Reconstructing a phylogenetic tree involves searching for the clues that tell you about the relationships among different species and then applying rigorous, and often quite complicated, analytical techniques to turn your pile of clues into your best hypothesis about the actual tree.
1. Identify and analyze characters shared by the species for which you're constructing the tree.
2. Use outgroup analysis to determine whether each character state is either ancestral or derived.
3. Group the species based on your analysis.
^ Identifying the characters: Character simply refers to an organism's specific, measurable traits. A character can be just about anything — hair, eye color, ability to digest milk, and resistance to antibiotics, for example.
^ Determining the different states the characters have: The character states could be hair present and hair absent.
^ Polarizing the characters: Polarizing characters refers to determining the direction of evolution with respect to each particular change. When you take all your clues and put them together into a tree, it helps to know which of the character states is the ancestral state (the one before the evolutionary event) and which is the derived state (the one after the evolutionary event).
^ Both the platypus and the rabbit have hair. ^ Both the platypus and the turtle lay eggs.
You know that both characters (hair and egg-laying) have two states: present and absent, as in "has hair/doesn't have hair" and "lays eggs/doesn't lay eggs."
¿jtiM^ When scientists decide to reconstruct a phylogenetic tree, they have to decide what characters, or traits, to use. These characters can be either morphological (visible physical characters) or molecular (genetic). In this section, I limit the discussion to morphological characters, such as wings, flippers, and eggs. Before scientists knew about DNA and DNA sequencing, they were limited to these types of characters as well. Today, the ever-increasing amount of information available about DNA sequences gives scientists the opportunity to use the organism's nucleotide sequence (basically, its genes; refer to Chapter 3) as a character.
Now you need to determine which character state is ancestral and which is derived. To do that, you use outgroup analysis. Collectively, the species you're trying to make a tree for — in this example, the platypus, turtle, and rabbit — are called the in group. The outgroup includes species that don't belong in the in group. In this example, they'd be things like clams, bumblebees, and mushrooms — things without backbones and four legs and stuff like that.
^ Almost all the animals in the outgroup lay eggs. From this character, you can conclude that live birth is the derived character state and that laying eggs is ancestral.
^ None of the organisms in the outgroup has hair. From this character, you can conclude that hair is a derived state.
After you've polarized the characters, you're able to conclude that the split between the platypus and the rabbit occurred after the split between the lineage that led to the turtle and the lineage that led to the ancestor of the platypus and the rabbit.
Based on analyzing the outgroups, you know that "laying eggs" is the ancestral state and "live birth" is the derived state. You also know that "no hair" is the ancestral state and "hair" is the derived state.
From this analysis, you know that in the past, farther down the tree of life, critters laid eggs but didn't have hair. Under this scenario, the change from egg-laying to live birth doesn't help you group any two species as having the most recent common ancestor. Only one of the species — the rabbit — has this derived character state. Because only one species has the derived state, you can't use it to form a group.
The situation is different for hair. Two species — the platypus and the rabbit — have the derived character state, which allows you to group these two species as having a more recent common ancestor than the common ancestor of all three species. You envision that the platypus and the bunny have a hairy ancestor, so they both have hair.
To reconstruct evolutionary history successfully, you need shared derived characters — evolutionary events that enable you to differentiate among different groups of organisms. In Figure 9-6, shown earlier in this chapter, the monophyletic-group mammals can be distinguished based on the presence of several novel traits, such as hair and the production of milk. The dolphin and the chimpanzee can be further classified as a monophyletic group based on the derived character of live birth. The platypus retains the ancestral condition of laying eggs.
When comparing character states among different species, you have to make certain that you're comparing two different states of the same character rather than two completely different characters.
Take the case of forelimbs. Humans have forelimbs; earthworms and bacteria don't. A scan through the diversity of life suggests that having forelimbs is a derived character state. Because the forelimbs in vertebrates share a common evolutionary origin, they are said to be homologous.
Homology, which is similarity as a result of common descent, helps scientists reconstruct the history of evolution and determine the best estimate of the tree of life. Although forelimbs themselves are homologous, they can evolve in several directions: into wings in birds and bats, flippers in dolphins, arms in humans, front legs in horses, and so on.
I Convergence: Different structures in two different organisms evolve to appear similar. The streamlined shape and fins of dolphins appear similar to those of sharks and fish, but closer analysis reveals that the structures are quite different — an example of convergence. Dolphins and sharks don't share fins because their common ancestor had fins; they share fins because fins evolved independently in the dolphin lineage.
i Reversal: A structure that previously evolved is subsequently lost. The moa is an example of a reversal. Moas, like most of the other vertebrates, don't fly; they're similar to chimpanzees in this regard. But the similarity is not the result of descent from a common ancestor that didn't fly (though it's true that the most recent common ancestor of the moa and chimp didn't have wings). Numerous structural characteristics of the moa skeleton place the moa firmly within the group of birds, and its ancestors had wings, but these wings were lost during the course of evolution to larger body size.
i Parallelism: Two organisms evolve to acquire the same trait, but — and this point is key — they don't share this trait because of a common ancestor. Rather, they share the trait because it evolved independently in both lineages. As the preceding section explains, the presence of fore-limbs in birds and bats is the result of homology; both species descended from an ancestor that had front limbs. The fact that the front limbs of bats and birds now function as wings is a homoplasy because wings evolved independently in the two different lineages. Most mammals don't fly; the evolution of flight in bats happened independently of the evolution of flight in birds.
The presence of homoplasy throws a wrench into the works, limiting your ability to reconstruct the tree accurately, because it's not always simple to tell when a character similarity is due to homology or homoplasy.
So how do you know that the phylogenetic trees you can create represent evolutionary relationships accurately? The answer is the way you know anything: Someone did an experiment and checked.
eSiv^*^. David Hillis and co-researchers set out to determine how well different meth-tjf ods of phylogenetic reconstruction worked at reconstructing the history of evolution. To do this, they needed to know the history of evolution, and they came up with a very clever method.
Taking advantage of the fact that viruses have extremely short generation times, they produced the tree of viral lineages in the laboratory. They started with one virus, from which they collected different mutants. Then they grew these slightly different viruses, collecting mutants from these viruses in turn. In this way they were able to generate a branching tree for which they knew the exact pattern of the branches, because they knew which viruses came out of which flasks.
The different strains of viruses didn't become different species over the course of the experiment (and as Chapter 8 explains, scientists aren't completely clear on what constitutes a species in viruses). But the descendant viruses were different enough from the parent virus to allow Hillis and his co-workers to construct an evolutionary tree.
Viruses don't have a lot of characters like eggs and hair, so the researchers determined what changes had occurred in the viral DNA. They were able to demonstrate that it's possible to measure the traits of the terminal taxa and accurately reconstruct their evolutionary history. In addition, they were able to use the character states of the terminal taxa to make good predictions about the character states of the ancestral taxa at the nodes of the tree. Because they had kept all these ancestral taxa in little vials in a freezer, they were able to show that their predictions were accurate.


text:
\section{Introduction}
A semi-Riemannian manifold $(M,g)$, $n = \dim M \geq 2$, is said to be
an {\sl Einstein manifold} \cite{Besse-1987},
or an {\sl Einstein space}, if at every point of $M$
its Ricci tensor $S$ is proportional to $g$,
i.e., $S\, =\, ( \kappa /n)\, g$ on $M$,
assumed that $\kappa$ is constant when $n = 2$.
According to {\cite[p. 432] {Besse-1987}}
this condition is called the {\sl Einstein metric condition}.
Einstein manifolds form a natural subclass
of several classes of semi-Riemannian manifolds which are determined by
curvature conditions imposed on their Ricci tensor
{\cite[Table, pp. 432-433] {Besse-1987}}.
These conditions are called
{\sl generalized Einstein metric conditions}
{\cite[Chapter XVI] {Besse-1987}}.

Let $(M,g)$, $n \geq 4$, be a semi-Riemannian manifold.
We define the following subsets of $M$:
${\mathcal U}_{R}
= \{x \in M|\, R \neq  \frac{\kappa }{2(n-1) n } g \wedge g
\ \mbox {at}\ x \}$,
${\mathcal U}_{S} = \{x \in M|\, S \neq \frac{\kappa }{n}\, g
\ \mbox {at}\ x \}$ and
${\mathcal U}_{C} = \{x \in M|\, C \neq 0\ \mbox {at}\ x \}$.
We note that  ${\mathcal U}_{S} \cup  {\mathcal U}_{C}
=  {\mathcal U}_{R}$ (see, e.g., {\cite[p. 151] {2013_DGHHY}}).
We refer to Section 2 of this paper for precise definitions
of the symbols used
(see also \cite{{2013_DGHHY}, {DGHSaw}, {R99}, {2018_DGZ}, {DHV2008}}).

The semi-Riemannian manifold $(M,g)$ is said to be
a {\sl quasi-Einstein manifold} (see, e.g., \cite{DGHS}),
or a {\sl quasi-Einstein space}, if
\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 base manifold $(\overline{M}, \overline{g})$ and
a $2$-dimensional manifold $(\widetilde{N}, \widetilde{g})$
or an $(n-1)$-dimensional Einstein manifold
$(\widetilde{N}, \widetilde{g})$, $n \geq 4$, and a warping function $F$,
is a  quasi-Einstein manifold (see, e.g., \cite{{Ch-DDGP}, {2016_DGJZ}}).
Evidently, (\ref{quasi02}) implies
{\cite[eq. (8)] {G6}}
\begin{eqnarray}
S^{2} &=& ( \kappa - (n-2) \alpha )\, S
+ \alpha ( (n-1) \alpha - \kappa  )\, g  .
\label{quasi0202}
\end{eqnarray}
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., \cite{{DGHSaw}, {2016_DGJZ}} and references therein.
Quasi-Einstein hypersurfaces
in semi-Riemannian space forms
were studied among others in
\cite{{DGHS}, {2016_DGHZhyper}, {2018_DGZ}, {R102}, {DHS02},
{DHS03}, {DHS105}, {G6}},
see also \cite{DGHSaw} and references therein.
Quasi-Einstein manifolds satisfying some pseudo\-sym\-met\-ry type
curvature conditions were investigated among others in
\cite{{Ch-DDGP}, {2013_DGHHY}, {2015_DGHZ},
{2016_DGJZ}, {DeHoJJKunSh}, {SDHJK}}.

The semi-Riemannian manifold $(M,g)$, $n \geq 3$, will be called
a {\sl{partially Einstein manifold}}, or
a {\sl{partially Einstein space}}
(cf. {\cite[Foreword] {CHEN-2017}},
{\cite[p. 20] {V2}}, \cite{LV3-Foreword}),
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 $T_{x} M$.
Evidently, (\ref{quasi0202})
is a special case of (\ref{partiallyEinstein}).
Thus every quasi-Einstein manifold is partially Einstein.
The converse statement is not true.
Contracting (\ref{partiallyEinstein}) we get
$ \mathrm{tr}_{g} (S^{2}) = \lambda \, \kappa + n\, \mu$.
This together with
(\ref{partiallyEinstein}) yields (cf. {\cite[Section 5] {2021-DGH}})
\begin{eqnarray}
S^{2} - \frac{  \mathrm{tr}_{g} (S^{2})} {n} \, g
&=&  \lambda \left( S - \frac{\kappa }{n} \, g \right) .
\label{partiallyEinstein.11}
\end{eqnarray}
In particular, a Riemannian manifold $(M,g)$, $n \geq 3$,
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.

Let $(M,g)$, $n \geq 4$, be a non-quasi-Einstein
and non-conformally flat semi-Riemannian manifold.
If the following equation is satisfied on
${\mathcal U}_{S} \cap {\mathcal U}_{C} \subset M$
\begin{eqnarray}
R &=& \frac{\phi}{2}\, S\wedge S + \mu\, g\wedge S
+ \frac{\eta}{2}\, g \wedge g ,
\label{eq:h7a}
\end{eqnarray}
where
$\phi$, $\mu $ and $\eta $ are some functions on
${\mathcal U}_{S} \cap {\mathcal U}_{C}$,
then $(M,g)$
is called a {\sl Roter type manifold}, or a {\sl Roter manifold},
or a {\sl Roter space}
{\cite[Section 15] {CHEN-2021}},
\cite{{P106}, {DGHSaw}, {DGP-TV02}, {2018_DGZ}, {DHV2008},
{2018_DH}, {G2007}} and
(\ref{eq:h7a}) a {\sl Roter equation}.
It is easy to check
that at every point of ${\mathcal U}_{S} \cap {\mathcal U}_{C}$
of a Roter space $(M,g)$ its tensor $S^{2}$
is a linear combination of the tensors $S$ and $g$,
i.e., (\ref{partiallyEinstein}) holds on
${\mathcal U}_{S} \cap {\mathcal U}_{C}$
(see, e.g., {\cite[Theorem 2.4] {2018_DGZ}},
{\cite[Theorem 2.1] {2018_DH}}).
Precisely, (\ref{partiallyEinstein.11})
with $\lambda = \phi ^{-1} ( (n-2) \mu + \phi \kappa -1)$,
is satisfied on
${\mathcal U}_{S} \cap {\mathcal U}_{C}$.
It is easy to verify that every Roter space
is a non-quasi Einstein manifold.
For further results on Roter spaces we refer to
\cite{{DecuDH}, {DecuP-TSVer}, {2021-DGH}, {2016_DGJZ},
{2011_DGP-TV}, {DK2003}, {DePlaScher}, {DeScher},
{Kow01}, {Kow2}, {SDHJK}}.

Let $(M,g)$, $n \geq 4$, be a non-partially Einstein
and non-conformally flat semi-Rieman\-nian manifold.
If its Riemann-Christoffel curvature tensor $R$
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 {\sl generalized Roter type manifold},
or a {\sl generalized Roter manifold},
or a {\sl generalized Roter type space},
or a {\sl generalized Roter space}.
An equation in which the tensor $R$ is expressed by
the mentioned above linear combination of the Kulkarni-Nomizu
products is called a {\sl{Roter type equation}}.
For instance, for $p = 2$,
we have the following Roter type equation
\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}
where
$\phi $,
$\phi _{1}$,
$\phi _{2}$,
$\mu_{1}$, $\mu$ and
$\eta$
are some functions on
${\mathcal U}_{S} \cap {\mathcal U}_{C}$.
Because $(M,g)$ is a non-partially Einstein manifold,
at least one of the functions
$\phi _{1}$, $\phi _{2}$ and $\mu_{1}$ is a non-zero function.
We refer to
\cite{{2013_DGJP-TZ}, {2016_DGJZ}, {DGP-TV02}, {DeHoJJKunSh},
{Saw-2006}, {Saw-2015}, {SDHJK}, {2016_SK}, {2019_SK}}
for results on manifolds (hypersurfaces) satisfying (\ref{B001simply}).

