail<EMAIL_ADDRESS>
\vspace{1mm}

\noindent
\footnotesize{Miroslava Petrovi\'{c}-Torga\v{s}ev\\
Department of Natural and Mathematical Sciences\\
State University of Novi Pazar\\
Vuka Karad\v{z}i\'{c}a bb, Novi Pazar, RS-36300, Serbia}\\
{\sf E-mail<EMAIL_ADDRESS>
\vspace{1mm}

\noindent
\footnotesize{Georges Zafindratafa\\
Laboratoire de Math\'{e}matiques pour l'Ing\'{e}nieur (LMI)\\
Universit\'{e} Polytechnique Hauts-de-France\\
59313 Va\-len\-cien\-nes cedex 9, France}\\
{\sf E-mail<EMAIL_ADDRESS>
\end{document}