ON SEMI-RIEMANNIAN MANIFOLDS SATISFYING SOME GENERALIZED EINSTEIN METRIC
CONDITIONS

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

Dedicated to the memory of Professor Oldřich Kowalski (1936–2021)

Abstract. The difference tensor $R\cdot C-C\cdot R$ of a semi-Riemannian
manifold $(M,g)$, $\dim M\geq 4$, formed by its Riemann-Christoffel curvature
tensor $R$ and the Weyl conformal curvature tensor $C$, under some
assumptions, can be expressed as a linear combination of $(0,6)$-Tachibana
tensors $Q(A,T)$, where $A$ is a symmetric $(0,2)$-tensor and $T$ a
generalized curvature tensor. These conditions form a family of generalized
Einstein metric conditions. In this survey paper we present recent results on
manifolds and submanifolds, and in particular hypersurfaces, satisfying such
conditions.1112020 Mathematics Subject Classification. Primary 53B20, 53B25;
Secondary 53C25.
Keywords and phrases. Warped product manifold; Einstein, quasi-Einstein,
2-quasi-Einstein and partially Einstein manifold; generalized Einstein metric
condition; pseudosymmetry type curvature condition; hypersurface; Chen ideal
submanifold.

##  1\. Introduction

Let $(M,g)$ be a semi-Riemannian manifold. We denote by $g$, $\nabla$, $R$,
$S$, $\kappa$ and $C$, the metric tensor, the Levi-Civita connection, the
Riemann-Christoffel curvature tensor, the Ricci tensor, the scalar curvature
and the Weyl conformal curvature tensor of $(M,g)$, respectively. Now we can
define the $(0,2)$-tensors $S^{2}$ and $S^{3}$, the $(0,4)$-tensors $R\cdot
S$, $C\cdot S$ and $Q(A,B)$, and the $(0,6)$-tensors $R\cdot R$, $R\cdot C$,
$C\cdot R$, $C\cdot C$ and $Q(A,T)$, where $A$ and $B$ are symmetric
$(0,2)$-tensors and $T$ a generalized curvature tensor. Thus, in particular,
we have the difference tensor $R\cdot C-C\cdot R$, or $C\cdot R-R\cdot C$,
[68, Section 1]. Furthermore for $A$ and $B$ we define their Kulkarni-Nomizu
product $A\wedge B$. For precise definitions of the symbols used, we refer to
Section 2 of this paper, as well as to [51, Section 1], [55, Section 1], [56,
Chapter 6] and [68, Sections 1 and 2].

A semi-Riemannian manifold $(M,g)$, $\dim M=n\geq 2$, is said to be an
Einstein manifold [4], or an Einstein space, if at every point of $M$ its
Ricci tensor $S$ is proportional to $g$, i.e.,

(1.1) $\displaystyle S=\frac{\kappa}{n}\,g$

on $M$, assuming that the scalar curvature $\kappa$ is constant when $n=2$.
According to [4, p. 432] this condition is called the 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 [4, Table, pp. 432-433]. These conditions are called
generalized Einstein metric conditions [4, Chapter XVI].

Semi-Riemannian manifolds of dimension $\geq 4$ and in particular,
hypersurfaces in spaces of constant curvature or Chen ideal submanifolds in
Euclidean spaces, satisfying curvature conditions of the form

$(\ast)$ the difference tensor $R\cdot C-C\cdot R$ and a finite sum of the
Tachibana tensors of the form $Q(A,T)$ are linearly dependent,

where $A=g$, $A=S$, or $A=S^{2}$, and $T=R$, $T=C$, $T=g\wedge g$, $T=g\wedge
S$, $T=g\wedge S^{2}$, $T=S\wedge S$, $T=S\wedge S^{2}$, or $T=S^{2}\wedge
S^{2}$, were studied in several papers, see, e.g., [3, 16, 48, 49, 51, 54, 55,
59, 63, 68]. Conditions of this form are also generalized Einstein metric
conditions, see, e.g., [43, Section 6] or [68, Section 1]. We refer to [43]
for a survey of results on manifolds satisfying conditions of the form
$(\ast)$.

Precisely, in that paper results obtained to 2010 are presented. In the next
years studies on manifolds and submanifolds, and in particular hypersurfaces,
satisfying such conditions are continued. As it was already mentioned in
Abstract, in this paper we present a survey of recent results on that subject.

On every semi-Riemannian Einstein manifold $(M,g)$, $\dim M=n\geq 4$, the
following identity is satisfied [63, Theorem 3.1] (see also [68, p. 107])

$\displaystyle R\cdot C-C\cdot R=\frac{\kappa}{(n-1)n}\,Q(g,C).$

This, by (1.1) and
$\frac{\kappa}{(n-1)n}=\frac{\kappa}{n-1}-\frac{\kappa}{n}$, turns into

(1.2) $\displaystyle C\cdot R-R\cdot C=Q(S,C)-\frac{\kappa}{n-1}\,Q(g,C).$

We mention that there are non-Einstein and non-conformally flat semi-
Riemannian manifolds satisfying (1.2). Namely, every Roter space satisfies
(1.2) (see Section 4).

Let $(M,g)$ be a semi-Riemannian manifold of dimension $n\geq 3$. We define
the subsets ${\mathcal{U}}_{R}$ and ${\mathcal{U}}_{S}$ of $M$ by
${\mathcal{U}}_{R}=\\{x\in M\,|\,R-\frac{\kappa}{2(n-1)n}\,g\wedge g\neq 0\
\mbox{at}\ x\\}$ and ${\mathcal{U}}_{S}=\\{x\in
M\,|\,S-\frac{\kappa}{n}\,g\neq 0\ \mbox{at}\ x\\}$, respectively. 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 $C\neq 0$. We note that on any semi-Riemannian
manifold $(M,g)$, $\dim M=n\geq 4$, we have (see, e.g., [40])

(1.3) $\displaystyle{\mathcal{U}}_{S}\cup{\mathcal{U}}_{C}$ $\displaystyle=$
$\displaystyle{\mathcal{U}}_{R}.$

In Section 3 we present a survey on semi-Riemannian manifolds satisfying
curvature conditions known as pseudosymmetry type curvature conditions. In
particular, curvature conditions of the form $(\ast)$ are conditions of this
kind.

An extension of the class of Einstein manifolds also form quasi-Einstein,
$2$-quasi-Einstein and partially Einstein manifolds. These manifolds satisfy
some pseudosymmetry type curvature conditions (see Section 4).

In Section 5 we consider warped product manifolds
$\overline{M}\times_{F}\widetilde{N}$ with a $2$-dimensional semi-Riemannian
manifold $(\overline{M},\overline{g})$, a warping function $F$ and an
$(n-2)$-dimensional semi-Riemannian manifold $(\widetilde{N},\widetilde{g})$,
$n\geq 4$, assuming that $(\widetilde{N},\widetilde{g})$ is a space of
constant curvature when $n\geq 5$. Such manifolds satisfy some pseudosymmetry
type curvature conditions (see Theorems 5.1 and 5.2). We mention that the
warped product manifolds $\overline{M}\times_{F}\widetilde{N}$, with a
$2$-dimensional Riemannian manifold $(\overline{M},\overline{g})$ and an
$(n-2)$-dimensional unit sphere ${\mathbb{S}}^{n-2}$, $n\geq 4$, where the
warping function $F$ is a solution of some second order quasilinear elliptic
partial differential equation in the plane are related to Chen ideal
submanifolds (see Section 11 for details).

Section 6 contains results on semi-Riemannian manifolds $(M,g)$ of dimension
$\geq 4$ satisfying the following generalized Einstein metric condition on
${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}\subset M$

(1.4) $\displaystyle R\cdot C-C\cdot R$ $\displaystyle=$ $\displaystyle
L\,Q(S,C),$

where $L$ is some function on this set.

Let $N_{s}^{n+1}(c)$, $n\geq 4$, 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 $M$ be a connected hypersurface isometrically immersed in
$N_{s}^{n+1}(c)$. We denote by ${\mathcal{U}}_{H}\subset M$ 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 can verify that
${\mathcal{U}}_{H}\subset{\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}\subset M$
(see, e.g., [81, p. 137]). We note that

(1.5) $\displaystyle H^{2}$ $\displaystyle=$ $\displaystyle\alpha\,H+\beta\,g$

on $({\mathcal{U}}_{S}\cap{\mathcal{U}}_{C})\setminus{\mathcal{U}}_{H}$, where
$\alpha$ and $\beta$ are some functions defined on this set.

As it was stated in [47, Corollary 4.1], for a hypersurface $M$ in
$N^{n+1}_{s}(c)$, $n\geq 4$, if at every point of ${\mathcal{U}}_{H}\subset M$
one of the tensors $R\cdot C$, $C\cdot R$ or $R\cdot C-C\cdot R$ is a linear
combination of the tensor $R\cdot R$ and a finite sum of the Tachibana tensors
of the form $Q(A,T)$, where $A$ is a symmetric $(0,2)$-tensor and $T$ a
generalized curvature tensor, then the following equation is satisfied on
${\mathcal{U}}_{H}$

(1.6) $\displaystyle H^{3}$ $\displaystyle=$
$\displaystyle\mathrm{tr}(H)\,H^{2}+\psi\,H+\rho\,g,$

where $\psi$ and $\rho$ are some functions on ${\mathcal{U}}_{H}$. We also
mention that if the condition

$\displaystyle R\cdot C-C\cdot R$ $\displaystyle=$ $\displaystyle Q(g,T),$

where $T$ is a generalized curvature tensor, holds on the set
${\mathcal{U}}_{H}$ of a hypersurface $M$ in $N^{n+1}_{s}(c)$, $n\geq 4$, then
the tensor $T$ is a linear combination of the tensors: $R$, $g\wedge g$,
$g\wedge S$, $S\wedge S$ and $g\wedge S^{2}$ [54, Theorem 5.2 (iii)].

In Section 7 we present results on $2$-quasi-umbilical hypersurfaces in
$N_{s}^{n+1}(c)$, $n\geq 4$, contained in [53]. In particular (see Theorem
7.3), we present curvature properties of a minimal $2$-quasi-umbilical
hypersurface $M$ in an Euclidean space $\mathbb{E}^{n+1}$, $n\geq 4$, having
exactly three distinct principal curvatures $\lambda_{1}$, $\lambda_{2}$ and
$\lambda_{3}$ satisfying at every point of $M$: $\lambda_{1}=0$,
$\lambda_{2}=-(n-2)\lambda$ and
$\lambda_{3}=\lambda_{4}=\ldots=\lambda_{n}=\lambda\neq 0$, where $\lambda$ is
a non-zero function on $M$. It is easy to check that $\mathrm{tr}(H)=0$ and

(1.7) $\displaystyle H^{3}$ $\displaystyle=$
$\displaystyle\phi\,H^{2}+\psi\,H$

on ${\mathcal{U}}_{H}\subset M$, where $\phi=-(n-3)\lambda$ and
$\psi=(n-2)\lambda^{2}$. Because $\mathrm{tr}(H)\neq\phi$ at every point of
${\mathcal{U}}_{H}$, (1.6) is not satisfied on ${\mathcal{U}}_{H}$. Now, in
view of the presented above result (i.e., [47, Corollary 4.1]), we conclude
that the difference tensor $R\cdot C-C\cdot R$ of the considered hypersurface
$M$ cannot be expressed on ${\mathcal{U}}_{H}$ as a linear combination of
certain Tachibana tensors of the form $Q(A,T)$, where $A$ is a symmetric
$(0,2)$-tensor and $T$ a generalized curvature tensor. The tensor $R\cdot
C-C\cdot R$ of that hypersurface we can express, for instance, by (7.7). In
this section we also present results on type number two hypersurfaces in
$N_{s}^{n+1}(c)$, $n\geq 3$.

Results on hypersurfaces in $N_{s}^{n+1}(c)$, $n\geq 4$, satisfying some
special generalized Einstein metric conditions of the form $(\ast)$ are given
in Sections 8, 9 and 10. In Section 8 we present results on hypersurfaces $M$
in a semi-Riemannian space of constant curvature $N_{s}^{n+1}(c)$, $n\geq 4$,
satisfying (1.6) on ${\mathcal{U}}_{H}\subset M$, for some functions $\psi$
and $\rho$ on ${\mathcal{U}}_{H}$. Among other things it was stated that some
generalized Einstein metric condition of the form $(\ast)$ is satisfied on
${\mathcal{U}}_{H}$ (see Theorem 8.3). Sections 9 and 10 contain results on
quasi-Einstein and non-quasi-Einstein hypersurfaces in semi-Riemannian spaces
of constant curvature $N_{s}^{n+1}(c)$, $n\geq 4$, satisfying on
${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}\subset M$ the conditions: (1.4) and

(1.8) $\displaystyle R\cdot C-C\cdot R$ $\displaystyle=$ $\displaystyle
L_{1}\,Q(S,C)+L_{2}\,Q(g,C),$

where $L$, $L_{1}$ and $L_{2}$ are some functions defined on this set.

In Section 11 we present results on non-Einstein and non-conformally flat Chen
ideal submanifolds satisfying some generalized Einstein metric conditions of
the form $(\ast)$. For instance, Theorems 11.5, 11.8 and 11.11 (see Theorems
5, 6 and 7 of [68]) contain results on Chen ideal submanifolds $M$ in
$\mathbb{E}^{n+m}$, $n\geq 4$, $m\geq 1$, satisfying the following conditions
of the form $(\ast)$: (1.8),

(1.9) $\displaystyle R\cdot C-C\cdot R$ $\displaystyle=$ $\displaystyle
L_{3}\,Q(g,R)+L_{4}\,Q(S,R),$ (1.10) $\displaystyle R\cdot C-C\cdot R$
$\displaystyle=$ $\displaystyle L_{5}\,Q(g,g\wedge S)+L_{6}\,Q(S,g\wedge S),$

respectively, where $L_{3}$, $L_{4}$, $L_{5}$ and $L_{6}$ are some functions
defined on $M$.

##  2\. 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

$\displaystyle(X\wedge_{A}Y)Z$ $\displaystyle=$ $\displaystyle
A(Y,Z)X-A(X,Z)Y,$ $\displaystyle{\mathcal{R}}(X,Y)Z$ $\displaystyle=$
$\displaystyle\nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{[X,Y]}Z,$

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

$\displaystyle S(X,Y)\ =\ \mathrm{tr}\\{Z\rightarrow{\mathcal{R}}(Z,X)Y\\},\ \
g({\mathcal{S}}X,Y)\ =\ S(X,Y),\ \ \kappa\ =\ \mathrm{tr}\,{\mathcal{S}},$

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

$\displaystyle{\mathcal{C}}(X,Y)Z$ $\displaystyle=$
$\displaystyle{\mathcal{R}}(X,Y)Z-\frac{1}{n-2}(X\wedge_{g}{\mathcal{S}}Y+{\mathcal{S}}X\wedge_{g}Y-\frac{\kappa}{n-1}X\wedge_{g}Y)Z.$

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

$\displaystyle G(X_{1},X_{2},X_{3},X_{4})$ $\displaystyle=$ $\displaystyle
g((X_{1}\wedge_{g}X_{2})X_{3},X_{4}),$ $\displaystyle
R(X_{1},X_{2},X_{3},X_{4})$ $\displaystyle=$ $\displaystyle
g({\mathcal{R}}(X_{1},X_{2})X_{3},X_{4}),$ $\displaystyle
C(X_{1},X_{2},X_{3},X_{4})$ $\displaystyle=$ $\displaystyle
g({\mathcal{C}}(X_{1},X_{2})X_{3},X_{4}),$

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

(2.1) $\displaystyle B(X_{1},X_{2},X_{3},X_{4})$ $\displaystyle=$
$\displaystyle g({\mathcal{B}}(X_{1},X_{2})X_{3},X_{4}).$

The tensor $B$ is said to be a generalized curvature tensor if the following
two conditions are fulfilled:

$\displaystyle B(X_{1},X_{2},X_{3},X_{4})\ =\ B(X_{3},X_{4},X_{1},X_{2}),$
$\displaystyle
B(X_{1},X_{2},X_{3},X_{4})+B(X_{2},X_{3},X_{1},X_{4})+B(X_{3},X_{1},X_{2},X_{4})\
=\ 0.$

For ${\mathcal{B}}$ as above, let $B$ be again defined by (2.1). 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

$\displaystyle(B\cdot T)(X_{1},\ldots,X_{k},X,Y)\ =\ ({\mathcal{B}}(X,Y)\cdot
T)(X_{1},\ldots,X_{k})$ $\displaystyle=$
$\displaystyle-T({\mathcal{B}}(X,Y)X_{1},X_{2},\ldots,X_{k})-\cdots-T(X_{1},\ldots,X_{k-1},{\mathcal{B}}(X,Y)X_{k}).$

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

$\displaystyle Q(A,T)(X_{1},\ldots,X_{k},X,Y)\ =\ (X\wedge_{A}Y\cdot
T)(X_{1},\ldots,X_{k})$ $\displaystyle=$
$\displaystyle-T((X\wedge_{A}Y)X_{1},X_{2},\ldots,X_{k})-\cdots-T(X_{1},\ldots,X_{k-1},(X\wedge_{A}Y)X_{k}).$

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 $E$ and a $(0,k)$-tensor $T$, $k\geq 2$, we
define their Kulkarni-Nomizu tensor $E\wedge T$ by (see, e.g., [40, Section
2])

$\displaystyle(E\wedge T)(X_{1},X_{2},X_{3},X_{4};Y_{3},\ldots,Y_{k})$
$\displaystyle=$ $\displaystyle
E(X_{1},X_{4})T(X_{2},X_{3},Y_{3},\ldots,Y_{k})+E(X_{2},X_{3})T(X_{1},X_{4},Y_{3},\ldots,Y_{k})$
$\displaystyle-E(X_{1},X_{3})T(X_{2},X_{4},Y_{3},\ldots,Y_{k})-E(X_{2},X_{4})T(X_{1},X_{3},Y_{3},\ldots,Y_{k}).$

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

(2.2) $\displaystyle C$ $\displaystyle=$ $\displaystyle
R-\frac{1}{n-2}\,g\wedge S+\frac{\kappa}{(n-2)(n-1)}\,G,$ (2.3) $\displaystyle
G$ $\displaystyle=$ $\displaystyle\frac{1}{2}\,g\wedge g,$

and (see, e.g., [40, Lemma 2.2(i)])

(2.4) $\displaystyle(a)$ $\displaystyle\ \ Q(E,E\wedge F)\ =\
-\frac{1}{2}\,Q(F,E\wedge E),$ $\displaystyle(b)$ $\displaystyle\ \ E\wedge
Q(E,F)\ =\ -\frac{1}{2}\,Q(F,E\wedge E).$

By an application of (2.4)(a) we obtain on $M$ the identities

(2.5) $\displaystyle Q(g,g\wedge S)$ $\displaystyle=$ $\displaystyle-Q(S,G),\
\ Q(S,g\wedge S)\ =\ -\frac{1}{2}\,Q(g,S\wedge S).$

Further, by making use of (2.2), (2.3) and (2.5), we get immediately

$\displaystyle Q(g,C)$ $\displaystyle=$ $\displaystyle
Q(g,R)-\frac{1}{n-2}\,Q(g,g\wedge S)+\frac{\kappa}{(n-2)(n-1)}\,Q(g,G)$
$\displaystyle=$ $\displaystyle Q(g,R)-\frac{1}{n-2}\,Q(g,g\wedge S),$
$\displaystyle Q(S,C)$ $\displaystyle=$ $\displaystyle
Q(S,R)-\frac{1}{n-2}\,Q(S,g\wedge S)+\frac{\kappa}{(n-2)(n-1)}\,Q(S,G)$
$\displaystyle=$ $\displaystyle Q(S,R)+\frac{1}{2(n-2)}\,Q(g,S\wedge
S)-\frac{\kappa}{(n-2)(n-1)}\,Q(g,g\wedge S).$

Let $E_{1}$, $E_{2}$ and $F$ be symmetric $(0,2)$-tensors. We have (see, e.g.,
[16, Lemma 2.1(i)] and references therein)

(2.6) $\displaystyle E_{1}\wedge Q(E_{2},F)+E_{2}\wedge
Q(E_{1},F)+Q(F,E_{1}\wedge E_{2})$ $\displaystyle=$ $\displaystyle 0.$

From (2.6) we get easily (see also [40, Lemma 2.2(iii)] and references
therein)

$\displaystyle Q(F,E\wedge E_{1})+Q(E,E_{1}\wedge F)+Q(E_{1},F\wedge E)$
$\displaystyle=$ $\displaystyle 0.$

We denote by $g_{ij}$, $g^{ij}$, $R_{hijk}$, $S_{ij}$, $C_{hijk}$, $(R\cdot
S)_{hklm}$, $Q(g,S)_{hklm}$, $Q(g,R)_{hijklm}$, $Q(S,R)_{hijklm}$, $(R\cdot
C)_{hijklm}$ and $(C\cdot R)_{hijklm}$, the local components of the tensors
$g$, $g^{-1}$, $R$, $S$, $C$, $R\cdot S$, $Q(g,S)$, $Q(g,R)$, $Q(S,R)$,
$R\cdot C$ and $C\cdot R$, respectively. On every semi-Riemannian manifold
$(M,g)$, $n\geq 4$, the following identity is satisfied (see, e.g., [43,
Section 2])

