# On warped product manifolds satisfying some pseudosymmetric type conditions

Absos Ali Shaikh and Haradhan Kundu
Department of Mathematics,
University of Burdwan, Golapbag,
Burdwan-713104,
West Bengal, India<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

The object of the present paper is to study the characterization of warped
product manifolds satisfying some pseudosymmetric type conditions, especially,
due to projective curvature tensor. For this purpose we consider a warped
product manifold satisfying the pseudosymmetric type condition $R\cdot
R=L_{1}Q(g,R)+L_{2}Q(S,R)$ and evaluate its characterization theorem. As
special cases of $L_{1}$ and $L_{2}$ we find out the necessary and sufficient
condition for a warped product manifold to satisfy various pseudosymmetric
type, such as pseudosymmetry, Ricci generalized pseudosymmetry, semisymmetry
due to projective curvature tensor ($P\cdot R=0$), pseudosymmetry due to
projective curvature tensor ($P\cdot R=LQ(g,R)$) etc. Finally we present some
suitable examples of warped product manifolds satisfying such pseudosymmetric
type conditions.

###### Key words and phrases:

Semisymmetric manifold, pseudosymmetric manifold, pseudosymmetric type
manifold, warped product manifold

###### 2010 Mathematics Subject Classification:

53C15, 53C25, 53C35

##  1\. Introduction

Let $\nabla$, $R$, $S$, $G$, $P$ and $\kappa$ be respectively the Levi-Civita
connection, the Riemann-Christoffel curvature tensor, Ricci tensor, the
Gaussian curvature tensor, the projective curvature tensor and the scalar
curvature of an $n$-dimensional $(n\geq 3)$ connected smooth semi-Riemanian
manifold $M$ equipped with the semi-Riemannian metric $g$. Symmetry is a very
important geometric property of a space. Cartan [3] introduced the notion of
symmetry (local and global) on a Riemannian manifold in terms of geodesic
symmetries. A semi-Riemannian manifold is said to be locally symmetric [3] if
its local geodesic symmetries at each point are all isometry. According to
Cartan-Ambrose-Hicks theorem a locally symmetric manifold can be characterized
by the curvature condition $\nabla R=0$, i.e., the curvature tensor is
covariantly constant (For the meaning and definition of various notations and
symbols used here, we refer the reader to Section 2 of this paper).
As a proper generalization of locally symmetric manifold, Cartan [4]
introduced the notion of semisymmetric manifold. A semi-Riemannian manifold is
said to be semisymmetric [4] (see also [30], [31], [32]) if $R\cdot R=0$,
where the first $R$ stands for the curvature operator acting as a derivation
on the second $R$. It may be noted that a semi-Riemannian manifold is
semisymmetric if and only if its sectional curvature function $k(p,\pi)$ is
invariant up to second order, under parallel transport of any plane $\pi$ at
any point $p$ of $M$ around any infinitesimal coordinate parallelogram
centered at $p$ (see [12], [15], [16]).
During the study of totally umbilical submanifolds of semisymmetric manifolds
as well as during the consideration of geodesic mappings on semisymmetric
manifolds, Adamow and Deszcz [1] (see [7] and also references therein)
introduced the notion of pseudosymmetric manifolds as a proper generalization
of semisymmetric manifolds. A semi-Riemannian manifold is said to be
pseudosymmetric if

$R\cdot R$ and $Q(g,R)$ are linearly dependent.

Let $M$ be not of constant curvature and $U$ denotes the set $\\{x\in
M:Q(g,R)\neq 0\\}$. Then at a point $p\in U$, a plane
$\pi_{1}(\vec{v_{p}},\vec{w_{p}})\subset T_{p}M$ is said to be curvature
dependent with respect to another plane
$\pi_{2}(\vec{x_{p}},\vec{y_{p}})\subset T_{p}M$ if
$Q(g,R)(\vec{v_{p}},\vec{w_{p}},\vec{v_{p}},\vec{w_{p}},\vec{x_{p}},\vec{y_{p}})\neq
0$. Now if $\pi_{1}$ is curvature dependent with respect to $\pi_{2}$, then
the scalar

$L(p,\pi_{1},\pi_{2})=\frac{R\cdot
R(\vec{v_{p}},\vec{w_{p}},\vec{v_{p}},\vec{w_{p}},\vec{x_{p}},\vec{y_{p}})}{Q(g,R)(\vec{v_{p}},\vec{w_{p}},\vec{v_{p}},\vec{w_{p}},\vec{x_{p}},\vec{y_{p}})}$

is called the double sectional curvature or Deszcz sectional curvature ([12],
[15], [16], [17]) of the plane $\pi_{1}$ with respect to $\pi_{2}$ at $p$. In
terms of Deszcz sectional curvature, a semi-Riemannian manifold is
pseudosymmetric if at each point $p\in U$, $L(p,\pi_{1},\pi_{2})$ is
independent of the planes $\pi_{1}$ and $\pi_{2}$ (see [12], [15], [16],
[17]).
Replacing $R$ by other curvature tensors and $g$ by other symmetric
$(0,2)$-tensor in the defining condition of semisymmetric manifold and
pseudosymmetric manifold one can get various curvature restricted geometric
structures, which are simply called as semisymmetric type and pseudosymmetric
type manifolds ([10], [11], [13], [22], [25], [28]). For the geometric meaning
of Weyl semisymmetric and Weyl pseudosymmetric spaces we refer the reader to
see [18]. One of the important semisymmetric type (resp., pseudosymmetric
type) manifold is semisymmetric (resp., pseudosymmetric) manifold due to
projective curvature tensor. A semi-Riemannian manifold is said to be
semisymmetric (resp., pseudosymmetric) due to projective curvature tensor if

$P\cdot R=0$ (resp., $P\cdot R$ and $Q(g,R)$ are linearly dependent).

Another important pseudosymmetric type manifold is Ricci generalized
pseudosymmetric manifold. A semi-Riemannian manifold is said to be Ricci
generalized pseudosymmetric ([5], [6]) if

$R\cdot R$ and $Q(S,R)$ are linearly dependent.

We refer the reader to see [26] for details about various curvature restricted
geometric structures due to projective curvature tensor.
Again the notion of warped product manifold ([2], [19]) is a generalization of
product manifold and this notion is important due to its applications in
general theory of relativity and cosmology. Various spacetimes are warped
product, e.g., Robertson-Walker spacetimes, asymptotically flat spacetimes,
Schwarzschild spacetimes, Kruskal space-times, Reissner-Nordström spacetimes
etc. The main purpose of this present paper is to study the characterization
of a warped product semi-Riemannian manifold realizing some pseudosymmetric
type curvature conditions, especially, due to the projective curvature tensor.
For this purpose we consider the pseudosymmetric type condition $R\cdot
R=L_{1}Q(g,R)+L_{2}Q(S,R)$, and evaluate the characterization of a warped
product manifold satisfying such curvature condition. As a special case we get
the characterization of a warped product manifold which is (i) semisymmetric,
(ii) pseudosymmetric, (iii) Ricci generalized pseudosymmetric, (iv) special
Ricci generalized pseudosymmetric, (v) semisymmetric due to projective
curvature tensor and (vi) pseudosymmetric due to projective curvature tensor
etc. It is shown that if a warped product manifold
$M=\overline{M}\times_{f}\widetilde{M}$ satisfies $R\cdot
R=L_{1}Q(g,R)+L_{2}Q(S,R)$ such that $L_{2}$ is nowhere zero, then at either
the base $\overline{M}$ is flat or the fiber $\widetilde{M}$ is Einstein.
Consequently for a warped product pseudosymmetric manifold due to projective
curvature tensor or special Ricci generalized pseudosymmetric manifold either
the base is flat or the fiber is Einstein.
The paper is organized as follows. After discussing various notations as
preliminaries in Section 2, we define various pseudosymmetric type curvature
restricted geometric structures in Section 3. Section 4 is devoted to the
study of warped product manifold and we state the curvature relation of a
warped product manifold with its base and fiber. In Section 5 we discuss about
the characterization theorems of various pseudosymmetric type warped product
manifolds. Finally to support our results we present some suitable examples of
warped product manifolds in the last section. It is interesting to mention
that notion of pseudosymmetric manifold arose during the study of totally
umbilical hypersurface of a semisymmetric manifold, and in Example 1 we
present a pseudosymmetric totally umbilical hypersurface of a semisymmetric
manifold.