A semi-Riemannian manifold $(M,g)$, $n \geq 3$,
is called a {\sl $2$-quasi-Einstein manifold},
or a {\sl $2$-quasi-Einstein space}, if
\begin{eqnarray}
\mathrm{rank}(S - \alpha \, g ) &\leq & 2
\label{quasi0202weak}
\end{eqnarray}
on ${\mathcal U}_{S}$ and $\mathrm{rank} (S - \alpha \, g ) = 2$
on some open non-empty subset of ${\mathcal U}_{S}$,
where $\alpha $ is some function on ${\mathcal U}_{S}$
(see, e.g., {\cite[Section 6] {2016_DGHZhyper}},
{\cite[Section 1] {2016_DGJZ}}, {\cite[Section 2] {DGP-TV02}}).
Every non-quasi-Einstein warped product manifold
$\overline{M} \times _{F} \widetilde{N}$
with a $2$-dimensional base manifold $(\overline{M}, \overline{g})$
and a $2$-dimensional manifold $(\widetilde{N}, \widetilde{g})$
or an $(n-2)$-dimensional Einstein semi-Riemannian manifold
$(\widetilde{N}, \widetilde{g})$, when $n \geq 5$,
and a war\-ping function $F$ satisfies (\ref{quasi0202weak})
(see, e.g., {\cite[Theorem 6.1] {2016_DGJZ}}).
Thus some exact solutions of the Einstein field equations are
non-conformally flat $2$-quasi-Einstein manifolds.
For instance, the Reissner-Nordstr\"{o}m spacetime, as well as
the Reissner-Nordstr\"{o}m-de Sitter type spacetimes
are such manifolds (see, e.g.,
\cite{Kow2}).
It seems that the Reissner-Nordstr\"{o}m spacetime
is the "oldest" example of
a non-conformally flat
$2$-quasi-Einstein warped pro\-duct manifold
{\cite[Section 1] {2016_DGJZ}}.
We also mention
that some $2$-quasi-umbilical hypersurfaces
in a semi-Riemannian space forms are $2$-quasi-Einstein manifolds
(see, e.g., \cite{DGP-TV02}).

Let $N^{n+1}_{s}(c)$ be a semi-Riemannian space forms
with signature $(s,n+1-s)$, $n \geq 3$, where
$c = \widetilde{\kappa}/(n(n+1))$ and $\widetilde{\kappa}$
is its scalar curvature.
Let $(M,g)$ be a connected hypersurface isometrically immersed
in $N_{s}^{n+1}(c)$,
where the metric tensor $g$ of $M$ is induced
by the metric tensor of the ambient space.
The Gauss equation of $M$ in $N_{s}^{n+1}(c)$ reads
(see, e.g., \cite{{2016_DGHZhyper}, {DGP-TV02},
{DGPSS}, {2018_DGZ}, {Saw-2015}})
\begin{eqnarray}
\ \
R_{hijk}
&=& \varepsilon  \, ( H_{hk}H_{ij} - H_{hj}H_{ik}) + c\, G_{hijk},\ \
\varepsilon \ =\ \pm 1 ,
\label{realC5}
\end{eqnarray}
where
$g_{hk}$, $R_{hijk}$, $G_{hijk} = g_{hk}g_{ij} - g_{hj}g_{ik}$ and $H_{hk}$
are the local components of the metric tensor $g$,
the curvature tensor $R$, the tensor $G = (1/2)\, g \wedge g$
and the second fundamental tensor $H$ of $M$, respectively.

Let $M$ be a hypersurface in $N_{s}^{n+1}(c)$, $n \geq 3$.
We denote by ${\mathcal U}_{H}$ the set of all points of $M$
at which the $(0,2)$-tensor $H^{2}$ is not expressed
by a linear combination of the second fundamental tensor
$H$ and the metric tensor $g$ of $M$.
It is known that
${\mathcal U}_{H} \subset {\mathcal{U}}_{S} \cap {\mathcal{U}}_{C}
\subset M$, provided that $n \geq 4$,
see, e.g., {\cite[Section 1] {2018_DGZ}}.

In Section 3 we present results on
non-Einstein and non-conformally flat hypersurfaces in
$N_{s}^{n+1}(c)$, $n \geq 4$,
satisfying at every point
$x \in {\mathcal{U}}_{S} \setminus {\mathcal U}_{H}$
\begin{eqnarray}
H^{2} &=& \alpha \, H + \beta \, g,\ \ \
\alpha , \beta \in \mathbb{R} .
\label{2021.10.25.a2}
\end{eqnarray}
The main result of that section (see Theorem 3.3) states that if
a hypersurface  $M$ in a Riemannian space forms $N^{n+1}(c)$,
$n \geq 4$, has at every point exactly two distinct
principal curvatures, $\lambda_{1}$ with multiplicity $p$ and
$\lambda_{2}$ with multiplicity $n-p$, $2 \leq p \leq n-2$,
such that
\begin{eqnarray}
(p-1) \lambda_{1} + (n-p-1) \lambda_{2} \neq 0
\label{2022.09.23.cc}
\end{eqnarray}
then ${\mathcal{U}}_{S}  \cap {\mathcal{U}}_{C} = M$
and an equation of the form (\ref{eq:h7a}) holds on $M$,
i.e., $M$ is a Roter space. In particular,
$M$ is a non-quasi-Einstein partially Einstein space.
In Example 3.4 we present a family of
non-Einstein and non-conformally flat hypersurfaces
having at every  point exactly two distinct
principal curvatures, $\lambda_{1}$ with multiplicity $p$ and
$\lambda_{2}$ with multiplicity $n-p$, $2 \leq p \leq n-2$,
and satisfying (\ref{2022.09.23.cc}).

In Sections 4, 5 and 6 we investigate
hypersurfaces $M$ in $N_{s}^{n+1}(c)$, $n \geq 4$,
with the tensor $H$ satisfying on ${\mathcal U}_{H} \subset M$
\begin{eqnarray}
H^{3} &=& \phi \, H^{2} + \psi \, H + \rho \, g ,
\label{real702}
\end{eqnarray}
where $\phi$, $\psi$ and $\rho$ are some functions on ${\mathcal U}_{H}$.
Curvature properties of pseudosymmetry type of such hypersurfaces
were investigated among others in
\cite{{51Y}, {DeGl}, {DG90}, {DGHS},
{R99},  {DGP-TV02}, {DGPSS},
{2018_DGZ}, {R102}, {DHS01}, {DHS105}, {P107}, {DeVerYap},
{G6}, {Saw-2004}, {Saw-2005}, {Saw-2006}, {Saw-2007}, {Saw-2015}}.

In Proposition 4.1 we state
that on the set ${\mathcal U}_{H}$
of every hypersurface $M$ in $N_{s}^{n+1}(c)$, $n \geq 4$,
satisfying (\ref{real702})
a family of curvature conditions is satisfied. Among other things
\begin{eqnarray}
\tau ^{2}\, R &=& \frac{\varepsilon}{2}\,
( S^{2} - \rho _{1}\, S - \rho _{0}\, g)
\wedge
( S^{2} - \rho _{1}\, S - \rho _{0}\, g)
+ \tau ^{2} c\, G ,
\label{2022.09.19.aa}
\end{eqnarray}
holds on ${\mathcal U}_{H}$,
where $\tau$, $\rho _{0}$ and $\rho _{1}$
are functions defined on ${\mathcal U}_{H}$ by
(\ref{real705}), (\ref{real707rho0}) and (\ref{real707rho1}),
respectively. Evidently, if $\tau$ is non-zero at every point
of ${\mathcal U}_{H}$ then from (\ref{2022.09.19.aa})
we obtain a Roter type equation of the form (\ref{B001simply}).

Let $M$ be a hypersurface in a Riemannian space forms
$N^{n+1}(c)$, $n \geq 3$,
having at every point $x \in {\mathcal U}_{H} \subset M$
three distinct principal curvatures
$\lambda _{0}$, $\lambda _{1}$ and $\lambda _{2}$,
with multiplicities $n_{0}$, $n_{1}$ and $n_{2}$, respectively.
Moreover, let the eigenvalues
$\kappa _{0}$, $\kappa _{1}$  and $\kappa _{2}$ of the Ricci operator
$\mathcal{S}$ of $M$
satisfy (\ref{2021-10-18-a}).
Then the function $\tau$, defined by (\ref{real705}),
is expressed on ${\mathcal U}_{H}$ by (see Theorem 5.2)
\begin{eqnarray}
\tau
&=&
\frac{(\kappa _{0} - \kappa _{1}) ( \kappa _{0} - \kappa _{2})
( \kappa _{1} - \kappa _{2})}{(\lambda _{0} - \lambda _{1})
( \lambda _{0} - \lambda _{2}) ( \lambda _{1} - \lambda _{2})} .
\label{2021-11-15-a}
\end{eqnarray}

If $M$ is a partially Einstein hypersurface in $N_{s}^{n+1}(c)$,
$n \geq 4$, satisfying (\ref{real702})
on ${\mathcal U}_{H} \subset M$
then
$S^{2} = \rho _{1}\, S + \rho _{0}\, g$ and $\tau = 0$
hold on ${\mathcal U}_{H}$ (Theorem 5.1).

An example of a partially Einstein hypersurface $M$ in a Riemannian space
forms $N^{n+1}(c)$, $n \geq 3$, having at every point exactly three
distinct principal curvatures is presented in Example 5.4
and Example 5.5 (ii).
In view of Theorem 5.1, $\tau = 0$ on those hypersurfaces.

Examples of hypersurfaces $M$ in an Euclidean space
$\mathbb{E}^{n+1}(c)$, $n \geq 4$, or, $n \geq 5$,
having at every point exactly three
distinct principal curvatures with non-zero
function $\tau$ are presented in Examples 6.2 (ii)
and 6.3 (ii). Thus
a Roter type equation of the form (\ref{B001simply})
is satisfied on those hypersurfaces.

\section{Preliminaries.}
Throughout this paper, all manifolds are assumed
to be connected paracompact
mani\-folds of class $C^{\infty }$. Let $(M,g)$, $n \geq 3$,
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
\begin{eqnarray*}
(X \wedge _{A} Y)Z
&=&
A(Y,Z)X - A(X,Z)Y, \\
{\mathcal R}(X,Y)Z
&=&
\nabla _X \nabla _Y Z - \nabla _Y \nabla _X Z - \nabla _{[X,Y]}Z ,
\end{eqnarray*}
respectively, where
$X, Y, Z \in \Xi (M)$
and
$A$ 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*}
respectively.
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} \left(X \wedge _{g} {\mathcal S}Y
+ {\mathcal S}X \wedge _{g} Y
- \frac{\kappa}{n-1}X \wedge _{g} Y \right) 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
\begin{eqnarray*}
G(X_1,X_2,X_3,X_4) &=& g((X_1 \wedge _{g} X_2)X_3,X_4) ,\\
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 $X_1,X_2,\ldots \in \Xi (M)$.

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 {\sl{generalized curvature tensor}}  if the
following two conditions are fulfilled:
\begin{eqnarray*}
& &
B(X_1,X_2,X_3,X_4)\ =\ B(X_3,X_4,X_1,X_2),\\
& &
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{DS5}).
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
${\mathcal{B}} = {\mathcal{R}}$ or ${\mathcal{B}} = {\mathcal{C}}$,
$T=R$ or $T=C$ or $T=S$, $A=g$ or $A=S$ in the above formulas,
we get the tensors $R\cdot R$, $R\cdot C$, $C\cdot R$, $R\cdot S$,
$Q(g,R)$, $Q(S,R)$, $Q(g,C)$ and $Q(g,S)$.

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 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, if $A = S$ and
${\mathcal{A}} = {\mathcal S}$,
we obtain the tensors $S^{p}$,
$p = 2, 3, \ldots $, assuming that $S^{1} = S$.

For a symmetric $(0,2)$-tensor $E$ and a $(0,k)$-tensor $T$, $k \geq 2$, we
define their {\sl{Kulkarni-Nomizu tensor}} $E \wedge T$ by
(see, e.g., {\cite[Section 2] {2013_DGHHY}})
\begin{eqnarray*}
& &(E \wedge T )(X_{1}, X_{2}, X_{3}, X_{4}; Y_{3}, \ldots , Y_{k})\\
&=&
E(X_{1},X_{4}) T(X_{2},X_{3}, Y_{3}, \ldots , Y_{k})
+ E(X_{2},X_{3}) T(X_{1},X_{4}, Y_{3}, \ldots , Y_{k} )\\
& &
- E(X_{1},X_{3}) T(X_{2},X_{4}, Y_{3}, \ldots , Y_{k})
- E(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
$E \wedge F$, where $E$ and $F$ are symmetric $(0,2)$-tensors.
We have
\begin{eqnarray}
G &=& \frac{1}{2}\, g \wedge g ,
\label{2020.12.08.a}\\
C &=& R - \frac{1}{n-2}\, g \wedge S + \frac{\kappa }{(n-2) (n-1)} \, G ,
\label{Weyl}\\
Q(E,F) &=& - Q(F,E)
\label{2022.09.16.a}
\end{eqnarray}
and (see, e.g., {\cite[Lemma 2.2(i)] {2013_DGHHY}})
\begin{eqnarray}
&(a)&
\ \
Q(E, E \wedge F) \ =\ - \frac{1}{2}\, Q(F, E \wedge E ) ,\nonumber\\
&(b)&
\ \
E \wedge Q(E,F) \ =\ - \frac{1}{2}\, Q(F, E \wedge E ).
\label{DS7}
\end{eqnarray}
We also have the following identity
(see, e.g.,
{\cite[Lemma 2.1(i)] {Ch-DDGP}} and references therein)
\begin{eqnarray}
E_{1} \wedge Q(E_{2},F) + E_{2} \wedge Q(E_{1},F)
+ Q(F, E_{1} \wedge E_{2})
&=& 0 ,
\label{2021.11.30.a4}
\end{eqnarray}
where $E_{1}$, $E_{2}$ and $F$
are symmetric $(0,2)$-tensors.
We note that
(\ref{2022.09.16.a}) and (\ref{2021.11.30.a4}) yield
\begin{eqnarray*}
Q(F, E_{1} \wedge E_{2} )
+
Q(E_{1}, E_{2} \wedge F )
+
Q(E_{2}, F \wedge E_{1} )
&=& 0 .
\end{eqnarray*}