(2.7) $\displaystyle(n-2)(R\cdot C-C\cdot R)$ $\displaystyle=$ $\displaystyle
Q(S,R)-\frac{\kappa}{n-1}\,Q(g,R)-g\wedge(R\cdot S)+P,$

where the $(0,6)$-tensor $P$ is defined by

$\displaystyle P(X_{1},X_{2},X_{3},X_{4};X,Y)$ $\displaystyle=$ $\displaystyle
g(X,X_{1})\,R({\mathcal{S}}Y,X_{2},X_{3},X_{4})-g(Y,X_{1})\,R({\mathcal{S}}X,X_{2},X_{3},X_{4})$
$\displaystyle+g(X,X_{2})R(X_{1},{\mathcal{S}}Y,X_{3},X_{4})-g(Y,X_{2})\,R(X_{1},{\mathcal{S}}X,X_{3},X_{4})$
$\displaystyle+g(X,X_{3})\,R(X_{1},X_{2},{\mathcal{S}}Y,X_{4})-g(Y,X_{3})\,R(X_{1},X_{2},{\mathcal{S}}X,X_{4})$
$\displaystyle+g(X,X_{4})\,R(X_{1},X_{2},X_{3},{\mathcal{S}}Y)-g(Y,X_{4})\,R(X_{1},X_{2},X_{3},{\mathcal{S}}X),$

and $X_{1},X_{2},X_{3},X_{4},X,Y$ are vector fields on $M$. The local
expression of the identity (2.7) reads

$\displaystyle(n-2)(R\cdot C-C\cdot R)_{hijklm}$ $\displaystyle=$
$\displaystyle Q(S,R)_{hijklm}-\frac{\kappa}{n-1}\,Q(g,R)_{hijklm}$
$\displaystyle+g_{hl}A_{mijk}-g_{hm}A_{lijk}-g_{il}A_{mhjk}+g_{im}A_{lhjk}$
$\displaystyle+g_{jl}A_{mkhi}-g_{jm}A_{lkhi}-g_{kl}A_{mjhi}+g_{km}A_{ljhi}$
$\displaystyle-g_{ij}(R\cdot S)_{hklm}-g_{hk}(R\cdot S)_{ijlm}+g_{ik}(R\cdot
S)_{hjlm}+g_{hj}(R\cdot S)_{iklm},$

where $A_{mijk}=g^{rs}S_{mr}R_{sijk}$ and

$\displaystyle(R\cdot S)_{hklm}$ $\displaystyle=$ $\displaystyle
g^{rs}(S_{hr}R_{sklm}+S_{kr}R_{shlm}),$ $\displaystyle Q(g,S)_{hklm}$
$\displaystyle=$ $\displaystyle
g_{hl}S_{km}+g_{kl}S_{hm}-g_{hm}S_{kl}-g_{km}S_{hl},$ $\displaystyle
Q(g,R)_{hijklm}$ $\displaystyle=$ $\displaystyle
g_{hl}R_{mijk}+g_{il}R_{hmjk}+g_{jl}R_{himk}+g_{kl}R_{hijm}$ $\displaystyle-
g_{hm}R_{lijk}-g_{im}R_{hljk}-g_{jm}R_{hilk}-g_{km}R_{hijl},$ $\displaystyle
Q(S,R)_{hijklm}$ $\displaystyle=$ $\displaystyle
S_{hl}R_{mijk}+S_{il}R_{hmjk}+S_{jl}R_{himk}+S_{kl}R_{hijm}$ $\displaystyle-
S_{hm}R_{lijk}-S_{im}R_{hljk}-S_{jm}R_{hilk}-S_{km}R_{hijl},$
$\displaystyle(R\cdot C)_{hijklm}$ $\displaystyle=$ $\displaystyle
g^{rs}(C_{rijk}R_{shlm}+C_{hrjk}R_{silm}+C_{hirk}R_{sjlm}+C_{hijr}R_{sklm}),$
$\displaystyle(C\cdot R)_{hijklm}$ $\displaystyle=$ $\displaystyle
g^{rs}(R_{rijk}C_{shlm}+R_{hrjk}C_{silm}+R_{hirk}C_{sjlm}+R_{hijr}C_{sklm}).$

Let $A$ be a symmetric $(0,2)$-tensor on a semi-Riemannian manifold $(M,g)$,
$\dim M=n\geq 3$. Let $A_{ij}$ be the local components of the tensor $A$.
Further, let $A^{2}$ and $A^{3}$ be the $(0,2)$-tensors with the local
components $A^{2}_{ij}=A_{ir}g^{rs}A_{sj}$ and
$A^{3}_{ij}=A^{2}_{ir}g^{rs}A_{sj}$, respectively. We have
$\mathrm{tr}_{g}(A)=\mathrm{tr}(A)=g^{rs}A_{rs}$,
$\mathrm{tr}_{g}(A^{2})=\mathrm{tr}(A^{2})=g^{rs}A^{2}_{rs}$ and
$\mathrm{tr}_{g}(A^{3})=\mathrm{tr}(A^{3})=g^{rs}A^{3}_{rs}$. We denote by
${\mathcal{U}}_{A}$ the set of points of $M$ at which
$A\neq\frac{\mathrm{tr}(A)}{n}\,g$.

######  Proposition 2.1.

Let $A$ be a symmetric $(0,2)$-tensor on a semi-Riemannian manifold $(M,g)$,
$\dim M=n\geq 3$, such that $\mathrm{rank}A=2$ on ${\mathcal{U}}_{A}\subset
M$. Then on this set we have [36, Lemma 2.1; eqs. (2.6) and (2.10)]

(2.8) $\displaystyle A^{3}$ $\displaystyle=$
$\displaystyle\mathrm{tr}(A)\,A^{2}+\frac{\mathrm{tr}(A^{2})-(\mathrm{tr}(A))^{2}}{2}\,A$
(2.9) $\displaystyle A\wedge A^{2}$ $\displaystyle=$
$\displaystyle\frac{\mathrm{tr}(A)}{2}\,A\wedge A.$

Moreover, the following identity is satisfied on ${\mathcal{U}}_{A}$

(2.10) $\displaystyle Q(A^{2},\frac{1}{2}\,A\wedge A)\ =\ -Q(A,A\wedge A^{2})\
=\ -\frac{\mathrm{tr}(A)}{2}\,Q(A,A\wedge A)\ =\ 0.$

######  Proposition 2.2.

Let $A$ be a symmetric $(0,2)$-tensor on a semi-Riemannian manifold $(M,g)$,
$\dim M=n\geq 4$.
(i) ( [44, Lemma 2.1], see also [42, Proposition 2.1 (i)]) If the following
condition is satisfied on ${\mathcal{U}}_{A}\subset M$

$\displaystyle\mathrm{rank}(A-\alpha\,g)=1$

then

(2.11) $\displaystyle g\wedge A^{2}+\frac{n-2}{2}\,A\wedge
A-\mathrm{tr}(A)\,g\wedge
A+\frac{(\mathrm{tr}(A))^{2}-\mathrm{tr}(A^{2})}{2(n-1)}\,g\wedge g=0$

and

$\displaystyle
A^{2}-\frac{\mathrm{tr}(A^{2})}{n}=(\mathrm{tr}(A)-(n-2)\alpha)\left(A-\frac{\mathrm{tr}(A)}{n}\,g\right)$

on ${\mathcal{U}}_{A}$, where $\alpha$ is some function on
${\mathcal{U}}_{A}$.
(ii) [42, Proposition 2.1 (ii)] If (2.11) is satisfied on
${\mathcal{U}}_{A}\subset M$ then

$\displaystyle
A^{2}-\frac{\mathrm{tr}(A^{2})}{n}\,g=\rho\left(A-\frac{\mathrm{tr}(A)}{n}\,g\right)$

and

$\displaystyle\left(A-\frac{\mathrm{tr}(A)-\rho}{n-2}\,g\right)\wedge\left(A-\frac{\mathrm{tr}(A)-\rho}{n-2}\,g\right)=0$

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

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

(2.12) $\displaystyle E=g\wedge S^{2}+\frac{n-2}{2}\,S\wedge S-\kappa\,g\wedge
S+\frac{\kappa^{2}-\mathrm{tr}(S^{2})}{2(n-1)}\,g\wedge g.$

It is easy to check that the tensor $E$ is a generalized curvature tensor. In
the same way, we define the $(0,4)$-tensor $E(A)$ corresponding to a symmetric
$(0,2)$-tensor $A$ [42, eq. (1.7])

(2.13) $\displaystyle E(A)=g\wedge A^{2}+\frac{n-2}{2}\,A\wedge
A-\mathrm{tr}_{g}(A)\,g\wedge
A+\frac{(\mathrm{tr}(A))^{2}-\mathrm{tr}(A^{2})}{2(n-1)}\,g\wedge g.$

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 [40, Section 2], [41, Section 3]
or [48, Section 3] for definitions of the considered tensors. In particular,
we have

$\displaystyle\mathrm{Weyl}(T)=T-\frac{1}{n-2}\,g\wedge\mathrm{Ric}(T)+\frac{\kappa(T)}{2(n-2)(n-1)}\,g\wedge
g.$

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 (2.13). 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
$T=g\wedge\mathrm{Ric}(T)-(\kappa(T)/4)g\wedge g$ (see [42, Section 2, p. 383]
and references therein). 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
$E=0$.

######  Proposition 2.3.

[42, Proposition 2.2] 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 $x\in M$

$\displaystyle T=\alpha_{1}\,R+\frac{\alpha_{2}}{2}\,S\wedge
S+\alpha_{3}\,g\wedge S+\alpha_{4}\,g\wedge
S^{2}+\frac{\alpha_{5}}{2}\,g\wedge g$

then

$\displaystyle\mathrm{Weyl}(T)=\alpha_{1}\,C+\frac{\alpha_{2}}{n-2}\,E$

at this point, where the tensor $E$ is defined by (2.12) and
$\alpha_{1},\ldots,\alpha_{5}\in\mathbb{R}$.

According to [36], a generalized curvature tensor $T$ on a semi-Riemannian
manifold $(M,g)$, $\dim M=n\geq 4$, is called a Roter type tensor if

(2.14) $\displaystyle T$ $\displaystyle=$
$\displaystyle\frac{\phi}{2}\,Ric(T)\wedge Ric(T)+\mu\,g\wedge Ric(T)+\eta\,G$

on ${\mathcal{U}}_{Ric(T)}\cap{\mathcal{U}}_{Weyl(T)}$, where $\phi$, $\mu$
and $\eta$ are some functions on this set. Manifolds admitting Roter type
tensors were investigated (e.g.) in [91]. We have

######  Proposition 2.4.

Let $T$ be a generalized curvature tensor on a semi-Riemannian manifold
$(M,g)$, $\dim M=n\geq 4$, satisfying (2.14) on
${\mathcal{U}}_{Ric(T)}\cap{\mathcal{U}}_{Weyl(T)}\subset M$. Then the
following relations hold on this set
(i) [48, Proposition 3.2]

$\displaystyle\ \ \ (Ric(T))^{2}\ =\ \alpha_{1}\,Ric(T)+\alpha_{2}\,g,$
$\displaystyle\ \ \ \alpha_{1}\ =\ \kappa(T)+\phi^{-1}((n-2)\mu-1),\ \ \
\alpha_{2}\ =\ \phi^{-1}(\mu\kappa(T)+(n-1)\eta),$ $\displaystyle(a)$
$\displaystyle\ \ \ T\cdot T\ =\ L_{T}\,Q(g,T),\ \ \ L_{T}\ =\
\phi^{-1}\left((n-2)(\mu^{2}-\phi\eta)-\mu\right),$ $\displaystyle(b)$
$\displaystyle\ \ \ T\cdot Weyl(T)\ =\ L_{T}\,Q(g,Weyl(T)),$
$\displaystyle(c)$ $\displaystyle\ \ \ T\cdot T\ =\
Q(Ric(T),T)+(L_{T}+\phi^{-1}\mu)\,Q(g,Weyl(T)),$ (2.15) $\displaystyle
Weyl(T)\cdot T$ $\displaystyle=$ $\displaystyle L_{Weyl(T)}\,Q(g,T),\ \ \
L_{Weyl(T)}\ =\
L_{T}+\frac{1}{n-2}\left(\frac{\kappa(T)}{n-1}-\alpha_{1}\right),$ (2.16)
$\displaystyle Weyl(T)\cdot Weyl(T)$ $\displaystyle=$ $\displaystyle
L_{Weyl(T)}\,Q(g,Weyl(T)),$ $\displaystyle Weyl(T)\cdot T$ $\displaystyle=$
$\displaystyle
Q(Ric(T),Weyl(T))+\left(L_{T}-\frac{\kappa(T)}{n-1}\right)Q(g,Weyl(T)).$

We also have

$\displaystyle T\cdot Weyl(T)-Weyl(T)\cdot T\ =\
\left(\frac{(n-1)\mu-1}{(n-2)\phi}+\frac{\kappa(T)}{n-1}\right)Q(g,T)$
$\displaystyle+\frac{1}{n-2}\,Q(Ric(T),T)+\frac{\mu((n-1)\mu-1)-(n-1)\phi\eta}{(n-2)\phi}\,Q(Ric(T),G),$

and, equivalently,

$\displaystyle T\cdot Weyl(T)-Weyl(T)\cdot T$ $\displaystyle=$
$\displaystyle\left(\phi^{-1}\left(\mu-\frac{1}{n-2}\right)+\frac{\kappa(T)}{n-1}\right)Q(g,T)$
$\displaystyle+\left(\phi^{-1}\mu\left(\mu-\frac{1}{n-2}\right)-\eta\right)Q(Ric(T),G).$

(ii) [91, Sections 1 and 4]

$\displaystyle Q(Ric(T),Weyl(T))$ $\displaystyle=$
$\displaystyle\phi^{-1}\left(\frac{1}{n-2}-\mu\right)Q(g,T)$
$\displaystyle+\frac{1}{n-2}\left(L_{T}-\frac{\kappa(T)}{n-1}\right)Q(g,g\wedge
Ric(T)).$

Moreover, if $L_{T}\,=\,\frac{\kappa(T)}{n-1}$, resp., $\kappa(T)\,=\,0$, then
we have

$\displaystyle Q(Ric(T),Weyl(T))$ $\displaystyle=$ $\displaystyle
L_{Weyl(T)}\,Q(g,T),$

and

$\displaystyle T\cdot Weyl(T)-Weyl(T)\cdot T$ $\displaystyle=$
$\displaystyle-Q(Ric(T),Weyl(T)),$

respectively. Moreover, if $L_{Weyl(T)}\,=\,0$ at a point of
${\mathcal{U}}_{Ric(T)}\cap{\mathcal{U}}_{Weyl(T)}$ then at this point we have

$\displaystyle\phi^{-1}\left(\frac{1}{n-2}-\mu\right)\,(T\cdot
Weyl(T)-Weyl(T)\cdot T)$ $\displaystyle=$
$\displaystyle(n-2)\left(\phi^{-1}\mu\left(\mu-\frac{1}{n-2}\right)-\eta\right)Q(Ric(T),Weyl(T)).$

For further results on generalizing curvature tensors satisfying some
conditions we refer to [41, Section 3].

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., [54]). 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 (2.4)(a) and (2.5), 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 $M$.

According to [51, eq. (1.13), Theorem 3.4 (i)] the following identity is
satisfied on any semi-Riemannian manifold $(M,g)$ of dimension $n\geq 4$

(2.17) $\displaystyle C\cdot R+R\cdot C$ $\displaystyle=$ $\displaystyle
R\cdot R+C\cdot C-\frac{1}{(n-2)^{2}}\,Q\left(g,g\wedge
S^{2}-\frac{\kappa}{n-1}\,g\wedge S\right).$

From (2.17), by a suitable contraction, we get (cf. [53, Lemma 2.3], [61, p.
217])

$\displaystyle C\cdot S$ $\displaystyle=$ $\displaystyle R\cdot
S-\frac{1}{n-2}Q\left(g,S^{2}-\frac{\kappa}{n-1}S\right).$

Let $(\overline{M},\overline{g})$ and $(\widetilde{N},\widetilde{g})$,
$\dim\overline{M}=p$, $\dim\widetilde{N}=n-p$, $1\leq p<n$, be semi-Riemannian
manifolds covered by systems of charts $\\{U;x^{a}\\}$ and
$\\{V;y^{\alpha}\\}$, respectively. Let $F$ be a positive smooth function on
$\overline{M}$. It is well known that the warped product
$\overline{M}\times_{F}\widetilde{N}$ of $(\overline{M},\overline{g})$ and
$(\widetilde{N},\widetilde{g})$ is the product manifold
$\overline{M}\times\widetilde{N}$ with the metric
$g=\overline{g}\times_{F}\widetilde{g}$ defined by (see, e.g., [104, 107])

$\displaystyle\overline{g}\times_{F}\widetilde{g}$ $\displaystyle=$
$\displaystyle{\pi}_{1}^{*}\overline{g}+(F\circ{\pi}_{1})\,{\pi}_{2}^{*}\widetilde{g},$

where ${\pi}_{1}:\overline{M}\times\widetilde{N}\longrightarrow\overline{M}$
and ${\pi}_{2}:\overline{M}\times\widetilde{N}\longrightarrow\widetilde{N}$
are the natural projections on $\overline{M}$ and $\widetilde{N}$,
respectively. Let $\\{U\times
V;x^{1},\ldots,x^{p},x^{p+1}=y^{1},\ldots,x^{n}=y^{n-p}\\}$ be a product chart
for $\overline{M}\times\widetilde{N}$. The local components $g_{ij}$ of the
metric $g=\overline{g}\times_{F}\widetilde{g}$ with respect to this chart are
the following $g_{ij}=\overline{g}_{ab}$ if $i=a$ and $j=b$,
$g_{ij}=F\,\widetilde{g}_{\alpha\beta}$ if $i=\alpha$ and $j=\beta$, and
$g_{ij}=0$ otherwise, where $a,b,c,d,f\in\\{1,\ldots,p\\}$,
$\alpha,\beta,\gamma,\delta\in\\{p+1,\ldots,n\\}$ and
$h,i,j,k,r,s\in\\{1,2,\ldots,n\\}$. We will denote by bars (resp., by tildes)
tensors formed from $\overline{g}$ (resp., $\widetilde{g}$). The local
components

$\displaystyle\Gamma^{h}_{ij}\ =\
\frac{1}{2}\,g^{hs}(\partial_{i}g_{js}+\partial_{j}g_{is}-\partial_{s}g_{ij}),\
\ \partial_{j}\ =\ \frac{\partial}{\partial x^{j}},$

of the Levi-Civita connection $\nabla$ of
$\overline{M}\times_{F}\widetilde{N}$ are the following (see, e.g., [104]):

$\displaystyle\Gamma^{a}_{bc}\ =\ \overline{\Gamma}^{a}_{bc},\ \
\Gamma^{\alpha}_{\beta\gamma}\ =\ \widetilde{\Gamma}^{\alpha}_{\beta\gamma},\
\ \Gamma^{a}_{\alpha\beta}\ =\
-\frac{1}{2}\bar{g}^{ab}F_{b}\widetilde{g}_{\alpha\beta},\ \
\Gamma^{\alpha}_{a\beta}\ =\ \frac{1}{2F}F_{a}\delta^{\alpha}_{\beta},$
$\displaystyle\Gamma^{a}_{\alpha b}\ =\ \Gamma^{\alpha}_{ab}\ =\ 0,\ \ F_{a}\
=\ \partial_{a}F\ =\ \frac{\partial F}{\partial x^{a}},\ \ \partial_{a}\ =\
\frac{\partial}{\partial x^{a}}.$

The local components

$\displaystyle R_{hijk}\ =\ g_{hs}R^{s}_{\,ijk}\ =\
g_{hs}(\partial_{k}\Gamma^{s}_{ij}-\partial_{j}\Gamma^{s}_{ik}+\Gamma^{r}_{ij}\Gamma^{s}_{rk}-\Gamma^{r}_{ik}\Gamma^{s}_{rj}),\
\ \partial_{k}\ =\ \frac{\partial}{\partial x^{k}},$

of the Riemann-Christoffel curvature tensor $R$ and the local components
$S_{ij}$ of the Ricci tensor $S$ of the warped product
$\overline{M}\times_{F}N$ which may not vanish identically are the following:

$\displaystyle R_{abcd}\ =\ \overline{R}_{abcd},\ \ \ R_{\alpha ab\beta}\ =\
-\frac{1}{2}\,T_{ab}\widetilde{g}_{\alpha\beta},\ \ \
R_{\alpha\beta\gamma\delta}\ =\
F\widetilde{R}_{\alpha\beta\gamma\delta}-\frac{1}{4}\,\Delta_{1}F\,\widetilde{G}_{\alpha\beta\gamma\delta}\,,$
$\displaystyle S_{ab}\ =\
\overline{S}_{ab}-\frac{n-p}{2}\,\frac{1}{F}\,T_{ab},\ \ \ S_{\alpha\beta}\ =\
\tilde{S}_{\alpha\beta}-\frac{1}{2}\,(tr(T)+\frac{n-p-1}{2F}\Delta_{1}F)\widetilde{g}_{\alpha\beta},$

where

$\displaystyle T_{ab}$ $\displaystyle=$
$\displaystyle\overline{\nabla}_{b}F_{a}-\frac{1}{2F}F_{a}F_{b},\ \ tr(T)\ =\
\overline{g}^{ab}T_{ab}\ =\ \Delta F-\frac{1}{2F}\,{\Delta}_{1}F,\ \ $
$\displaystyle\Delta F$ $\displaystyle=$
$\displaystyle{\Delta}_{\overline{g}}F\ =\ \overline{g}^{ab}\nabla_{a}F_{b},\
\ {\Delta}_{1}F\ =\ {\Delta}_{1\overline{g}}F\ =\
\overline{g}^{ab}F_{a}F_{b}\,,$

and $T$ is the $(0,2)$-tensor with the local components $T_{ab}$. We can
express the scalar curvature $\kappa$ of $\overline{M}\times_{F}\widetilde{N}$
by

$\displaystyle\kappa$ $\displaystyle=$
$\displaystyle\overline{\kappa}+\frac{1}{F}\,\widetilde{\kappa}-\frac{n-p}{F}\,(tr(T)+\frac{n-p-1}{4F}\Delta_{1}F)\
=\ \overline{\kappa}+\frac{1}{F}\,\widetilde{\kappa}-\frac{n-p}{F}\,(\Delta
F+\frac{n-p-3}{4F}\Delta_{1}F).$

##  3\. 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., [105, Chapter 1.5]).
This implies the following integrability condition ${\mathcal{R}}(X,Y)\cdot
R=0$, in short

(3.1) $\displaystyle R\cdot R$ $\displaystyle=$ $\displaystyle 0.$

Semi-Riemannian manifold satisfying (3.1) is called semisymmetric [117] (see
also [6, Chapter 8.5.3], [10, Chapter 20.7], [105, Chapter 1.6], [118, 119,
124]).