##  2\. Preliminaries

Let $M$ be a connected $n$-dimensional smooth manifold equipped with the semi-
Riemannian metric $g$. Let us consider the following notations related to
$(M,g)$:
$C^{\infty}(M)=$ the algebra of all smooth functions on $M$,
$\mathcal{T}^{r}_{k}(M)=$ the space of all smooth tensor fields of type
$(r,k)$ on $M$ and
$\chi(M)=\mathcal{T}^{1}_{0}(M)=$ the Lie algebra of all smooth vector fields
on $M$.
The Kulkarni-Nomizu product ([9], [14], [29]) $A\wedge
E\in\mathcal{T}^{0}_{4}(M)$ of $A$ and $E\in\mathcal{T}^{0}_{2}(M)$, is given
by

$\displaystyle(A\wedge E)(X_{1},X_{2},X_{3},X_{4})$ $\displaystyle=$
$\displaystyle A(X_{1},X_{4})E(X_{2},X_{3})+A(X_{2},X_{3})E(X_{1},X_{4})$
$\displaystyle-$ $\displaystyle
A(X_{1},X_{3})E(X_{2},X_{4})-A(X_{2},X_{4})E(X_{1},X_{3}),$

where $X_{1},X_{2},X_{3},X_{4}\in\chi(M)$. Throughout the paper we consider
$X,Y,X_{1},X_{2},\cdots\in\chi(M)$.
Now for $D\in\mathcal{T}^{0}_{4}(M)$, $A\in\mathcal{T}^{0}_{2}(M)$ and
$X,Y\in\chi(M)$, we get two endomorphisms $\mathscr{D}(X,Y)$ (called the
associated curvature operator of $D$) and $X\wedge_{A}Y$ defined by

$\mathscr{D}(X,Y)(X_{1})=\mathcal{D}(X,Y)X_{1}\ \mbox{and}$
$(X\wedge_{A}Y)X_{1}=A(Y,X_{1})X-A(X,X_{1})Y,$

where $\mathcal{D}\in\mathcal{T}^{1}_{3}(M)$ such that
$g(\mathcal{D}(X,Y)X_{1},X_{2})=D(X,Y,X_{1},X_{2})$, called the associated
$(1,3)$ tensor of $D$.
A tensor $D\in\mathcal{T}^{0}_{4}(M)$ is said to be a generalized curvature
tensor ([9], [20], [22]) if

$D(X_{1},X_{2},X_{3},X_{4})+D(X_{2},X_{3},X_{1},X_{4})+D(X_{3},X_{1},X_{2},X_{4})=0,$
$D(X_{1},X_{2},X_{3},X_{4})+D(X_{2},X_{1},X_{3},X_{4})=0\ \ \mbox{and}$
$D(X_{1},X_{2},X_{3},X_{4})=D(X_{3},X_{4},X_{1},X_{2}).$

The Gaussian curvature tensor $G$, Weyl conformal curvature tensor $C$,
concircular curvature tensor $W$ and conharmonic curvature tensor $K$are all
generalized curvature tensors and respectively given by

$G=\frac{1}{2}g\wedge g,$ $C=R-\frac{1}{n-2}g\wedge
S+\frac{\kappa}{2(n-1)(n-2)}g\wedge g,$ $W=R-\frac{\kappa}{2n(n-1)}g\wedge g\
\ \mbox{and}$ $K=R-\frac{1}{n-2}g\wedge S.$

The projective curvature tensor $P$ of type $(0,4)$ , given by

$P(X_{1},X_{2},X_{3},X_{4})=R(X_{1},X_{2},X_{3},X_{4})-\frac{1}{n-2}\left[S(X_{2},X_{3})g(X_{1},X_{4})-S(X_{1},X_{3})g(X_{2},X_{4})\right],$

is not a generalized curvature tensor.
Again an endomorphism $\mathscr{L}$ can be operate on a $(0,k)$-tensor $H$ and
obtain $\mathscr{L}H$ as follows:

$\displaystyle(\mathscr{L}H)(X_{1},X_{2},\cdots,X_{k})$ $\displaystyle=$
$\displaystyle-H(\mathscr{L}X_{1},X_{2},\cdots,X_{k})-\cdots-H(X_{1},X_{2},\cdots,\mathscr{L}X_{k}).$

In particular, for $\mathscr{L}=\mathscr{D}(X,Y)$ and $X\wedge_{A}Y$ we get
two $(0,k+2)$ tensors $D\cdot H$ and $Q(A,H)$ defined as ([23], [28], [11] and
also references therein)

$\displaystyle D\cdot H(X_{1},X_{2},\ldots,X_{k},X,Y)=(\mathscr{D}(X,Y)\cdot
H)(X_{1},X_{2},\ldots,X_{k})$
$\displaystyle=-H(\mathcal{D}(X,Y)X_{1},X_{2},\ldots,X_{k})-\cdots-H(X_{1},X_{2},\ldots,\mathcal{D}(X,Y)X_{k}),$
$\displaystyle Q(A,H)(X_{1},X_{2},\cdots,X_{k},X,Y)=((X\wedge_{A}Y)\cdot
H)(X_{1},X_{2},\ldots,X_{k})$
$\displaystyle=A(X,X_{1})H(Y,X_{2},\cdots,X_{k})+\cdots+A(X,X_{k})H(X_{1},X_{2},\cdots,Y)$
$\displaystyle-A(Y,X_{1})H(X,X_{2},\cdots,X_{k})-\cdots-A(Y,X_{k})H(X_{1},X_{2},\cdots,X).$

######  Lemma 2.1.

Let $M$ be a connected warped product manifold with base $\overline{M}$ and
fiber $\widetilde{M}$. If $f_{1}\in C^{\infty}(\overline{M})$ and $f_{2}\in
C^{\infty}(\widetilde{M})$ satisfies $f_{1}~{}f_{2}\equiv 0$, then either
$f_{1}\equiv 0$ or $f_{2}\equiv 0$.

######  Lemma 2.2.

[5] Let $A,E\in\mathcal{T}^{0}_{2}(M)$ be symmetric. Then $Q(A,E)=0$ if and
only if $A$ and $E$ are linearly dependent [5]. Moreover $Q(A,E)=0$ if and
only if $M$ is Einstein, i.e., $S=\frac{\kappa}{n}g$.

##  3\. Pseudosymmetric type curvature restricted geometric structures

######  Definition 3.1.

([22], [30], [31], [32]) For $H\in\mathcal{T}^{0}_{k}(M)$ and
$D\in\mathcal{T}^{0}_{4}(M)$, a semi-Riemannian manifold $M$ is said to be
$H$-semisymmetric type if $D\cdot H=0$.

In particular, a semi-Riemannian manifold with the semisymmetric type
conditions $R\cdot R=0$, $R\cdot S=0$, $R\cdot P=0$, $P\cdot R=0$ and $P\cdot
S=0$ is respectively called semisymmetric, Ricci semisymmetric, projective
semisymmetric, semisymmetric due to projective curvature tensor and Ricci
semisymmetric due to projective curvature tensor respectively.

######  Definition 3.2.

$($[1], [7], [9], [22]$)$ For $H\in\mathcal{T}^{0}_{k}(M)$ and
$D_{i}\in\mathcal{T}^{0}_{4}(M)$, $i=1,2,\cdots r,r\geq 2$, a semi-Riemannian
manifold is said to be $H$-pseudosymmetric type if $D_{i}\cdot H$ are linearly
dependent, i.e., $\sum\limits_{i=1}^{r}c_{i}(D_{i}\cdot H)=0$ for some
$c_{i}\in C^{\infty}(M)$, called the associated scalars. Moreover if the
associated scalars are constant then the manifold is called pseudosymmetric
type manifold of constant type.

In particular, a semi-Riemannian manifold satisfying

$R\cdot C=L_{R}Q(g,C)\ \ \mbox{on}\ \ \left\\{x\in M:C_{x}\neq 0\right\\},$
$R\cdot S=L_{S}Q(g,S)\ \ \mbox{on}\ \ \left\\{x\in
M:\left(S-\frac{\kappa}{n}g\right)_{x}\neq 0\right\\},$ $R\cdot P=L_{P}Q(g,P)\
\ \mbox{on}\ \ \left\\{x\in M:Q(g,P)_{x}\neq 0\right\\},$ $P\cdot
R=L_{1}Q(g,R)\ \ \mbox{on}\ \ \left\\{x\in
M:\left(R-\frac{\kappa}{n(n-1)}G\right)_{x}\neq 0\right\\}\ \mbox{and}$
$P\cdot S=L_{2}Q(g,S)\ \ \mbox{on}\ \ \left\\{x\in
M:\left(S-\frac{\kappa}{n}g\right)_{x}\neq 0\right\\}$

is called conformally pseudosymmetric, Ricci pseudosymmetric, projective
pseudosymmetric, pseudosymmetric due to projective curvature tensor and Ricci
pseudosymmetric due to projective curvature tensor respectively, where
$L_{R},L_{S},L_{P},L_{1},L_{2}$ are the associated scalars.
In this paper we are mainly interested on the pseudosymmetric type condition
$R\cdot R=L_{1}Q(g,R)+L_{2}Q(S,R)$. For particular values of $L_{1}$ and
$L_{2}$, we get various pseudosymmetric type conditions as follows:

Values of $L_{1}$ & $L_{2}$ | Structure name | Curvature condition
---|---|---
$L_{1}=L_{2}=0$ | Semisymmetric manifold | $R\cdot R=0$
$L_{2}=0$ | Pseudosymmetric manifold | $R\cdot R=L_{1}Q(g,R)$
$L_{1}=0$ | $\begin{array}[]{c}\mbox{Ricci generalized}\\\ \text{Pseudosymmetric manifold}\end{array}$ | $R\cdot R=L_{2}Q(S,R)$
$L_{1}=0,L_{2}=1$ | $\begin{array}[]{c}\mbox{Special Ricci generalized}\\\ \text{Pseudosymmetric manifold}\end{array}$ | $R\cdot R=Q(S,R)$
$L_{1}=\frac{\kappa}{n(n-1)},L_{2}=0$ | $\begin{array}[]{c}\mbox{Semisymmetric due to}\\\ \text{concircular curvature tensor}\end{array}$ | $W\cdot R=0$
$L_{1}=\frac{\kappa}{n(n-1)}$ | $\begin{array}[]{c}\mbox{Ricci generalized Pseudosymmetric}\\\ \text{due to concircular curvature tensor}\end{array}$ | $W\cdot R=L_{2}Q(S,R)$
$L_{1}=0,L_{2}=\frac{1}{n-2}$ | $\begin{array}[]{c}\mbox{Semisymmetric due to}\\\ \text{projective curvature tensor}\end{array}$ | $P\cdot R=0$
$L_{2}=\frac{1}{n-2}$ | $\begin{array}[]{c}\mbox{Pseudosymmetric due to}\\\ \text{projective curvature tensor}\end{array}$ | $P\cdot R=L_{1}Q(g,R)$