Let
$T$ be a $(0,k)$-tensor, $k = 2, 3, \ldots $.
The tensor $Q(A,T)$ is called the {\sl{Tachibana tensor}} of $A$ and $T$,
or the Tachibana tensor for short (see, e.g., \cite{DGPSS}).
By an application of (\ref{DS7})(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}

On every semi-Riemannian manifold
$(M,g)$, $n \geq 4$,
the following identity is satisfied
{\cite[eq. (1.13), Theorem 3.4(i)] {2016_DGJZ}}
\begin{eqnarray}
C \cdot R + R \cdot C &=& C \cdot C  + R \cdot R
- \frac{1}{(n-2)^{2}} \,
Q\left( g, g \wedge \left( S^{2}
- \frac{\kappa }{n-1}\, S \right) \right) .
\label{2021.11.30.a1}
\end{eqnarray}
From (\ref{2021.11.30.a1}), by a suitable contraction, we get
(cf.  {\cite[Lemma 2.3] {DGP-TV02}}, {\cite[ p. 217] {DHS01}})
\begin{eqnarray}
C \cdot S
&=& R \cdot S - \frac{1}{n-2}
Q\left( g, S^{2} - \frac{\kappa }{n-1} S \right) .
\label{uuWeyl}
\end{eqnarray}
In particular, if $(M,g)$, $n \geq 4$,
is a conformally flat manifold then
by making use of
(\ref{2020.12.08.a}) and (\ref{DS7})(a)
identities
(\ref{2021.11.30.a1})
and
(\ref{uuWeyl}) reduce to
(cf. {\cite[Section 10.1] {DHV2008}})
\begin{eqnarray*}
R \cdot R &=&
\frac{1}{(n-2)^{2}} \,
Q\left( g, g \wedge \left( S^{2} - \frac{\kappa }{n-1}\, S \right) \right)
\nonumber\\
&=&
- \frac{1}{(n-2)^{2}} \,
Q \left(
 S^{2} - \frac{\kappa }{n-1}\, S , G \right) ,\\
R \cdot S &=&
\frac{1}{n-2} \,
Q\left( g,  S^{2} - \frac{\kappa }{n-1}\, S \right) ,
\end{eqnarray*}
respectively.
We mention that the following condition
(see, e.g., {\cite[eq. (14)] {DGP-TV02}})
\begin{eqnarray}
R \cdot R &=& Q(S,R) - (n-2)c\, Q(g,C)
\label{2021.10.25.a1}
\end{eqnarray}
is satisfied on every hypersurface $M$
in $N_{s}^{n+1}(c)$, $n \geq 4$.
Thus
on every hypersurface $M$
in $N_{s}^{n+1}(c)$, $n \geq 4$,
by making use of (\ref{2021.10.25.a1}), the
identity (\ref{2021.11.30.a1})
turns into (cf. {\cite[Theorem 3.4(ii)] {2016_DGJZ}})
\begin{eqnarray}
C \cdot R + R \cdot C
&=&
C \cdot C + Q(S,R)
- (n-2)c\, Q(g,C)\nonumber\\
& &
- \frac{1}{(n-2)^{2}} \,
Q\left( g, g \wedge
\left( S^{2} - \frac{\kappa }{n-1}\, S \right) \right) .
\label{2021.11.30.a2}
\end{eqnarray}

Let  $(M,g)$, $n \geq 4$, be a Roter space.
We can check that at every point of
${\mathcal U}_{S} \cap {\mathcal U}_{C} \subset M$
the tensors $R \cdot R$ and $Q(g,R)$ are lineary dependent
(see, e.g., {\cite[Theorem 6.7] {DGHSaw}},
{\cite[Theorem 3.2] {2016_DGJZ}}, {\cite[Theorem 2.1] {2018_DH}}).
Semi-Riemannian manifolds with that property are called pseudosymmetric
manifolds. More precisely,
a semi-Riemannian manifold $(M,g)$, $n \geq 3$, is said to be
{\sl pseudosymmetric}
if the tensors $R \cdot R$ and $Q(g,R)$
are linearly dependent at every point of $M$
(see, e.g.,
{\cite[Chapter 8.5.3] {M7}},
{\cite[Section 15] {CHEN-2021}},
{\cite[Chapter 12.4] {M6}},
\cite{{2013_DGHHY}, {DGHSaw}, {DHV2008}, {DVV1991}, {HaVerSigma},
{SDHJK}, {LV1}, {V2}, {LV2}, {LV3-Foreword}, {LV-2020}}
and references therein).
This is equivalent to
\begin{eqnarray}
R \cdot R &=& L_{R}\, Q(g,R)
\label{pseudo}
\end{eqnarray}
on ${\mathcal{U}}_{R}$,
where $L_{R}$ is some function on ${\mathcal U}_{R}$.
We note that (\ref{pseudo}) implies
\begin{eqnarray*}
(a)\ \ R \cdot S \ =\ L_{R}\, Q(g,S)\ \ &\mbox{and}& \ \
(b)\ \ R \cdot C \ =\  L_{R}\, Q(g,C) .
\end{eqnarray*}

We recall that if a semi-Riemannian manifold $(M,g)$, $n \geq 3$,
is {\sl{locally symmetric}} then $\nabla R = 0$ on $M$. This
implies the integrability condition
${\mathcal{R}}(X,Y ) \cdot R = 0$. In short,
\begin{eqnarray}
R \cdot R &=& 0 .
\label{semisymmetry}
\end{eqnarray}
Semi-Riemannian manifolds satisfying (\ref{semisymmetry})
are called {\sl semisymmetric} (see, e.g.,
{\cite[Chapter 8.5.3] {M7}} and
{\cite[Chapter 1.6] {M6}}).
Thus every semisymmetric manifold is pseudosymmetric.
The converse statement is not true (see, e.g., \cite{{DVV1991}}).
For examples of semisymmetric Roter spaces we refer to
{\cite[Example 5.4] {2015_DGHZ}},
{\cite[Example 5.2] {DH-2003}},  {\cite[Section 3] {G5}}
and {\cite[Section 4] {Kow01}}.

A semi-Riemannian manifold $(M,g)$, $\dim M = n \geq 3$,
is called {\sl Ricci-pseudosymmetric}
if the tensors $R \cdot S$ and $Q(g,S)$ are linearly dependent
at every point of $M$
(see, e.g., {\cite[Chapter 8.5.3] {M7}},
{\cite[Section 15.1] {CHEN-2021}}, \cite{DGHSaw}).
This is equivalent on ${\mathcal{U}}_{S}$ to
\begin{eqnarray*}
R \cdot S &=& L_{S}\, Q(g,S) ,
\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 $(\overline{M}, \overline{g})$ manifold 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., {\cite[Section 1] {Ch-DDGP}}
and {\cite[Example 4.1] {2016_DGJZ}}).

It seems that ({\ref{pseudo}) is the most important curvature condition
between conditions called pseudosymmetry type conditions.
We refer to
{\cite[Section 4] {2013_DGHHY}},
{\cite[Sections 3 and 4] {DGHSaw}},
{\cite[Section 1] {2016_DGJZ}},
{\cite[Section 2] {2018_DH}
and
{\cite[Section 1] {2016_DP-TVZ}
for a survey of results on semi-Riemannian manifolds
satisfying such conditions.
We mention that Roter spaces satisfy  ({\ref{pseudo})
and some other pseudosymmetry type conditions
(see, e.g., {\cite[Theorem 6.7] {DGHSaw}},
{\cite[Theorem 3.2] {2016_DGJZ}}, {\cite[Theorem 2.1] {2018_DH}}).

\begin{lemma}
If $(M,g)$, $n \geq 4$, is a quasi-Einstein space satisfying
(\ref{quasi02})
on ${\mathcal U}_{S} \cap  {\mathcal U}_{C} \subset M$
then the following equation is satisfied on this set
\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}
\end{lemma}
{\bf{Proof.}} Equation (\ref{quasi02}),
in view of {\cite[Proposition 2.2] {G6}},
yields
\begin{eqnarray}
\frac{1}{2}\, S \wedge S &=& \alpha \, g \wedge S
- \frac{\alpha ^{2}}{2}\, g \wedge g .
\label{2022.11.20.bb}
\end{eqnarray}
From (\ref{2022.11.20.bb}), by suitable contractions, we get
\begin{eqnarray}
S^{2} - \kappa \, S &=& - (n-2) \alpha \, S - \alpha \kappa \, g
+ (n-1) \alpha ^{2}\, g ,
\label{2022.11.20.cc}\\
\mathrm{tr}_{g} (S^{2}) - \kappa ^{2} &=&
- 2 (n-1)\alpha \kappa + n (n-1) \alpha ^{2} .
\label{2022.11.20.dd}
\end{eqnarray}
Now using (\ref{2022.11.20.bb}), (\ref{2022.11.20.cc}) and
(\ref{2022.11.20.dd}) we can easily check that
(\ref{2022.11.20.aa})
holds on ${\mathcal U}_{S} \cap {\mathcal U}_{C}$.
Our lemma is thus proved.

\begin{lemma}
If $(M,g)$, $n \geq 4$, is a Roter space satisfying
(\ref{eq:h7a})
on ${\mathcal U}_{S} \cap  {\mathcal U}_{C} \subset M$
then the following equation is satisfied 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}
\end{lemma}
{\bf{Proof.}} Equation (\ref{eq:h7a}),
by making use of (\ref{2020.12.08.a}) and (\ref{Weyl}),
turns into
\begin{eqnarray}
C &=& \frac{\phi}{2}\, S\wedge S
+ \left(\mu - \frac{1}{n-2} \right) g\wedge S + \frac{1}{2}
\left( \eta + \frac{\kappa }{(n-2) (n-1)} \right) g \wedge g .
\label{2022.11.22.bb}
\end{eqnarray}
From (\ref{2022.11.22.bb}), by suitable contractions, we get
\begin{eqnarray}
& &
\frac{\phi}{n-2}\, ( S^{2} - \kappa \, S)
- \left(\mu - \frac{1}{n-2} \right) S\nonumber\\
& &
- \frac{\kappa }{n-2} \left(\mu - \frac{1}{n-2} \right) g
- \frac{n-1}{n-2} \left( \eta + \frac{\kappa }{(n-2) (n-1)} \right) g
\ =\ 0 ,
\label{2022.11.22.cc}\\
& &
\frac{\phi\, ( \kappa ^{2} - \mathrm{tr}_{g}(S^{2}))}{(n-2)(n-1)}
\ =\
- \frac{2 \kappa }{n-2} \left(\mu - \frac{1}{n-2} \right)\nonumber\\
& &
- \frac{n}{n-2} \left( \eta + \frac{\kappa }{(n-2) (n-1)} \right) .
\label{2022.11.22.dd}
\end{eqnarray}
Now (\ref{2022.11.22.bb}), (\ref{2022.11.22.cc})
and (\ref{2022.11.22.dd}) yield
\begin{eqnarray*}
C &=& \frac{\phi}{2}\, S\wedge S
+ \left(\mu - \frac{1}{n-2} \right) g\wedge S + \frac{1}{2}
\left( \eta + \frac{\kappa }{(n-2) (n-1)} \right) g \wedge g\nonumber\\
& &
+ \frac{\phi}{n-2}\, g \wedge ( S^{2} - \kappa \, S)
- \left(\mu - \frac{1}{n-2} \right) g \wedge S \nonumber\\
& &
- \frac{\kappa}{n-2} \left(\mu - \frac{1}{n-2} \right) g \wedge g
- \frac{n-1}{n-2} \left( \eta + \frac{\kappa }{(n-2) (n-1)} \right)
g \wedge g \nonumber\\
&=& \frac{\phi}{2}\, S\wedge S
+ \frac{\phi}{n-2}\, g \wedge ( S^{2} - \kappa \, S)
- \frac{2 \kappa}{2 (n-2)} \left(\mu - \frac{1}{n-2} \right) g \wedge g
\nonumber\\
& &
- \frac{n}{2 (n-2)} \left( \eta + \frac{\kappa }{(n-2) (n-1)} \right)
g \wedge g \nonumber\\
&=& \phi \left( \frac{1}{2}\, S\wedge S
+ \frac{1}{n-2}\, g \wedge S^{2} - \frac{\kappa}{n-2}\, g \wedge S
+ \frac{  \kappa ^{2} - \mathrm{tr}_{g} (S^{2})}{2 (n-2)(n-1)}\,
g \wedge g
\right)
\end{eqnarray*}
and we get (\ref{2022.11.22.aa}), completing the proof.

Lemma 2.1 and Lemma 2.2 imply
\begin{proposition}
If $(M,g)$, $n \geq 4$, 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{proposition}

Proposition 2.3, {\cite[Theorem 7.1 (ii)] {2016_DGJZ}} and
{\cite[Theorem 4.1] {DK2003}} imply
\begin{theorem}
Let $\overline{M} \times _{F} \widetilde{N}$
be the warped product manifold
with a $2$-dimensional semi-Riemannian manifold
$(\overline{M},\overline{g})$,
an $(n-2)$-dimensional semi-Riemannian manifold
$(\widetilde{N},\widetilde{g})$,
$n \geq 4$, 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{theorem}

\begin{example}
(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
(ii)
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
(iii)
{\cite[Example 4.1] {DK2003}}
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}}.
Evidently,
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 2.4,
we can state that (\ref{2022.11.22.jj}) holds on
${\mathcal U}_{S} \cap  {\mathcal U}_{C}$.
\end{example}

\section{Hypersufaces in space forms satisfying $H^{2} = \alpha H + \beta g$.}

Let $N^{n+1}_{s}(c)$ be a semi-Riemannian space forms
with signature $(s,n+1-s)$, $n \geq 3$, where
$c = \widetilde{\kappa}/(n(n+1))$ and $\widetilde{\kappa}$
is its scalar curvature.
Let $M$ be a connected hypersurface isometrically immersed
in $N_{s}^{n+1}(c)$.
From (\ref{realC5}), by contraction with $g^{ij}$, we obtain
\begin{eqnarray}
S_{hk} - (n-1) c\, g_{hk}
&=&  \varepsilon \, ( \mathrm{tr} (H)\, H_{hk} - H^{2}_{hk}) ,
\label{realC6}\\
\kappa - n (n-1) c
&=&  \varepsilon \, ( ( \mathrm{tr} (H))^{2} - \mathrm{tr} (H^{2}) ) ,
\label{realC6bb}
\end{eqnarray}
where
$S_{hk}$ and $g^{hk}$ are the local components of the Ricci tensor $S$
and the tensor $g^{-1}$ of $M$, res\-pectively,
and
$\mathrm{tr} (H) = g^{ij}H_{ij}$,
$H^{2}_{hk} = g^{ij} H_{hi}H_{jk}$,
$\mathrm{tr} (H^{2}) = g^{ij}H^{2}_{ij}$.
Further, from (\ref{realC6}) we get immediately
\begin{eqnarray}
\ \ \ \
S^{2}_{hk} - 2 (n-1) c\, S_{hk} + (n-1)^{2} c^{2}\, g_{hk}
\ =\
H^{4}_{hk} - 2 \mathrm{tr} (H)\, H^{3}_{hk}
+
(\mathrm{tr} (H))^{2} \, H^{2}_{hk} ,
\label{realC7}
\end{eqnarray}
where $S^{2}_{hk} = g^{ij} S_{hi}S_{jk}$,
$H^{3}_{hk} = g^{ij} H^{2}_{hi}H_{jk}$ and
$H^{4}_{hk} = g^{ij} H^{3}_{hi}H_{jk}$,
are the local components of the $(0,2)$-tensors
$S^{2}$, $H^{3}$ and $H^{4}$, respectively.