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., [6, Chapter 8.5.3], [10, Chapter 20.7], [12,
Section 15.1], [56, Chapter 6], [105, Chapter 12.4], [35, 40, 43, 56, 72, 84,
85, 114, 121, 123, 124, 125] and references therein). This is equivalent to

(3.2) $\displaystyle R\cdot R$ $\displaystyle=$ $\displaystyle L_{R}\,Q(g,R)$

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., [72]). A non-semisymmetric
pseudosymmetric manifold is called properly pseudosymmetric. A pseudosymmetric
manifold $(M,g)$, $\dim M=n\geq 3$, is called a pseudosymmetric manifold of
constant type if the function $L_{R}$ is constant on ${\mathcal{U}}_{R}$ [99]
(see also [86, 88, 100, 101, 102, 103]).

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., [6, Chapter 8.5.3], [12, Section 15.1],
[43]). This is equivalent on ${\mathcal{U}}_{S}\subset M$ to

(3.3) $\displaystyle R\cdot S$ $\displaystyle=$ $\displaystyle L_{S}\,Q(g,S),$

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., [16,
Section 1] and [51, Example 4.1]. According to [83], a Ricci-pseudosymmetric
manifold $(M,g)$, $\dim M=n\geq 3$, is called a Ricci-pseudosymmetric manifold
of constant type if the function $L_{S}$ is constant on ${\mathcal{U}}_{S}$.

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 $M$ [40, 43]. This is equivalent on
${\mathcal{U}}_{C}\subset M$ to

(3.4) $\displaystyle R\cdot C$ $\displaystyle=$ $\displaystyle L_{1}\,Q(g,C),$

where $L_{1}$ is some function on ${\mathcal{U}}_{C}$. We can easily check
that on every Einstein manifold $(M,g)$, $\dim M\geq 4$, (3.4) turns into

$\displaystyle R\cdot R$ $\displaystyle=$ $\displaystyle L_{1}\,Q(g,R).$

For a presentation of results on the problem of the equivalence of
pseudosymmetry, Ricci-pseudosymmetry and Weyl-pseudosymmetry we refer to [43,
Section 4]. Inclusions between mentioned above semi-Riemannian manifolds
$(M,g)$, i.e., pseudosymmetric, Ricci-pseudosymmetric and Weyl-pseudosymmetric
manifolds, can be presented in the following diagram [43, Section 4]

$\begin{array}[]{ccccc}\framebox{\makebox[85.92293pt]{\rule[-10.24275pt]{0.0pt}{25.6073pt}$\displaystyle
R\cdot
S=L_{S}Q(g,S)$\rule[-10.24275pt]{0.0pt}{25.6073pt}}}&\supset&\framebox{\makebox[85.92293pt]{\rule[-10.24275pt]{0.0pt}{25.6073pt}$\displaystyle
R\cdot
R=L_{R}Q(g,R)$\rule[-10.24275pt]{0.0pt}{25.6073pt}}}&\subset&\framebox{\makebox[85.92293pt]{\rule[-10.24275pt]{0.0pt}{25.6073pt}$\displaystyle
R\cdot C=L_{C}Q(g,C)$\rule[-10.24275pt]{0.0pt}{25.6073pt}}}\\\
\rotatebox{90.0}{$\displaystyle{}\;\subset\;\;\;{}$}&&\rotatebox{90.0}{$\displaystyle{}\;\subset\;\;\;{}$}&&\rotatebox{90.0}{$\displaystyle{}\;\subset\;\;\;{}$}\\\
\framebox{\makebox[85.92293pt]{\rule[-10.24275pt]{0.0pt}{25.6073pt}$\displaystyle
R\cdot
S=0$\rule[-10.24275pt]{0.0pt}{25.6073pt}}}&\supset&\framebox{\makebox[85.92293pt]{\rule[-10.24275pt]{0.0pt}{25.6073pt}$\displaystyle
R\cdot
R=0$\rule[-10.24275pt]{0.0pt}{25.6073pt}}}&\subset&\framebox{\makebox[85.92293pt]{\rule[-10.24275pt]{0.0pt}{25.6073pt}$\displaystyle
R\cdot C=0$\rule[-10.24275pt]{0.0pt}{25.6073pt}}}\\\
\rotatebox{90.0}{$\displaystyle{}\;\subset\;\;\;{}$}&&\rotatebox{90.0}{$\displaystyle{}\;\subset\;\;\;{}$}&&\rotatebox{90.0}{$\displaystyle{}\;\subset\;\;\;{}$}\\\
\framebox{\makebox[85.92293pt]{\rule[-10.24275pt]{0.0pt}{25.6073pt}$\displaystyle\nabla
S=0$\rule[-10.24275pt]{0.0pt}{25.6073pt}}}&\supset&\framebox{\makebox[85.92293pt]{\rule[-10.24275pt]{0.0pt}{25.6073pt}$\displaystyle\nabla
R=0$\rule[-10.24275pt]{0.0pt}{25.6073pt}}}&\subset&\framebox{\makebox[85.92293pt]{\rule[-10.24275pt]{0.0pt}{25.6073pt}$\displaystyle\nabla
C=0$\rule[-10.24275pt]{0.0pt}{25.6073pt}}}\\\
\rotatebox{90.0}{$\displaystyle{}\;\subset\;\;\;{}$}&&\rotatebox{90.0}{$\displaystyle{}\;\subset\;\;\;{}$}&&\rotatebox{90.0}{$\displaystyle{}\;\subset\;\;\;{}$}\\\
\framebox{\makebox[85.92293pt]{\rule[-10.24275pt]{0.0pt}{25.6073pt}$\displaystyle
S=\frac{\kappa}{n}g$\rule[-10.24275pt]{0.0pt}{25.6073pt}}}&\supset&\framebox{\makebox[85.92293pt]{\rule[-10.24275pt]{0.0pt}{25.6073pt}$\displaystyle
R=\frac{\kappa}{(n-1)n}G$\rule[-10.24275pt]{0.0pt}{25.6073pt}}}&\subset&\framebox{\makebox[85.92293pt]{\rule[-10.24275pt]{0.0pt}{25.6073pt}$\displaystyle
C=0$\rule[-10.24275pt]{0.0pt}{25.6073pt}}}\end{array}$

All inclusions in the above presentation are strict, provided that $\dim
M=n\geq 4$.

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., [12, Section 15.1], [40, 43, 51]). This is equivalent
on ${\mathcal{U}}_{C}\subset M$ to

(3.5) $\displaystyle C\cdot C$ $\displaystyle=$ $\displaystyle L_{C}\,Q(g,C),$

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 (3.5) (see, e.g., [40, 43,
51] and references therein). Thus in particular, the Schwarzschild spacetime,
the Kottler spacetime and the Reissner-Nordström spacetime satisfy (3.5).
Semi-Riemannian manifolds with pseudosymmetric Weyl tensor were investigated
among others in [40, 60, 73, 76].

We can show that $C\cdot R=L\,Q(g,R)$ implies $C\cdot C=L\,Q(g,C)$ (see [106,
Proposition 2.1]). Let $(M,g)$, $\dim M=n\geq 4$, be a semi-Riemannian
manifold satisfying $C\cdot R=L\,Q(g,R)$ on ${\mathcal{U}}_{C}\subset M$. From
this we get easily $C\cdot S=L\,Q(g,S)$ on ${\mathcal{U}}_{C}$. Further, we
have

$\displaystyle C\cdot C$ $\displaystyle=$ $\displaystyle
C\cdot\left(R-\frac{1}{n-2}\,g\wedge S+\frac{\kappa}{(n-2)(n-1)}\,G\right)$
$\displaystyle=$ $\displaystyle C\cdot R-\frac{1}{n-2}\,g\wedge(C\cdot
S)+\frac{\kappa}{(n-2)(n-1)}\,C\cdot G$ $\displaystyle=$ $\displaystyle
L\,Q(g,R)-\frac{L}{n-2}\,g\wedge Q(g,S)$ $\displaystyle=$ $\displaystyle
L\,Q(g,R)-\frac{L}{n-2}\,Q(g,g\wedge S)\ =\ L\,Q(g,R-\frac{1}{n-2}\,g\wedge
S)$ $\displaystyle=$ $\displaystyle L\,Q(g,R-\frac{1}{n-2}\,g\wedge
S+\frac{\kappa}{(n-2)(n-1)}\,G)\ =\ L\,Q(g,C).$

We also have the following diagram

$\begin{array}[]{ccccc}\framebox{\makebox[85.92293pt]{\rule[-10.24275pt]{0.0pt}{25.6073pt}$\displaystyle
C\cdot
S=L_{S}Q(g,S)$\rule[-10.24275pt]{0.0pt}{25.6073pt}}}&\supset&\framebox{\makebox[85.92293pt]{\rule[-10.24275pt]{0.0pt}{25.6073pt}$\displaystyle
C\cdot
R=L_{R}Q(g,R)$\rule[-10.24275pt]{0.0pt}{25.6073pt}}}&\subset&\framebox{\makebox[85.92293pt]{\rule[-10.24275pt]{0.0pt}{25.6073pt}$\displaystyle
C\cdot C=L_{C}Q(g,C)$\rule[-10.24275pt]{0.0pt}{25.6073pt}}}\\\
\rotatebox{90.0}{$\displaystyle{}\;\subset\;\;\;{}$}&&\rotatebox{90.0}{$\displaystyle{}\;\subset\;\;\;{}$}&&\rotatebox{90.0}{$\displaystyle{}\;\subset\;\;\;{}$}\\\
\framebox{\makebox[85.92293pt]{\rule[-10.24275pt]{0.0pt}{25.6073pt}$\displaystyle
C\cdot
S=0$\rule[-10.24275pt]{0.0pt}{25.6073pt}}}&\supset&\framebox{\makebox[85.92293pt]{\rule[-10.24275pt]{0.0pt}{25.6073pt}$\displaystyle
C\cdot
R=0$\rule[-10.24275pt]{0.0pt}{25.6073pt}}}&\subset&\framebox{\makebox[85.92293pt]{\rule[-10.24275pt]{0.0pt}{25.6073pt}$\displaystyle
C\cdot C=0$\rule[-10.24275pt]{0.0pt}{25.6073pt}}}\\\
\rotatebox{90.0}{$\displaystyle{}\;\subset\;\;\;{}$}&&\rotatebox{90.0}{$\displaystyle{}\;\subset\;\;\;{}$}&&\rotatebox{90.0}{$\displaystyle{}\;\subset\;\;\;{}$}\\\
\framebox{\makebox[85.92293pt]{\rule[-10.24275pt]{0.0pt}{25.6073pt}$\displaystyle
S=\frac{\kappa}{n}g$\rule[-10.24275pt]{0.0pt}{25.6073pt}}}&\supset&\framebox{\makebox[85.92293pt]{\rule[-10.24275pt]{0.0pt}{25.6073pt}$\displaystyle
R=\frac{\kappa}{(n-1)n}G$\rule[-10.24275pt]{0.0pt}{25.6073pt}}}&\subset&\framebox{\makebox[85.92293pt]{\rule[-10.24275pt]{0.0pt}{25.6073pt}$\displaystyle
C=0$\rule[-10.24275pt]{0.0pt}{25.6073pt}}}\end{array}$

All inclusions in the above presentation are strict, provided that $\dim
M=n\geq 4$.

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

(3.6) $\displaystyle R\cdot R-Q(S,R)$ $\displaystyle=$ $\displaystyle
L\,Q(g,C),$

where $L$ is some function on ${\mathcal{U}}_{C}$, were studied among others
in [21]. In that paper necessary and sufficient conditions for
$\overline{M}\times_{F}\widetilde{N}$ to be a manifold satisfying (3.6) 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 (3.6) [21, Theorem 4.1].

We refer to [16, 40, 43, 48, 51, 56, 60, 114] for details on semi-Riemannian
manifolds satisfying (3.2) and (3.3)-(3.6), as well other conditions of this
kind, named pseudosymmetry type curvature conditions. We also refer to [60,
Section 3] for a recent survey on manifolds satisfying such curvature
conditions. It seems that the condition (3.2) is the most important condition
of that family of curvature conditions (see, e.g., [51]). 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., [51, 56, 72, 114]).

##  4\. 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 if

(4.1) $\displaystyle\mathrm{rank}\,(S-\alpha\,g)$ $\displaystyle=$
$\displaystyle 1$

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
$(\widetilde{N},\widetilde{g})$ and a warping function $F$, $n\geq 4$, is a
quasi-Einstein manifold (see, e.g., [16, 51]). A Riemannian manifold $(M,g)$,
$\dim M=n\geq 3$, whose Ricci tensor has an eigenvalue of multiplicity $n-1$
is a quasi-Einstein manifold (cf. [37, Introduction]). Evidently, the converse
statement is also true. 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., [43, 51] and references therein). Quasi-Einstein hypersurfaces in
semi-Riemannian spaces of constant curvature were studied among others in [45,
58, 64, 83] (see also [43] and references therein). Quasi-Einstein manifolds
satisfying some pseudosymmetry type curvature conditions were investigated
recently in [3, 16, 40, 48, 60]. We mention that an example of a non-
semisymmetric ($R\cdot R\neq 0$) Ricci-semisymmetric ($R\cdot S=0$) quasi-
Einstein hypersurface $M$ in an Euclidean space $\mathbb{E}^{n+1}$, $n\geq 5$,
was constructed in [2].

A semi-Riemannian manifold $(M,g)$, $\dim M=n\geq 3$, is called a $2$-quasi-
Einstein manifold if

(4.2) $\displaystyle\mathrm{rank}\,(S-\alpha\,g)$ $\displaystyle\leq$
$\displaystyle 2$

on ${\mathcal{U}}_{S}\subset M$ 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., [53]).

Every non-Einstein and 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
warping function $F$ satisfies (4.2) (see, e.g., [51, Theorem 6.1]). Thus some
exact solutions of the Einstein field equations are non-conformally flat
$2$-quasi-Einstein manifolds. For instance, the Reissner-Nordström spacetime,
as well as the Reissner-Nordström-de Sitter type spacetimes are such manifolds
(see, e.g., [91]). It seems that the Reissner-Nordström spacetime is the
"oldest" example of a non-conformally flat $2$-quasi-Einstein warped product
manifold. It is easy to see that every $2$-quasi-umbilical hypersurface in a
semi-Riemannian space of constant curvature is a $2$-quasi-Einstein manifold
(see, e.g., [53]).

We mention that Einstein warped product manifolds
$\overline{M}\times_{F}\widetilde{N}$, with $\dim\overline{M}=1,2$, were
studied in [4, Chapter 9.J] (see also [4, Chapter 3.F]).

The semi-Riemannian manifold $(M,g)$, $\dim M=n\geq 3$, will be called a
partially Einstein manifold, or a partially Einstein space (cf. [11,
Foreword], [122, p. 20]), 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,

(4.3) $\displaystyle S^{2}$ $\displaystyle=$ $\displaystyle\lambda\,S+\mu\,g,$

where $\lambda,\mu\in{\mathbb{R}}$ and $I\\!d_{x}$ is the identity
transformation of $T_{x}M$. Thus every quasi-Einstein manifold is a partially
Einstein manifold. The converse statement is not true. Contracting (4.3) we
get $\mathrm{tr}(S^{2})=\lambda\,\kappa+n\,\mu$. This together with (4.3)
yields (cf. [41, Section 5])

(4.4) $\displaystyle S^{2}-\frac{\mathrm{tr}(S^{2})}{n}\,g$ $\displaystyle=$
$\displaystyle\lambda\left(S-\frac{\kappa}{n}\,g\right).$

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 (principal Ricci
curvatures) $\rho_{1}=\rho_{2}=\ldots=\rho_{p}$ and
$\rho_{p+1}=\rho_{p+2}=\ldots=\rho_{n}$ with multiplicities $p$ and $n-p$,
respectively, where $1\leq p\leq n-1$. Now (4.3) and (4.4) yield

(4.5) $\displaystyle S^{2}\ =\ (\rho_{1}+\rho_{p+1})\,S-\rho_{1}\rho_{p+1}\,g\
\ \ \mbox{and}\ \ \ S^{2}-\frac{\mathrm{tr}(S^{2})}{n}\,g\ =\
(\rho_{1}+\rho_{p+1})\left(S-\frac{\kappa}{n}\,g\right),$

respectively. Evidently, if $p=1$, or $p=n-1$, then $(M,g)$ is a quasi-
Einstein manifold.

######  Remark 4.1.

(i) Let $(M,g)$, $n=\dim M\geq 3$, be a semi-Riemannian manifold. In addition,
let $(M,g)$ be a conformally flat manifold when $n\geq 4$. Thus the set
${\mathcal{U}}_{C}\subset M$ is an empty set. Now, by (1.3), the subsets
${\mathcal{U}}_{S}$ and ${\mathcal{U}}_{R}$ of $M$ satisfy
${\mathcal{U}}_{S}={\mathcal{U}}_{R}$. In view of [22, Lemma 1.2] (see also
[66, Lemma 2.1]) we can state that on ${\mathcal{U}}_{S}$ any of the following
three conditions is equivalent to each other: $R\cdot R=\rho\,Q(g,R)$, $R\cdot
S=\rho\,Q(g,S)$ and

(4.6) $\displaystyle S^{2}-\frac{\mathrm{tr}(S^{2})}{n}\,g$ $\displaystyle=$
$\displaystyle\left(\frac{\kappa}{n-1}+(n-2)\rho\right)\left(S-\frac{\kappa}{n}\,g\right).$

where $\rho$ is some function on ${\mathcal{U}}_{S}$.
(ii) Let $(M,g)$, $n=\dim M\geq 3$, be a Riemannian manifold with vanishing
Weyl conformal curvature tensor $C$ such that its Ricci operator $\mathcal{S}$
has at every point of $M$ exactly two Ricci principal curvatures $\rho_{1}$
and $\rho_{p+1}$ with multiplicities $p$ and $n-p$, respectively, where $1\leq
p<n$. Using (4.5) and (4.6) we can easily check that $(M,g)$ is a
pseudosymmetric manifold satisfying (3.2) with

(4.7) $\displaystyle L_{R}\ =\ \rho$ $\displaystyle=$
$\displaystyle\frac{1}{(n-2)(n-1)}((n-1)(\rho_{1}+\rho_{p+1})-\kappa).$

Evidently, if $p=1$ then (4.7) turns into

$\displaystyle L_{R}\ =\ \rho$ $\displaystyle=$
$\displaystyle\frac{1}{n-1}\rho_{1}.$

(iii) As an immediate consequence of the above presented results we obtain
[124, Foreword, p. xviii]:
Riemannian spaces of dimensions $\geq 3$ with wanishing Weyl conformal
curvature tensor are Deszcz symmetric if and only if they are Einstein or
"partially Einstein", - partially Einstein spaces being defined by the
condition that their Ricci tensor has precisely two distinct eigenvalues -.
(iv) (cf. [83, Proposition 2.1]) Let $(M,g)$, $\dim M=n\geq 3$, be a semi-
Riemannian manifold. We note that (4.1) 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.,

(4.8) $\displaystyle\frac{1}{2}\,S\wedge S-\alpha\,g\wedge
S+\frac{\alpha^{2}}{2}\,g\wedge g$ $\displaystyle=$ $\displaystyle 0.$

From (4.8), by a suitable contraction, we get immediately

(4.9) $\displaystyle S^{2}$ $\displaystyle=$
$\displaystyle(\kappa-(n-2)\alpha)\,S+\alpha((n-1)\alpha-\kappa)\,g.$

Evidently, (4.9) is a special case of (4.3).
(v) Let $\rho_{1}$, $\rho_{2}$ and $\rho_{3}$ be principal Ricci curvatures of
a $3$-dimensional Riemannian manifold $(M,g)$. Moreover, let the condition

(4.10) $\displaystyle\rho_{1}=\rho_{2}\neq\rho_{3}$

be satisfied at every point of the set
${\mathcal{U}}_{S}={\mathcal{U}}_{R}\subset M$. Thus

$\displaystyle R\cdot R$ $\displaystyle=$
$\displaystyle\frac{\rho_{3}}{2}\,Q(g,R)$

on ${\mathcal{U}}_{R}$. Evidently, if $\rho_{3}=0$ on ${\mathcal{U}}_{R}$ then
$(M,g)$ is semisymmetric.
(vi) $3$-dimensional Riemannian manifolds satisfying (4.10) were studied,
among others, in the following papers: [95, 97, 98, 99, 100, 101, 102]. We
mention that an explicit classification of $3$-dimensional semisymmetric
Riemannian manifolds is given in [96].