##  4\. Warped product manifolds

Warped product is an important notion in semi-Riemannian geometry and it has a
great applications in general theory of relativity and cosmology. To impose a
semi-Riemannian structure on a product smooth manifold, the notion of warped
product metric arose as a generalization of Riemannian product metric. This
notion was independently introduced by
Kru$\breve{\mbox{c}}$kovi$\breve{\mbox{c}}$ [19] (as semi-decomposable metric)
as well as Bishop and O’Neill [2]. The most important example of non-Riemann
product but a warped product is surface of revolution. Many well-known
spacetimes, e.g., Schwarzschild, Kottler, Reissner-Nordström, Reissner-
Nordström-de Sitter, Vaidya, Robertson-Walker spacetimes are all warped
products.
Let $(\overline{M},\overline{g})$ and $(\widetilde{M},\widetilde{g})$ be two
semi-Riemannian manifolds of dimension $p$ and $(n-p)$ respectively ($1\leq
p\leq n-1$) and $M=\overline{M}\times\widetilde{M}$. The warped product metric
$g$ on $M$ is given by

$g=\pi^{*}(\overline{g})+(f\circ\pi)\sigma^{*}(\widetilde{g}),$

where $f$ is a positive smooth function on $\overline{M}$ and
$\pi:M\rightarrow\overline{M}$ (resp., $\sigma:M\rightarrow\widetilde{M}$) is
the natural projection on $\overline{M}$ (resp., $\widetilde{M}$). The
manifold $\overline{M}$ is called the base, $\widetilde{M}$ is called the
fiber and $f$ is called the warping function of $M$. If $f=1$, then the warped
product reduces to the Riemann product. If we consider a product chart
$\left(U\times
V;x^{1},x^{2},...,x^{p},x^{p+1}=y^{1},x^{p+2}=y^{2},...,x^{n}=y^{n-p}\right)$
on $M$, then in terms of local coordinates, $g$ can be expressed as

(4.4) $\displaystyle g_{ij}=\left\\{\begin{array}[]{lll}&\overline{g}_{ij}&\ \
\ \ \mbox{for}\ i=a\ \mbox{and}\ j=b,\\\ &f\widetilde{g}_{ij}&\ \ \ \
\mbox{for $i=\alpha$ and $j=\beta$,}\\\ &0&\ \ \ \ \mbox{otherwise,}\\\
\end{array}\right.$

where $a,b\in\\{1,2,...,p\\}$ and $\alpha,\beta\in\\{p+1,p+2,...,n\\}$. We
note that throughout the paper we consider $a,b,c,d,e,s,t\in\\{1,2,...,p\\}$;
$\alpha,\beta,\gamma,\delta,\epsilon,\mu,\eta\in\\{p+1,p+2,...,n\\}$ and
$i,j,k,l,q,u,v\in\\{1,2,...,n\\}$. Moreover, when $\Box$ is a quantity formed
with respect to $g$, we denote by $\overline{\Box}$ and $\widetilde{\Box}$,
the similar quantities formed with respect to $\overline{g}$ and
$\widetilde{g}$ respectively.
By a straightforward calculation we can evaluate the components of various
necessary tensors of a warped product manifold in terms of the base and fiber
components. The non-zero local components $R_{hijk}$ of the Riemann-
Christoffel curvature tensor $R$, $S_{jk}$ of the Ricci tensor $S$ and the
scalar curvature $\kappa$ of $M$ are respectively given by

$\displaystyle R_{abcd}=\overline{R}_{abcd},\,\,\,\,R_{a\alpha
b\beta}=fT_{ab}\widetilde{g}_{\alpha\beta},\,\,\,R_{\alpha\beta\gamma\delta}=f\widetilde{R}_{\alpha\beta\gamma\delta}-f^{2}\Delta\widetilde{G}_{\alpha\beta\gamma\delta},$
$\displaystyle
S_{ab}=\overline{S}_{ab}-(n-p)T_{ab},\,\,\,\,S_{\alpha\beta}=\widetilde{S}_{\alpha\beta}+\Omega\widetilde{g}_{\alpha\beta}$
$\displaystyle\mbox{and
}\kappa=\overline{\kappa}+\frac{\widetilde{\kappa}}{f}-(n-p)[(n-p-1)\Delta-2\;tr(T)],$

where $G_{ijkl}=g_{il}g_{jk}-g_{ik}g_{jl}$ are the components of Gaussian
curvature and

$T_{ab}=\frac{1}{2f}(f_{a,b}-\frac{1}{2f}f_{a}f_{b}),\ \ \ \
tr(T)=g^{ab}T_{ab},$ $\Delta=\frac{1}{4f^{2}}g^{ab}f_{a}f_{b},\ \ \ \
\Omega=-f((n-p-1)\Delta+tr(T)),\ \ \ f_{a}=\partial_{a}f=\frac{\partial
f}{\partial x^{a}}.$

The non-identically zero local components of $R\cdot R$, $Q(g,R)$ and $Q(S,R)$
of $M$ are given by

(4.5) $\displaystyle(R\cdot R)_{abcdst}$ $\displaystyle=$
$\displaystyle(\overline{R}\cdot\overline{R})_{abcdst},$ $\displaystyle(R\cdot
R)_{a\alpha b\beta st}$ $\displaystyle=$ $\displaystyle
f\widetilde{g}_{\alpha\beta}(\overline{R}\cdot T)_{abst},$
$\displaystyle(R\cdot R)_{abc\alpha s\eta}$ $\displaystyle=$ $\displaystyle
f\widetilde{g}_{\alpha\eta}\left(T_{as}T_{bc}-T_{ac}T_{bs}+T^{t}_{s}\overline{R}_{abct}\right),$
$\displaystyle(R\cdot R)_{a\alpha\beta\gamma s\eta}$ $\displaystyle=$
$\displaystyle-
fT_{as}[\widetilde{R}_{\eta\alpha\beta\gamma}-f\Delta\widetilde{G}_{\eta\alpha\beta\gamma}]-f^{2}T^{2}_{as}\widetilde{G}_{\eta\alpha\beta\gamma},$
$\displaystyle(R\cdot R)_{\alpha\beta\gamma\delta\mu\eta}$ $\displaystyle=$
$\displaystyle
f[(\widetilde{R}\cdot\widetilde{R})_{\alpha\beta\gamma\delta\mu\eta}-f\Delta
Q(\widetilde{g},\widetilde{R})_{\alpha\beta\gamma\delta\mu\eta}];$ (4.6)
$\displaystyle Q(g,R)_{abcdst}$ $\displaystyle=$ $\displaystyle
Q(\overline{g},\overline{R})_{abcdst},$ $\displaystyle Q(g,R)_{a\alpha b\beta
st}$ $\displaystyle=$ $\displaystyle
f\widetilde{g}_{\alpha\beta}Q(\overline{g},T)_{abst},$ $\displaystyle
Q(g,R)_{abc\alpha s\eta}$ $\displaystyle=$ $\displaystyle
f\widetilde{g}_{\alpha\eta}\left(\overline{R}_{abct}-g_{bs}T_{ac}+g_{as}T_{bc}\right),$
$\displaystyle Q(g,R)_{a\alpha\beta\gamma s\eta}$ $\displaystyle=$
$\displaystyle-f[\overline{g}_{as}\widetilde{R}_{\eta\alpha\beta\gamma}+f(T_{as}-\Delta\overline{g}_{as})\widetilde{G}_{\eta\alpha\beta\gamma}],$
$\displaystyle Q(g,R)_{\alpha\beta\gamma\delta\mu\eta}$ $\displaystyle=$
$\displaystyle
f^{2}Q(\widetilde{g},\widetilde{R})_{\alpha\beta\gamma\delta\mu\eta};$ (4.7)
$\displaystyle Q(S,R)_{abcdst}$ $\displaystyle=$ $\displaystyle
Q(\overline{S},\overline{R})_{abcdst}-(n-p)Q(\overline{S},\overline{R})_{abcdst},$
$\displaystyle Q(S,R)_{a\alpha b\beta st}$ $\displaystyle=$ $\displaystyle
f\widetilde{g}_{\alpha\beta}Q(\overline{S},T)_{abst},$ $\displaystyle
Q(S,R)_{abc\alpha s\eta}$ $\displaystyle=$
$\displaystyle\overline{R}_{abcs}[\widetilde{S}_{\alpha\beta}+\Omega\widetilde{g}_{\alpha\beta}]$
$\displaystyle+f\widetilde{g}_{\alpha\eta}[T_{bc}(\overline{S}_{as}-(n-p)T_{as})-T_{ac}(\overline{S}_{bs}-(n-p)T_{bs})],$
$\displaystyle Q(S,R)_{a\alpha\beta\gamma s\eta}$ $\displaystyle=$
$\displaystyle-f(\overline{S}_{as}-(n-p)T_{as})(\widetilde{R}_{\eta\alpha\beta\gamma}-f\Delta\widetilde{G}_{\eta\alpha\beta\gamma})$
$\displaystyle-
fT_{as}\left[\widetilde{g}_{\alpha\beta}(\widetilde{S}_{\gamma\eta}+\Omega\widetilde{g}_{\gamma\eta})-\widetilde{g}_{\alpha\gamma}(\widetilde{S}_{\beta\eta}+\Omega\widetilde{g}_{\beta\eta})\right],$
$\displaystyle Q(S,R)_{a\alpha b\beta\mu\eta}$ $\displaystyle=$
$\displaystyle-fT_{ab}Q(\widetilde{g},\widetilde{S})_{\alpha\beta\mu\eta},$
$\displaystyle Q(S,R)_{\alpha\beta\gamma\delta\mu\eta}$ $\displaystyle=$
$\displaystyle
f[Q(\widetilde{S},\widetilde{R})_{\alpha\beta\gamma\delta\mu\eta}-\Delta
Q(\widetilde{S},\widetilde{G})_{\alpha\beta\gamma\delta\mu\eta}+f\Omega
Q(\widetilde{g},\widetilde{R})_{\alpha\beta\gamma\delta\mu\eta}].$