Let $M$ be a non-Einstein hypersurface in
a semi-Riemannian space forms $N_{s}^{n+1}(c)$, $n \geq 3$.
We assume that at a point $x \in {\mathcal{U}}_{S} \subset M$
the tensor $H^{2}$ is a linear combination of the tensors
$g$ and $H$, i.e.,
$x \in {\mathcal{U}}_{S} \setminus {\mathcal U}_{H}$.
Evidently, the tensor $S - (\kappa /n)\, g$ is a non-zero tensor at $x$
and (\ref{2021.10.25.a2}) holds at $x$.

Now (\ref{realC6}), by (\ref{2021.10.25.a2}), turns into
\begin{eqnarray}
S &=& \varepsilon ( \mathrm{tr} (H) - \alpha )\, H
+ ( (n-1)c - \varepsilon \beta )\, g .
\label{2021.10.25.a3}
\end{eqnarray}
This together with (\ref{2021.10.25.a2}) yields
\begin{eqnarray}
S^{2} &=& ( \mathrm{tr} (H) - \alpha )^{2} \, H^{2}
+ 2 \varepsilon ( \mathrm{tr} (H) - \alpha )
( (n-1)c - \varepsilon \beta )\, H \nonumber\\
& &
+ ( (n-1)c - \varepsilon \beta )^{2}\, g \nonumber\\
&=&
( \mathrm{tr}\, (H) - \alpha )^{2} \,
(  \alpha \, H + \beta \, g)\nonumber\\
& &
+ 2 \varepsilon ( \mathrm{tr} (H) - \alpha )
( (n-1)c - \varepsilon \beta ) \, H
+
( (n-1)c - \varepsilon \beta )^{2} \, g
\nonumber\\
&=& ( \mathrm{tr} (H) - \alpha )
( \alpha \, ( \mathrm{tr}\, (H) - \alpha )
+ 2 \varepsilon
( (n-1)c - \varepsilon \beta ) ) \, H \nonumber\\
& &
+
( \beta \, ( \mathrm{tr}\, (H) - \alpha )^{2}
+
( (n-1)c - \varepsilon \beta )^{2} )\, g .
\label{2021.10.25.a4}
\end{eqnarray}
If we would have
$\mathrm{tr} (H) - \alpha = 0$ at $x$
then
(\ref{2021.10.25.a3}) leads to
$S = ( (n-1)c - \varepsilon \beta )\, g$,
and in a consequence we get
$S = ( \kappa / n)\, g$, a contradiction.
Therefore
\begin{eqnarray}
\mathrm{tr} (H) - \alpha \neq 0
\label{2021.10.25.a55}
\end{eqnarray}
at this point.
Now from
(\ref{2021.10.25.a3}), by (\ref{2021.10.25.a55}), we obtain
\begin{eqnarray}
H &=&  \varepsilon ( \mathrm{tr} (H) - \alpha )^{-1}
( S - ( (n-1)c - \varepsilon \beta )\, g ).
\label{2021.10.25.a5}
\end{eqnarray}
Next, using
(\ref{2021.10.25.a4}), (\ref{2021.10.25.a55})
and
(\ref{2021.10.25.a5})
we can check that $M$ is partially Einstein at $x$.
We also note that (\ref{realC5}), by making use of
(\ref{2021.10.25.a5}), turns into
\begin{eqnarray}
\ \ \ \ \ \
R &=& \frac{ \varepsilon }{2 ( \mathrm{tr} (H) - \alpha )^{2} }\,
( S - ( (n-1)c - \varepsilon \beta )\, g ) \wedge
( S - ( (n-1)c - \varepsilon \beta )\, g ) + c\, G.
\label{2021.10.25.a6}
\end{eqnarray}
Thus we have
\begin{proposition}
Let $M$ be a hypersurface in a semi-Riemannian space forms
$N_{s}^{n+1}(c)$, $n \geq 3$. Then
(\ref{2021.10.25.a4}), (\ref{2021.10.25.a55}) and (\ref{2021.10.25.a5})
are satisfied
at every point
of ${\mathcal{U}}_{S} \setminus {\mathcal U}_{H} \subset M$.
\end{proposition}
\begin{proposition}
$\mathrm{ ( }$cf. {\cite[Theorem 3.3] {2018_DGZ}},
{\cite[Proposition 3.3] {G5}}$\mathrm{ ) }$
Let $M$ be a hypersurface in a semi-Riemannian space forms
$N_{s}^{n+1}(c)$, $n \geq 4$.
Then (\ref{2021.10.25.a6}) is satisfied
at every point of
$({\mathcal{U}}_{S} \cap {\mathcal{U}}_{C})
\setminus {\mathcal U}_{H}   \subset M$, i.e.,
an equation of the form (\ref{eq:h7a}).
\end{proposition}

Let $M$ be a hypersurface in a Riemannian space forms $N^{n+1}(c)$,
$n \geq 3$. We assume that $M$ has at every point exactly two distinct
principal curvatures $\lambda_{1}$ and $\lambda_{2}$, with multiplicities
$p$ and $n-p$, respectively, where $1 \leq p \leq n-1$.
Thus (\ref{2021.10.25.a2}) and
\begin{eqnarray}
\ \ \
\alpha \ =\ \lambda_{1} + \lambda_{2} ,\ \ \
\beta \ =\ - \lambda_{1} \lambda_{2},\ \ \
\mathrm{tr} (H) - \alpha \ =\ (p-1) \lambda_{1} + (n-p-1) \lambda_{2} ,
\label{2022.09.16.bb}
\end{eqnarray}
hold on $M$.
Further, from (\ref{realC6}) it follows that the eigenvalues
$\kappa_{1}$ and $\kappa_{2}$ of the Ricci opertator ${\mathcal{S}}$
of $M$ are expressed by
\begin{eqnarray}
\kappa_{1} &=& - \lambda_{1}^{2}
+ \mathrm{tr}(H) \lambda_{1} + (n-1) c ,\nonumber\\
\kappa_{2} &=& - \lambda_{2}^{2}
+ \mathrm{tr}(H) \lambda_{2} + (n-1) c .
\label{2022.01.12.a2}
\end{eqnarray}
Now from
(\ref{2022.09.16.bb}) and (\ref{2022.01.12.a2})
we get immediately
\begin{eqnarray}
\kappa_{1} - \kappa_{2} &=&
- \lambda_{1}^{2} + \lambda_{2}^{2}
+ \mathrm{tr}(H) (\lambda_{1} - \lambda_{2})\nonumber\\
&=&
(\lambda_{1} - \lambda_{2})
(\mathrm{tr}(H) - \alpha )\nonumber\\
&=&
(\lambda_{1} - \lambda_{2})
( (p-1) \lambda_{1} + (n-p-1) \lambda_{2} ) .
\label{2022.01.12.a3}
\end{eqnarray}
Evidently, if at every point of $M$ we have
$\mathrm{tr}(H) - \alpha \neq 0$,
or, equivalently,
(\ref{2022.09.23.cc}) is satisfied,
then
${\mathcal{U}}_{S} = M$
and
\begin{eqnarray*}
S^{2} &=& ( \kappa_{1} + \kappa_{2})\, S - \kappa_{1} \kappa_{2} \, g
\end{eqnarray*}
on $M$,
i.e., such hypersurface is partially Einstein.
In addition, we assume that
$n \geq 4$ and $2 \leq p \leq n-2$.
Thus $M$ is a non-quasi-umbilical at every point,
and in a consequence,
$M$ is a non-conformally flat hypersurface. Thus we have
${\mathcal{U}}_{S} \cap {\mathcal{U}}_{C} = M$.
Now from Proposition 3.2 it follows that
$M$ is a Roter hypersurface. Precisely, we have
\begin{theorem}
Let $M$ be a hypersurface in a Riemannian space forms $N^{n+1}(c)$,
$n \geq 4$, such that at every point of $M$ has exactly two distinct
principal curvatures, $\lambda_{1}$ with multiplicity $p$ and
$\lambda_{2}$ with multiplicity $n-p$, $2 \leq p \leq n-2$.
If (\ref{2022.09.23.cc}) is satisfied
at every point of $M$
then $M = {\mathcal{U}}_{S}  \cap {\mathcal{U}}_{C}$
and an equation of the form (\ref{eq:h7a}) holds on $M$,
i.e., $M$ is a Roter space. In particular,
$M$ is a non-quasi-Einstein partially Einstein space.
\end{theorem}
\begin{example}
We present now a family
of non-Einstein and non-conformally flat hypersurfaces $M$ in
$N^{n+1}(c)$, $n \geq 4$,
having at every point exactly two distinct principal curvatures.
We consider some Clifford hypersurfaces
(see, e.g., {\cite[Example 1.1] {Wu-2011}}).
Let
\begin{eqnarray*}
N^{n+1}(c) \ =\ \mathbb{S}^{n+1}(c) &=&
\{ y \in \mathbb{R}^{n+2}\, : \, < y,y > \ =\ \frac{1}{c} ,\, c > 0 \},
\end{eqnarray*}
where $< \cdot , \cdot >$ is the standard inner product on
$\mathbb{R}^{n+2}$, $n \geq 3$.
Let $M_{p,n-p} (c,t)$ be hypersurfaces in $\mathbb{S}^{n+1}(c)$
defined by
$M_{p,n-p} (c,t)  =
\mathbb{S}^{p}( c / \sin ^{2} t )
\times
\mathbb{S}^{n-p} ( c / \cos ^{2} t )$,
where $t \in ( 0, \frac{\pi }{2})$ and $1 \leq p \leq n-1$.
The principal curvatures
$\lambda_{1}$ and $\lambda_{2}$ of
every hypersurface $M_{p,n-p} (c,t)$ are the following:
$\lambda_{1}\, =\, \sqrt{c}\, \cot t > 0$
and
$\lambda_{2}\, =\, - \sqrt{c}\, \tan t < 0$.
Hypersurfaces
$M_{p,n-p} (c,t)$ are non-conformally flat,
provided that  $n \geq 4$ and $2 \leq p \leq n-2$. In addition,
from (\ref{2022.01.12.a3}) it follows that a hypersurface
$M_{p,n-p} (c,t)$ is a non-Einstein hypersurface
if and only if
(\ref{2022.09.23.cc}) is satisfied
at every point of $M_{p,n-p} (c,t)$.
The last condition is satisfied when
$t \neq \arctan \sqrt{ ( p-1 ) / ( n -p -1) }$.
\end{example}
\begin{remark}
Hypersurfaces
in space forms having at every point
two distinct principal curvatures
also satisfy some curvature conditions of pseudosymmetry type,
see, e.g., {\cite[Section 5] {2016_DGHZhyper}}
and {\cite[Section 3] {2018_DGZ}}.
\end{remark}

\section{Hypersufaces in space forms satisfying $H^{3} = \phi H^{2} + \psi H + \rho g$. The general case.}

Let $(M,g)$, $\dim M \geq 4$, be a hypersurface in an $(n+1)$-dimensional
semi-Riemannian manifold $(N,\tilde{g})$,
where the metric tensor $g$ of $M$ is induced by the metric tensor $\tilde{g}$ of the ambient space $N$.
The hypersurface $(M,g)$ is said to be {\sl{quasi-umbilical}}
(see, e.g., \cite{{DV_1991_GTS}, {G6}})
resp., $2$-{\sl{quasi-umbilical}}
(see, e.g., \cite{{DeVerYap}, {G6}})
at a point $x \in M$
if $\mathrm{rank} (H - \alpha \, g) = 1$,
resp., $\mathrm{rank} (H - \alpha \, g) = 2$ holds at $x$,
for some $\alpha \in \mathbb{R}$.
If at a point  $x \in M$ we have $H =  \alpha \, g$,
for some $\alpha \in \mathbb{R}$, then $M$ is called
{\sl{umbilical}} at $x$.

If $(M,g)$, $\dim M \geq 4$, is a hypersurface in an $(n+1)$-dimensional
Riemannian manifold $(N,\tilde{g})$ then the above presented definitions of
a  quasi-umbilical and $2$-quasi-umbilical points of the hypersurface
$M$ are equivalent to the following definitions.
Namely,
according to
{\cite{{CHEN-1973}, {CHEN-2000}, {CHEN-VERSTRAELEN-1976}, {PRV_2002},
{V2}} (see also \cite{DGP-TV02}),
the hypersurface $M$ is said to be {\sl{quasi-umbilical}},
resp., $2$-{\sl{quasi-umbilical}}, at a point $x \in M$
if it has at $x$ a principal curvature with multiplicity $n-1$,
resp., $n-2$,
i.e., when the principal curvatures of $M$ at $x$ are the following:
$\lambda _{1} \neq \lambda _{2} = \lambda _{3} = \ldots = \lambda _{n}$,
resp.,
$\lambda _{1}$, $\lambda _{2}$,
$\lambda _{3} = \lambda _{4} = \ldots = \lambda _{n}$,
with
$\lambda _{1} \neq \lambda _{3}$ and $\lambda _{2} \neq \lambda _{3}$.

We recall that a hypersurface $(M,g)$, $\dim M = n \geq 4$,
in a semi-Riemannian conformally flat manifold $(N,\tilde{g})$,
where $g$ is induced by $\tilde{g}$,
is quasi-umbilical
at a point $x \in M$ if and only if its Weyl conformal curvature tensor
$C$ vanishes at this point {\cite[Theorem 4.1] {DV_1991_GTS}}.
Thus $M$ is non-quasi-umbilical at a point $x \in M$
if and only if its Weyl conformal curvature tensor $C$
is non-zero at $x$, i.e., $x \in \mathcal{U}_{C} \subset M$.