######  Example 4.2.

The warped product manifold $\overline{M}\times_{F}\widetilde{N}$ with a
$1$-dimensional manifold $(\overline{M},\overline{g})$, $\overline{g}_{11}=\pm
1$, and an $(n-1)$-dimensional semi-Riemannian Einstein manifold
$(\widetilde{N},\widetilde{g})$, $n\geq 5$, assumed that it is not of constant
curvature, and a warping function $F$, satisfies on
${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}\subset\overline{M}\times_{F}\widetilde{N}$
[16, Theorem 4.1]

$\displaystyle\mathrm{rank}(S-\alpha\,g)\ =\ 1,\ \ \ R\cdot S\ =\
L_{S}\,Q(g,S),$ $\displaystyle(n-2)\,(R\cdot C-C\cdot R)\ =\
Q(S,R)-L_{S}\,Q(g,R),$ $\displaystyle\alpha\ =\ \frac{\kappa}{n-1}-L_{S},\ \ \
L_{S}\ =\ -\frac{\mathrm{tr}T}{2F}.$

Furthermore, using

$\displaystyle Q(g,R)$ $\displaystyle=$ $\displaystyle
Q(g,C)-\frac{1}{n-2}\,Q(S,G),$ $\displaystyle Q(S,R)$ $\displaystyle=$
$\displaystyle
Q(S,C)+\frac{1}{n-2}\left(\alpha-\frac{\kappa}{n-1}\right)Q(S,G),$

we obtain [51, Example 4.1]

$\displaystyle(n-2)\,(R\cdot C-C\cdot R)$ $\displaystyle=$ $\displaystyle
Q(S,C)-L_{S}\,Q(g,C).$

######  Example 4.3.

Let $M$ be an open connected non-empty subset of $\mathbb{R}^{5}$ endowed with
the metric $g$ of the form [92, 93, 94]

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle g_{ij}dx^{i}dx^{j}\ =\
dx^{2}+dy^{2}+du^{2}+dv^{2}+\rho^{2}\,(xdu-ydv+dz)^{2}\,,\ \ \ \rho\ =\
\mbox{const.}\neq 0\,.$

The manifold $(M,g)$ is a non-conformally flat manifold satisfying the
following conditions [114]:

$\displaystyle\nabla_{X}S(Y,Z)+\nabla_{Y}S(Z,X)+\nabla_{Z}S(X,Y)\ =\ 0,$
$\displaystyle S=\frac{\kappa}{2}\,g-\frac{3\kappa}{2}\,\eta\otimes\eta,\ \ \
\eta\ =\ (0,0,-\rho,-x\rho,y\rho),\ \ \ \kappa\ =\ \rho^{2},\ \ \ S^{2}\ =\
-\frac{\kappa}{2}\,S+\frac{\kappa^{2}}{2}\,g,$ $\displaystyle S\cdot R\ =\
2\kappa\,R-\frac{\kappa}{2}g\wedge S+\frac{\kappa^{2}}{4}\,g\wedge g,\ \ \
C\cdot S\ =\ 0,\ \ \ R\cdot S\ =\ -\frac{\kappa}{4}\,Q(g,S),$ $\displaystyle
R\cdot R\ =\ -\frac{\kappa}{4}\,Q(g,R),\ \ \ R\cdot C\ =\
-\frac{\kappa}{4}\,Q(g,C),$ $\displaystyle C\cdot R\ =\
-\frac{1}{3}\,Q(S,C)-\frac{\kappa}{3}\,Q(g,C),\ \ \ C\cdot C\ =\ C\cdot R,$
$\displaystyle R\cdot C+C\cdot R\ =\
-\frac{1}{3}\,Q(S,C)-\frac{7\kappa}{12}\,Q(g,C)$

and the condition of the form $(\ast)$

$\displaystyle R\cdot C-C\cdot R$ $\displaystyle=$
$\displaystyle\frac{1}{3}\,Q(S,C)+\frac{\kappa}{12}\,Q(g,C).$

The $(0,4)$-tensor $S\cdot R$ is defined by

$\displaystyle(S\cdot R)(X,Y,W,Z)\ =\
R({\mathcal{S}}X,Y,W,Z)+R(X,{\mathcal{S}}Y,W,Z)+R(X,Y,{\mathcal{S}}W,Z)+R(X,Y,W,{\mathcal{S}}Z).$

An important subclass of the class of partially Einstein manifolds, of
dimension $\geq 4$, form some non-conformally flat and non-quasi-Einstein
manifolds called Roter spaces. A semi-Riemannian manifold $(M,g)$, $\dim
M=n\geq 4$, satisfying on ${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}\subset M$
the following equation

(4.11) $\displaystyle R$ $\displaystyle=$
$\displaystyle\frac{\phi}{2}\,S\wedge S+\mu\,g\wedge S+\frac{\eta}{2}\,g\wedge
g,$

where $\phi$, $\mu$ 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., [12,
Section 15.5], [36, 51, 53, 56]). Equation (4.11) is called a Roter equation
(see, e.g., [44, Section 1]). Roter spaces and in particular Roter
hypersurfaces in semi-Riemannian spaces of constant curvature were studied in:
[18, 36, 40, 48, 57, 58, 65, 69, 70, 79, 82, 90, 91]. We only mention that
every Roter space $(M,g)$, $\dim M=n\geq 4$, satisfies on
${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}\subset M$ among others the following
conditions: (1.2), (3.2) and (3.3)-(3.6) (see, e.g., [43, 51] and references
therein). We also refer to [42, Section 4] for a more detailed presentation of
results on Roter spaces.

Let $(M,g)$, $n\geq 4$, 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 $S^{1}=S,S^{2},\ldots,S^{p-1},S^{p}$, 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

(4.12) $\displaystyle R$ $\displaystyle=$
$\displaystyle\frac{\phi_{2}}{2}\,S^{2}\wedge S^{2}+\phi_{1}\,S\wedge
S^{2}+\frac{\phi}{2}\,S\wedge S+\mu_{1}\,g\wedge S^{2}+\mu\,g\wedge
S+\frac{\eta}{2}\,g\wedge g,$

where $\phi$, $\phi_{1}$, $\phi_{2}$, $\mu_{1}$, $\mu$ and $\eta$ 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 (4.12) is called a
Roter type equation (see, e.g., [44, Section 1]). We refer to [44, 50, 51, 53,
60, 111, 113, 114, 115, 116] for results on manifolds (hypersurfaces)
satisfying (4.12).

If $(M,g)$, $\dim M=n\geq 4$, is a semi-Riemannian manifold satisfying (3.5)
on ${\mathcal{U}}_{C}\subset M$, i.e., $C\cdot C=L_{C}\,Q(g,C)$, then (2.17)
yields

(4.13) $\displaystyle C\cdot R+R\cdot C$ $\displaystyle=$ $\displaystyle
R\cdot R+L\,Q(g,C)-\frac{1}{(n-2)^{2}}\,Q\left(g,g\wedge S^{2}-\kappa\,g\wedge
S\right)$

on ${\mathcal{U}}_{C}$. In addition, if (3.6) is satisfied on
${\mathcal{U}}_{C}$, i.e., $R\cdot R-Q(S,R)=L\,Q(g,C)$, then (4.13) turns into
[51, Theorem 3.4 (ii)-(iii)]

(4.14) $\displaystyle C\cdot R+R\cdot C$ $\displaystyle=$ $\displaystyle
Q(S,C)+(L+L_{C})\,Q(g,C)$ $\displaystyle-\frac{1}{(n-2)^{2}}\,Q\left(g,g\wedge
S^{2}+\frac{n-2}{2}\,S\wedge S-\kappa\,g\wedge S\right).$

Evidently, (4.14), by making use of (2.12) and the identity
$Q(g,\frac{1}{2}\,g\wedge g)=Q(g,G)=0$ takes the form

$\displaystyle C\cdot R+R\cdot C$ $\displaystyle=$ $\displaystyle
Q(S,C)+(L+L_{C})\,Q(g,C)-\frac{1}{(n-2)^{2}}\,Q(g,E).$

We also have (cf. [51, Theorem 3.4 (iv)-(v)]): if $(M,g)$, $\dim M=n\geq 4$,
is a quasi-Einstein semi-Riemannian manifold satisfying
$\mathrm{rank}\,(S-\alpha\,g)=1$ (i.e. (4.1)), (3.5) and (3.6) then (4.14), by
making use of (4.8) and (4.9), yields

(4.15) $\displaystyle C\cdot R+R\cdot C$ $\displaystyle=$ $\displaystyle
Q(S,C)+(L+L_{C})\,Q(g,C).$

In particular, if $(M,g)$ is the Gödel spacetime then (4.15) turns into

$\displaystyle C\cdot R+R\cdot C$ $\displaystyle=$ $\displaystyle
Q(S,C)+\frac{\kappa}{6}\,Q(g,C).$

######  Theorem 4.4.

(cf. [40, Proposition 3.2, Theorem 3.3, Theorem 4.4]) If $(M,g)$, $\dim
M=n\geq 4$, is a semi-Riemannian manifold satisfying on
${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}\subset M$ conditions (3.5), (3.6) and

(4.16) $\displaystyle R\cdot S$ $\displaystyle=$ $\displaystyle Q(g,D),$

where $D$ is a symmetric $(0,2)$-tensor, then $R\cdot R=L_{R}\,Q(g,R)$ on this
set, where $L_{R}$ is some function on this set. Moreover, at every point of
${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}$ we have
(i) $\mathrm{rank}\,(S-\alpha_{1}\,g)=1$ and
$\alpha_{1}=\frac{1}{2}\left(\frac{\kappa}{n-1}-L+L_{C}\right)$, or
(ii) $\mathrm{rank}\,(S-\alpha_{1}\,g)\geq 2$ and
$\alpha_{1}=\frac{1}{2}\left(\frac{\kappa}{n-1}-L+L_{C}\right)$ and (4.11).

######  Corollary 4.5.

[51, Corollary 3.6] If $(M,g)$, $\dim M=n\geq 4$, is a semi-Riemannian
manifold satisfying on ${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}\subset M$
conditions (3.5), (3.6) and (4.16) then

$\displaystyle C\cdot R+R\cdot C$ $\displaystyle=$ $\displaystyle
Q(S,C)+L_{2}\,Q(g,C)$

on ${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}$, where $L_{2}$ is some function on
this set.

######  Theorem 4.6.

[51, Theorem 3.1] If $(M,g)$, $\dim M=n\geq 4$, is a pseudosymmetric Einstein
semi-Riemannian manifold satisfying $R\cdot R=L_{R}\,Q(g,R)$ (i.e. (3.2)) on
${\mathcal{U}}_{R}\subset M$, then on this set

$\displaystyle R\cdot R-Q(S,R)$ $\displaystyle=$
$\displaystyle\left(L_{R}-\frac{\kappa}{n}\right)Q(g,C),$ $\displaystyle
C\cdot C$ $\displaystyle=$
$\displaystyle\left(L_{R}-\frac{\kappa}{(n-1)n}\right)Q(g,C),$ $\displaystyle
C\cdot R+R\cdot C$ $\displaystyle=$ $\displaystyle
Q(S,C)+\left(2L_{R}-\frac{\kappa}{n-1}\right)Q(g,C).$

######  Theorem 4.7.

[51, Theorem 4.1] Let $\overline{M}\times_{F}\widetilde{N}$ be the warped
product manifold with a $1$-dimensional manifold
$(\overline{M},\overline{g})$, $\overline{g}_{11}=\pm 1$, and a
$3$-dimensional semi-Riemannian manifold $(\widetilde{N},\widetilde{g})$, and
a warping function $F$. If $(\widetilde{N},\widetilde{g})$ is not a space of
constant curvature then (3.6) and (4.13) hold on
${\mathcal{U}}_{C}\subset\overline{M}\times_{F}\widetilde{N}$. Moreover, if
$(\widetilde{N},\widetilde{g})$ is a quasi-Einstein manifold then (3.5) and
(4.14) hold on
${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}\subset\overline{M}\times_{F}\widetilde{N}$.

The last theorem leads to the following results.

######  Theorem 4.8.

[51, Theorem 4.3] Let $\overline{M}\times_{F}\widetilde{N}$ be the warped
product manifold with a $1$-dimensional manifold
$(\overline{M},\overline{g})$, $\overline{g}_{11}=\pm 1$, and an
$(n-1)$-dimensional quasi-Einstein semi-Riemannian manifold
$(\widetilde{N},\widetilde{g})$, $n\geq 4$, and a warping function $F$, and
let $(\widetilde{N},\widetilde{g})$ be a conformally flat manifold, when
$n\geq 5$. Then conditions (3.5), (3.6) and (4.14) are satisfied on
${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}\subset\overline{M}\times_{F}\widetilde{N}$.

######  Theorem 4.9.

[51, Theorem 4.4] Let $\overline{M}\times\widetilde{N}$ be the product
manifold with a $1$-dimensional manifold $(\overline{M},\overline{g})$,
$\overline{g}_{11}=\pm 1$, and an $(n-1)$-dimensional quasi-Einstein semi-
Riemannian manifold $(\widetilde{N},\widetilde{g})$, $n\geq 4$, satisfying
$\mathrm{rank}\,(\widetilde{S}-\rho\,\widetilde{g})=1$ on
${\mathcal{U}}_{\widetilde{S}}\subset\widetilde{M}$, where $\rho$ is some
function on ${\mathcal{U}}_{\widetilde{S}}$, and let
$(\widetilde{N},\widetilde{g})$ be a conformally flat manifold, when $n\geq
5$. Then on
${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}\subset\overline{M}\times\widetilde{N}$
we have

$\displaystyle(n-3)(n-2)\rho\,C$ $\displaystyle=$ $\displaystyle g\wedge
S^{2}+\frac{n-2}{2}\,S\wedge S-\kappa\,g\wedge
S+(n-2)\rho\left(\frac{2\kappa}{n-1}-\rho\right)G,$ $\displaystyle C\cdot
R+R\cdot C$ $\displaystyle=$ $\displaystyle
Q(S,C)+\left(\frac{\kappa}{(n-2)(n-1)}-\rho\right)Q(g,C).$

We present now the following results on Einstein and quasi-Einstein manifolds.
Using (1.1) we can easily check that equation

(4.17) $\displaystyle g\wedge S^{2}+\frac{n-2}{2}\,S\wedge S-\kappa\,g\wedge
S+\frac{\kappa^{2}-\mathrm{tr}(S^{2})}{2(n-1)}\,g\wedge g$ $\displaystyle=$
$\displaystyle 0$

is satisfied on any Einstein manifold $(M,g)$. Moreover, we also have

######  Theorem 4.10.

[44, Lemma 2.1] If $(M,g)$, $n\geq 4$, is a quasi-Einstein manifold satisfying
(4.1) on ${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}\subset M$ then (4.17) holds
on this set.

We finish this section with the following results.

######  Theorem 4.11.

[44, Lemma 2.2] If $(M,g)$, $n\geq 4$, is a Roter space satisfying (4.11) on
${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}\subset M$ then

$\displaystyle C$ $\displaystyle=$ $\displaystyle\frac{\phi}{n-2}\left(g\wedge
S^{2}+\frac{n-2}{2}\,S\wedge S-\kappa\,g\wedge
S+\frac{\kappa^{2}-\mathrm{tr}(S^{2})}{2(n-1)}\,g\wedge g\right)$

on this set.

######  Remark 4.12.

(i) In view of [35, Lemma 3.2 (ii)], we can state that the following identity
is satisfied on every semi-Riemannian manifold $(M,g)$, $n=\dim M\geq 3$, with
vanishing Weyl conformal curvature tensor $C$

(4.18) $\displaystyle R\cdot R-Q(S,R)$ $\displaystyle=$
$\displaystyle\frac{1}{(n-2)^{2}}\,Q(g,g\wedge S^{2}+\frac{n-2}{2}\,S\wedge
S-\kappa\,g\wedge S).$

(ii) On every $3$-dimensional semi-Riemannian manifold $(M,g)$ the identity
$R\cdot R=Q(S,R)$ is satisfied [35, Theorem 3.1]. Thus (4.18) reduces to

$\displaystyle Q(g,g\wedge S^{2}+\frac{n-2}{2}\,S\wedge S-\kappa\,g\wedge S)$
$\displaystyle=$ $\displaystyle 0.$

From this, by suitable contractions we easily get (4.17).
(iii) From (i) we easily deduce that on every semi-Riemannian conformally flat
manifold $(M,g)$, $n=\dim M\geq 4$, conditions: $R\cdot R=Q(S,R)$ and (4.17)
are equivalent.

######  Remark 4.13.

Warped product manifolds $\overline{M}\times_{F}\widetilde{N}$, with a
$1$-dimensional base manifold $(\overline{M},\overline{g})$,
$\overline{g}_{11}=\pm 1$, and an $(n-1)$-dimensional Einsteinian or non-
Einsteinian fiber $(\widetilde{N},\widetilde{g})$, $n\geq 4$, satisfying the
condition

$\displaystyle R\cdot C-C\cdot R$ $\displaystyle=$ $\displaystyle L\,Q(S,R),$

were studied in [3].

##  5\. Warped product manifolds with 2-dimensional base manifold

Let $\overline{M}\times_{F}\widetilde{N}$ be the warped product manifold with
a $2$-dimensional semi-Riemannian manifold $(\overline{M},\overline{g})$ and
an $(n-2)$-dimensional semi-Riemannian manifold
$(\widetilde{N},\widetilde{g})$, $n\geq 4$, and a warping function $F$, and
let $(\widetilde{N},\widetilde{g})$ be a space of constant curvature when
$n\geq 5$.

Let $S_{hk}$ and $C_{hijk}$ be the local components of the Ricci tensor $S$
and the Weyl conformal curvature tensor $C$ of
$\overline{M}\times_{F}\widetilde{N}$, respectively. We have

(5.1) $\displaystyle S_{ad}$ $\displaystyle=$
$\displaystyle\frac{\overline{\kappa}}{2}\,g_{ab}-\frac{n-2}{2F}\,T_{ab},\ \
S_{\alpha\beta}\ =\ \tau_{1}\,g_{\alpha\beta},\ \ S_{a\alpha}\ =\ 0,$ (5.2)
$\displaystyle\tau_{1}$ $\displaystyle=$
$\displaystyle\frac{\widetilde{\kappa}}{(n-2)F}-\frac{\mathrm{tr}(T)}{2F}-(n-3)\,\frac{\Delta_{1}F}{4F^{2}},\
\ \ \Delta_{1}F\ =\ {\Delta}_{1\,\overline{g}}F\ =\
\overline{g}^{ab}F_{a}F_{b},$ $\displaystyle T_{ab}$ $\displaystyle=$
$\displaystyle\overline{\nabla}_{a}F_{b}-\frac{1}{2F}F_{a}F_{b},\ \
\mathrm{tr}(T)\ =\ \overline{g}^{ab}T_{ab},$

where $T$ is the $(0,2)$-tensor with the local components $T_{ab}$. We also
have [51, eqs. (5.10)-(5.14)]

(5.3) $\displaystyle C_{abcd}$ $\displaystyle=$
$\displaystyle\frac{n-3}{n-1}\,\rho_{1}\,G_{abcd}\ =\
\frac{n-3}{n-1}\,\rho_{1}\,(g_{ad}g_{bc}-g_{ac}g_{bd}),$ (5.4) $\displaystyle
C_{\alpha bc\beta}$ $\displaystyle=$
$\displaystyle-\frac{n-3}{(n-2)(n-1)}\,\rho_{1}\,G_{\alpha bc\beta}\ =\
-\frac{n-3}{(n-2)(n-1)\rho_{1}}\,\rho_{1}\,g_{bc}g_{\alpha\beta},$ (5.5)
$\displaystyle C_{\alpha\beta\gamma\delta}$ $\displaystyle=$
$\displaystyle\frac{2\rho_{1}}{(n-2)(n-1)}\,G_{\alpha\beta\gamma\delta}\ =\
\frac{2\rho_{1}}{(n-2)(n-1)}\,(g_{\alpha\delta}g_{\beta\gamma}-g_{\alpha\gamma}g_{\beta\delta}),$
(5.6) $\displaystyle C_{abc\delta}$ $\displaystyle=$ $\displaystyle
C_{ab\alpha\beta}\ =\ C_{a\alpha\beta\gamma}\ =\ 0,$

where $G_{hijk}=g_{hk}g_{ij}-g_{hj}g_{ik}$ and

$\displaystyle\rho_{1}$ $\displaystyle=$
$\displaystyle\frac{\overline{\kappa}}{2}+\frac{\widetilde{\kappa}}{(n-3)(n-2)F}+\frac{1}{2F}\left(\Delta
F-\frac{\Delta_{1}F}{F}\right),\ \ \ \Delta F\ =\
\overline{g}^{ab}\overline{\nabla}_{a}F_{b}\,.$

If we set [51, eqs. (5.13)]

(5.7) $\displaystyle\rho$ $\displaystyle=$
$\displaystyle\frac{2(n-3)}{n-1}\,\rho_{1}\ =\
\frac{2(n-3)}{n-1}\left(\frac{\overline{\kappa}}{2}+\frac{\widetilde{\kappa}}{(n-3)(n-2)F}+\frac{1}{2F}\left(\Delta
F-\frac{\Delta_{1}F}{F}\right)\right)$

then (5.3)-(5.6) turn into [51, eqs. (5.14)]

$\displaystyle C_{abcd}$ $\displaystyle=$
$\displaystyle\frac{\rho}{2}\,G_{abcd},\ \ \ C_{\alpha bc\beta}\ =\
-\frac{\rho}{2(n-2)}\,G_{\alpha bc\beta},$ $\displaystyle
C_{\alpha\beta\gamma\delta}$ $\displaystyle=$
$\displaystyle\frac{\rho}{(n-3)(n-2)}\,G_{\alpha\beta\gamma\delta},\ \ \
C_{abc\delta}\ =\ C_{ab\alpha\beta}\ =\ C_{a\alpha\beta\gamma}\ =\ 0.$

Further, by making use of the formulas for the local components $(C\cdot
C)_{hijklm}$ and $Q(g,C)_{hijklm}$ of the tensors $C\cdot C$ and $Q(g,C)$,
i.e.