We refer the readers to see [5], [21], [24], [27] and also references therein
for detail information about warped product components of various tensors on
$M$.

##  5\. Main results

######  Theorem 5.1.

Let $M^{n}=\overline{M}^{p}\times_{f}\widetilde{M}^{n-p}$ be a warped product
manifold. Then $M$ satisfies the pseudosymmetric type condition

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

if and only if the following conditions hold simultaneously:
$(I)$
$\overline{R}\cdot\overline{R}=L_{1}Q(\overline{g},\overline{R})+L_{2}Q(\overline{S},\overline{R})-L_{2}(n-p)Q(T,\overline{R}),$
$(II)$
$f\widetilde{g}_{\alpha\beta}\left(T_{as}T_{bc}-T_{ac}T_{bs}+T^{t}_{s}\overline{R}_{abct}\right)=L_{1}f\widetilde{g}_{\alpha\beta}\left(\overline{R}_{abcs}-T_{ac}\overline{g}_{bs}+T_{bc}\overline{g}_{as}\right)+L_{2}\overline{R}_{abcs}\left(\widetilde{S}_{\alpha\beta}+\Omega\widetilde{g}_{\alpha\beta}\right)\\\
\indent\hskip
187.90244pt+L_{2}f\widetilde{g}_{\alpha\beta}\left[T_{bc}\left(\overline{S}_{as}-(n-p)T_{as}\right)-T_{ac}\left(\overline{S}_{bs}-(n-p)T_{bs}\right)\right]$,
$(III)$
$T_{as}\left(\widetilde{R}_{\eta\alpha\beta\gamma}-f\Delta\widetilde{G}_{\eta\alpha\beta\gamma}\right)+fT^{2}_{as}\widetilde{G}_{\eta\alpha\beta\gamma}=L_{1}\left[\overline{g}_{as}\widetilde{R}_{\eta\alpha\beta\gamma}+f(T_{as}-\Delta\overline{g}_{as})\widetilde{G}_{\eta\alpha\beta\gamma}\right]\\\
\indent\hskip
238.49121pt+L_{2}(\overline{S}_{as}-(n-p)T_{as})(\widetilde{R}_{\eta\alpha\beta\gamma}-f\Delta\widetilde{G}_{\eta\alpha\beta\gamma})\\\
\indent\hskip
238.49121pt+L_{2}T_{as}\left[\widetilde{g}_{\alpha\beta}(\widetilde{S}_{\gamma\eta}+\Omega\widetilde{g}_{\gamma\eta})-\widetilde{g}_{\alpha\gamma}(\widetilde{S}_{\beta\eta}+\Omega\widetilde{g}_{\beta\eta})\right]$,
$(IV)$ $L_{2}T_{ab}Q(\widetilde{g},\widetilde{S})_{\alpha\beta\gamma\delta}=0$
and
$(V)$ $\widetilde{R}\cdot\widetilde{R}=f\left(\Delta+L_{1}+\Omega
L_{2}\right)Q(\widetilde{g},\widetilde{R})+L_{2}Q(\widetilde{S},\widetilde{R})-L_{2}\Delta
Q(\widetilde{S},\widetilde{G})$.

Proof: In terms of local coordinates, (5.1) can be written as

(5.2) $(R\cdot R)_{ijkluv}=L_{1}Q(g,R)_{ijkluv}+L_{2}Q(S,R)_{ijkluv}.$

Now from (4.5), (4.6) and (4.7) we see that the non-zero possibilities of
(5.2) are
(i) $i=a,j=b,k=c,l=d,u=s,v=t$,
(ii) $i=a,j=\alpha,k=b,l=\beta,u=s,v=t$,
(iii) $i=a,j=b,k=c,l=\alpha,u=s,v=\eta$,
(iv) $i=a,j=\alpha,k=\beta,l=\gamma,u=s,v=\eta$,
(v) $i=a,j=\alpha,k=b,l=\beta,u=\mu,v=\eta$ and
(vi) $i=\alpha,j=\beta,k=\gamma,l=\delta,u=\mu,v=\eta$.
So it is obvious that to prove the theorem we have only to show that the
conditions (I)-(V) are the simplified form of the possibilities (i)-(vi).
Putting (i), (iii), (iv), (v) and (vi) in (5.2) and simplifying we get (I) to
(V) respectively. Again putting (ii) in (5.2), we get $\overline{R}\cdot
T=L_{1}Q(\overline{g},T)+L_{2}Q(\overline{S},T)$, which obviously follows from
(II). This completes the proof.

######  Corollary 5.1.

If a warped product manifold
$M^{n}=\overline{M}^{p}\times_{f}\widetilde{M}^{n-p}$ satisfies the
pseudosymmetric type condition $R\cdot R=L_{1}Q(g,R)+L_{2}Q(S,R)$, then
$\overline{R}\cdot T=L_{1}Q(\overline{g},T)+L_{2}Q(\overline{S},T)$.

Proof: From condition (II) of Theorem 5.1 we get the result easily.

######  Theorem 5.2.

If $M^{n}=\overline{M}^{p}\times_{f}\widetilde{M}^{n-p}$ is a warped product
manifold satisfying (5.1), then

$M=\\{x\in M:\overline{R}|_{\pi(x)}=0\\}\cup\\{x\in
M:(T-L_{1}\overline{g})|_{\pi(x)}=0\\}\cup\\{x\in
M:(\widetilde{S}-\mbox{$\frac{\widetilde{\kappa}}{n-p}$}\widetilde{g})|_{\sigma(x)}=0\\}.$

Proof: Since $M$ satisfies (5.1), then from the condition (IV) of Theorem 5.1,
at every point $x\in M$ we have the following three cases:
Case 1: $L_{2}=0$. Therefore from (III) of Theorem 5.1, we get

$(T-L_{1}g)\widetilde{R}=f\left[L_{1}(T-\Delta\overline{g})+\Delta
T-T^{2}\right]\widetilde{G}$

$\Rightarrow$ either $T-L_{1}\overline{g}=0$ or $\widetilde{R}$ is a scalar
multiple of $\widetilde{G}$ and thus
$\widetilde{S}=\frac{\widetilde{\kappa}}{n-p}\widetilde{g}$ at $x$.
Case 2: $T=0$. In this case putting the value of $T$ in (II) of Theorem 5.1,
we get

$\overline{R}(L_{1}f\widetilde{g}+L_{2}\widetilde{S}+L_{2}\Omega\widetilde{g})=0$
$\Rightarrow\mbox{ either }\overline{R}=0\mbox{ or
}L_{2}\widetilde{S}+(L_{1}f+L_{2}\Omega)\widetilde{g}=0\mbox{ at }x$
$\Rightarrow\mbox{ either }\overline{R}=0\mbox{ or }L_{2}=0\mbox{ or
}\widetilde{S}=\frac{\widetilde{\kappa}}{n-p}\widetilde{g}\mbox{ at }x.$

Case 3: $Q(\widetilde{g},\widetilde{S})=0$. Then from Lemma 2.1 of [5],
$\widetilde{S}=\frac{\widetilde{\kappa}}{n-p}~{}\widetilde{g}$ at $x$.
Now $\overline{R}=0$ at $x$ means $\overline{R}|_{\pi(x)}=0$,
$T-L_{1}\overline{g}=0$ at $x$ means $(T-L_{1}\overline{g})|_{\pi(x)}=0$ and
$\widetilde{S}-\frac{\widetilde{\kappa}}{n-p}\widetilde{g}=0$ at $x$ means
$(\widetilde{S}-\frac{\widetilde{\kappa}}{n-p}\widetilde{g})|_{\sigma(x)}=0$.
Now combining the resulting condition of above cases we get our assertion.