Let now $M$ be a hypersurface in $N_{s}^{n+1}(c)$, $n \geq 3$,
with the tensor $H$ satisfying (\ref{real702})  on ${\mathcal U}_{H}$.
From (\ref{real702}) we get on ${\mathcal U}_{H}$
\begin{eqnarray}
H^{4} &=& \phi \, H^{3} + \psi \, H^{2} + \rho \, H
\ =\ ( \phi ^{2} + \psi ) \, H^{2}
+ (\phi \psi + \rho)\, H + \phi \rho \, g .
\label{real703}
\end{eqnarray}
Now (\ref{realC7}), with the aid of (\ref{real702}),  (\ref{realC6}) and (\ref{real703}), turns into
\begin{eqnarray}
A &=& \tau \, H ,
\label{real70456}
\end{eqnarray}
where the $(0,2)$-tensor $A$ and the function $\tau$ are defined by
\begin{eqnarray}
A &=&  S^{2} - \rho _{1}\, S - \rho _{0}\, g ,
\label{real707}\\
\tau &=& \rho
+ ( \phi - \mathrm{tr} (H) )
( \psi  + \mathrm{tr} (H) ( \phi - \mathrm{tr} (H) ) )  ,
\label{real705}
\end{eqnarray}
respectively, where
\begin{eqnarray}
\rho _{0} &=&
- ( (n-1)^{2} c^{2}
- \varepsilon  ( \psi + ( \phi - \mathrm{tr} (H) )^{2} )  (n-1) c
+ \rho (\phi - 2 \mathrm{tr} (H) ) ) ,
\label{real707rho0}\\
\rho _{1} &=& 2 (n-1) c
- \varepsilon ( \psi +  ( \phi - \mathrm{tr} (H) )^{2} )  .
\label{real707rho1}
\end{eqnarray}
We note that (\ref{realC5}),
by (\ref{real70456}) and (\ref{real707}),
leads to
\begin{eqnarray}
\tau ^{2}\, R &=& \frac{\varepsilon}{2}\, A \wedge A + \tau ^{2} c\, G ,
\label{2022.01.11.a1}
\end{eqnarray}
and (\ref{2022.09.19.aa}).
Further, from
(\ref{realC6}),
(\ref{real70456}) and (\ref{real707}) we get
\begin{eqnarray}
\rho _{0} &=& \frac{1}{n}
\left(
\mathrm{tr} (S^{2}) - \rho_{1} \kappa - \tau \mathrm{tr}(H) \right) ,
\nonumber\\
S^{2} - \rho _{1}\, S - \tau \, H
&=&
\frac{1}{n}
\left( \mathrm{tr} (S^{2}) - \rho _{1}\, \kappa
- \tau \, \mathrm{tr} (H) \right) g ,
\label{2021.12.17.a6}
\end{eqnarray}
\begin{eqnarray}
Q(g,S^{2})  &=& \rho _{1}\, Q(g,S) + \tau \, Q(g,H) ,
\label{2021.12.17.a5}\\
\tau \, Q(H,S)  &=&
Q(\tau \, H, S) \ =\ Q(A,S)
\ =\  Q( S^{2} - \rho _{1}\, S - \rho _{0}\, g  ,S)
\nonumber\\
&=&
Q(S^{2}, S) - \rho _{0} \, Q(g ,S)
\ = \ -  Q(S, S^{2}) - \rho _{0} \, Q(g ,S) ,
\label{2022.01.07.a7}\\
\tau \, Q(g,H)  &=&
Q(g,\tau \, H) \ =\ Q(g, A)
\ =\
Q(g, S^{2} - \rho _{1}\, S - \rho _{0}\, g )
\nonumber\\
&=&
Q(g, S^{2} ) - \rho _{1}\, Q(g,S) ,
\label{2022.01.07.77}
\end{eqnarray}
and
\begin{eqnarray}
\tau \, Q(H,H^{2})  &=&
\tau \, Q(H, \mathrm{tr} (H)\, H -\varepsilon \, S
+ (n-1) \varepsilon c\, g ) \nonumber\\
&=& \varepsilon \tau \, Q(H, - S + (n-1) c\, g )\nonumber\\
&=&
\varepsilon  \, Q( A , - S + (n-1) c\, g )
\nonumber\\
&=& \varepsilon
\left( Q(S, S^{2})
+ (\rho_{0} + (n-1)c \rho_{1})\, Q(g, S)
- (n-1) c\, Q(g, S^{2})
\right) .
\label{2022.01.07.88}
\end{eqnarray}
From (\ref{real70456}) and (\ref{real707}) we also obtain
\begin{eqnarray}
S^{2}_{ij} &=& \rho _{1}\, S_{ij} + \rho _{0}\, g_{ij} + \tau\, H_{ij} ,
\label{new01-real706} \\
S^{3}_{ij} &=& \rho _{1}\, S^{2}_{ij} + \rho _{0}\, S_{ij}
+ \tau H_{ir}g^{rs}S_{sj} ,
\label{new02-real706}
\end{eqnarray}
where $S^{3}_{ij} = S_{ih}g^{hk}S^{2}_{kj}$ are the local components
of the $(0,2)$-tensor $S^{3}$.
Now applying
(\ref{real702}), (\ref{realC6}) and (\ref{real70456}) into (\ref{new02-real706})
we get
\begin{eqnarray*}
S^{3}_{ij} &=& \rho _{1}\, S^{2}_{ij} + \rho _{0}\, S_{ij}
+ \tau H_{ir} g^{rs} (
\varepsilon \, (\mathrm{tr} (H)\, H_{sj} - H^{2}_{sk} )
+ (n-1) c\, g_{sj} ) \nonumber\\
&=& \rho _{1}\, S^{2}_{ij} + \rho _{0}\, S_{ij}
+ \tau (
\varepsilon \, ( \mathrm{tr} (H)\, H^{2}_{ij} - H^{3}_{ij})
+ (n-1) c\, H_{ij} ) \nonumber\\
&=& \rho _{1}\, S^{2}_{ij} + \rho _{0}\, S_{ij}
+ \varepsilon \, \tau ( \mathrm{tr} (H)\, H^{2}_{ij} - \phi H^{2}_{ij}
- \psi \, H_{ij} - \rho \, g_{ij}
+ \varepsilon (n-1) c\, H_{ij} ) \nonumber\\
&=& \rho _{1}\, S^{2}_{ij} + \rho _{0}\, S_{ij}
+ \varepsilon \,
 \tau ( ( \mathrm{tr} (H) - \phi ) H^{2}_{ij}
+ (- \psi + \varepsilon  (n-1) c)\, H_{ij}
- \rho\, g_{ij} ) ,
\end{eqnarray*}
i.e.,
\begin{eqnarray}
S^{3} &=& \rho _{1}\, S^{2} + \rho _{0}\, S
+ \varepsilon
 \tau ( ( \mathrm{tr} (H) - \phi )\, H^{2}
+ ( \varepsilon (n-1) c - \psi )\, H
- \rho \, g) .
\label{new04-real7077}
\end{eqnarray}

Let $M$ be a hypersurface in $N_{s}^{n+1}(c)$, $n \geq 4$,
with the tensor $H$ satisfying (\ref{real702})
on ${\mathcal U}_{H} \subset M$.
We define on ${\mathcal U}_{H}$ the following functions
({\cite[eq. (34)] {Saw-2004}}, {\cite[Section 4] {Saw-2007}}):
\begin{eqnarray}
\beta_{1} &=& \varepsilon (\phi - \mathrm{tr}(H)) ,\nonumber\\
\beta_{2} &=& - \frac{\varepsilon }{n-2}
( \phi \, (2 \mathrm{tr}(H) - \phi)
- (\mathrm{tr}(H))^{2} - \psi - (n-2) \varepsilon \mu ) \nonumber\\
&=&
\mu +
\frac{\varepsilon }{n-2}
\left( \beta_{1}^{2} + \psi \right) ,\nonumber\\
\beta_{3} &=& \varepsilon \mu \mathrm{tr}(H)
+ \frac{1}{n-2}\, (\psi \, (2 \mathrm{tr}(H) - \phi)
 + (n-3) \rho ) ,
\label{ZZ23}\\
\beta_{4} &=& \beta_{3} -  \varepsilon \beta_{2} \mathrm{tr}(H)
+ \frac{ (n-1) \widetilde{\kappa} \beta_{1} }{n(n+1)}  ,\nonumber\\
\beta_{5} &=&
\frac{\kappa }{n-1}
+ \varepsilon \psi
- \frac{ (n^{2} - 3n + 3) \widetilde{\kappa} }{n(n+1)}
+ \beta_{1} \mathrm{tr}(H) ,
\nonumber\\
\beta_{6} &=& \beta_{2}
- \frac{ (n-3) \widetilde{\kappa} }{n(n+1)} ,\nonumber
\end{eqnarray}
and
\begin{eqnarray}
\mu
&=&
\frac{1}{n-2}
\left( \frac{\kappa }{n-1}
- \frac{\widetilde{\kappa} }{n+1}
\right) ,
\label{newZZ01}
\end{eqnarray}
where $\phi$, $\psi $ and $\rho $
are defined by (\ref{real702}).
We also have on ${\mathcal U}_{H}$
{\cite[eq. (24)] {Saw-2007}}
\begin{eqnarray}
R \cdot S
&=&
\frac{\widetilde{\kappa}}{n (n+1)}\, Q(g,S)
+ \rho \, Q(g,H)
- \varepsilon \beta _{1}\, Q(H, H^{2}) ,
\label{900r1}
\end{eqnarray}
and
{\cite[eqs. (52), (45), (46), (47)] {Saw-2004}}
\begin{eqnarray}
C \cdot S
&=&
\beta _{1}\, Q(H,S)
+ \beta _{2}\, Q(g,S)
+ \beta _{4} \, Q(g,H) ,
\label{newZZ04}
\end{eqnarray}
\begin{eqnarray}
(n-2)\, R \cdot C  &=& (n-2)\, Q(S,R)
- \frac{(n-2)^{2} \widetilde{\kappa }}{n (n+1)}\, Q(g,R)
- \frac{(n-3) \widetilde{\kappa} }{n (n+1)}\, Q(S,G)
\nonumber\\
& &
+ \rho \, Q(H,G)
+ \varepsilon \beta _{1} \, g \wedge Q(H,H^{2}) ,
\label{ZZ1}
\end{eqnarray}
\begin{eqnarray}
(n-2)\, C \cdot R &=&
(n-3)\, Q(S,R) + \varepsilon \beta _{1} \, H \wedge Q(g,H^{2})
- \frac{(n - 3) \widetilde{\kappa }}{n (n+1)}\, Q(S,G)
\nonumber\\
& &
+
 \left( \frac{\kappa }{n-1} +  \varepsilon  \psi
- \frac{(n^{2} - 3n + 3) \widetilde{\kappa }}{n(n+1)}
\right) Q(g,R) ,
\label{ZZ2}
\end{eqnarray}
\begin{eqnarray}
(n-2)\, C \cdot C &=&
(n-3)\, Q(S,R)
+ \beta _{5}\, Q(g,R) + \beta _{6} Q(S,G)
\nonumber\\
& &
+ \beta _{1} \, Q(S, g \wedge H)
+ \beta _{4} \, Q(H, G) ,
\label{2021.11.30.a5}
\end{eqnarray}
where $\beta_{1}, \ldots , \beta_{6}$
are defined by (\ref{ZZ23}).

From (\ref{900r1}), by making use of
(\ref{2022.01.07.77}) and (\ref{2022.01.07.88}), we get
\begin{eqnarray}
\tau \, R \cdot S
&=&
- \beta _{1} \, Q(S, S^{2})
+ ( \rho + (n-1) c \beta_{1} )\, Q(g, S^{2})\nonumber\\
& &
+ ( \tau c -  \rho  \rho _{1}
- \beta _{1} ( \rho _{0} + (n-1) c  \rho _{1}) )\, Q(g,S) .
\label{2022.01.10.99}
\end{eqnarray}
We also note that (\ref{newZZ04}),
by (\ref{2021.12.17.a5}) and (\ref{2022.01.07.a7}), turns into
\begin{eqnarray}
\tau \, C \cdot S
&=&
- \beta _{1}\, Q(S,S^{2})
+ \beta _{4} \, Q(g,S^{2})
+
(\beta _{2} \tau - \beta _{1} \rho _{0} - \beta _{4} \rho _{1} )
\, Q(g,S)  .
\label{2022.01.10.j}
\end{eqnarray}
Thus we have
\begin{proposition}
Let $M$ be a hypersurface in a semi-Riemannian space forms
$N_{s}^{n+1}(c)$, $n \geq 4$,
with the tensor $H$ satisfying (\ref{real702})
on ${\mathcal U}_{H} \subset M$.
Then
(\ref{2022.09.19.aa}),
(\ref{real70456}),
(\ref{real705}) and
(\ref{2022.01.11.a1})--(\ref{2022.01.10.j})
hold
on ${\mathcal U}_{H}$.
\end{proposition}