$\displaystyle(C\cdot C)_{hijklm}$ $\displaystyle=$ $\displaystyle
g^{rs}(C_{rijk}C_{shlm}+C_{hrjk}C_{silm}+C_{hirk}C_{sjlm}+C_{hijr}C_{sklm}),$
$\displaystyle Q(g,C)_{hijklm}$ $\displaystyle=$ $\displaystyle
g_{hl}C_{mijk}+g_{il}C_{hmjk}+g_{jl}C_{himk}+g_{kl}C_{hijm}$ $\displaystyle-
g_{hm}C_{lijk}-g_{im}C_{hljk}-g_{jm}C_{hilk}-g_{km}C_{hijl},$

we obtain [51, eqs. (7.7)-(7.8)]

$\displaystyle(C\cdot C)_{\alpha abcd\beta}\ =\
-\frac{(n-1)\rho^{2}}{4(n-2)^{2}}\,g_{\alpha\beta}G_{dabc},\ \ \ (C\cdot
C)_{a\alpha\beta\gamma d\delta}\ =\
\frac{(n-1)\rho^{2}}{4(n-2)^{2}(n-3)}\,g_{ad}G_{\delta\alpha\beta\gamma},$
$\displaystyle Q(g,C)_{\alpha abcd\beta}\ =\
\frac{(n-1)\rho}{2(n-2)}\,g_{\alpha\beta}G_{dabc},\ \ \
Q(g,C)_{a\alpha\beta\gamma d\delta}\ =\
-\frac{(n-1)\rho}{2(n-2)(n-3)}\,g_{ad}G_{\delta\alpha\beta\gamma}.$

######  Theorem 5.1.

[51, Theorem 7.1 (i)] Let $\overline{M}\times_{F}\widetilde{N}$ be the warped
product manifold with a $2$-dimensional semi-Riemannian manifold
$(\overline{M},\overline{g})$ and an $(n-2)$-dimensional semi-Riemannian
manifold $(\widetilde{N},\widetilde{g})$, $n\geq 4$, and a warping function
$F$, and let $(\widetilde{N},\widetilde{g})$ be a space of constant curvature,
when $n\geq 5$.
(i) The following equation is satisfied on
${\mathcal{U}}_{C}\subset\overline{M}\times_{F}\widetilde{N}$

$\displaystyle C\cdot C$ $\displaystyle=$ $\displaystyle L_{C}\,Q(g,C)\,,$
(5.8) $\displaystyle L_{C}$ $\displaystyle=$
$\displaystyle-\frac{\rho}{2(n-2)}\ =\
-\frac{n-3}{(n-2)(n-1)}\left(\frac{\overline{\kappa}}{2}+\frac{\widetilde{\kappa}}{(n-3)(n-2)F}+\frac{1}{2F}\left(\Delta
F-\frac{\Delta_{1}F}{F}\right)\right),$

where the function $\rho$ is defined by (5.7).
(ii) Equation (3.6) is satisfied on
${\mathcal{U}}_{C}\subset\overline{M}\times_{F}\widetilde{N}$, where the
functions $\tau_{1}$ and $L$ are defined by (5.2) and

(5.9) $\displaystyle L$ $\displaystyle=$
$\displaystyle-\frac{n-2}{(n-1)\rho}\left(\overline{\kappa}\left(\tau_{1}+\frac{\mathrm{tr}(T)}{2F}\right)+\frac{n-3}{4F^{2}}\left(\mathrm{tr}(T^{2})-(\mathrm{tr}(T))^{2}\right)\right),$

$T$ is the $(0,2)$-tensor with the local components
$T_{ab}=\overline{\nabla}_{a}F_{b}-\frac{1}{2F}F_{a}F_{b}$, $\mathrm{tr}(T)\
=\ \overline{g}^{ab}T_{ab}$, $T^{2}_{ad}=T_{ac}\overline{g}^{cd}T_{db}$ and
$\mathrm{tr}(T^{2})=\overline{g}^{ab}T^{2}_{ab}$.
(iii) The following equation is satisfied on
${\mathcal{U}}_{C}\subset\overline{M}\times_{F}\widetilde{N}$

$\displaystyle C\cdot R+R\cdot C$ $\displaystyle=$ $\displaystyle
Q(S,C)+(L_{C}+L)\,Q(g,C)-\frac{1}{(n-2)^{2}}\,Q\left(g,g\wedge
S^{2}+\frac{n-2}{2}\,S\wedge S-\kappa\,g\wedge S\right),$

where $L_{C}$ and $L$ are functions defined by (5.8) and (5.9).

We have (see, eq. (5.1))

$\displaystyle S_{ad}$ $\displaystyle=$
$\displaystyle\frac{\overline{\kappa}}{2}\,g_{ab}-\frac{n-2}{2F}\,T_{ab},\ \
S_{\alpha\beta}\ =\ \tau_{1}\,g_{\alpha\beta},\ \ S_{a\alpha}\ =\ 0,$

where $\tau_{1}$ is defined by (5.2).

We define now on ${\mathcal{U}}_{S}\subset\overline{M}\times_{F}\widetilde{N}$
the $(0,2)$-tensor $A$ by $A=S-\tau_{1}\,g$. We can check that
$\mathrm{rank}(A)=2$ at a point of ${\mathcal{U}}_{S}$ if and only if
$\mathrm{tr}(A^{2})-(\mathrm{tr}(A))^{2}\neq 0$ at this point [51, Section 6].
At all points of ${\mathcal{U}}_{S}$, at which $\mathrm{rank}(A)=2$, we set

(5.10) $\displaystyle\tau_{2}$ $\displaystyle=$
$\displaystyle(\mathrm{tr}(A^{2})-(\mathrm{tr}(A))^{2})^{-1}\,.$

Let $V$ be the set of all points of ${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}$
at which: $\mathrm{rank}(A)=2$ and $S_{ad}$ is not proportional to $g_{ad}$.

######  Theorem 5.2.

[51, Theorem 7.1 (ii)] Let $\overline{M}\times_{F}\widetilde{N}$ be the warped
product manifold with a $2$-dimensional semi-Riemannian manifold
$(\overline{M},\overline{g})$ and an $(n-2)$-dimensional semi-Riemannian
manifold $(\widetilde{N},\widetilde{g})$, $n\geq 4$, and a warping function
$F$, and let $(\widetilde{N},\widetilde{g})$ be a space of constant curvature
when $n\geq 5$. Then on the set
$V\subset{\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}$ (defined above) we have:

$\displaystyle C$ $\displaystyle=$
$\displaystyle-\frac{(n-1)\rho\tau_{2}}{(n-3)(n-2)}\left(g\wedge
S^{2}+\frac{n-2}{2}S\wedge S-\kappa g\wedge
S+\frac{\kappa^{2}-\mathrm{tr}(S^{2})}{n-1}G\right),$ $\displaystyle R\cdot
C+C\cdot R$ $\displaystyle=$ $\displaystyle
Q(S,C)+\left(L-\frac{\rho}{2(n-2)}+\frac{n-3}{(n-2)(n-1)\rho\tau_{2}}\right)Q(g,C),$

where $\rho$, $\tau_{1}$ and $\tau_{2}$ are defined by (5.7), (5.2) and
(5.10), respectively.

######  Theorem 5.3.

[51, Theorem 6.2] Let $\overline{M}\times_{F}\widetilde{N}$ be the warped
product manifold with a $2$-dimensional semi-Riemannian manifold
$(\overline{M},\overline{g})$ and an $(n-2)$-dimensional semi-Riemannian
manifold $(\widetilde{N},\widetilde{g})$, $n\geq 4$, and a warping function
$F$, and let $(\widetilde{N},\widetilde{g})$ be an Einstein space, when $n\geq
5$. On the set $V\subset{\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}$ we have

$\displaystyle R\cdot S$ $\displaystyle=$
$\displaystyle(\phi_{1}-2\tau_{1}\phi_{2}+\tau_{1}^{2}\phi_{3})\,Q(g,S)+(\phi_{2}-\tau_{1}\phi_{3})\,Q(g,S^{2})+\phi_{3}\,Q(S,S^{2}),$
$\displaystyle\phi_{1}$ $\displaystyle=$
$\displaystyle\frac{2\tau_{1}-\overline{\kappa}}{2(n-2)},\ \ \
\phi_{2}=\frac{1}{n-2},\ \ \
\phi_{3}=\frac{\tau_{2}(2\kappa-\overline{\kappa}-2(n-1)\tau_{1})}{n-2},$

where $\tau_{1}$ is defined by (5.2).

Theorems 4.10 and 4.11 imply

######  Theorem 5.4.

[44, Proposition 2.3] If $(M,g)$, $n\geq 4$, is a semi-Riemannian manifold
satisfying (4.1) or (4.11) at every point of
${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}\subset M$ then the following equation
is satisfied on this set

(5.11) $\displaystyle\tau\,C$ $\displaystyle=$ $\displaystyle g\wedge
S^{2}+\frac{n-2}{2}\,S\wedge S-\kappa\,g\wedge
S+\frac{\kappa^{2}-\mathrm{tr}(S^{2})}{2(n-1)}\,g\wedge g,$

where $\tau$ is some function on ${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}$.

Theorem 5.4, [51, Theorem 7.1 (ii)] and [65, Theorem 4.1] imply

######  Theorem 5.5.

[44, Theorem 2.4] 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 (5.11) holds on
${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}\subset\overline{M}\times_{F}\widetilde{N}$.

######  Remark 5.6.

Let $\overline{M}\times_{F}\widetilde{N}$ be the warped product manifold with
a $2$-dimensional semi-Riemannian manifold $(\overline{M},\overline{g})$ and
an $(n-2)$-dimensional semi-Riemannian manifold
$(\widetilde{N},\widetilde{g})$, $n\geq 4$, and a warping function $F$, and
let $(\widetilde{N},\widetilde{g})$ be a space of constant curvature when
$n\geq 5$.
(i) From [65, Theorem 4.1] it follows that at all points of the set
${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}$, at which $S_{ad}$ is proportional to
$g_{ad}$ and $\mathrm{rank}(A)=2$, the Riemann-Christoffel curvature tensor
$R$ is a linear combination of the Kulkarni-Nomizu products $S\wedge S$,
$g\wedge S$ and $g\wedge g$, i.e., (4.11) is satisfied. Thus, in view of [43,
Theorem 6.7] (see also [51, Theorem 3.2]), we have $R\cdot R=L_{R}\,Q(g,R)$
and in a consequence $R\cdot S=L_{R}\,Q(g,S)$, for some function $L_{R}$.
(ii) On the set $V\subset{\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}$ we also have
[51, Section 7]

$\displaystyle R\cdot C$ $\displaystyle=$ $\displaystyle
Q(S,C)+\left(L+\frac{n-3}{(n-2)(n-1)\rho\tau_{2}}\right)Q(g,C)+\frac{(n-1)\,\rho\tau_{2}}{(n-2)^{2}}\,g\wedge
Q(S,S^{2})$
$\displaystyle+\frac{1}{(n-2)^{2}}\,Q\left(\left(\frac{\rho}{2}+(n-1)\,\rho\tau_{1}^{2}\tau_{2}\right)S-(n-1)\,\rho\tau_{1}\tau_{2}\,S^{2},G\right),$
$\displaystyle C\cdot R$ $\displaystyle=$
$\displaystyle-\frac{\rho}{2(n-2)}\,Q(g,C)-\frac{(n-1)\,\rho\tau_{2}}{(n-2)^{2}}\,g\wedge
Q(S,S^{2})$
$\displaystyle-\frac{1}{(n-2)^{2}}\,Q\left(\left(\frac{\rho}{2}+(n-1)\,\rho\tau_{1}^{2}\tau_{2}\right)S-(n-1)\,\rho\tau_{1}\tau_{2}\,S^{2},G\right),$

and in a consequence

$\displaystyle R\cdot C-C\cdot R$ $\displaystyle=$ $\displaystyle
Q(S,C)+\left(L+\frac{n-3}{(n-2)(n-1)\rho\tau_{2}}+\frac{\rho}{2(n-2)}\right)Q(g,C)$
$\displaystyle+\frac{2}{(n-2)^{2}}\,Q\left(\left(\frac{\rho}{2}+(n-1)\,\rho\tau_{1}^{2}\tau_{2}\right)S-(n-1)\,\rho\tau_{1}\tau_{2}\,S^{2},G\right)$
$\displaystyle+\frac{2(n-1)\,\rho\tau_{2}}{(n-2)^{2}}\,g\wedge Q(S,S^{2}),$

where $\rho$, $\tau_{1}$ and $\tau_{2}$ are defined by (5.7), (5.2) and
(5.10), respectively.
(iii) A short presentation on quasi-Einstein and $2$-quasi-Einstein warped
product manifolds satisfying conditions of the form $(\ast)$ is given in [46]
(see also [39]).

##  6\. Semi-Riemannian manifolds satisfying the condition $R\cdot C-C\cdot
R=L\,Q(S,C)$

We can check that on any Einstein manifold $(M,g)$, $\dim M=n\geq 4$, the
tensors $Q(g,R)$, $Q(S,R)$, $Q(g,C)$ and $Q(S,C)$ satisfy

(6.1) $\displaystyle\frac{\kappa}{n}\,Q(g,R)\ =\ Q(S,R)$ $\displaystyle=$
$\displaystyle Q(S,C)\ =\ \frac{\kappa}{n}\,Q(g,C).$

Further, in [63, Theorem 3.1] it was stated that on every Einstein manifold
$(M,g)$, $\dim M=n\geq 4$, the following identity is satisfied

(6.2) $\displaystyle R\cdot C-C\cdot R$ $\displaystyle=$
$\displaystyle\frac{\kappa}{(n-1)n}\,Q(g,R).$

The remarks above lead to the problem of investigation of curvature properties
of non-Einstein and non-conformally flat semi-Riemannian manifolds $(M,g)$,
$\dim M=n\geq 4$, satisfying at every point of $M$ the curvature condition of
the following form: the difference tensor $R\cdot C-C\cdot R$ is proportional
to $Q(g,R)$, $Q(S,R)$, $Q(g,C)$ and $Q(S,C)$. Such conditions are strongly
related to some pseudosymmetry type curvature conditions, see, e.g., [43] and
references therein.

Semi-Riemannian manifolds $(M,g)$, $\dim M=n\geq 4$, satisfying at every point
of $M$ the following condition

(6.3) $\displaystyle\mbox{the tensors}\ \ R\cdot C-C\cdot R\ \ \mbox{and}\ \
Q(S,C)\ \ \mbox{are linearly dependent},$

were investigated in [48]. It is obvious that (6.3) is satisfied at every
point of $M$ at which $C$ vanishes. It is also clear that (6.1) and (6.2)
imply that

$\displaystyle R\cdot C-C\cdot R$ $\displaystyle=$
$\displaystyle\frac{1}{n-1}\,Q(S,C)$

holds on any Einstein manifold $(M,g)$, $\dim M=n\geq 4$. Therefore we will
restrict our considerations to manifolds $(M,g)$, $\dim M=n\geq 4$, satisfying
(6.3) on the set ${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}\subset M$. Thus on
${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}$ we have (1.4), i.e., $R\cdot C-C\cdot
R=L\,Q(S,C)$, where $L$ is some function on this set. We mention that if the
tensor $R\cdot C-C\cdot R$ vanishes on
${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}$ then on this set [63, Theorem 4.1]:

(6.4) $\displaystyle R\cdot C\ =\ C\cdot R$ $\displaystyle=$ $\displaystyle
0,$ $\displaystyle Q(S,C)$ $\displaystyle=$ $\displaystyle 0.$

On the other hand, if $Q(S,C)$ vanishes on
${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}$ then at every point
$x\in{\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}$ we have:
(i) if $\mbox{rank}\,S=1$ at $x$ then $\kappa=0$ and (6.4) hold at $x$ [48,
Section 3], or
(ii) if $\mbox{rank}\,S>1$ at $x$ then $C\cdot R=0$ and $R\cdot
C=\frac{\kappa}{n-1}\,Q(g,C)$ hold at $x$ [48, Section 4].
Thus we see that in the case (ii), if a manifold satisfies (1.4) then its
scalar curvature must vanish on ${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}$.

The main result of [48, Section 3] states that pseudosymmetric manifolds
satisfying some additional curvature conditions are quasi-Einstein manifolds
satisfying the conditions: $C\cdot C=0$, $C\cdot R=0$, and (1.4) with
$L=\frac{1}{n-1}$. Precisely we have

######  Theorem 6.1.

[48, Theorem 3.4] Let $(M,g)$, $\dim M=n\geq 4$, be a semi-Riemannian
manifold. If the following conditions:

$\displaystyle R\cdot R$ $\displaystyle=$
$\displaystyle\frac{\kappa}{(n-1)n}\,Q(g,R),$ $\displaystyle R\cdot R-Q(S,R)$
$\displaystyle=$ $\displaystyle-\frac{(n-2)\kappa}{(n-1)n}\,Q(g,C),$
$\displaystyle R\cdot C$ $\displaystyle=$ $\displaystyle\frac{1}{n-1}\,Q(S,C)$

are satisfied on ${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}\subset M$, then on
this set we have:

$\displaystyle C\cdot C$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\mbox{rank}\,(S-\frac{\kappa}{n}\,g)$ $\displaystyle=$
$\displaystyle 1,$ $\displaystyle C\cdot R$ $\displaystyle=$ $\displaystyle
0,$ $\displaystyle(n-1)\,(R\cdot C-C\cdot R)$ $\displaystyle=$ $\displaystyle
Q(S,C).$

######  Theorem 6.2.

[48, Proposition 3.9] Let $T$ be a generalized curvature tensor on a semi-
Riemannian manifold $(M,g)$, $\dim M=n\geq 4$.
(i) If the conditions

$\displaystyle Q(Ric(T),T)$ $\displaystyle=$ $\displaystyle 0,$ (6.5)
$\displaystyle\mbox{rank}\,(Ric(T))$ $\displaystyle=$ $\displaystyle 1$

are satisfied on ${\mathcal{U}}_{Ric(T)}\cap{\mathcal{U}}_{Weyl(T)}\subset M$,
then on this set we have

(6.6) $\displaystyle\kappa(T)$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle T\cdot Weyl(T)\ =\ Weyl(T)\cdot T$ $\displaystyle=$
$\displaystyle Q(Ric(T),Weyl(T))\ =\ 0.$

(ii) If the conditions (6.5) and

$\displaystyle Q(Ric(T),Weyl(T))$ $\displaystyle=$ $\displaystyle 0$

are satisfied on ${\mathcal{U}}_{Ric(T)}\cap{\mathcal{U}}_{Weyl(T)}\subset M$,
then on this set we have (6.6) and

$\displaystyle T\cdot Weyl(T)$ $\displaystyle=$ $\displaystyle Weyl(T)\cdot T\
=\ 0.$

In [48, Section 3] an example of warped product manifolds satisfying
assumptions of [48, Theorem 3.4] is also given.