######  Theorem 5.3.

If a warped product manifold
$M^{n}=\overline{M}^{p}\times_{f}\widetilde{M}^{n-p}$ satisfies the
pseudosymmetric type condition (5.1) and $L_{2}$ is nowhere zero, then
either (i) the base $\overline{M}$ is flat
or (ii) the fiber $\widetilde{M}$ is Einstein.

Proof: Since $M$ satisfies (5.1) and $L_{2}$ is nowhere zero, then from the
condition (IV) of Theorem 5.1,

$TQ(\widetilde{g},\widetilde{S})\equiv 0$ $\Rightarrow T\equiv 0\mbox{ or
}Q(\widetilde{g},\widetilde{S})\equiv 0\ \ \mbox{ [by Lemma \ref{lem1}].}$

Now if $T$ is identically zero, then from Condition (II),
$\overline{R}(L_{1}f\widetilde{g}+L_{2}\widetilde{S}+L_{2}\Omega\widetilde{g})\equiv
0$.

$\Rightarrow\overline{R}\equiv 0\mbox{ or
}L_{1}f\widetilde{g}+L_{2}\widetilde{S}+L_{2}\Omega\widetilde{g}\equiv 0.\ \
\mbox{ [by Lemma \ref{lem1}]}$ $\Rightarrow\mbox{base is flat or fiber is
Einstein \ \ \ [as $L_{2}$ is nowhere zero].}$

Again if $Q(\widetilde{g},\widetilde{S})\equiv 0$ then from Lemma 2.2, the
fiber $\widetilde{M}$ is Einstein. This completes the proof.
Now from Theorem 5.1 we can easily get the characterization for a warped
product semisymmetric and various pseudosymmetric type manifolds as follows:

######  Corollary 5.2.

Let $M^{n}=\overline{M}^{p}\times_{f}\widetilde{M}^{n-p}$ be a warped product
manifold. Then $M$ satisfies $R\cdot R=0$ if and only if the following
conditions hold simultaneously:
(I) $\overline{R}\cdot\overline{R}=0,$
(II) $T_{as}T_{bc}-T_{ac}T_{bs}+fT^{t}_{s}\overline{R}_{abct}=0,$
(III)
$T_{as}\left(\widetilde{R}_{\eta\alpha\beta\gamma}-f\Delta\widetilde{G}_{\eta\alpha\beta\gamma}\right)+T^{2}_{as}\widetilde{G}_{\eta\alpha\beta\gamma}=0,$
(IV) $\widetilde{R}\cdot\widetilde{R}=f\Delta Q(\widetilde{g},\widetilde{R})$.

######  Corollary 5.3.

Let $M^{n}=\overline{M}^{p}\times_{f}\widetilde{M}^{n-p}$ be a warped product
semisymmetric manifold. Then
(i) the base $\overline{M}$ is semisymmetric,
(ii) the fiber $\widetilde{M}$ is pseudosymmetric,
(iii) the fiber $\widetilde{M}$ is of constant curvature if $T\neq 0$.
(iv) $R\cdot T=0$.

######  Corollary 5.4.

([6], [8], [11]) Let $M^{n}=\overline{M}^{p}\times_{f}\widetilde{M}^{n-p}$ be
a warped product manifold. Then $M$ satisfies $R\cdot R=L_{R}Q(g,R)$ if and
only if the following conditions hold simultaneously:
(I) $\overline{R}\cdot\overline{R}=L_{R}Q(\overline{g},\overline{R}),$
(II)
$T^{t}_{s}\left(\overline{R}_{abct}-T_{ac}\overline{g}_{bt}+T_{bc}\overline{g}_{at}\right)=L_{R}\left(\overline{R}_{abcs}-T_{ac}\overline{g}_{bs}+T_{bc}\overline{g}_{as}\right),$
(III)
$T_{as}\left(\widetilde{R}_{\eta\alpha\beta\gamma}-f\Delta\widetilde{G}_{\eta\alpha\beta\gamma}\right)+fT^{2}_{as}\widetilde{G}_{\eta\alpha\beta\gamma}=L_{R}\left[\overline{g}_{as}\widetilde{R}_{\eta\alpha\beta\gamma}+f(T_{as}-\Delta\overline{g}_{as})\widetilde{G}_{\eta\alpha\beta\gamma}\right]$,
(IV)
$\widetilde{R}\cdot\widetilde{R}=\left(f\Delta+fL_{R}\right)Q(\widetilde{g},\widetilde{R})$.

######  Corollary 5.5.

Let $M^{n}=\overline{M}^{p}\times_{f}\widetilde{M}^{n-p}$ be a warped product
pseudosymmetric manifold $(R\cdot R=L_{R}Q(g,R))$. Then
(i) the base and fiber both are pseudosymmetric,
(ii) $R\cdot T=L_{R}Q(\overline{g},T)$.

######  Corollary 5.6.

Let $M^{n}=\overline{M}^{p}\times_{f}\widetilde{M}^{n-p}$ be a warped product
manifold. Then $M$ satisfies $R\cdot R=LQ(S,R)$ if and only if the following
conditions hold simultaneously:
(I)
$\overline{R}\cdot\overline{R}=LQ(\overline{S},\overline{R})-L(n-p)Q(T,\overline{R}),$
(II)
$f\widetilde{g}_{\alpha\beta}\left(T_{as}T_{bc}-T_{ac}T_{bs}+T^{t}_{s}\overline{R}_{abct}\right)=L\overline{R}_{abcs}\left(\widetilde{S}_{\alpha\beta}+\Omega\widetilde{g}_{\alpha\beta}\right)\\\
\indent\hskip
187.90244pt+Lf\widetilde{g}_{\alpha\beta}\left[T_{bc}\left(\overline{S}_{as}-(n-p)T_{as}\right)-T_{ac}\left(\overline{S}_{bs}-(n-p)T_{bs}\right)\right]$,
(III)
$T_{as}\left(\widetilde{R}_{\eta\alpha\beta\gamma}-f\Delta\widetilde{G}_{\eta\alpha\beta\gamma}\right)+fT^{2}_{as}\widetilde{G}_{\eta\alpha\beta\gamma}=L(\overline{S}_{as}-(n-p)T_{as})(\widetilde{R}_{\eta\alpha\beta\gamma}-f\Delta\widetilde{G}_{\eta\alpha\beta\gamma})\\\
\indent\hskip
238.49121pt+LT_{as}\left[\widetilde{g}_{\alpha\beta}(\widetilde{S}_{\gamma\eta}+\Omega\widetilde{g}_{\gamma\eta})-\widetilde{g}_{\alpha\gamma}(\widetilde{S}_{\beta\eta}+\Omega\widetilde{g}_{\beta\eta})\right]$,
(IV) $LT_{ab}Q(\widetilde{g},\widetilde{S})_{\alpha\beta\gamma\delta}=0$ and
(V) $\widetilde{R}\cdot\widetilde{R}=\left(f\Delta+\Omega
L\right)Q(\widetilde{g},\widetilde{R})+LQ(\widetilde{S},\widetilde{R})-L\Delta
Q(\widetilde{S},\widetilde{G})$.

######  Corollary 5.7.

If a warped product manifold
$M^{n}=\overline{M}^{p}\times_{f}\widetilde{M}^{n-p}$ satisfies $R\cdot
R=LQ(S,R)$, then

$M=\\{x\in M:\overline{R}|_{\pi(x)}=0\\}\cup\\{x\in
M:T|_{\pi(x)}=0\\}\cup\\{x\in
M:(\widetilde{S}-\mbox{$\frac{\widetilde{\kappa}}{n-p}$}\widetilde{g})|_{\sigma(x)}=0\\}.$

######  Corollary 5.8.

[6] Let $M^{n}=\overline{M}^{p}\times_{f}\widetilde{M}^{n-p}$ be a warped
product manifold. Then $M$ satisfies the special Ricci generalized
pseudosymmetric condition $R\cdot R=Q(S,R)$ if and only if the following
conditions hold simultaneously:
(I) $\overline{R}\cdot\overline{R}=Q(\overline{S}-(n-p)T,\overline{R}),$
(II)
$f\widetilde{g}_{\alpha\beta}\left(T_{as}T_{bc}-T_{ac}T_{bs}+T^{t}_{s}\overline{R}_{abct}\right)=\overline{R}_{abcs}\left(\widetilde{S}_{\alpha\beta}+\Omega\widetilde{g}_{\alpha\beta}\right)\\\
\indent\hskip
187.90244pt+f\widetilde{g}_{\alpha\beta}\left[T_{bc}\left(\overline{S}_{as}-(n-p)T_{as}\right)-T_{ac}\left(\overline{S}_{bs}-(n-p)T_{bs}\right)\right]$,
(III)
$T_{as}\left(\widetilde{R}_{\eta\alpha\beta\gamma}-f\Delta\widetilde{G}_{\eta\alpha\beta\gamma}\right)+fT^{2}_{as}\widetilde{G}_{\eta\alpha\beta\gamma}=(\overline{S}_{as}-(n-p)T_{as})(\widetilde{R}_{\eta\alpha\beta\gamma}-f\Delta\widetilde{G}_{\eta\alpha\beta\gamma})\\\
\indent\hskip
238.49121pt+T_{as}\left[\widetilde{g}_{\alpha\beta}(\widetilde{S}_{\gamma\eta}+\Omega\widetilde{g}_{\gamma\eta})-\widetilde{g}_{\alpha\gamma}(\widetilde{S}_{\beta\eta}+\Omega\widetilde{g}_{\beta\eta})\right]$,
(IV) $T_{ab}Q(\widetilde{g},\widetilde{S})_{\alpha\beta\gamma\delta}=0$ and
(V)
$\widetilde{R}\cdot\widetilde{R}=\left(f\Delta+\Omega\right)Q(\widetilde{g},\widetilde{R})+Q(\widetilde{S},\widetilde{R})-\Delta
Q(\widetilde{S},\widetilde{G})$.