\begin{remark} Conditions (\ref{real70456}),
(\ref{real705}) and
(\ref{2022.01.11.a1})
were already presented in the last section of a preliminary version of
\cite{2021-DGH}. Precisely, see eqs. (8.6)--(8.9)
in Section 8 of arXiv:1911.02482v2 [math.DG] 27 Jan 2020.
However that section was not included to \cite{2021-DGH}.
\end{remark}
\begin{remark}
We also can obtain (\ref{2021.12.17.a6}) in other ways.
\newline
(i) From (\ref{uuWeyl}), (\ref{900r1}) and (\ref{newZZ04}) we get
\begin{eqnarray}
& &
\beta _{1}\, Q(H,S)
+ \beta _{2}\, Q(g,S)
+ \beta _{4} \, Q(g,H)
+ \frac{1}{n-2}
Q\left( g, S^{2} - \frac{\kappa }{n-1} S \right)
\nonumber\\
&=&
\frac{\widetilde{\kappa}}{n (n+1)}\, Q(g,S)
+ \rho \, Q(g,H)
- \varepsilon \beta _{1}\, Q(H, H^{2}) .
\label{2022.01.10.k}
\end{eqnarray}
Furthermore, (\ref{realC6}) yields
\begin{eqnarray*}
Q(H,S) - \frac{(n - 1) \widetilde{\kappa }}{n (n+1)}\, Q(H, g)
&=&
\varepsilon \mathrm{tr} (H)\,
Q(H,H)
- \varepsilon \, Q(H,H^{2})
\end{eqnarray*}
and
\begin{eqnarray}
Q(H,S) + \varepsilon \, Q(H, H^{2})
&=&
- \frac{(n - 1) \widetilde{\kappa }}{n (n+1)}\, Q(g,H) .
\label{2022.01.10.l}
\end{eqnarray}
Now (\ref{2022.01.10.k}), by (\ref{2022.01.10.l}), turns into
\begin{eqnarray}
Q(g,S^{2}) &=& (n-2)
\left(
\frac{\widetilde{\kappa }}{n (n+1)}
+ \frac{\kappa }{(n-2)(n-1)} - \beta_{2} \right) Q(g,S)\nonumber\\
& &
+ (n-2)
\left(\rho + \frac{(n-1) \widetilde{\kappa } \beta _{1}}{n (n+1)}
- \beta_{4} \right) Q(g,H) .
\label{2021.12.17.a1}
\end{eqnarray}
We note that
\begin{eqnarray}
& &
(n-2)
\left(
\rho + \frac{(n-1) \widetilde{\kappa }\beta _{1}}{n (n+1)} - \beta_{4}
\right)\nonumber\\
&=&
(n-2)
\left(
\rho + \frac{(n-1) \widetilde{\kappa } \beta _{1}}{n (n+1)} - \beta_{3}
 + \varepsilon \beta_{2}  \mathrm{tr}(H)
 - \frac{(n-1) \widetilde{\kappa } \beta _{1}}{n (n+1)}
\right)\nonumber\\
&=&
(n-2)
\left( \rho - \varepsilon \mu \mathrm{tr}(H)
- \frac{1}{n-2}\, (\psi \, (2 \mathrm{tr}(H) - \phi) + (n-3) \rho )
\right.\nonumber\\
& &\left.
- \frac{1}{n-2}
( \phi \, (2 \mathrm{tr}(H) - \phi)
- (\mathrm{tr}(H))^{2} - \psi - (n-2) \varepsilon \mu ) \,  \mathrm{tr}(H)
\right)\nonumber\\
&=&
\rho -  \psi \, (2 \mathrm{tr}(H) - \phi)
-
( \phi \, (2 \mathrm{tr}(H) - \phi)
- (\mathrm{tr}(H))^{2} - \psi ) \,  \mathrm{tr}(H)\nonumber\\
&=&
\rho
 + ( \phi - \mathrm{tr}(H) )\, \psi
+  \mathrm{tr}(H)\, ( \phi - \mathrm{tr}(H) )^{2} \ =\ \tau ,
\label{2021.12.17.a3}
\end{eqnarray}
and
\begin{eqnarray}
& &
 (n-2) \left( \frac{\kappa }{(n-2)(n-1)}
- \frac{(n-4) \widetilde{\kappa }}{n (n+1)}
- \beta_{6} \right)\nonumber\\
&=&
 (n-2) \left( \frac{\kappa }{(n-2)(n-1)}
- \frac{(n-4) \widetilde{\kappa }}{n (n+1)}
- \beta_{2}
+ \frac{(n-3) \widetilde{\kappa }}{n (n+1)}
\right)\nonumber\\
&=&
 (n-2) \left( \frac{\kappa }{(n-2)(n-1)}
+ \frac{ \widetilde{\kappa }}{n (n+1)}
- \beta_{2}
\right)\nonumber\\
&=&
\frac{\kappa }{n-1} + \frac{(n-2) \widetilde{\kappa }}{n (n+1)}
+
\varepsilon
( \phi \, (2 \mathrm{tr}(H) - \phi)
- (\mathrm{tr}(H))^{2} - \psi - (n-2) \varepsilon \mu )\nonumber\\
&=&
\frac{2 (n-1) \widetilde{\kappa }}{n (n+1)}
+
\varepsilon ( \phi \, (2 \mathrm{tr}(H) - \phi)
- (\mathrm{tr}(H))^{2} - \psi)\nonumber\\
&=&
\frac{2 (n-1) \widetilde{\kappa }}{n (n+1)}
-\varepsilon ( \psi + ( \phi - \mathrm{tr}(H) )^{2} ) \ =\ \rho_{1} .
\label{2021.12.17.a4}
\end{eqnarray}
Now (\ref{2021.12.17.a1}), by (\ref{2021.12.17.a3})
and (\ref{2021.12.17.a4}), turns into
(\ref{2021.12.17.a5}),
which by suitable contractions yields (\ref{2021.12.17.a6}).
\newline
(ii) From (\ref{2021.11.30.a2}),
by making use of
(\ref{dghz01}), (\ref{ZZ1}), (\ref{ZZ2}) and (\ref{2021.11.30.a5}),
we get
\begin{eqnarray}
& &
(\rho - \beta_{4} )\, Q(H,G)
+ \varepsilon \beta _{1}
\left( g \wedge Q(H,H^{2}) + H \wedge Q(g,H^{2})  \right) \nonumber\\
&=&
\left( \frac{\kappa }{(n-2)(n-1)}
- \frac{(n-4) \widetilde{\kappa }}{n (n+1)} - \beta_{6} \right)
Q( g, g \wedge S)  \nonumber\\
& &
+ \left( \beta_{5} - \left(  \frac{\kappa }{n-1}
- \frac{(n^{2} - 3n + 3) \widetilde{\kappa }}{n (n+1)}
+ \varepsilon \psi
\right) \right) Q(g,R) \nonumber\\
& &
- \frac{1}{n-2}\, Q(g, g \wedge S^{2})
+ \beta_{1} \, Q(S, g \wedge H) .
\label{2021.11.30.a6}
\end{eqnarray}
Condition (\ref{2021.11.30.a6}),
by an application of (\ref{2021.11.30.a4}), (\ref{realC5}), (\ref{ZZ23})
and identities
\begin{eqnarray*}
Q(g, G)  &=& 0 ,\nonumber\\
Q(H,G) &=& - Q(g, g \wedge H) ,\nonumber\\
Q(g, \frac{1}{2}\, H \wedge H) &=& - Q(H, g \wedge H) ,
\end{eqnarray*}
yields
\begin{eqnarray*}
& &
(\rho - \beta_{4} )\, Q(H,G)
- \varepsilon \beta _{1} \, Q( H^{2} , g \wedge H) \nonumber\\
&=&
\left( \frac{\kappa }{(n-2)(n-1)}
- \frac{(n-4) \widetilde{\kappa }}{n (n+1)} - \beta_{6} \right)
Q( g, g \wedge S)  \nonumber\\
& &
+ \varepsilon \beta_{1} \mathrm{tr}(H)\, Q(g, \frac{1}{2}\, H \wedge H)
- \frac{1}{n-2}\, Q(g, g \wedge S^{2})
+ \beta_{1} \, Q(S, g \wedge H) .
\end{eqnarray*}
Thus we have
\begin{eqnarray*}
& &
- (\rho - \beta_{4} )\, Q(g , g \wedge H)
- \varepsilon \beta _{1} \, Q( H^{2} , g \wedge H) \nonumber\\
&=&
\left( \frac{\kappa }{(n-2)(n-1)}
- \frac{(n-4) \widetilde{\kappa }}{n (n+1)} - \beta_{6} \right)
Q( g, g \wedge S)  \nonumber\\
& &
- \varepsilon \beta_{1} \mathrm{tr}(H)\, Q(H, g \wedge H)
- \frac{1}{n-2}\, Q(g, g \wedge S^{2})
+ \beta_{1} \, Q(S, g \wedge H)
\end{eqnarray*}
and
\begin{eqnarray}
& &
- (\rho - \beta_{4} )\, Q(g , g \wedge H)  \nonumber\\
& &
+ \beta_{1}
\left( \varepsilon  \mathrm{tr}(H)\, Q(H, g \wedge H)
- \varepsilon  \, Q( H^{2} , g \wedge H)
-  Q(S, g \wedge H) \right) \nonumber\\
&=&
\left( \frac{\kappa }{(n-2)(n-1)}
- \frac{(n-4) \widetilde{\kappa }}{n (n+1)} - \beta_{6} \right)
Q( g, g \wedge S)
- \frac{1}{n-2}\, Q(g, g \wedge S^{2}) .
\label{2021.11.30.a7}
\end{eqnarray}
Using now (\ref{realC6}) we can verify that
\begin{eqnarray*}
& &
\varepsilon \mathrm{tr}(H)\, Q(H, g \wedge H)
- \varepsilon  \, Q( H^{2} , g \wedge H)
-  Q(S, g \wedge H)
\nonumber\\
&=&
Q (  \varepsilon  ( \mathrm{tr} (H)\, H -  H^{2}) , g \wedge H)
- Q(S, g \wedge H) \nonumber\\
&=&
Q(S - \frac{(n-1) \widetilde{\kappa }}{n (n+1)}\, g ,  g \wedge H)
- Q(S, g \wedge H) \nonumber\\
&=&
 Q( - \frac{(n-1) \widetilde{\kappa }}{n (n+1)}\, g , g \wedge H)
\ =\
-  \frac{(n-1) \widetilde{\kappa }}{n (n+1)}\, Q(g , g \wedge H) ,
\end{eqnarray*}
i.e.,
\begin{eqnarray*}
\varepsilon \mathrm{tr}(H)\, Q(H, g \wedge H)
- \varepsilon  \, Q( H^{2} , g \wedge H)
-  Q(S, g \wedge H)
&=&
-  \frac{(n-1) \widetilde{\kappa }}{n (n+1)}\, Q(g , g \wedge H) .
\end{eqnarray*}
Next from (\ref{2021.11.30.a7}) we obtain
\begin{eqnarray*}
& &
(  \beta_{4}  -  \frac{(n-1) \widetilde{\kappa } \beta _{1}}{n (n+1)}
- \rho)\,
Q(g , g \wedge H)
+ \frac{1}{n-2}\, Q(g, g \wedge S^{2}) \nonumber\\
& &
- \left( \frac{\kappa }{(n-2)(n-1)}
- \frac{(n-4) \widetilde{\kappa }}{n (n+1)} - \beta_{6} \right)
Q( g, g \wedge S)
 = 0
\end{eqnarray*}
and
\begin{eqnarray}
& &
Q(g,\left(\beta_{4} - \frac{(n-1) \widetilde{\kappa } \beta _{1}}{n (n+1)}
- \rho  \right)  g \wedge H + \frac{1}{n-2}\,  g \wedge S^{2} \nonumber\\
& &
- \left( \frac{\kappa }{(n-2)(n-1)}
- \frac{(n-4) \widetilde{\kappa }}{n (n+1)} - \beta_{6} \right)
g \wedge S)
 = 0 .
\label{2021.12.01.a2}
\end{eqnarray}
From (\ref{2021.12.01.a2}), by suitable contractions, we get
\begin{eqnarray*}
& &
 \frac{1}{n-2}\, S^{2}
- \left( \frac{\kappa }{(n-2)(n-1)}
- \frac{(n-4) \widetilde{\kappa }}{n (n+1)}
- \beta_{6} \right)  S
\nonumber\\
& &
+
\left( \beta_{4} - \frac{(n-1) \widetilde{\kappa } \beta _{1}}{n (n+1)}
- \rho \right) H \ =\ \widetilde{\rho _{0}}\, g
\end{eqnarray*}
and
\begin{eqnarray}
& &
S^{2}
-  (n-2) \left( \frac{\kappa }{(n-2)(n-1)}
- \frac{(n-4) \widetilde{\kappa }}{n (n+1)}
- \beta_{6} \right)  S
-  (n-2)  \widetilde{\rho _{0}}\, g
\nonumber\\
&=&
 (n-2) \left( \rho + \frac{(n-1) \widetilde{\kappa }}{n (n+1)}
- \beta_{4} \right) H ,
\label{2021.12.04.a1}
\end{eqnarray}
where
\begin{eqnarray}
 \widetilde{\rho _{0}} &=& \frac{1}{n}
\left(
 \frac{1}{n-2}\, \mathrm{tr} (S^{2})
-
\left( \frac{\kappa }{(n-2)(n-1)}
- \frac{(n-4) \widetilde{\kappa }}{n (n+1)} - \beta_{6} \right)
\kappa \right.  \nonumber\\
& &
+
\left.
\left(  \beta_{4}  -  \frac{(n-1) \widetilde{\kappa }}{n (n+1)}
- \rho \right)  \mathrm{tr} (H) \right) .
\label{2021.12.04.a2}
\end{eqnarray}
Now (\ref{2021.12.04.a1}),
by (\ref{2021.12.17.a3}), (\ref{2021.12.17.a4}) and (\ref{2021.12.04.a2}),
turns into
\begin{eqnarray}
S^{2} - \rho _{1} \, S - \tau \, H
&=& (n-2) \widetilde{\rho _{0}} \, g .
\label{2021.12.15.a}
\end{eqnarray}
From (\ref{2021.12.15.a}) by contraction we obtain
\begin{eqnarray}
(n-2) \widetilde{\rho _{0}}
&=&
\frac{1}{n} \left(
\mathrm{tr}(S^{2}) - \rho _{1} \kappa - \tau \mathrm{tr}(H)
\right)  .
\label{2021.12.15.b}
\end{eqnarray}
Now (\ref{2021.12.15.a}) and (\ref{2021.12.15.b}) yield
(\ref{2021.12.17.a6}).
\end{remark}

\section{Hypersufaces in space forms satisfying $H^{3} = \phi H^{2} + \psi H + \rho g$. The partially Einstein case.}

Let $M$ be a hypersurface in $N_{s}^{n+1}(c)$, $n \geq 3$,
with the tensor $H$ satisfying (\ref{real702})
on ${\mathcal U}_{H} \subset M$.
We assume that the hypersurface $M$ is a partially Einstein
at a point
$x \in {\mathcal U}_{H}$, i.e., let
$S^{2} = \widetilde{\alpha }\, S + \widetilde{\beta }\, g$ at $x$,
where $\widetilde{\alpha }, \, \widetilde{\beta} \in \mathbb{R}$.
Now (\ref{real70456}) turns into
\begin{eqnarray}
(\widetilde{\alpha } - \rho _{1}) \, S
+ (\widetilde{\beta } - \rho _{0} ) \, g  &=& \tau \, H  .
\label{2022.01.15.c}
\end{eqnarray}
If we would have $\widetilde{\alpha } \neq \rho _{1}$ then
$S = (\widetilde{\alpha } - \rho _{1})^{-1}
\, (\tau \, H - ( \widetilde{\beta } - \rho _{0} ) \, g)$
at $x$. This by (\ref{realC6}) leads to an equation of the form
(\ref{2021.10.25.a2}), a contradiction.
Therefore $\widetilde{\alpha } = \rho _{1}$ at $x$.
Now (\ref{2022.01.15.c}) reduces to
$(\widetilde{\beta } - \rho _{0} ) \, g  = \tau \, H$.
From this it follows that
$\widetilde{\beta } = \rho _{0}$ and $\tau = 0$ at $x$.
Thus we have
\begin{theorem}
Let $M$ be a hypersurface in a semi-Riemannian space forms
$N_{s}^{n+1}(c)$, $n \geq 3$,
with the tensor $H$ satisfying (\ref{real702})
on ${\mathcal U}_{H} \subset M$.
If $M$ is partially Einstein at every point of ${\mathcal U}_{H}$
then
$S^{2} = \rho _{1}\, S + \rho _{0}\, g$ and $\tau = 0$, i.e.,
\begin{eqnarray*}
\rho
+ ( \phi - \mathrm{tr} (H) )
( \psi  + \mathrm{tr} (H) ( \phi - \mathrm{tr} (H) ) ) &=& 0
\end{eqnarray*}
on ${\mathcal U}_{H}$, where $\rho _{0}$, $\rho _{1}$ and $\tau $ are defined by
(\ref{real707rho0}), (\ref{real707rho1})
and (\ref{real705}), respectively.
\end{theorem}