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
essentially conformally symmetric manifolds, e.c.s. manifolds/metrics, or ECS
manifolds/metrics, in short (see, e.g., [23, 24, 26, 30, 31]). E.c.s.
manifolds are semisymmetric manifolds ($R\cdot R=0$ [23, Theorem 9])
satisfying $\kappa=0$ and $Q(S,C)\,=\,0$ ([23, Theorems 7 and 8]). In
addition,

(6.7) $\displaystyle F\,C$ $\displaystyle=$ $\displaystyle\frac{1}{2}\,S\wedge
S$

holds on $M$, where $F$ is some function on $M$, called the fundamental
function [24]. At every point of $M$ we also have $\mbox{rank}\,S\leq 2$ [24,
Theorem 5]. We mention that the local structure of e.c.s. manifolds is already
determined. We refer to [25, 28] for results related to this subject. We also
mention that certain e.c.s. metrics are realized on compact manifolds [27, 29,
30, 31, 32, 33, 34].

Equation (6.7), 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 (6.7) we get

$\displaystyle F\,C$ $\displaystyle=$
$\displaystyle\frac{n-2}{2(n-2)}\,S\wedge S\ =\ \frac{1}{n-2}\left(g\wedge
S^{2}+\frac{n-2}{2}\,S\wedge S-\kappa\,g\wedge
S+\frac{\kappa^{2}-\mathrm{tr}(S^{2})}{2(n-1)}\,g\wedge g\right).$

Thus we have

######  Theorem 6.3.

[42, Theorem 6.1] Condition (5.11), with $\tau=(n-2)F$, is satisfied on every
essentially conformally symmetric manifold $(M,g)$.

We assume that $F=0$ at $x\in M$. Now (6.7) implies $\mbox{rank}\,S\leq 1$ at
$x$. It is clear that if $S$ vanishes then

(6.8) $\displaystyle R\cdot C$ $\displaystyle=$ $\displaystyle C\cdot R\ =\
Q(S,C)\ =\ 0$

holds at $x$. If $\mbox{rank}\,S=1$ then in view of [48, Proposition 3.9(ii)]
we also have (6.8) at $x$. Next, we assume that $F$ is non-zero at $x\in M$.
Thus $\mbox{rank}\,S=2$ at $x$. Now (6.7) turns into (2.14) with $T=R$,
$Ric(T)=S$, $\phi=F^{-1}$, $\mu=\frac{1}{n-2}$ and $\eta=0$. Therefore (2.15)
and (2.16) reduce to $C\cdot R=0$ and $C\cdot C=0$, respectively.
Consequently, (6.8) holds at $x$. Thus we have

######  Theorem 6.4.

[48, Theorem 4.1] Condition (6.8) is satisfied on every essentially
conformally symmetric manifold $(M,g)$.

Thus we see that e.c.s. manifolds satisfy (1.4). We also mention that the
tensor $C\cdot C$ of every e.c.s. manifold is the zero tensor ([48, Remark
4.2(ii)]).

E.c.s. warped product manifolds were investigated in [89]. In that paper
examples of such manifolds are given [48, Remark 4.2(i)].

In [48, Section 5] Roter type manifolds satisfying (1.4) are investigated. In
[48, Theorem 5.2] it was stated that if $(M,g)$, $\dim M=n\geq 4$, is a Roter
type manifold with vanishing scalar curvature $\kappa$ on
${\mathcal{U}}\subset M$ then (1.4), with $L=-1$, holds on this set.

######  Theorem 6.5.

[48, Theorem 5.2] Let $(M,g)$, $\dim M=n\geq 4$, be a a semi-Riemannian
manifold satisfying (4.11) on ${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}\subset
M$. If $\kappa=0$ on ${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}$ then (1.4), with
$L\,=\,-1$, holds on this set.

This result is also an immediate consequence of the fact that every Roter
space $(M,g)$ satisfies (1.2) on
${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}\subset M$ (see Section 4). However in
the proof of the last theorem formula (1.2) was not applied.

We also have

######  Theorem 6.6.

[48, Theorem 5.3] Let $(M,g)$, $\dim M=n\geq 4$, be a semi-Riemannian manifold
satisfying (4.11) and (1.4) on ${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}\subset
M$, and let ${\mathcal{U}}_{1}\subset{\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}$
be the set of all points at which the functions $L$ and $L_{C}$, defined by
(1.4), (2.15) and (2.16) (for $T=R$), respectively, are nowhere zero on this
set. Then we have on ${\mathcal{U}}_{1}$: $L=-1$ and $\kappa=0$.

In [48, Example 5.4] it was shown that under some conditions the Cartesian
product of two semi-Riemannian spaces of constant curvature satisfies
assumptions of Theorem 6.6 (i.e., [48, Theorem 5.3]).

##  7\. 2-quasi-umbilical hypersurfaces

Let $N_{s}^{n+1}(c)$, $n\geq 3$, be a semi-Riemannian space of constant
curvature $c=\frac{\widetilde{\kappa}}{n(n+1)}$ with signature $(s,n+1-s)$,
where $\widetilde{\kappa}$ is its scalar curvature. Let $M$ be a connected
hypersurface isometrically immersed in $N_{s}^{n+1}(c)$. We have (see, e.g.,
[47])

(7.1) $\displaystyle R_{hijk}$ $\displaystyle=$
$\displaystyle\varepsilon\,(H_{hk}H_{ij}-H_{hj}H_{ik})+\frac{\widetilde{\kappa}}{n(n+1)}\,G_{hijk}\,,\
\ \ \varepsilon\ =\ \pm 1,$

where $R_{hijk}$, $G_{hijk}$ and $H_{hk}$, respectively, are the local
components of the curvature tensor $R$ of $M$, the tensor $G$ and the second
fundamental tensor $H$, respectively. Contracting (7.1) with $g^{ij}$ and
$g^{kh}$, respectively, we obtain

$\displaystyle S_{hk}$ $\displaystyle=$
$\displaystyle\varepsilon\,(\mathrm{tr}(H)\,H_{hk}-H^{2}_{hk})+\frac{(n-1)\widetilde{\kappa}}{n(n+1)}\,g_{hk},$
$\displaystyle\frac{\kappa}{n-1}$ $\displaystyle=$
$\displaystyle\frac{\varepsilon}{n-1}\,((\mathrm{tr}(H))^{2}-\mathrm{tr}(H^{2}))+\frac{\widetilde{\kappa}}{n+1},$

respectively, where $\mathrm{tr}(H)=g^{hk}H_{hk}$,
$\mathrm{tr}(H^{2})=g^{hk}H^{2}_{hk}$ and $S_{hk}$ are the local components of
the Ricci tensor $S$ of $M$ and $\kappa$ is the scalar curvature of $M$. It is
known that (3.6) holds on $M$. Precisely,

(7.2) $\displaystyle R\cdot R-Q(S,R)$ $\displaystyle=$
$\displaystyle-\frac{(n-2)\widetilde{\kappa}}{n(n+1)}\,Q(g,C)$

on $M$ [71, Lemma 2.1] (see also [19, Lemma 2.1 (ii)]). Evidently, if $n=3$
then (7.2) turns into

(7.3) $\displaystyle R\cdot R$ $\displaystyle=$ $\displaystyle Q(S,R).$

If the ambient space is a semi-Euclidean space $\mathbb{E}^{n+1}_{s}$, $n\geq
3$, then (7.2) also reduces to (7.3). As it was proved in [20, Lemma 4.1], we
also have the following identity on $M$ in $N_{s}^{n+1}(c)$, $n\geq 3$,

(7.4) $\displaystyle R\cdot R-\frac{\widetilde{\kappa}}{n(n+1)}\,Q(g,R)$
$\displaystyle=$ $\displaystyle-Q(H^{2},\frac{1}{2}\,H\wedge H).$

Further we have

######  Theorem 7.1.

[51, Theorem 3.7] Let $M$ be a hypersurface isometrically immersed in
$N_{s}^{n+1}(c)$, $n\geq 4$. Then

$\displaystyle C\cdot R+R\cdot C$ $\displaystyle=$ $\displaystyle
Q(S,C)-\frac{(n-2)\widetilde{\kappa}}{n(n+1)}\,Q(g,C)+C\cdot C$
$\displaystyle-\frac{1}{(n-2)^{2}}\,Q\left(g,g\wedge
S^{2}+\frac{n-2}{2}\,S\wedge S-\kappa\,g\wedge S\right)$

holds on $M$. Moreover, if the condition $C\cdot C=L_{C}Q(g,C)$ (i.e., (3.5))
is satisfied on ${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}\subset M$ then on this
set we have

(7.5) $\displaystyle C\cdot R+R\cdot C$ $\displaystyle=$ $\displaystyle
Q(S,C)+\left(L_{C}-\frac{(n-2)\widetilde{\kappa}}{n(n+1)}\right)Q(g,C)$
$\displaystyle-\frac{1}{(n-2)^{2}}\,Q\left(g,g\wedge
S^{2}+\frac{n-2}{2}\,S\wedge S-\kappa\,g\wedge S\right),$

and in addition, if $M$ is a quasi-Einstein hypersurface satisfying the
condition $\mathrm{rank}\,(S-\alpha\,g)=1$ (i.e., (4.1)), on
${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}$ then on this set we have

$\displaystyle C\cdot R+R\cdot C$ $\displaystyle=$ $\displaystyle
Q(S,C)+\left(L_{C}-\frac{(n-2)\widetilde{\kappa}}{n(n+1)}\right)Q(g,C).$

It is known that every $2$-quasi-umbilical hypersurface in a semi-Riemannian
space of constant curvature $N_{s}^{n+1}(c)$, $n\geq 4$, satisfies (3.5) (see,
e.g., [51, Theorem 3.8]). We have

######  Theorem 7.2.

[51, Theorem 3.8] If $M$ is a $2$-quasi-umbilical hypersurface isometrically
immersed in $N_{s}^{n+1}(c)$, $n\geq 4$, then (7.5) holds on
${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}\subset M$.

######  Theorem 7.3.

[53, Proposition 4.2] Let $M$ be a hypersurface in an Euclidean space
$\mathbb{E}^{n+1}$, $n\geq 4$, having exactly three distinct principal
curvatures $\lambda_{1}$, $\lambda_{2}$ and $\lambda_{3}$ satisfying at every
point of $M$: $\lambda_{1}=0$, $\lambda_{2}=-(n-2)\lambda$ and
$\lambda_{3}=\lambda_{4}=\ldots=\lambda_{n}=\lambda\neq 0$. Then $M$ is a
minimal, $2$-quasi-umbilical and $2$-quasi-Einstein hypersurface satisfying:
(1.7), (7.3) and

$\displaystyle\mathrm{rank}\,\left(S-\frac{\kappa}{(n-2)(n-1)}\,g\right)$
$\displaystyle=$ $\displaystyle 2,$ $\displaystyle S$ $\displaystyle=$
$\displaystyle-H^{2},\ \ \ \kappa\ =\ -tr(H^{2})\ =\ -(n-2)(n-1)\lambda^{2},$
$\displaystyle S^{2}$ $\displaystyle=$
$\displaystyle-(\phi^{2}+\psi)\,S+\phi\psi\,H,$ $\displaystyle S^{3}$
$\displaystyle=$ $\displaystyle-(\phi^{2}+2\psi)\,S^{2}-\psi^{2}S,$ (7.6)
$\displaystyle R$ $\displaystyle=$
$\displaystyle\frac{1}{2(\phi\psi)^{2}}\left(S^{2}+(\phi^{2}+\psi)S\right)\wedge\left(S^{2}+(\phi^{2}+\psi)S\right),$
$\displaystyle R\cdot S$ $\displaystyle=$ $\displaystyle\phi\,Q(H,S)\ =\
\frac{n-1}{\kappa}\,Q(S,S^{2}),$ $\displaystyle C\cdot S$ $\displaystyle=$
$\displaystyle\phi\,Q(H,S)+\frac{\phi^{2}}{n-2}\,Q(g,S)-\frac{\phi\psi}{n-2}\,Q(g,H)$
$\displaystyle=$
$\displaystyle\frac{\kappa}{(n-2)(n-1)}\,Q(g,S)-\frac{1}{n-2}\,Q(g,S^{2})+\frac{n-1}{\kappa}\,Q(S,S^{2}),$
$\displaystyle(n-2)\,R\cdot C$ $\displaystyle=$
$\displaystyle(n-2)\,Q(S,R)-\phi\,g\wedge Q(H,S),$ $\displaystyle(n-2)\,C\cdot
R$ $\displaystyle=$ $\displaystyle(n-3)\,Q(S,R)-\phi\,H\wedge Q(g,S),$
$\displaystyle C\cdot C$ $\displaystyle=$ $\displaystyle 0,$

where $\phi=-(n-3)\lambda$ and $\psi=(n-2)\lambda^{2}$. Moreover, we have

(7.7) $\displaystyle(n-2)(R\cdot C-C\cdot R)$ $\displaystyle=$ $\displaystyle
Q(S,R)+\phi\,(H\wedge Q(g,S)-g\wedge Q(H,S)).$

We note that (7.6) is a particular form of the Roter type equation (4.12).

Biharmonic hypersurfaces with three distinct principal curvatures in an
Euclidean $5$-space $\mathbb{E}^{5}$ were investigated in [78]. The main
result of [78, Theorem 3.2] states that every biharmonic hypersurface $M$ with
three distinct principal curvatures in $\mathbb{E}^{5}$ is minimal. The
principal curvatures of $M$ are the following: $\lambda_{1}=0$,
$\lambda_{2}=-2\lambda$ and $\lambda_{3}=\lambda_{4}=\lambda\neq 0$, where
$\lambda$ is some function on $M$. Curvature properties of such hypersurfaces
are expressed in the last theorem, provided that $n=4$ (see also [53, Theorem
4.3]).

We refer to [108] for a survey of results on biharmonic hypersurfaces.

Let now $M$ be a type number two hypersurface in $N_{s}^{n+1}(c)$, $n\geq 3$,
i.e., let $\mathrm{rank}(H)=2$ at every point of $M$ (see, e.g., [14]). Using
(2.8), (2.9), (2.10) and (7.4) we obtain on $M$ (cf. [20, Theoem 4.2])

(7.8) $\displaystyle H^{3}\ =\
\mathrm{tr}(H)\,H^{2}+\frac{\mathrm{tr}(H^{2})-(\mathrm{tr}(H))^{2}}{2}\,H,$
(7.9) $\displaystyle Q(H,H\wedge H^{2})\ =\ 0,$ (7.10) $\displaystyle R\cdot
R\ =\ \frac{\widetilde{\kappa}}{n(n+1)}\,Q(g,R).$

Thus we have

######  Theorem 7.4.

(cf. [20, Theorem 4.2]) Every type number two hypersurface $M$ in
$N_{s}^{n+1}(c)$, $n\geq 3$, is a pseudosymmetric manifold of constant type
satisfying (7.8), (7.9) and (7.10).

We mention that type number two hypersurfaces in $4$-dimensional Riemannian
space of constant curvature were investigated in [86]. We also note that in
[74, Section 5] it was stated that the Cartan hypersurface in a
$4$-dimensional sphere is a pseudosymmetric manifold of constant type.
Evidently, (7.8) is a special form of the equation

(7.11) $\displaystyle H^{3}$ $\displaystyle=$
$\displaystyle\mathrm{tr}(H)\,H^{2}+\psi\,H,$

where $\psi$ is a function on $M$. We refer to Remark 8.7 (ii) of this paper
for results on hypersurfaces in $N_{s}^{n+1}(c)$, $n\geq 4$, satisfying
(7.11).

As an immediate consequence of Proposition 2.1 and [42, Remark 7.1 (iii)] we
obtain the following

######  Theorem 7.5.

If $M$, $\dim M=n\geq 4$, is a type number two hypersurface isometrically
immersed in a semi-Riemannian conformally flat manifold $N$, $\dim N=n+1$,
then $C\cdot C=L_{C}Q(g,C)$ (i.e., (3.5)) holds on ${\mathcal{U}}_{C}\subset
M$, where $L_{C}$ is some function on this set.

Let $M$ be a type number two hypersurface in $N_{s}^{n+1}(c)$, $n\geq 4$.
Thus, in view of the last theorem, (3.5) holds on ${\mathcal{U}}_{C}\subset
M$. Moreover, in view of Theorem 7.1, (7.5) is satisfied on
${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}\subset M$. In addition, we note that
(7.10) implies

(7.12) $\displaystyle R\cdot C\ =\ \frac{\widetilde{\kappa}}{n(n+1)}\,Q(g,C).$

Now using Theorem 7.1, Theorem 7.5 and (7.12) we get easily the following
result.

######  Theorem 7.6.

Let $M$ be a type number two hypersurface in $N_{s}^{n+1}(c)$, $n\geq 4$. Then
the following condition of the form $(\ast)$ is satisfied on
${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}$

(7.13) $\displaystyle\ \ \ C\cdot R-R\cdot C$ $\displaystyle=$ $\displaystyle
Q(S,C)+L_{1}\,Q(g,C)-\frac{1}{(n-2)^{2}}\,Q\left(g,g\wedge
S^{2}+\frac{n-2}{2}\,S\wedge S-\kappa\,g\wedge S\right),$

where $L_{1}$ is some function on this set. Moreover, if $M$ is a quasi-
Einstein hypersurface satisfying the condition
$\mathrm{rank}\,(S-\alpha\,g)=1$ (i.e., (4.1)) on
${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}$ then on this set we have

(7.14) $\displaystyle\ \ \ C\cdot R-R\cdot C$ $\displaystyle=$ $\displaystyle
Q(S,C)+L_{1}\,Q(g,C).$

We refer to [58, Section 5] for further results on type number two
hypersurfaces in $N_{s}^{n+1}(c)$, $n\geq 4$, satisfying curvatures conditions
of pseudosymmetry type.

##  8\. The condition $H^{3}=\mathrm{tr}(H)\,H^{2}+\psi\,H+\rho\,g$

Let $M$ be a hypersurface isometrically immersed in $N_{s}^{n+1}(c)$, $n\geq
4$, satisfying on ${\mathcal{U}}_{H}\subset M$ a curvature condition of the
kind: the tensor $R\cdot C$, $C\cdot R$ or $R\cdot C-R\cdot C$ is a linear
combination of the tensor $R\cdot R$ and of a finite sum of the Tachibana
tensors of the form $Q(A,T)$, where $A$ is a symmetric $(0,2)$-tensor and $T$
a generalized curvature tensor. As it was mentioned in Introduction, if such
condition is satisfied on ${\mathcal{U}}_{H}$ then (1.6) holds on this set. We
present now the following results on hypersurfaces $M$ in $N_{s}^{n+1}(c)$,
$n\geq 4$, satisfying (1.6) on ${\mathcal{U}}_{H}\subset M$.