######  Corollary 5.9.

If a warped product manifold
$M^{n}=\overline{M}^{p}\times_{f}\widetilde{M}^{n-p}$ satisfies $R\cdot
R=Q(S,R)$, then

$M=\\{x\in M:\overline{R}|_{\pi(x)}=0\\}\cup\\{x\in
M:(\widetilde{S}-\mbox{$\frac{\widetilde{\kappa}}{n-p}$}\widetilde{g})|_{\sigma(x)}=0\\}.$

######  Theorem 5.4.

If a warped product manifold
$M^{n}=\overline{M}^{p}\times_{f}\widetilde{M}^{n-p}$ satisfies $R\cdot
R=Q(S,R)$, then
either (i) the base $\overline{M}$ is flat
or (ii) the fiber $\widetilde{M}$ is Einstein.

Proof: Since the condition $R\cdot R=Q(S,R)$ is a special case of $R\cdot
R=L_{1}Q(g,R)+L_{2}Q(S,R)$, for $L_{1}=0$ and $L_{2}=1$. Therefore as $L_{2}$
is nowhere zero, so from Theorem 5.3, we get our assertion.

######  Corollary 5.10.

Let $M^{n}=\overline{M}^{p}\times_{f}\widetilde{M}^{n-p}$ be a warped product
manifold. Then $M$ satisfies $W\cdot R=0$ if and only if the following
conditions hold simultaneously:
(I)
$\overline{R}\cdot\overline{R}=\frac{\kappa}{n(n-1)}Q(\overline{g},\overline{R}),$
(II)
$T^{t}_{s}\left(\overline{R}_{abct}-T_{ac}\overline{g}_{bt}+T_{bc}\overline{g}_{at}\right)=\frac{\kappa}{n(n-1)}\left(\overline{R}_{abcs}-T_{ac}\overline{g}_{bs}+T_{bc}\overline{g}_{as}\right),$
(III)
$T_{as}\left(\widetilde{R}_{\eta\alpha\beta\gamma}-f\Delta\widetilde{G}_{\eta\alpha\beta\gamma}\right)+fT^{2}_{as}\widetilde{G}_{\eta\alpha\beta\gamma}=\frac{\kappa}{n(n-1)}\left[\overline{g}_{as}\widetilde{R}_{\eta\alpha\beta\gamma}+f(T_{as}-\Delta\overline{g}_{as})\widetilde{G}_{\eta\alpha\beta\gamma}\right]$,
(IV)
$\widetilde{R}\cdot\widetilde{R}=f\left(\Delta+\frac{\kappa}{n(n-1)}\right)Q(\widetilde{g},\widetilde{R})$.

######  Corollary 5.11.

Let $M^{n}=\overline{M}^{p}\times_{f}\widetilde{M}^{n-p}$ be a warped product
manifold. Then $M$ satisfies $W\cdot R=L_{2}Q(S,R)$ if and only if the
following conditions hold simultaneously:
$(I)$
$\overline{R}\cdot\overline{R}=\frac{\kappa}{n(n-1)}Q(\overline{g},\overline{R})+L_{2}Q(\overline{S},\overline{R})-L_{2}(n-p)Q(T,\overline{R}),$
$(II)$
$f\widetilde{g}_{\alpha\beta}\left(T_{as}T_{bc}-T_{ac}T_{bs}+T^{t}_{s}\overline{R}_{abct}\right)=\frac{\kappa}{n(n-1)}f\widetilde{g}_{\alpha\beta}\left(\overline{R}_{abcs}-T_{ac}\overline{g}_{bs}+T_{bc}\overline{g}_{as}\right)\\\
\indent\hskip
187.90244pt+L_{2}\overline{R}_{abcs}\left(\widetilde{S}_{\alpha\beta}+\Omega\widetilde{g}_{\alpha\beta}\right)\\\
\indent\hskip
187.90244pt+L_{2}f\widetilde{g}_{\alpha\beta}\left[T_{bc}\left(\overline{S}_{as}-(n-p)T_{as}\right)-T_{ac}\left(\overline{S}_{bs}-(n-p)T_{bs}\right)\right]$,
$(III)$
$T_{as}\left(\widetilde{R}_{\eta\alpha\beta\gamma}-f\Delta\widetilde{G}_{\eta\alpha\beta\gamma}\right)+fT^{2}_{as}\widetilde{G}_{\eta\alpha\beta\gamma}=\frac{\kappa}{n(n-1)}\left[\overline{g}_{as}\widetilde{R}_{\eta\alpha\beta\gamma}+f(T_{as}-\Delta\overline{g}_{as})\widetilde{G}_{\eta\alpha\beta\gamma}\right]\\\
\indent\hskip
238.49121pt+L_{2}(\overline{S}_{as}-(n-p)T_{as})(\widetilde{R}_{\eta\alpha\beta\gamma}-f\Delta\widetilde{G}_{\eta\alpha\beta\gamma})\\\
\indent\hskip
238.49121pt+L_{2}T_{as}\left[\widetilde{g}_{\alpha\beta}(\widetilde{S}_{\gamma\eta}+\Omega\widetilde{g}_{\gamma\eta})-\widetilde{g}_{\alpha\gamma}(\widetilde{S}_{\beta\eta}+\Omega\widetilde{g}_{\beta\eta})\right]$,
$(IV)$ $L_{2}T_{ab}Q(\widetilde{g},\widetilde{S})_{\alpha\beta\gamma\delta}=0$
and
$(V)$
$\widetilde{R}\cdot\widetilde{R}=f\left(\Delta+\frac{\kappa}{n(n-1)}+\Omega
L_{2}\right)Q(\widetilde{g},\widetilde{R})+L_{2}Q(\widetilde{S},\widetilde{R})-L_{2}\Delta
Q(\widetilde{S},\widetilde{G})$.

######  Corollary 5.12.

Let $M^{n}=\overline{M}^{p}\times_{f}\widetilde{M}^{n-p}$ be a warped product
manifold. Then $M$ satisfies $P\cdot R=0$ if and only if the following
conditions hold simultaneously:
(I)
$\overline{R}\cdot\overline{R}=\frac{1}{n-2}Q(\overline{S}-(n-p)T,\overline{R}),$
(II)
$f\widetilde{g}_{\alpha\beta}\left(T_{as}T_{bc}-T_{ac}T_{bs}+T^{t}_{s}\overline{R}_{abct}\right)=\frac{1}{n-2}\overline{R}_{abcs}\left(\widetilde{S}_{\alpha\beta}+\Omega\widetilde{g}_{\alpha\beta}\right)\\\
\indent\hskip
187.90244pt+\frac{1}{n-2}f\widetilde{g}_{\alpha\beta}\left[T_{bc}\left(\overline{S}_{as}-(n-p)T_{as}\right)-T_{ac}\left(\overline{S}_{bs}-(n-p)T_{bs}\right)\right]$,
(III)
$T_{as}\left(\widetilde{R}_{\eta\alpha\beta\gamma}-f\Delta\widetilde{G}_{\eta\alpha\beta\gamma}\right)+fT^{2}_{as}\widetilde{G}_{\eta\alpha\beta\gamma}=\frac{1}{n-2}(\overline{S}_{as}-(n-p)T_{as})(\widetilde{R}_{\eta\alpha\beta\gamma}-f\Delta\widetilde{G}_{\eta\alpha\beta\gamma})\\\
\indent\hskip
238.49121pt+\frac{1}{n-2}T_{as}\left[\widetilde{g}_{\alpha\beta}(\widetilde{S}_{\gamma\eta}+\Omega\widetilde{g}_{\gamma\eta})-\widetilde{g}_{\alpha\gamma}(\widetilde{S}_{\beta\eta}+\Omega\widetilde{g}_{\beta\eta})\right]$,
(IV) $T_{ab}Q(\widetilde{g},\widetilde{S})_{\alpha\beta\gamma\delta}=0$ and
(V)
$\widetilde{R}\cdot\widetilde{R}=f\left(\Delta+\frac{\Omega}{n-2}\right)Q(\widetilde{g},\widetilde{R})+\frac{1}{n-2}Q(\widetilde{S},\widetilde{R})-\frac{1}{n-2}\Delta
Q(\widetilde{S},\widetilde{G})$.

######  Theorem 5.5.

If a warped product manifold
$M^{n}=\overline{M}^{p}\times_{f}\widetilde{M}^{n-p}$ satisfies $P\cdot R=0$,
then either (i) the base $\overline{M}$ is flat or (ii) the fiber
$\widetilde{M}$ is Einstein.