Let $M$ be a hypersurface in a Riemannian space forms
$N^{n+1}(c)$, $n \geq 3$,
having at every point $x \in {\mathcal U}_{H} \subset M$
three distinct principal curvatures
$\lambda _{0}$, $\lambda _{1}$ and $\lambda _{2}$,
with multiplicities $n_{0}$, $n_{1}$ and $n_{2}$, respectively,
where $n_{0} + n_{1} + n_{2} = n$.
Evidently,
(\ref{real702}) holds on ${\mathcal U}_{H}$ with
\begin{eqnarray}
\phi \ =\ \lambda _{0} + \lambda _{1} + \lambda _{2},\ \ \
\psi \ =\  - \lambda _{0}
 ( \lambda _{1} + \lambda _{2}) - \lambda _{1} \lambda _{2},\ \ \
\rho \ =\ \lambda _{0} \lambda _{1} \lambda _{2}.
\label{tau-real702}
\end{eqnarray}
Further, by making use of (\ref{realC6}),
we define the following functions on ${\mathcal U}_{H}$:
\begin{eqnarray}
\kappa _{0} &=&  \mathrm{tr} (H)\, \lambda _{0}
- \lambda _{0}^{2} + (n-1)\, c ,\nonumber\\
\kappa _{1} &=&  \mathrm{tr} (H)\, \lambda _{1}
- \lambda _{1}^{2} + (n-1)\, c , \label{2021-10-18-a}\\
\kappa _{2} &=&  \mathrm{tr} (H)\, \lambda _{2}
- \lambda _{2}^{2} + (n-1)\, c .\nonumber
\end{eqnarray}
We note that
$\kappa _{0}$, $\kappa _{1}$  and $\kappa _{2}$
are eigenvalues of the Ricci operator $\mathcal{S}$ of $M$.
Now, using (\ref{real705}), (\ref{tau-real702})
and (\ref{2021-10-18-a}) we can easy check that the following
equations are satisfied:
\begin{eqnarray}
\kappa _{0} - \kappa _{1}
&=&
( \lambda _{0} - \lambda _{1}) \,
\left( \lambda _{2} - ( \phi -  \mathrm{tr} (H) ) \right) ,
\nonumber\\
\kappa _{0} - \kappa _{2}
&=&
( \lambda _{0} - \lambda _{2}) \,
\left( \lambda _{1} - ( \phi -  \mathrm{tr} (H) ) \right) ,
\label{2022-01-17-a}\\
\kappa _{1} - \kappa _{2}
&=&
( \lambda _{1} - \lambda _{2}) \,
\left( \lambda _{0} - ( \phi -  \mathrm{tr} (H) ) \right) , \nonumber
\end{eqnarray}
\begin{eqnarray}
& &
\left( \lambda _{0} - ( \phi -  \mathrm{tr} (H) ) \right)
\left( \lambda _{1} - ( \phi -  \mathrm{tr} (H) ) \right)
\left( \lambda _{2} - ( \phi -  \mathrm{tr} (H) ) \right)\nonumber\\
&=&
\rho + \psi \, ( \phi - \mathrm{tr} (H))
+ \phi \, ( \phi - \mathrm{tr} (H))^{2}
- ( \phi - \mathrm{tr} (H))^{3}\label{2022-01-17-b}\\
&=&
\rho + ( \phi - \mathrm{tr} (H)) \,
( \psi +  \mathrm{tr} (H) ( \phi - \mathrm{tr} (H)) )
\ =\ \tau .\nonumber
\end{eqnarray}
Now (\ref{2022-01-17-a}) and (\ref{2022-01-17-b}) lead to
\begin{eqnarray}
( \kappa _{0} - \kappa _{1})
  ( \kappa _{0} - \kappa _{2})
\, ( \kappa _{1} - \kappa _{2})
&=&
(\lambda _{0} - \lambda _{1}) \, ( \lambda _{0} - \lambda _{2})
\, ( \lambda _{1} - \lambda _{2}) \, \tau .
\label{2022-01-17-c}
\end{eqnarray}
Thus we have
\begin{theorem}
Let $M$ be a hypersurface in a Riemannian space forms
$N^{n+1}(c)$, $n \geq 3$,
having at every point $x \in {\mathcal U}_{H} \subset M$
three distinct principal curvatures
$\lambda _{0}$, $\lambda _{1}$ and $\lambda _{2}$,
with multiplicities $n_{0}$, $n_{1}$ and $n_{2}$, respectively.
Moreover, let the eigenvalues
$\kappa _{0}$, $\kappa _{1}$  and $\kappa _{2}$
of the Ricci operator $\mathcal{S}$ of $M$
satisfy (\ref{2021-10-18-a}).
Then the function $\tau$, defined by (\ref{real705}), satisfies
(\ref{2021-11-15-a})
on
${\mathcal U}_{H}$.
\end{theorem}

\begin{remark}
Let $M$ be a hypersurface in a Riemannian space forms
$N^{n+1}(c)$, $n \geq 3$, satisfying assumptions of the last theorem.
\newline
(i)
If we would have $\kappa _{0} = \kappa _{1} = \kappa _{2}$ at
a point $x \in {\mathcal U}_{H}$ then $S = (\kappa / n)\, g$ at $x$,
a contradiction.
\newline
(ii) If $\kappa _{0} = \kappa _{1} \neq \kappa _{2}$,
or $\kappa _{0} = \kappa _{2} \neq \kappa _{1}$,
or $\kappa _{1} = \kappa _{2} \neq \kappa _{0}$,
at a point $x \in {\mathcal U}_{H}$
then $\tau = 0$ and $M$ is partially Einstein at $x$.
\newline
(iii) If $\kappa _{0} \neq \kappa _{1}$
and $\kappa _{0} \neq \kappa _{2}$
and $\kappa _{1} \neq \kappa _{2}$ at a point $x \in {\mathcal U}_{H}$
then $\tau \neq 0$ at every point of a some neighbourhood
$U \subset {\mathcal U}_{H}$ of $x$.
\end{remark}

\begin{example} (i)
Let $M$ be an austere hypersurface in a Riemannian space forms
$N^{n+1}(c)$, $n \geq 3$, with three distinct principal curvatures
$\lambda _{0}$, $\lambda _{1}$ and $\lambda _{2}$.
It is known
that at every point of such hypersurface there are exactly three
distinct principal curvatures
$\lambda _{0} = 0$, $\lambda _{1} = \lambda$
and $\lambda _{2} = - \lambda$,
with multiplicities $n - 2p$, $p$ and $p$, respectively,
where $1 \leq p < n$ and $\lambda$ is a function on $M$
(see, e.g., \cite{Bryant-1991}).
From  (\ref{2021-11-15-a})  and (\ref{2021-10-18-a})
we get immedia\-tely $\kappa _{0} = (n-1)c$
and $\kappa _{1} = \kappa _{2} = (n-1)c - \lambda^{2}$,
$\kappa _{0} - \kappa _{1} = \lambda^{2} >0$,
and in a consequence $\tau = 0$
at every point of $M$.
Now, by making use of (\ref{real702}),
(\ref{realC6bb}) and  (\ref{real703})--(\ref{2021.11.30.a5}),
we obtain
\begin{eqnarray*}
\phi &=&  \rho = 0,\ \ \ \psi \ =\ \lambda ^{2},\ \ \
\mathrm{tr} (H)\ =\ 0,\ \ \
\mathrm{tr} (H^{2})\ =\ 2p \lambda ^{2},\ \ \
H^{3} \ =\ \lambda ^{2} H,\\
\beta _{1} &=& \beta _{3}\ =\   \beta _{4}\ =\ 0,\ \ \
\beta_{2}\ =\ \frac{n - 2p - 1}{(n-2)(n-1)}\, \lambda ^{2},\\
\beta_{5} &=&  (n-2) \beta_{2} - (n-3)(n-1) c,\ \ \
\beta_{6}\ =\ \beta_{2} - (n-3) c , \\
\kappa &=& n (n-1) c - 2p \lambda ^{2},\ \ \
\mu \ =\ - \frac{2p}{(n-2)(n-1)}\, \lambda ^{2},\\
S^{2} &=& \left(2(n-1)c - \lambda^{2} \right) S
+ (n-1)c \left( \lambda^{2} - (n-1)c  \right) g ,\\
R \cdot S &=&  c\, Q(g,S).
\end{eqnarray*}
Thus $M$ is a Ricci-pseudosymmetric partially Einstein hypersurface.
Moreover,
\begin{eqnarray*}
C \cdot S &=&  \beta _{2}\, Q(g,S),\\
(n-2)\, R \cdot C  &=&   (n-2)\, Q(S,R)
- (n-2)^{2}c\, Q(g,R)
- (n-3)c \, Q(S,G) ,\\
(n-2)\, C \cdot R &=&
(n-3)\, Q(S,R) - (n - 3)c \, Q(S,G)
\nonumber\\
& &
+  \left( \frac{\kappa }{n-1} +  \lambda ^{2}
-(n^{2} - 3n + 3) c \right) Q(g,R) ,\\
(n-2)\, C \cdot C &=&
(n-3)\, Q(S,R)
+ \beta _{5}\, Q(g,R) + \beta _{6} Q(S,G),
\end{eqnarray*}
provided that $n \geq 4$.
\newline
(ii) (cf. {\cite[Section 10.2] {{DHV2008}}})
If $p = 1$ then, in view of results of \cite{1995_DDDVY},
(\ref{pseudo}) with $L_{R} = c$
holds on $M$.
Thus $M$ is a pseudosymmetric manifold.
In particular,
generalized Cartan hypersurfaces \cite{CHEN-1994}
as well as
the $3$-dimensional Cartan hypersurfaces are pseudosymmetric
(see also {\cite[Example 1] {DVY-1994}}).
If $p \geq 2$ then  $R \cdot S = c\, Q(g,S)$ holds on $M$,
i.e., $M$ is a non-pseudosymmetric Ricci-pseudosymmetric manifold.
In particular, Cartan hypersurfaces with
$p = 2$ and $n = 6$, or
$p = 4$ and $n = 12$, or
$p = 8$ and $n = 24$,
are such manifolds.
Pseudosymmetry type curvature properties of Cartan hypersurfaces
are presented in {\cite[Theorem 4.3] {DG90}}.
\newline
(iii) In {\cite[Example 3] {2005-LSchS}}
an example of a hypersurface $M$ in a $5$-dimensional unit
sphere with four distinct principal curvatures is presented.
We can easy verify that $M$ is a partially Einstein hypersurface.
\end{example}

Let (\ref{pseudo}) be satisfied on the set $\mathcal{U}_{R}$
of a hypersurface $M$ in a Riemannian space forms $N^{n+1}(c)$,
$n \geq 4$. From {\cite[Theorem 5.1] {P39}} it follows that
at every point $x \in \mathcal{U}_{R}$ we have:
(i) $H^{2} = \alpha \, H + \beta \, g$, for some
$\alpha , \beta \in \mathbb{R}$, or, (ii) $\mathrm{rank} (H) = 2$.
In other words, at every point $x \in \mathcal{U}_{R}$ we have:
(i) $M$ has exactly two distinct principal curvatures
$\lambda _{0}$ and $\lambda _{1}$, or,
(ii) $M$ has principal curvatures
$\lambda _{0} \neq 0$, $\lambda _{1} \neq 0$
and $\lambda _{2} = \lambda _{3}  =  \ldots = \lambda _{n-1} = 0$.