######  Theorem 8.1.

[110, Proposition 5.1, eq. (29)] If $M$ is a hypersurface in a semi-Riemannian
space of constant curvature $N_{s}^{n+1}(c)$, $n\geq 4$, satisfying (1.6) on
${\mathcal{U}}_{H}\subset M$, for some functions $\psi$ and $\rho$ on
${\mathcal{U}}_{H}$, then on ${\mathcal{U}}_{H}$ we have

$\displaystyle R\cdot C$ $\displaystyle=$ $\displaystyle
Q(S,R)-\frac{(n-2)\widetilde{\kappa}}{n(n+1)}\,Q(g,R)+\alpha_{2}\,Q(S,G)+\frac{\rho}{n-2}\,Q(H,G),$
$\displaystyle C\cdot R$ $\displaystyle=$
$\displaystyle\frac{n-3}{n-2}\,Q(S,R)+\alpha_{1}\,Q(g,R)+\alpha_{2}\,Q(S,G),$
(8.1) $\displaystyle(n-2)\,(R\cdot C-C\cdot R)$ $\displaystyle=$
$\displaystyle
Q(S,R)+\rho\,Q(H,G)+\left(\frac{(n-1)\widetilde{\kappa}}{n(n+1)}-\frac{\kappa}{n-1}-\varepsilon\psi\right)Q(g,R),$
$\displaystyle(n-2)\,C\cdot C$ $\displaystyle=$
$\displaystyle(n-3)\,Q(S,R)+(n-2)\alpha_{1}\,Q(g,R)$
$\displaystyle+(\alpha_{1}-\alpha_{2})\,Q(S,G)+\frac{n-3}{n-2}\rho\,Q(H,G),$
$\displaystyle R\cdot S$ $\displaystyle=$
$\displaystyle\frac{\widetilde{\kappa}}{n(n+1)}\,Q(g,S)+\rho\,Q(g,H),$ (8.2)
$\displaystyle\alpha_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{n-2}\left(\frac{\kappa}{n-1}+\varepsilon\psi-\frac{(n^{2}-3n+3)\widetilde{\kappa}}{n(n+1)}\right),$
(8.3) $\displaystyle\alpha_{2}$ $\displaystyle=$
$\displaystyle-\frac{(n-3)\widetilde{\kappa}}{(n-2)n(n+1)}.$

######  Theorem 8.2.

[55, Proposition 4.2] Let $M$ be a hypersurface in a semi-Riemannian space of
constant curvature $N_{s}^{n+1}(c)$, $n\geq 4$, satisfying (1.6) on
${\mathcal{U}}_{H}\subset M$, for some functions $\psi$ and $\rho$ on
${\mathcal{U}}_{H}$.
(i) The following conditions are satisfied on ${\mathcal{U}}_{H}$:

$\displaystyle Q(\rho\,H-\alpha_{3}\,S-S^{2},G)$ $\displaystyle=$
$\displaystyle 0,$ (8.4) $\displaystyle\rho\,H$ $\displaystyle=$
$\displaystyle S^{2}+\alpha_{3}\,S+\frac{\lambda}{n}\,g,\ \ \ \lambda\ =\
\rho\,\mathrm{tr}(H)-\kappa\,\alpha_{3}-\mathrm{tr}(S^{2}),$ (8.5)
$\displaystyle\alpha_{3}$ $\displaystyle=$
$\displaystyle(n-2)^{2}\left(\frac{1}{n-2}(\alpha_{1}-\alpha_{2})-2\alpha_{2}-\frac{\widetilde{\kappa}}{n(n+1)}\right)-\frac{\kappa}{n-1}\
=\ \varepsilon\psi-\frac{2(n-1)\widetilde{\kappa}}{n(n+1)},$

where $\alpha_{1}$ and $\alpha_{2}$ are defined by (8.2) and (8.3),
respectively. Moreover,

$\displaystyle R\cdot S$ $\displaystyle=$ $\displaystyle
Q(g,S^{2})+\left(\varepsilon\psi-\frac{(2n-3)\widetilde{\kappa}}{n(n+1)}\right)Q(g,S),$
$\displaystyle R\cdot S^{2}$ $\displaystyle=$ $\displaystyle
Q(S,S^{2})+\rho_{1}\,Q(g,S^{2})+\rho_{2}\,Q(g,S),$ $\displaystyle S^{3}$
$\displaystyle=$
$\displaystyle\left(-2\varepsilon\psi+\frac{3(n-1)\widetilde{\kappa}}{n(n+1)}\right)S^{2}+\rho_{2}\,S+\rho_{3}\,g,$

hold on ${\mathcal{U}}_{H}$, where the functions $\rho_{1}$, $\rho_{2}$ and
$\rho_{3}$ are defined by

$\displaystyle\rho_{1}$ $\displaystyle=$
$\displaystyle-\frac{(n-2)\widetilde{\kappa}}{n(n+1)}-\alpha_{3},$
$\displaystyle\rho_{2}$ $\displaystyle=$
$\displaystyle-\frac{\lambda}{n}-\left(\frac{(n-1)\widetilde{\kappa}}{n(n+1)}+\alpha_{3}\right)\alpha_{3},$
$\displaystyle\rho_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{n}\left(\mathrm{tr}(S^{3})+\left(2\varepsilon\psi-\frac{3(n-1)\widetilde{\kappa}}{n(n+1)}\right)\mathrm{tr}(S^{2})-\kappa\rho_{2}\right).$

(ii) If at a point $x\in{\mathcal{U}}_{H}$ we have
$S^{2}=\beta_{1}\,S+\beta_{2}\,g$, for some
$\beta_{1},\,\beta_{2}\in{\mathbb{R}}$, then $\rho=0$, $\beta_{1}=\alpha_{3}$
and $\beta_{2}=-(\lambda/n)$ at this point.

Theorems 8.1 and 8.2 (precisely, (8.1), (8.4) and (8.5)) lead to the following
result.

######  Theorem 8.3.

If $M$ is a hypersurface in a semi-Riemannian space of constant curvature
$N_{s}^{n+1}(c)$, $n\geq 4$, satisfying (1.6) on ${\mathcal{U}}_{H}\subset M$,
for some functions $\psi$ and $\rho$ on ${\mathcal{U}}_{H}$ then

$\displaystyle(n-2)\,(R\cdot C-C\cdot R)$ $\displaystyle=$ $\displaystyle
Q(S,R)+\left(\frac{(n-1)\widetilde{\kappa}}{n(n+1)}-\frac{\kappa}{n-1}-\varepsilon\psi\right)Q(g,R)$
$\displaystyle+Q(S^{2},G)+\left(\varepsilon\psi-\frac{2(n-1)\widetilde{\kappa}}{n(n+1)}\right)Q(S,G)$

on ${\mathcal{U}}_{H}$.

######  Theorem 8.4.

[55, Theorem 4.5] Let $M$ be a hypersurface in a semi-Riemannian space of
constant curvature $N_{s}^{n+1}(c)$, $n\geq 4$, satisfying (1.6) on
${\mathcal{U}}_{H}\subset M$, for some functions $\psi$ and $\rho$ on
${\mathcal{U}}_{H}$. If the tensor $C\cdot C$ and a generalized curvature
tensor $T$ satisfy $C\cdot C=Q(g,T)$ on ${\mathcal{U}}_{H}$, then on this set
we have

$\displaystyle T$ $\displaystyle=$
$\displaystyle\left(\frac{\kappa}{n-1}+\frac{2\varepsilon\psi}{n-1}-\frac{\widetilde{\kappa}}{n+1}\right)C+\lambda\,G-\frac{n-3}{(n-2)^{2}(n-1)}\left(g\wedge
S^{2}+\frac{n-2}{2}\,S\wedge S-\kappa\,g\wedge S\right),$

where $\lambda$ is some function on ${\mathcal{U}}_{H}$.

Further, we also have the following results obtained in [55, Section 4].

######  Theorem 8.5.

Let $M$ be a hypersurface isometrically immersed in $N_{s}^{n+1}(c)$, $n\geq
4$, satisfying (1.6) on ${\mathcal{U}}_{H}\subset M$.
(i) [55, Theorem 4.6] If on ${\mathcal{U}}_{H}$ the tensor $Q(S,R)$ is equal
to the Tachibana tensor $Q(g,T_{1})$, where $T_{1}$ is a generalized curvature
tensor, then any of the tensors: $R\cdot R$, $R\cdot C$, $C\cdot R$, $R\cdot
C-C\cdot R$ and $C\cdot C$ is equal to some Tachibana tensor $Q(g,T_{2})$,
where $T_{2}$ is a linear combination of the tensors $R$, $g\wedge g$,
$g\wedge S$, $g\wedge S^{2}$ and $S\wedge S$.
(ii) [55, Theorem 4.7] The following conditions are satisfied on
${\mathcal{U}}_{H}$

$\displaystyle C\cdot C$ $\displaystyle=$
$\displaystyle\frac{n-3}{n-2}\,R\cdot
C+\frac{1}{n-2}\left(\frac{\kappa}{n-1}+\varepsilon\psi-\frac{(2n-3)\widetilde{\kappa}}{n(n+1)}\right)Q(g,C),$
$\displaystyle(n-2)\,C\cdot R+R\cdot C$ $\displaystyle=$
$\displaystyle(n-2)\,Q(S,C)+\left(\frac{\kappa}{n-1}+\varepsilon\psi-\frac{(n-1)^{2}\widetilde{\kappa}}{n(n+1)}\right)Q(g,C)$
$\displaystyle-\frac{1}{(n-2)}\,Q\left(g,g\wedge S^{2}+\frac{n-2}{2}\,S\wedge
S-\kappa\,g\wedge S\right).$

######  Theorem 8.6.

[55, Theorem 4.8] Let $M$ be a hypersurface isometrically immersed in
$N_{s}^{n+1}(c)$, $n\geq 4$, such that (4.1) and (7.11) are satisfied on
${\mathcal{U}}_{H}\subset M$, where $\psi$ is a function on this set. Then
(1.2) holds on ${\mathcal{U}}_{H}$ if and only if the following two conditions
hold on this set

(8.6) $\displaystyle(a)\ \ \frac{\kappa}{n-1}\ =\
\frac{\widetilde{\kappa}}{n+1}\ \ \ $ and $\displaystyle\ \ \ (b)\ \
Q\left(S-\frac{\kappa}{n}\,g,C\right)\ =\ 0.$

For further results related to Theorem 8.5 (i) we refer to [42, Section 8].

######  Remark 8.7.

(i) Let $M$ be a hypersurface in $N_{s}^{n+1}(c)$, $n\geq 4$. As it was proved
in [20, Proposition 3.1 (ii), Proposition 3.2] the conditions (7.11) and

(8.7) $\displaystyle R\cdot S$ $\displaystyle=$
$\displaystyle\frac{\widetilde{\kappa}}{n(n+1)}\,Q(g,S)$

are equivalent on the set ${\mathcal{U}}_{H}\subset M$. If (7.11) holds on
${\mathcal{U}}_{H}\subset M$ then on this set we have [38, Theorem 3.1)]

(8.8) $\displaystyle(n-2)(R\cdot C-C\cdot R)$ $\displaystyle=$ $\displaystyle
Q(S,R)+\left(\frac{(n-1)\widetilde{\kappa}}{n(n+1)}-\frac{\kappa}{n-1}-\varepsilon\psi\right)Q(g,R).$

(ii) Let now $M$ be a hypersurface in a Riemannian space of constant curvature
$N^{n+1}(c)$, $n\geq 4$, such that at every point of $M$ there are principal
curvatures $0,\ldots,0,\lambda,\ldots,\lambda,-\lambda,\ldots,-\lambda$, with
the same multiplicity of $\lambda$ and $-\lambda$, and $\lambda$ is a positive
function on $M$. Thus we have on $M$: $\mathrm{tr}(H)=0$ and
$H^{3}=\lambda^{2}\,H$, and, in a consequence, we also have (8.7) and (8.8).
We mention that the Cartan hypersurfaces, as well as generalized Cartan
hypersurfaces [8, Section 6] have at every point principal curvatures
$0,\ldots,0,\lambda,\ldots,\lambda,-\lambda,\ldots,-\lambda$, with the same
multiplicity of $\lambda$ and $-\lambda$, and $\lambda$ is a positive function
on $M$. Curvature properties of pseudosymmetry type of Cartan hypersurfaces
(see, e.g., [6, Chapter 3], [108, Chapter 7.5]) are given in [38, 74, 75].
Both classes of the considered hypersurfaces are austere hypersurfaces [5].

##  9\. Hypersurfaces satisfying the condition $R\cdot C-C\cdot R=LQ(S,C)$

In [43] a survey on some family of generalized Einstein metric conditions was
given. Those curvature conditions are strongly related to pseudosymmetry. In
particular, [43, Section 6] contains results of non-Einstein and non-
conformally flat semi-Riemannian manifolds $(M,g)$, of dimension $n\geq 4$,
satisfying conditions of the form: the tensor $R\cdot C-C\cdot R$ is
proportional to the Tachibana tensor: $Q(g,R)$, $Q(S,R)$, $Q(g,C)$ or
$Q(S,C)$. More precisely, those conditions are considered on
${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}\subset M$. Among other results in that
section it was shown that some hypersurfaces $M$ isometrically immersed in a
semi-Riemannian space of constant curvature $N^{n+1}_{s}(c)$, $n\geq 4$,
satisfy (1.4) on ${\mathcal{U}}_{S}\cap{\mathcal{U}}_{C}\subset M$. We recall
that an example of a hypersurface having mentioned properties was constructed
in [58, Section 5]. We also mention that semi-Riemannian manifolds satisfying
(1.4) were investigated in [48].

Let $M$ be a hypersurface isometrically immersed in $N^{n+1}_{s}(c)$, $n\geq
4$, satisfying (1.4) on
$({\mathcal{U}}_{S}\cap{\mathcal{U}}_{C})\setminus{\mathcal{U}}_{H}$. We
recall that (1.5) holds on
$({\mathcal{U}}_{S}\cap{\mathcal{U}}_{C})\setminus{\mathcal{U}}_{H}$, where
$\alpha$ and $\beta$ are some functions defined on this set.

According to [81, Proposition 3.3], the Riemann-Christoffel curvature tensor
$R$ of $M$ is expressed on
$({\mathcal{U}}_{S}\cap{\mathcal{U}}_{C})\setminus{\mathcal{U}}_{H}$ by a
linear combination of the Kulkarni-Nomizu products $S\wedge S$, $g\wedge S$
and $G=\frac{1}{2}\,g\wedge g$ formed by the Ricci tensor $S$ and the metric
tensor $g$ of $M$. Precisely, we have (4.11), i.e.,

$\displaystyle R$ $\displaystyle=$ $\displaystyle\frac{\phi}{2}\,S\wedge
S+\mu\,g\wedge S+\eta\,G,$

where $\phi$, $\mu$ and $\eta$ are some functions on
$({\mathcal{U}}_{S}\cap{\mathcal{U}}_{C})\setminus{\mathcal{U}}_{H}$ (see [49,
eqs. (5.2)]). We also can express the tensors $C\cdot C$, $Q(g,C)$ and
$Q(S,C)$ by some linear combinations of the Tachibana tensors formed by the
tensors $g$ and $H$. In [49, Theorem 5.1] it was stated that if the scalar
curvature $\kappa$ of a hypersurface $M$ in $N^{n+1}_{s}(c)$, $n\geq 4$,
vanishes on
$({\mathcal{U}}_{S}\cap{\mathcal{U}}_{C})\setminus{\mathcal{U}}_{H}\subset M$
then

(9.1) $\displaystyle R\cdot C-C\cdot R$ $\displaystyle=$
$\displaystyle-Q(S,C)$

on this set. From that theorem it follows immediately [49, Corollary 5.3] that
if $M$ is a hypersurface in a Riemannian space of constant curvature
$N^{n+1}(c)$, $n\geq 4$, having at every point exactly two distinct principal
curvatures, and if its scalar curvature $\kappa$ vanishes on
$({\mathcal{U}}_{S}\cap{\mathcal{U}}_{C})\setminus{\mathcal{U}}_{H}\subset M$
then (9.1) holds on this set. In [49, Examples 5.4, 5.5 and 5.7] examples of
non-conformally flat and non-Einstein hypersurfaces, with $\kappa=0$, having
at every point exactly two distinct principal curvatures are presented.

As it was mentioned in Introduction, if at every point of ${\mathcal{U}}_{H}$
of a hypersurface $M$ in $N^{n+1}_{s}(c)$, $n\geq 4$, one of the tensors
$R\cdot C$, $C\cdot R$ or $R\cdot C-C\cdot R$ is a linear combination of the
tensor $R\cdot R$ and a finite sum of the Tachibana tensors of the form
$Q(A,T)$, where $A$ is a symmetric $(0,2)$-tensor and $T$ a generalized
curvature tensor, then (1.6) holds on ${\mathcal{U}}_{H}$ [47, Corollary 4.1].
Thus in particular, if (1.4) is satisfied on ${\mathcal{U}}_{H}$ then (1.6)
holds on this set. Hypersurfaces in $N^{n+1}_{s}(c)$, $n\geq 4$, satisfying
(1.6), or in particular (1.6) with $\rho=0$, i.e., (7.11), were studied in
several papers: [20, 38, 45, 47, 54, 58, 62, 64, 65, 70, 74, 80, 83, 109, 110,
111, 112, 113]. Section 6 of [49] contains some results on hypersurfaces
satisfying (1.6). In Section 7 of [49] hypersurfaces $M$ in $N^{n+1}_{s}(c)$,
$n\geq 4$, satisfying (1.4) on ${\mathcal{U}}_{H}\subset M$ were investigated.
The main result of this section (see, [49, Theorem 7.3]) states that if $M$ is
a hypersurface in $N^{n+1}_{s}(c)$, $n\geq 4$, satisfying on
${\mathcal{U}}_{H}\subset M$ the equalities: (7.11) and

(9.2) $\displaystyle\mathrm{rank}\,(S-\alpha\,g)$ $\displaystyle=$
$\displaystyle 1,$

for some function $\alpha$ on $\mathcal{U}_{H}$, then on this set

(9.3)
$\displaystyle\mathrm{rank}\left(S-\left(\frac{\kappa}{n-1}-\frac{\widetilde{\kappa}}{n(n+1)}\right)g\right)\
=\ 1,$ (9.4) $\displaystyle(n-2)\,(R\cdot C-C\cdot R)\ =\
Q(S,C)-\frac{\widetilde{\kappa}}{n(n+1)}\,Q(g,C).$

In particular, if the ambient space is a semi-Euclidean space
$\mathbb{E}_{s}^{n+1}$, $n\geq 4$, then (9.3) and (9.4) turn into

(9.5) $\displaystyle\mathrm{rank}\left(S-\frac{\kappa}{n-1}\,g\right)$
$\displaystyle=$ $\displaystyle 1,$ (9.6) $\displaystyle(n-2)\,(R\cdot
C-C\cdot R)$ $\displaystyle=$ $\displaystyle Q(S,C)$

respectively.

Let $M$ be a hypersurface in an Euclidean space $\mathbb{E}^{n+1}$, $n=2p+1$,
$p\geq 2$, having at every point three principal curvatures
$\lambda_{1}=\lambda\neq 0$, $\lambda_{2}=-\lambda$ and $\lambda_{3}=0$,
provided that the multiplicity of $\lambda_{1}$, as well as of $\lambda_{2}$
is $p$. Clearly, $M$ is an austere hypersurface [87, p. 102]. Evidently,
$M=\mathcal{U}_{H}$ and (7.11), (9.5) and (9.6) hold on $M$ (see, [49, Example
7.5(i)] for details). We recall that in [2] it was stated that $M$ is a non-
semisymmetric ($R\cdot R\neq 0$) Ricci-symmetric ($R\cdot S=0$) hypersurface.
Further results on hypersurfaces $M$ in $N^{n+1}_{s}(c)$, $n\geq 4$,
satisfying (1.4) on ${\mathcal{U}}_{H}\subset M$, with $L\neq-1$, are given in
[49, Proposition 7.1] and [49, Example 7.5(ii)-(iv)]. We also have

######  Theorem 9.1.

[49, Theorem 7.4] If $M$ is a hypersurface in $N^{n+1}_{s}(c)$, $n\geq 4$,
satisfying (1.4) with $L=-1$ on ${\mathcal{U}}_{H}$ then on this set we have:
$\kappa=0$ and

$\displaystyle(n-1)\,(R\cdot C-C\cdot R)$ $\displaystyle=$ $\displaystyle
Q\left(g,\left(\frac{(n-1)\widetilde{\kappa}}{n(n+1)}-\varepsilon\psi\right)R+\left(\frac{2(n-1)\widetilde{\kappa}}{n(n+1)}-\varepsilon\psi\right)g\wedge
S-\frac{1}{2(n-2)}\,S\wedge S-g\wedge S^{2}\right),$

where $\widetilde{\kappa}$ is the scalar curvature of $N^{n+1}_{s}(c)$,
$\kappa$ the scalar curvature of $M$ and the function $\psi$ is defined by
(1.6).

In [49, Example 7.6] it was stated that some tubular hypersurfaces with
vanishing scalar curvature satisfy (1.4) with $L=-1$.

##  10\. The condition $R\cdot C-C\cdot R=L_{1}\,Q(S,C)+L_{2}\,Q(g,C)$

In Section 5 of [55] we consider hypersurfaces $M$ in $N_{s}^{n+1}(c)$, $n\geq
4$, satisfying (1.8) on ${\mathcal{U}}_{H}\subset M$.

We recall that a semi-Riemannian manifold $(M,g)$, $n\geq 3$, is said to be a
quasi-Einstein manifold (see Section 4) if (4.1) holds on
${\mathcal{U}}_{S}\subset M$, where $\alpha$ is some function on this set. Let
$M$ be a quasi-Einstein hypersurface in $N_{s}^{n+1}(c)$, $n\geq 4$,
satysfying (1.8). We have

######  Theorem 10.1.

[55, Theorem 5.1] If $M$ be a hypersurface in a semi-Riemannian space of
constant curvature $N_{s}^{n+1}(c)$, $n\geq 4$, satisfying (1.8) and (4.1) on
${\mathcal{U}}_{H}\subset M$, for some functions $\alpha$, $L_{1}$ and
$L_{2}$, then on this set we have (7.11) and

$\displaystyle(n-2)\,(R\cdot C-C\cdot R)$ $\displaystyle=$ $\displaystyle
Q(S,C)-\frac{\widetilde{\kappa}}{n(n+1)}\,Q(g,C),$ $\displaystyle\alpha$
$\displaystyle=$
$\displaystyle\frac{\kappa}{n-1}-\frac{\widetilde{\kappa}}{n(n+1)},$
$\displaystyle R\cdot C-C\cdot R$ $\displaystyle=$
$\displaystyle\frac{\kappa}{n-1}\,Q(g,C)-Q(S,C)+B,$

where $\psi$ is some function on ${\mathcal{U}}_{H}$, and the function
$\alpha$ and the $(0,6)$-tensor $B$ are defined by (4.1) and

$\displaystyle B$ $\displaystyle=$ $\displaystyle
Q\left(S-\frac{1}{n-1}\left(\frac{(n-2)\kappa}{n-1}+\frac{\widetilde{\kappa}}{n(n+1)}\right)g,C\right),$

respectively. Moreover, (1.2) is satisfied on ${\mathcal{U}}_{H}$ if and only
if (8.6) holds on this set.