######  Corollary 5.13.

Let $M^{n}=\overline{M}^{p}\times_{f}\widetilde{M}^{n-p}$ be a warped product
manifold. Then $M$ satisfies the pseudosymmetric type condition

(5.3) $P\cdot R=L_{1}Q(g,R)$

if and only if the following conditions hold simultaneously:
(I)
$\overline{R}\cdot\overline{R}=L_{1}Q(\overline{g},\overline{R})+\frac{1}{n-2}Q(\overline{S},\overline{R})-\frac{n-p}{n-2}Q(T,\overline{R}),$
(II)
$f\widetilde{g}_{\alpha\beta}\left(T_{as}T_{bc}-T_{ac}T_{bs}+T^{t}_{s}\overline{R}_{abct}\right)=L_{1}f\widetilde{g}_{\alpha\beta}\left(\overline{R}_{abcs}-T_{ac}\overline{g}_{bs}+T_{bc}\overline{g}_{as}\right)+\frac{1}{n-2}\overline{R}_{abcs}\left(\widetilde{S}_{\alpha\beta}+\Omega\widetilde{g}_{\alpha\beta}\right)\\\
\indent\hskip
187.90244pt+\frac{f}{n-2}\widetilde{g}_{\alpha\beta}\left[T_{bc}\left(\overline{S}_{as}-(n-p)T_{as}\right)-T_{ac}\left(\overline{S}_{bs}-(n-p)T_{bs}\right)\right]$,
(III)
$T_{as}\left(\widetilde{R}_{\eta\alpha\beta\gamma}-f\Delta\widetilde{G}_{\eta\alpha\beta\gamma}\right)+fT^{2}_{as}\widetilde{G}_{\eta\alpha\beta\gamma}=L_{1}\left[\overline{g}_{as}\widetilde{R}_{\eta\alpha\beta\gamma}+f(T_{as}-\Delta\overline{g}_{as})\widetilde{G}_{\eta\alpha\beta\gamma}\right]\\\
\indent\hskip
238.49121pt+\frac{1}{n-2}(\overline{S}_{as}-(n-p)T_{as})(\widetilde{R}_{\eta\alpha\beta\gamma}-f\Delta\widetilde{G}_{\eta\alpha\beta\gamma})\\\
\indent\hskip
238.49121pt+\frac{1}{n-2}T_{as}\left[\widetilde{g}_{\alpha\beta}(\widetilde{S}_{\gamma\eta}+\Omega\widetilde{g}_{\gamma\eta})-\widetilde{g}_{\alpha\gamma}(\widetilde{S}_{\beta\eta}+\Omega\widetilde{g}_{\beta\eta})\right]$,
(IV) $T_{ab}Q(\widetilde{g},\widetilde{S})_{\alpha\beta\gamma\delta}=0$ and
(V)
$\widetilde{R}\cdot\widetilde{R}=\left(f\Delta+fL_{1}+\frac{\Omega}{n-2}\right)Q(\widetilde{g},\widetilde{R})+\frac{1}{n-2}Q(\widetilde{S},\widetilde{R})-\frac{1}{n-2}\Delta
Q(\widetilde{S},\widetilde{G})$.

######  Theorem 5.6.

If a warped product manifold
$M^{n}=\overline{M}^{p}\times_{f}\widetilde{M}^{n-p}$ satisfies $P\cdot
R=L_{1}Q(g,R)$, then either (i) the base $\overline{M}$ is flat or (ii) the
fiber $\widetilde{M}$ is Einstein.

##  6\. Examples

Example 1: Consider the warped product
$M=\overline{M}\times_{f}\widetilde{M}$, where $\overline{M}$ is an open
interval of $\mathbb{R}$ with the metric
$d\overline{s}^{2}=\frac{1}{1+a(1+x^{1})^{2}}(dx^{1})^{2}$ in local coordinate
$x^{1}$, $\widetilde{M}$ is a 4-dimensional manifold equipped with a semi-
Riemannian metric

$d\widetilde{s}^{2}=-(dx^{2})^{2}+e^{2x^{2}}(x^{5})^{2}(dx^{3})^{2}+2e^{2x^{2}}dx^{3}dx^{4}+e^{2x^{2}}(dx^{5})^{2}$

in local coordinates $(x^{2},x^{3},x^{4},x^{5})$ and the warping function
$f=(x^{1}+1)^{2}$. We can easily evaluate the local components of necessary
tensors of $\widetilde{M}$. The non-zero components of the Riemann-Christoffel
curvature tensor $\widetilde{R}$ and the Ricci tensor $\widetilde{S}$ of
$\widetilde{M}$ upto symmetry are

$\widetilde{R}_{1213}=\widetilde{R}_{1414}=-e^{2x^{2}},\ \
-\widetilde{R}_{2323}=\widetilde{R}_{2434}=e^{4x^{2}},$
$\widetilde{R}_{1212}=-e^{2x^{2}}(x^{5})^{2},\ \
\widetilde{R}_{2424}=e^{2x^{2}}(e^{x^{2}}x^{5}-1)(e^{x^{2}}x^{5}+1)$

and

$\widetilde{S}_{11}=3,\ \ \widetilde{S}_{22}=-3e^{2x^{2}}(x^{5})^{2}-1,\ \
\widetilde{S}_{23}=\widetilde{S}_{44}=-3e^{2x^{2}}.$

Scalar curvature of $\widetilde{M}$ is $(-12)$. Again the non-zero components
of $\widetilde{R}\cdot\widetilde{R}$, $Q(\widetilde{g},\widetilde{R})$ and
$Q(\widetilde{S},\widetilde{R})$ are

$\widetilde{R}\cdot\widetilde{R}_{122414}=-\widetilde{R}\cdot\widetilde{R}_{142412}=e^{2x^{2}},\
\
2\widetilde{R}\cdot\widetilde{R}_{232424}=-\widetilde{R}\cdot\widetilde{R}_{242423}=2e^{4x^{2}},$
$-Q(\widetilde{g},\widetilde{R})_{122414}=Q(\widetilde{g},\widetilde{R})_{142412}=e^{2x^{2}},\
\
-2Q(\widetilde{g},\widetilde{R})_{232424}=Q(\widetilde{g},\widetilde{R})_{242423}=2e^{4x^{2}},$
$-Q(\widetilde{S},\widetilde{R})_{121223}=-2Q(\widetilde{S},\widetilde{R})_{121424}=2Q(\widetilde{S},\widetilde{R})_{122312}=\frac{2}{3}Q(\widetilde{S},\widetilde{R})_{122414}=-Q(\widetilde{S},\widetilde{R})_{142412}=2e^{2x^{2}},$
$2Q(\widetilde{S},\widetilde{R})_{232424}=-Q(\widetilde{S},\widetilde{R})_{242423}=4e^{4x^{2}}.$

Then we can easily check that $\widetilde{M}$ satisfies
$\widetilde{R}\cdot\widetilde{R}=-Q(\widetilde{g},\widetilde{R})$, i.e.,
$\widetilde{M}$ is a pseudosymmetric manifold of constant type.
Now by a straightforward calculation we can evaluate the components of various
necessary tensors corresponding to $M$. The non-zero local components of the
Riemann-Christoffel curvature tensor $R$ and the Ricci tensor $S$ of $M$ upto
symmetry are

$R_{1212}=\frac{a(x^{1}+1)^{2}}{a(x^{1})^{2}+2ax^{1}+a+1},\ \
R_{1313}=-\frac{ae^{2x^{2}}(x^{1}+1)^{2}(x^{5})^{2}}{a(x^{1})^{2}+2ax^{1}+a+1},$
$R_{1314}=R_{1515}=-\frac{ae^{2x^{2}}(x^{1}+1)^{2}}{a(x^{1})^{2}+2ax^{1}+a+1},\
\ R_{2323}=ae^{2x^{2}}(x^{1}+1)^{4}(x^{5})^{2},$
$R_{2324}=R_{2525}=ae^{2x^{2}}(x^{1}+1)^{4},\ \
R_{3434}=-R_{3545}=ae^{4x^{2}}(x^{1}+1)^{4},$
$-R_{3535}=e^{2x^{2}}(x^{1}+1)^{2}(ae^{2x^{2}}(x^{5})^{2}+ae^{2x^{2}}(x^{1})^{2}(x^{5})^{2}+2ae^{2x^{2}}x^{1}(x^{5})^{2}+1)$

and

$S_{11}=\frac{4a}{a(x^{1})^{2}+2ax^{1}+a+1},\ \ S_{22}=-4a(x^{1}+1)^{2},$
$S_{33}=4ae^{2x^{2}}(x^{5})^{2}+4ae^{2x^{2}}(x^{1})^{2}(x^{5})^{2}+8ae^{2x^{2}}x^{1}(x^{5})^{2}+1,\
\ S_{34}=S_{55}=4ae^{2x^{2}}(x^{1}+1)^{2}.$

The scalar curvature of $M$ is $20a$. Now the non-zero components (upto
symmetry) of $R\cdot R$, $Q(g,R)$ and $Q(S,R)$ are