\begin{example}
Let $M$ be a hypersurface in a Riemannian space forms $N^{n+1}(c)$,
$n \geq 4$, having at every point principal curvatures
$\lambda _{0} \neq 0$, $\lambda _{1} \neq 0$
and
$\lambda _{2} = \lambda _{3} = \ldots = \lambda _{n-1} = 0$.
At every point of $M$ we have $\mathrm{rank} (H) = 2$. Thus
$M$ is a type number two hypersurface
(see, e.g., \cite{CHEN-YILDIRIM-2015}).
In view of {\cite[Theorem 4.2] {1995_DDDVY}},
we also have
$R \cdot R = c\, Q(g,R)$
on $M$.
\newline
(i) Let $\lambda _{0} = \lambda _{1}$ at a point $x \in M$. Using now
suitable formulas of Section 3 we get immediately
$H^{2} = \lambda _{0} \, H$
($\alpha = \lambda _{0}$ and $\beta = 0$),
$\mathrm{tr} (H) - \alpha
= 2 \lambda _{0} - \lambda _{0} = \lambda _{0} \neq 0$,
\begin{eqnarray*}
\kappa _{0} &=&  \mathrm{tr} (H)\, \lambda _{0}
- \lambda _{0}^{2} + (n-1)\, c
\ = \ \lambda _{0}^{2} + (n-1)c , \\
\kappa _{1} &=&  \mathrm{tr} (H)\, \lambda _{1}
- \lambda _{1}^{2} + (n-1)\, c
\ = \ \lambda _{0}^{2} + (n-1)c ,\\
\kappa _{2} &=&  \mathrm{tr} (H)\, \lambda _{2}
- \lambda _{2}^{2} + (n-1)\, c
\ = \ (n-1)c ,
\end{eqnarray*}
\begin{eqnarray*}
H
&=&
\lambda _{0}^{-1} ( S - (n-1) c \, g) ,\\
R &=& \frac{1}{2 \lambda _{0}}
( S - (n-1) c \, g) \wedge ( S - (n-1) c \, g) + c \, G ,\\
S^{2}
&=&
\lambda _{0} ( \lambda _{0}^{2} + 2 (n-1)c)\, H + (n-1)^{2}c^{2} \, g\\
&=&
( \lambda _{0}^{2} + 2 (n-1)c)\, S
-
(n-1)c ( \lambda _{0}^{2} +  (n-1)c)\, g .
\end{eqnarray*}
We assume now that $\lambda _{0} = \lambda _{1}$ at every point of $M$.
Thus $M$ is a $2$-quasi-Einstein and partially Einstein hypersurface.
Moreover,
in view of Proposition 3.2, the hypersurface $M$ is a Roter space.
Precisely,
\begin{eqnarray}
R  &=& \frac{\phi }{2} \, ( S - (n-1) c \, g) \wedge ( S - (n-1) c \, g) +
c\, G
\label{2022.09.21.aa}
\end{eqnarray}
on $M$, where $\phi$ is defined by $\phi = \lambda _{0}^{-2}$.
\newline
(ii)  Let $\lambda _{0} \neq \lambda _{1}$ at a point $x \in M$.
We note that the case $\lambda _{1} = - \lambda _{0}$
was considered in Example 5.4 (i) (the case $p = 1$).
Using
suitable formulas of Sections 4 and 5 we get immediately:
$\mathrm{tr} (H) = \lambda _{0} + \lambda _{1}$,
$\phi = \lambda _{0} + \lambda _{1}$, $\psi = - \lambda _{0} \lambda _{1}$,
$\rho = 0$,
$H^{3} = \mathrm{tr} (H)\, H^{2} + \psi \, H$,
$\tau = 0$, $A = 0$,
\begin{eqnarray*}
S^{2} &=& \rho _{1} \, S + \rho _{0}\, g,\\
\rho _{0} &=& - (n-1)c \,
(  \lambda _{0} \lambda _{1} + (n-1)^{2}c^{2} ) ,\\
\rho _{1} &=&
 \lambda _{0} \lambda _{1} + 2 (n-1) c ,
\end{eqnarray*}
\begin{eqnarray*}
\kappa _{0} &=&  \mathrm{tr} (H)\, \lambda _{0}
- \lambda _{0}^{2} + (n-1)\, c
\ = \ \lambda _{0} \lambda _{1} + (n-1)c , \\
\kappa _{1} &=&  \mathrm{tr} (H)\, \lambda _{1}
- \lambda _{1}^{2} + (n-1)\, c
\ = \ \lambda _{0} \lambda _{1} + (n-1)c ,\\
\kappa _{2} &=&  \mathrm{tr} (H)\, \lambda _{2}
- \lambda _{2}^{2} + (n-1)\, c
\ = \ (n-1)c .
\end{eqnarray*}
We assume that $\lambda _{0} \neq \lambda _{1}$ at every point of $M$.
We note that $\mathrm{rank} (S - (n-1)c\, g) = 2$
at every point of $M$.
Thus $M$ is a $2$-quasi-Einstein and partially Einstein hypersurface.
Furthermore, in view of {\cite[Theorem 3.2 (iii)] {2018_DGZ}},
(\ref{2022.09.21.aa}) holds on $M$,
for some function $\phi$.
As it was stated in {\cite[Remark 5.2] {P39}}),
if
$\lambda _{1} = - \lambda _{0} \neq 0$
and $\lambda _{2} = \lambda _{3} = \ldots = \lambda _{n-1} = 0$
at every point of a hypersurface $M$ in $N^{n+1}(c)$, $n \geq 4$,
then it is a minimal hypersurface realizing Chen's equality
{\cite[Lemma 3.2] {CHEN-1993}}.
\end{example}

\section{Hypersufaces in space forms satisfying $H^{3} = \phi H^{2} + \psi H + \rho g$. The non-partially Einstein case.}

Let $M$ be a hypersurface in $N_{s}^{n+1}(c)$, $n \geq 4$,
with the tensor $H$ satisfying (\ref{real702})
on ${\mathcal U}_{H} \subset M$.
From (\ref{real70456}) and (\ref{real707}) it follows that
if the function $\tau$,
defined by (\ref{real705}),
vanishes at a point $x \in {\mathcal U}_{H}$
then $M$ is partially Einstein at $x$.
We assume now that $\tau $ is non-zero at a point
$x \in {\mathcal U}_{H}$.
From
(\ref{real70456})--(\ref{2022.01.11.a1}), (\ref{2022.01.10.99})
and (\ref{2022.01.10.j})
it follows that
\begin{eqnarray}
H &=& \tau ^{-1} \, A
\ =\ \tau ^{-1} \, (S^{2} - \rho_{1} S - \rho _{0} g) ,
\label{new04-real7088} \\
R &=& \frac{1}{2}\tau ^{-2} \, A \wedge A + c\, G \nonumber\\
&=& \frac{1}{2} \tau ^{-2} \,
(S^{2} - \rho_{1} S - \rho _{0} g)
\wedge (S^{2} - \rho_{1} S - \rho _{0} g) + \frac{c}{2}\, g \wedge g ,
\label{real706.dd}\\
R \cdot S
&=&
- \tau ^{-1} \beta _{1} \, Q(S, S^{2})
+ \tau ^{-1} ( \rho + (n-1) c \beta_{1} )\, Q(g, S^{2})\nonumber\\
& &
+ ( c - \tau ^{-1} \rho  \rho _{1}
- \tau ^{-1} \beta _{1} ( \rho _{0} + (n-1) c  \rho _{1}) )\, Q(g,S) ,
\label{2022.01.10.99new}\\
C \cdot S
&=&
- \tau ^{-1} \beta _{1}\, Q(S,S^{2})
+ \tau ^{-1} \beta _{4} \, Q(g,S^{2})\nonumber\\
& &
+
(\beta _{2} - \tau ^{-1} ( \beta _{1} \rho _{0} + \beta _{4} \rho _{1} ) )
\, Q(g,S)
\label{2022.01.10.j.dd}
\end{eqnarray}
on some neighbourhood $U \subset {\mathcal U}_{H}$ of $x$.
Furthermore, (\ref{realC6}) and (\ref{new04-real7088}) give
\begin{eqnarray}
H^{2} &=& \mathrm{tr}(H)\, H
+ \varepsilon (n-1) c\, g - \varepsilon\, S \nonumber\\
&=&
\tau^{-1} \mathrm{tr}(H)\, (S^{2} - \rho _{1}\, S - \rho _{0}\, g)
+ \varepsilon (n-1) c\, g - \varepsilon \, S  \nonumber\\
&=& \tau^{-1} \mathrm{tr}(H) \, S^{2}
- ( \tau^{-1} \mathrm{tr}(H) \rho _{1} + \varepsilon )\, S
+ (\varepsilon (n-1) c - \tau^{-1} \rho _{0} \mathrm{tr}(H) )\,
g \nonumber\\
&=&
\tau^{-1} \left(
 \mathrm{tr}(H) S^{2}
- ( \mathrm{tr}(H) \rho _{1} + \varepsilon \tau ) S
+ (\varepsilon (n-1) c \tau  - \rho _{0} \mathrm{tr}(H) )\, g \right) .
\label{new04-real7099}
\end{eqnarray}
Now
(\ref{new04-real7077}),
by making use of (\ref{new04-real7088}) and (\ref{new04-real7099}),
turns into
\begin{eqnarray*}
S^{3} &=& \tau _{1}\, S^{2} + \tau _{1}\, S + \tau _{0}\, g ,
\end{eqnarray*}
where $\tau _{0}, \tau _{1}, \tau _{2}$
are some functions on $U$.
Thus we have
\begin{theorem}
Let $M$ be a hypersurface in a semi-Riemannian space forms
$N_{s}^{n+1}(c)$, $n \geq 4$,
with the tensor $H$ satisfying on ${\mathcal U}_{H}$
equation (\ref{real702})
and let  $\tau$ be the function defined on ${\mathcal U}_{H}$
by (\ref{real705}). We have:
\newline
(i) if $\tau$ vanishes at a point
$x \in {\mathcal U}_{H}$
then $M$ is partially Einstein at $x$,
\newline
(ii)
if $\tau$ is non-zero at a point
$x \in {\mathcal U}_{H}$ then
(\ref{new04-real7088})--(\ref{2022.01.10.j.dd})
hold on some
neighbourhood $U \subset {\mathcal U}_{H}$ of $x$
and at every point of $U$ the tensor $S^{3}$
is a linear combination of the tensors
$g$, $S$ and $S^{2}$.
\end{theorem}

\begin{example}
(i)
Let $M$ be a hypersurface in a Riemannian space forms
$N^{n+1}(c)$, $n \geq 3$,
having at every point $x \in {\mathcal U}_{H} \subset M$
three distinct principal curvatures
$\lambda _{0}$, $\lambda _{1}$
and $\lambda _{2}$,
with multiplicities
$1$, $1$ and $(n-2)$,
respectively, i.e.,
$M$ is $2$-quasi-umbilical at every point
of ${\mathcal U}_{H}$
satisfying
(\ref{real702})
(see, e.g., \cite{{CHEN-VERSTRAELEN-1976}, {DeVerYap}, {V2}}).
Further, from (\ref{2022-01-17-a}) and (\ref{2022-01-17-c})
we get immediately
\begin{eqnarray}
\tau &=&
(n-2) ( \lambda _{0} + (n-3) \lambda _{2})
 ( \lambda _{1} + (n-3) \lambda _{2})  \lambda _{2} .
\label{2022.02.04.a}
\end{eqnarray}
(ii) Curvature properties of pseudosymmetry type of some particular
$2$-quasi-umbilical hypersurfaces $M$ in an Euclidean space
$\mathbb{E}^{n+1}$, $n \geq 4$, are given in
{\cite[Section 4] {DGP-TV02}}.
At every point of the set ${\mathcal U}_{H}$
of those hypersufaces
principal curvatures are the following
$\lambda _{0} = 0$, $\lambda _{1} = - (n-2) \lambda$
and $\lambda _{2} = \lambda$, where $\lambda \neq 0$.
Therefore (\ref{2022.02.04.a}) turns into
\begin{eqnarray*}
\tau &=& - (n-3) (n-2) \lambda ^{3} .
\end{eqnarray*}
Thus $M$ is a non-partially Einstein hypersurface.
\newline
(iii)
Let $M$ be a hypersurface in a Riemannian space forms
$N^{n+1}(c)$, $n \geq 4$,
having at every point $x \in {\mathcal U}_{H} \subset M$
three distinct principal curvatures
$\lambda _{0} = a$, $\lambda _{1} = b$ and $\lambda _{2} = a+b$,
with multiplicities $1$, $1$ and $n - 2$, respectively.
Now the function $\tau$ takes the form
\begin{eqnarray*}
\tau &=&
(n-2) ((n-2) a + (n-3) b)
 ( (n-3) a +  (n-2) b )  (a+b) .
\end{eqnarray*}
Evidently, if
$a \neq - b$ and
$(n-2) a \neq - (n-3) b$
and
$(n-3) a \neq -  (n-2) b$
at a point $x \in {\mathcal U}_{H}$
then $\tau \neq 0$ at $x$
and
$M$ is non-partially Einstein at $x$.
\end{example}

Let $M$ be a hypersurface in a Riemannian space forms
$N^{n+1}(c)$, $n \geq 4$,
having at every point $x \in {\mathcal U}_{H} \subset M$
three distinct principal curvatures
$\lambda _{0}$, $\lambda _{1}$ and $\lambda _{2}$,
with multiplicities $1$, $n_{1}$ and $n_{2}$, respectively.
We can check that
(\ref{real705}) and (\ref{tau-real702}) yield
(cf. {\cite[Section 7, eq. (7.22)] {2021-DGH}})
\begin{eqnarray}
\tau &=&
\left( \lambda _{0}
 + (n_{1} - 1) \lambda _{1} + (n_{2} - 1) \lambda _{2} \right)
\cdot \nonumber\\
& &\cdot \left(
\lambda _{1} \lambda_{2}
+ (n_{1} \lambda _{1} + n_{2} \lambda _{2})
((n_{1} - 1) \lambda _{1} + (n_{2} - 1) \lambda _{2} )
\right) .
\label{tau-real705}
\end{eqnarray}

\begin{example} (i)
Hypersurfaces isometrically immersed
in ${\mathbb{E}}^{n+1}$, $n \geq 5$,
having at every point three distinct principal curvatures $\lambda _{0}$,
$\lambda _{1}$ and $\lambda _{2}$,
with multiplicities $1$, $n_{1}$ and $n_{2}$, respectively,
were investigated among others in
\cite{{ABPRS}, {Fu}, {Sato-2002}, {Sato-Neto-2006}, {Turgay-2015}}.
\newline
(ii) Curvature properties of pseudosymmetry type
of hypersurfaces in ${\mathbb{E}}^{n+1}$, $n \geq 5$,
having at every point three distinct principal curvatures $\lambda _{0}$,
$\lambda _{1}$ and $\lambda _{2}$,
with multiplicities $1$, $n_{1} = n_{2} \geq 2$,
are given
in {\cite[Section 4] {Saw-2015}}.
Moreover, in {\cite[Section 5, Example 5.1] {Saw-2015}},
by an application of results of {\cite[Section 2] {Sato-2002}}
(see also {\cite[Section 2] {ABPRS}} or
{\cite[Section 2] {Sato-Neto-2006}})
a class of hypersurfaces
$M$ in ${\mathbb{E}}^{n+1}$, $n \geq 5$,
having at every point
three distinct non-zero principal curvatures $\lambda _{0}$,
$\lambda _{1}$ and $\lambda _{2}$,
with multiplicities $1$, $n_{1}$, $n_{2} \geq 2$, respectively,
was determined.
In that class of hypersurfaces there are hypersurfaces
satisfying:
$n_{1} \neq  n_{2}$,
$( n_{1} - 1 ) \lambda_{1} + ( n_{2} - 1 ) \lambda_{2} = 0$
and
$\phi = \lambda _{0} + \lambda _{1} + \lambda _{2} = \mathrm{tr} (H)$
(see {\cite[Section 5, Example 5.1 (ii)] {Saw-2015}}).
In addition,
(\ref{tau-real702})
and
(\ref{tau-real705})
yield
$\rho = \tau = \lambda_{0} \lambda_{1} \lambda_{2}$.
On certain hypersurfaces of that class of hypersurfaces we also have:
$n_{1} = n_{2}$,
$\lambda_{1} = - \lambda_{2}$ and
$\phi = \lambda _{0}  = \mathrm{tr} (H)$
(see {\cite[Section 5, Example 5.1 (iii)] {Saw-2015}}).
In addition,
(\ref{tau-real702})
and
(\ref{tau-real705})
yield
$\rho = \tau = - \lambda_{0} \lambda_{1}^{2}$.
\end{example}

\noindent
{\bf{Acknowledgments.}}
The first two authors of this paper are supported
by the Wroc\l aw University of Environmental and Life Sciences (Poland).

redpajama_set_name: RedPajamaArXiv