We consider non-quasi-Einstein hypersurfaces satisfying (1.8). Precisely, we
consider (1.8) at all points of ${\mathcal{U}}_{H}$ at which the following
condition is satisfied

(10.1) $\displaystyle\mathrm{rank}(S-\alpha\,g)>1,\ \ \mbox{for any}\ \
\alpha\in{\mathbb{R}}.$

We have

######  Theorem 10.2.

[55, Theorem 5.2] Let $M$ be a hypersurface in a semi-Riemannian space of
constant curvature $N_{s}^{n+1}(c)$, $n\geq 4$, satisfying (1.8) on
${\mathcal{U}}_{H}\subset M$. If at a point $x\in{\mathcal{U}}_{H}$ (10.1) is
satisfied then at this point we have (1.2) and

$\displaystyle(n-1)\,Q(S,R)$ $\displaystyle=$ $\displaystyle
Q\left(g,\left(\varepsilon\psi+\kappa-\frac{(n-1)\widetilde{\kappa}}{n(n+1)}\right)\right)R+\left(\varepsilon\psi-\frac{2(n-1)\widetilde{\kappa}}{n(n+1)}\right)g\wedge
S+g\wedge S^{2}-\frac{1}{2}\,S\wedge S\,.$

######  Theorem 10.3.

[55, Theorem 5.4] If at every point of a non-quasi-Einstein hypersurface $M$
in a semi-Riemannian space of constant curvature $N_{s}^{n+1}(c)$, $n\geq 4$,
the difference tensor $R\cdot C-C\cdot R$ is a linear combination of the
Tachibana tensors $Q(g,C)$ and $Q(S,C)$, then (1.2) holds on $M$.

##  11\. Chen ideal submanifolds

Let $M$ be a submanifold of dimension $n$ in the Euclidean space
$\mathbb{E}^{n+m}$, $n\geq 2$, $m\geq 1$. Let $g$ be the Riemannian metric
induced on $M$ from the standard metric on $\mathbb{E}^{n+m}$, $\nabla$ the
corresponding Levi-Civita connection on $M$, and $R$, $S$, $\tau$ and $C$, the
Riemann-Christoffel curvature tensor, the Ricci tensor, the scalar curvature
and the Weyl conformal curvature tensor of $g$, respectively. For the scalar
curvature $\tau$ of $(M,g)$ we use the calibration

$\tau(p)\ =\ \sum_{i<j}K\left(p,e_{i}(p)\wedge e_{j}(p)\right)$

where $K\left(p,\pi\right)$ denotes the Riemannian sectional curvature of
$(M,g)$ at the point $p$ for a plane section $\pi$ in the tangent space
$T_{p}M$. For each point $p$ in $M$, considering the number

$\left(\mbox{\rm inf}\,K\right)(p):=\mbox{\rm inf}\left\\{K(p,\pi)|\pi\
\text{is a plane section in}\ T_{p}M\right\\},$

B.-Y. Chen (see [7, 10]) introduced the $\delta(2)$-curvature by

$\left(\delta(2)\right)(p)=\delta(p):=\tau(p)-\left(\mbox{\rm
inf}\,K\right)(p).$

This $\delta(2)$-, for short, $\delta$-curvature of Chen thus is a well
defined real function on $M$ which clearly is a Riemannian invariant of
$(M,g)$.

From [10] (see also [7, 9, 13]), we have the following basic result which, in
particular, answered a question raised by S.S. Chern [15] long before,
concerning intrinsic obstructions on Riemannian manifolds in view of minimal
immersibility in Euclidean spaces.

######  Theorem 11.1.

[7] (see also [68, Theorem 1]) For any submanifold $M$ of dimension $n$ in the
Euclidean space $\mathbb{E}^{n+m}$, $n\geq 2$, $m\geq 1$,

(11.1) $\displaystyle\delta\leq\dfrac{n^{2}(n-2)}{2(n-1)}H^{2},$

and in (11.1) equality holds at a point $p\in M$ if and only if, with respect
to some suitable adapted orthonormal frame
$\left\\{e_{i},\xi_{\alpha}\right\\}$ around $p$ on $M$ in $\mathbb{E}^{n+m}$,
the shape operators are given by

$A_{1}=\begin{pmatrix}a&0&0&\cdots&0\\\ 0&b&0&\cdots&0\\\ 0&0&z&\cdots&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\ 0&0&0&\cdots&z\\\ \end{pmatrix},\quad
A_{\beta}=\begin{pmatrix}c_{\beta}&d_{\beta}&0&\cdots&0\\\
d_{\beta}&-c_{\beta}&0&\cdots&0\\\ 0&0&0&\cdots&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\ 0&0&0&\cdots&0\\\ \end{pmatrix},\
\beta>1,$

where $z=a+b$ and $\inf
K=ab-\sum_{\beta>1}\left(c^{2}_{\beta}+d^{2}_{\beta}\right):M\longrightarrow\mathbb{R}$.

Evidently, if $m=1$ then $\inf K=ab$.

With respect to the above theorem, one has the following definition [7, 9, 52,
67, 120], also see [68, Definition 1]. Let $M$ be a submanifold of dimension
$n$ in the Euclidean space $\mathbb{E}^{n+m}$, $n\geq 2$, $m\geq 1$. It is
called a Chen ideal submanifold if, at each of its points, the Chen’s basic
inequality (11.1) in the Theorem 11.1 is actually an equality.

Let $M$ be a Chen ideal submanifold of dimension $n$ in the Euclidean space
$\mathbb{E}^{n+m}$, $n\geq 4$, $m\geq 1$. We use the notations as in Theorem
11.1. The Riemann-Christoffel curvature tensor $R$ satisfies:

(11.2) $\left\\{\begin{aligned} R\left(e_{1},e_{2},e_{2},e_{1}\right)&=\inf
K=ab-\sum_{\beta=2}^{m}\left(c^{2}_{\beta}+d^{2}_{\beta}\right);\\\
R\left(e_{1},e_{i},e_{i},e_{1}\right)&=az\quad\text{\rm for}\quad i\geq 3;\\\
R\left(e_{2},e_{i},e_{i},e_{2}\right)&=bz\quad\text{\rm for}\quad i\geq 3;\\\
R\left(e_{i},e_{j},e_{j},e_{i}\right)&=z^{2}\quad\text{\rm for}\quad 3\leq
i<j\leq n.\\\ \end{aligned}\right.$

The other values of $R\left(e_{u},e_{v},e_{w},e_{t}\right)$ are null. The
Ricci tensor $S$ satisfies:

(11.3) $\left\\{\begin{aligned} S\left(e_{1},e_{1}\right)&=\inf K+(n-2)az;\\\
S\left(e_{2},e_{2}\right)&=\inf K+(n-2)bz;\\\
S\left(e_{i},e_{i}\right)&=(n-2)z^{2}\quad\text{\rm for}\quad 3\leq i\leq
n;\\\ S\left(e_{u},e_{v}\right)&=0\quad\text{\rm for}\quad 1\leq u<v\leq n.\\\
\end{aligned}\right.$

The scalar curvature $\tau$ is given by

(11.4) $\tau=\sum_{i=1}^{n}\mbox{\rm S}(e_{i},e_{i})=2\inf K+(n-1)(n-2)z^{2}.$

The Weyl conformal curvature tensor $C$ of $M$ is determined by the following
relations:

(11.5) $\left\\{\begin{aligned}
C\left(e_{1},e_{2},e_{2},e_{1}\right)&=\dfrac{(n-3)\inf K}{n-1};\\\
C\left(e_{1},e_{i},e_{i},e_{1}\right)&=-\dfrac{(n-3)\inf K}{(n-1)(n-2)}\ \
\mbox{for}\ \ \geq 3\\\
C\left(e_{2},e_{i},e_{i},e_{2}\right)&=-\dfrac{(n-3)\inf K}{(n-1)(n-2)}\ \
\mbox{for}\ \ \geq 3\\\ C\left(e_{i},e_{j},e_{j},e_{i}\right)&=\dfrac{2\inf
K}{(n-1)(n-2)}\quad\text{\rm for}\quad 3\leq i<j\leq n.\\\
\end{aligned}\right.$

From (11.5) it follows (cf. [52], [67, Theorem F]) that every Chen ideal
submanifold $M$ of dimension $n$ in the Euclidean space ${\mathbb{E}}^{n+m}$,
$n\geq 4$, $m\geq 1$, has a pseudosymmetric Weyl conformal curvature $C$,
i.e., it satisfies the identity

(11.6) $\displaystyle C\cdot C$ $\displaystyle=$ $\displaystyle
L_{C}\,Q(g,C),\ \ \ L_{C}\ =\ -\dfrac{(n-3)\inf K}{(n-1)(n-2)}.$

As it was mentioned in Section 8, at every point of a hypersurface $M$ in a
space form $\widetilde{N}^{n+1}(c)$, $n\geq 4$, the tensors $R\cdot R-Q(S,R)$
and $Q(g,C)$ are linearly dependent. Precisely, (7.2) holds on $M$. Thus, in
particular, $R\cdot R-Q(S,R)=0$ on every Chen ideal hypersurface $M$ in
$\mathbb{E}^{n+1}$, $n\geq 4$.

Now let us compute the difference $R\cdot R-Q(S,R)$ on the Chen ideal
submanifold $M$ of codimension $m$ in $\mathbb{E}^{n+m}$, $n\geq 4$, $m\geq
1$. With respect to the notations in Theorem 11.1 and from the equalities
(11.2), (11.3) and (11.5), we can prove the following

######  Theorem 11.2.

[68, Theorem 2] The identity

$\displaystyle R\cdot R-Q(S,R)=\dfrac{ab-\inf K}{\inf K}(n-2)z^{2}\,Q(g,C)$

holds on the subset ${\mathcal{U}}_{C}$ of every Chen ideal submanifold $M$ of
dimension $n$ in the Euclidean space $\mathbb{E}^{n+m}$, $n\geq 4$, $m\geq 1$.
In addition, at each point $p\in M$ where $C$ vanishes ($\inf K=0$), the
following equalities hold:

$\displaystyle\left\\{\begin{aligned} \left(R\cdot
R-Q(S,R)\right)\left(e_{1},e_{2},Z,W;e_{1},e_{i}\right)&=(n-3)abz^{2}\left<\left(e_{2}\wedge_{g}e_{i}\right)(Z),W\right>,\\\
\left(R\cdot
R-Q(S,R)\right)\left(e_{1},e_{j},Z,W;e_{1},e_{i}\right)&=-abz^{2}\left<\left(e_{j}\wedge_{g}e_{i}\right)(Z),W\right>,\\\
\left(R\cdot
R-Q(S,R)\right)\left(e_{1},e_{2},Z,W;e_{2},e_{i}\right)&=-(n-3)abz^{2}\left<\left(e_{1}\wedge_{g}e_{i}\right)(Z),W\right>,\\\
\left(R\cdot
R-Q(S,R)\right)\left(e_{2},e_{j},Z,W;e_{2},e_{i}\right)&=-abz^{2}\left<\left(e_{j}\wedge_{g}e_{i}\right)(Z),W\right>,\\\
\end{aligned}\right.$

and the other values of $\left(R\cdot
R-Q(S,R)\right)\left(e_{u},e_{v},Z,W;e_{w},e_{t}\right)$ being null.

As it was proved in [51], every warped product manifold
$\overline{M}\times_{F}\widetilde{N}$ of a $2$-dimensional base manifold
$(\overline{M},\overline{g})$ and an $(n-2)$-dimensional fibre, which is a
space of constant curvature $(\widetilde{N},\widetilde{g})$, $n\geq 4$, with
the warping function $F$, satisfies

(11.7) $C\cdot C=-\frac{(n-3)\rho}{(n-2)(n-1)}\,Q(g,C),\ \ \ \
\rho=\frac{\overline{\tau}}{2}+\frac{\widetilde{\tau}}{(n-3)(n-2)F}+\frac{\Delta
F}{2F}-\frac{\Delta_{1}F}{2F^{2}},$

where $\Delta F=g^{ab}\nabla_{b}F_{a}$, $\Delta_{1}F=g^{ab}F_{a}F_{b}$, and
$\overline{\tau}$, $\widetilde{\tau}$ are the scalar curvatures of the base
and the fibre, respectively, see to [51] for details.

As we noted in Section 1, some warped product manifolds
$\overline{M}\times_{F}\widetilde{N}$ with a $2$-dimensional Riemannian
manifold $(\overline{M},\overline{g})$ and an $(n-2)$-dimensional unit sphere
${\mathbb{S}}^{n-2}$, are related to Chen ideal submanifolds. Namely,
according to [17], every non-trivial and non-minimal Chen ideal submanifold
$M$ of dimension $n$ in the Euclidean space ${\mathbb{E}}^{n+m}$, $n\geq 4$,
$m\geq 1$ is isometric to an open subset of a warped product
$\overline{M}\times_{F}{\mathbb{S}}^{n-2}$ of a $2$-dimensional base manifold
$(\overline{M},\overline{g})$ and an $(n-2)$-dimensional unit sphere
${\mathbb{S}}^{n-2}$, where the warping function $F$ is a solution of some
second order quasilinear elliptic partial differential equation in the plane.
Thus we see that (11.7) holds on $M$. Furthermore, from (11.6) and (11.7) it
follows that $\inf K$ is expressed on $M$ by

$\displaystyle\inf K$ $\displaystyle=$
$\displaystyle\frac{\overline{\tau}}{2}+\frac{1}{F}+\frac{\Delta
F}{2F}-\frac{\Delta_{1}F}{2F^{2}}.$

Since the scalar curvature $\tau$ of $M$ is given by (11.4) and satisfies
(see, e.g., [51])

$\displaystyle\tau$ $\displaystyle=$
$\displaystyle\overline{\tau}+\frac{(n-3)(n-2)}{F}-\frac{(n-2)\Delta
F}{F}-\frac{(n-2)(n-5)\Delta_{1}F}{4F^{2}},$

we get

$\displaystyle(n-2)z^{2}$ $\displaystyle=$
$\displaystyle\frac{n-4}{F}-\frac{\Delta
F}{F}-\frac{(n-6)\Delta_{1}F}{4F^{2}}.$

In [51], among other things, it was shown that the Hessian of the function
$f=\sqrt{F}$ is proportional to the metric $\overline{g}$ on $\overline{M}$.

We mention that Chen ideal submanifolds which are semisymmetric were
classified in [77]. We have

######  Theorem 11.3.

[77] (see also [68, Theorem 3]) A Chen ideal submanifold $M$ of dimension $n$
in the Euclidean space ${\mathbb{E}}^{n+m}$, $n\geq 3$, $m\geq 1$, is
semisymmetric if and only if $M$ is minimal (in which case $M$ is
$(n-2)$-ruled) or $M$ is a round hypercone in some totally geodesic subspace
${\mathbb{E}}^{n+1}$ of ${\mathbb{E}}^{n+m}$.

Chen ideal pseudosymmetric submanifolds were classified in [52, 67]. We have

######  Theorem 11.4.

[52, 67] (see also [68, Theorem 4]) A Chen ideal submanifold $M$ of
codimension $m$ in ${\mathbb{E}}^{n+m}$ ($n\geq 3$, $m\geq 1$) is
pseudosymmetric if and only if
(i) either $M$ is semisymmetric (see Theorem 11.3),
(ii) or at every point $p$ of $M$ where $R\cdot R\neq 0$, the $2D$ normal
section $\Sigma^{2}_{\tilde{\pi}}\subset{\mathbb{E}}^{2+m}$ of $M^{n}$ at $p$
in the direction of the tangent plane ${\tilde{\pi}}\subset T_{p}M^{n}$ for
which the sectional curvature function $K(p,\pi)$ at $p$ attains its minimal
value $\left(\inf K\right)(p)$ is pseudo-umbilical at $p$, or equivalently, if
$p$ is a spherical point of the projection
$\overline{\Sigma}^{2}_{\tilde{\pi}}\subset{\mathbb{E}}^{3}$ of this $2D$
normal section $\overline{\Sigma}^{2}_{\tilde{\pi}}$ on the space
${\mathbb{E}}^{3}$ spanned by ${\tilde{\pi}}$ and the mean curvature vector
$\stackrel{{\scriptstyle\rightarrow}}{{H}}(p)$ of $M^{n}$ in
${\mathbb{E}}^{n+m}$ at $p$ (and in this case
$L_{R}=\dfrac{n^{2}}{2(n-1)^{2}}H^{2}$, where $H$ is the mean curvature of
$M^{n}$ in ${\mathbb{E}}^{n+m}$).

The above presented part of this section based on [68, Section 3]) and the
remains part of this section on [68, Section 5]. We present now results on
Chen ideal submanifolds in Euclidean spaces whose difference tensor $R\cdot
C-C\cdot R$ can be expressed in terms of some of the Tachibana tensors
$Q(g,R)$, $Q(S,R)$, $Q(g,C)$, $Q(S,C)$, $Q(g,g\wedge S)$, $Q(S,g\wedge S)$.

######  Theorem 11.5.

[68, Theorem 6] Let $M$ be a non-conformally flat Chen ideal submanifold of
dimension $n$ in the Euclidean space $\mathbb{E}^{n+m}$, $n\geq 4$, $m\geq 1$.
Then there exist two real valued functions $L_{1}$ and $L_{2}$ on $M$ such
that (1.8), i.e.,

$\displaystyle R\cdot C-C\cdot R=L_{1}Q(S,C)+L_{2}Q(g,C),$

if and only if there exists an orthonormal tangent framefield
$\left\\{e_{1},\cdots,e_{n}\right\\}$ and an orthonormal normal framefield
$\left\\{\xi_{1},\cdots,\xi_{m}\right\\}$ such that the shape operators
$A_{\alpha}:=A_{\xi_{\alpha}},\quad 1\leq\alpha\leq m$, are given by:

$A_{1}=\begin{pmatrix}a&0&0&\cdots&0\\\ 0&\epsilon a&0&\cdots&0\\\
0&0&(1+\epsilon)a&\cdots&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
0&0&0&\cdots&(1+\epsilon)a\\\ \end{pmatrix},\quad
A_{\beta}=\begin{pmatrix}c_{\beta}&d_{\beta}&0&\cdots&0\\\
d_{\beta}&-c_{\beta}&0&\cdots&0\\\ 0&0&0&\cdots&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\ 0&0&0&\cdots&0\\\ \end{pmatrix},\
\beta\geq 2,$

where $\epsilon=\pm 1$, $a$, $c_{\beta}$ , $d_{\beta}$ (for $2\leq\beta\leq
m$) are real functions on $M$ such that

$\sum_{\beta=2}^{m}\left(c^{2}_{\beta}+d^{2}_{\beta}\right)=\epsilon
a^{2}-\inf K,\ \ \ L_{1}=-1,\quad L_{2}=-\dfrac{2\,\inf
K+2(n-1)(n-2)(1+\epsilon)a^{2}}{n-1}.$

In this case, $M$ is a Roter space. In addition one has one of the following
two cases:
(i) either $\epsilon=-1$ and $M$ is a semisymmetric and minimal submanifold
(see Theorem 11.3) such that

$R\cdot C-C\cdot R=\dfrac{2\,\inf K}{n-1}Q(g,C)-Q(S,C),$

(ii) or $\epsilon=+1$ and $M$ is a properly pseudosymmetric and non-minimal
submanifold (see Theorem 11.4) such that

$R\cdot C-C\cdot R=-\dfrac{2\,\inf K+4(n-1)(n-2)a^{2}}{n-1}Q(g,C)-Q(S,C).$

######  Corollary 11.6.

[68, Corollary 3] Let $M$ be a Chen ideal submanifold of dimension $n$ in the
Euclidean space $\mathbb{E}^{n+m}$, $n\geq 4$, $m\geq 1$. Then the difference
tensor $R\cdot C-C\cdot R$ and the Tachibana tensor $Q(g,C)$ are linearly
dependent if and only if $M$ is conformally flat.

######  Corollary 11.7.

[68, Corollary 4] Let $M$ be a non-conformally flat Chen ideal submanifold of
dimension $n$ in the Euclidean space $\mathbb{E}^{n+m}$, $n\geq 4$, $m\geq 1$.
Then there exists a real valued function $L$ on $M$ such that

$R\cdot C-C\cdot R=LQ(S,C),$

if and only if $M$ is not minimal, and there exists an orthonormal tangent
framefield $\left\\{e_{1},\cdots,e_{n}\right\\}$ and an orthonormal normal
framefield $\left\\{\xi_{1},\cdots,\xi_{m}\right\\}$ such that the shape
operators $A_{\alpha}:=A_{\xi_{\alpha}},\quad 1\leq\alpha\leq m$, are given by

$A_{1}=\begin{pmatrix}a&0&0&\cdots&0\\\ 0&a&0&\cdots&0\\\ 0&0&2a&\cdots&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\ 0&0&0&\cdots&2a\\\ \end{pmatrix},\quad
A_{\beta}=\begin{pmatrix}c_{\beta}&d_{\beta}&0&\cdots&0\\\
d_{\beta}&-c_{\beta}&0&\cdots&0\\\ 0&0&0&\cdots&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\ 0&0&0&\cdots&0\\\ \end{pmatrix},\
\beta\geq 2,$

where $a$, $c_{\beta}$ , $d_{\beta}$ (for $2\leq\beta\leq m$) are real
functions on $M$ such that

$\sum_{\beta=2}^{m}\left(c^{2}_{\beta}+d^{2}_{\beta}\right)=\left(2n^{2}-6n+5\right)a^{2}>0$

and $L=-1$. In this case, $M^{n}$ is properly pseudosymmetric (see Theorem
11.4).