$R\cdot R_{133515}=-R\cdot
R_{153513}=\frac{ae^{2x^{2}}(x^{1}+1)^{2}}{a(x^{1})^{2}+2ax^{1}+a+1},$
$-R\cdot R_{233525}=R\cdot R_{253523}=ae^{2x^{2}}(x^{1}+1)^{4},$ $-2R\cdot
R_{343535}=R\cdot R_{353534}=2ae^{4x^{2}}(x^{1}+1)^{4},$
$Q(g,R)_{133515}=-Q(g,R)_{153513}=\frac{e^{2x^{2}}(x^{1}+1)^{2}}{a(x^{1})^{2}+2ax^{1}+a+1},$
$-Q(g,R)_{233525}=Q(g,R)_{253523}=e^{2x^{2}}(x^{1}+1)^{4},$
$-2Q(g,R)_{343535}=Q(g,R)_{353534}=2e^{4x^{2}}(x^{1}+1)^{4}$

and

$-Q(S,R)_{121323}=Q(S,R)_{122313}=\frac{a(x^{1}+1)^{2}}{a(x^{1})^{2}+2ax^{1}+a+1},$
$-\frac{1}{2}Q(S,R)_{131334}=-Q(S,R)_{131535}=Q(S,R)_{133413}=\frac{1}{4}Q(S,R)_{133515}$
$=-\frac{1}{3}Q(S,R)_{153513}=\frac{ae^{2x^{2}}(x^{1}+1)^{2}}{a(x^{1})^{2}+2ax^{1}+a+1},$
$\frac{1}{2}Q(S,R)_{232334}=Q(S,R)_{232535}=-Q(S,R)_{233423}=-\frac{1}{4}Q(S,R)_{233525}$
$=\frac{1}{3}Q(S,R)_{253523}=ae^{2x^{2}}(x^{1}+1)^{4},$
$-2Q(S,R)_{343535}=Q(S,R)_{353534}=6ae^{4x^{2}}(x^{1}+1)^{4}.$

Then from the values of $R\cdot R$ and $Q(g,R)$, we see that $M$ satisfies
$R\cdot R=aQ(g,R)$. Therefore $M$ is a warped product pseudosymmetric manifold
of constant type. In particular if $a=0$, then $M$ is a warped product
semisymmetric manifold. Now $R\cdot R=aQ(g,R)\Rightarrow(R-aG)\cdot R=0$ and
$W=R-\frac{\kappa}{5\times 4}G=R-aG$ hence $W\cdot R=0$. Therefore $M$ is
semisymmetric type due to concircular curvature tensor.

######  Remark 6.1.

We know that fibers of a warped product manifold are totally umbilical
submanifold of it. In the above example, $\widetilde{M}$ is a totally
umbilical hypersurface of $M$ for $a\neq 0$ and for $a=0$, the above example
is an example of pseudosymmetric totally umbilical hypersurface of a
semisymmetric manifold.

Example 2: [26] Let $M_{2}=\mathbb{R}\times_{f}\mathbb{R}^{3}$ be the
$4$-dimensional connected semi-Riemannian warped product manifold, where the
base metric is given by $d\overline{s}^{2}=(1+2e^{x^{1}})(dx^{1})^{2}$, the
fiber metric is the usual Euclidean metric and the warping function
$f=(1+2e^{x^{1}})$. Hence the metric of $M_{2}$ is given by

(6.1)
$ds^{2}=(1+2e^{x^{1}})(dx^{1})^{2}+(1+2e^{x^{1}})\left[(dx^{2})^{2}+(dx^{3})^{2}+(dx^{4})^{2}\right].$

Therefore the non-zero components (upto symmetry) of $R$, $S$, $\kappa$ and
$P$ are given by

$R_{1212}=R_{1313}=R_{1414}=-\frac{e^{x^{1}}}{2e^{x^{1}}+1},\ \
R_{2323}=R_{2424}=R_{3434}=-\frac{e^{2x^{1}}}{2e^{x^{1}}+1};$
$S_{11}=\frac{3e^{x^{1}}}{\left(2e^{x^{1}}+1\right){}^{2}},\ \
S_{22}=S_{33}=S_{44}=\frac{e^{x^{1}}}{2e^{x^{1}}+1};\ \
\kappa=\frac{6e^{x^{1}}(1+e^{x^{1}})}{(1+2e^{x^{1}})^{3}};$
$\frac{1}{2}P_{1221}=\frac{1}{2}P_{1331}=\frac{1}{2}P_{1441}=P_{2323}=-P_{2332}=P_{2424}=-P_{2442}=P_{3434}=-P_{3443}=-\frac{e^{2x^{1}}-e^{x^{1}}}{6e^{x^{1}}+3}.$

Using above we can easily calculate the non-zero components (upto symmetry) of
$R\cdot R$, $Q(g,R)$, $Q(S,R)$ and $P\cdot R$ as follows:

$R\cdot R_{122313}=R\cdot R_{122414}=-R\cdot R_{132312}=R\cdot
R_{133414}=-R\cdot R_{142412}=-R\cdot
R_{143413}=\frac{e^{2x^{1}}\left(e^{x^{1}}-1\right)}{\left(2e^{x^{1}}+1\right)^{3}};$
$Q(g,R)_{122313}=Q(g,R)_{122414}=-Q(g,R)_{132312}=Q(g,R)_{133414}=-Q(g,R)_{142412}=-Q(g,R)_{143413}$
$=e^{x^{1}}\left(e^{x^{1}}-1\right);$
$Q(S,R)_{122313}=Q(S,R)_{122414}=-Q(S,R)_{132312}=Q(S,R)_{133414}=-Q(S,R)_{142412}=-Q(S,R)_{143413}$
$=\frac{e^{2x^{1}}\left(e^{x^{1}}-1\right)}{\left(2e^{x^{1}}+1\right)^{3}};$
$P\cdot R_{122313}=P\cdot R_{122414}=-P\cdot R_{132312}=P\cdot
R_{133414}=-P\cdot R_{142412}=-P\cdot
R_{143413}=\frac{2e^{2x^{1}}\left(e^{x^{1}}-1\right)}{3\left(2e^{x^{1}}+1\right)^{3}}.$

In view of above results we see that $M_{2}$ satisfies the following
pseudosymmetric type conditions:
(i) $R\cdot R=\frac{e^{x^{1}}}{\left(2e^{x^{1}}+1\right)^{3}}Q(g,R)=Q(S,R)$,
(ii) $P\cdot R=\frac{2e^{x^{1}}}{3\left(2e^{x^{1}}+1\right)^{3}}Q(g,R)$ and
(iii) $R\cdot
R=Q(g,R)+\left[1-e^{-x^{1}}\left(2e^{x^{1}}+1\right)^{3}\right]Q(S,R)$.

##  7\. Conclusions

In this present paper we have found out the necessary and sufficient condition
for which a warped product semi-Riemannian manifold
$M^{n}=\overline{M}^{p}\times_{f}\widetilde{M}^{n-p}$ satisfies the
pseudosymmetric type condition $R\cdot R=L_{1}Q(g,R)+L_{2}Q(S,R)$. It is shown
that on each point $x\in M$ of such manifold either $\overline{R}=0$ or
$T=L_{1}g$ or $\widetilde{S}=\frac{\widetilde{\kappa}}{(n-p)}\widetilde{g}$.
Moreover if $L_{2}$ is nowhere zero, then either the base $\overline{M}$ is
flat or the fiber $\widetilde{M}$ is Einstein. As a special case of the main
theorem we get the necessary and sufficient condition for which a warped
product semi-Riemannian manifold
$M^{n}=\overline{M}^{p}\times_{f}\widetilde{M}^{n-p}$ satisfies the
pseudosymmetric type condition (i) $R\cdot R=0$, (ii) $R\cdot R=L_{1}Q(g,R)$,
(iii) $R\cdot R=L_{2}Q(S,R)$, (iv) $R\cdot R=Q(S,R)$, (v) $P\cdot R=0$, (vi)
$P\cdot R=LQ(g,R)$.
It is proved that the base of a semisymmetric warped product manifold is
semisymmetric and fiber is pseudosymmetric, whereas both the base and fiber of
a pseudosymmetric warped product manifold are pseudosymmetric. It is also
proved that on a warped product pseudosymmetric or semisymmetric manifold due
to projective curvature tensor either the base is flat or the fiber is
Einstein. By using the fact that the fiber of a semisymmetric warped product
manifold is pseudosymmetric, we have established an example which ensures that
a pseudosymmetric manifold is a totally umbilical hypersurface of a
semisymmetric manifold.

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#  Learning to Compress Unmanned Aerial Vehicle (UAV) Captured Video:
Benchmark and Analysis

Chuanmin Jia† Feng Ye† Huifang Sun‡ Siwei Ma†,‡ Wen Gao†,‡
†Peking University, Beijing, China, ‡Peng Cheng Laboratory, Shenzhen, China

###### Abstract

During the past decade, the Unmanned-Aerial-Vehicles (UAVs) have attracted
increasing attention due to their flexible, extensive, and dynamic space-
sensing capabilities. The volume of video captured by UAVs is exponentially
growing along with the increased bitrate generated by the advancement of the
sensors mounted on UAVs, bringing new challenges for on-device UAV storage and
air-ground data transmission. Most existing video compression schemes were
designed for natural scenes without consideration of specific texture and view
characteristics of UAV videos. In this work, we first contribute a detailed
analysis of the current state of the field of UAV video coding. Then we
propose to establish a novel task for learned UAV video coding and construct a
comprehensive and systematic benchmark for such a task, present a thorough
review of high quality UAV video datasets and benchmarks, and contribute
extensive rate-distortion efficiency comparison of learned and conventional
codecs after. Finally, we discuss the challenges of encoding UAV videos. It is
expected that the benchmark will accelerate the research and development in
video coding on drone platforms.

##  1 Introduction

Drones (or Unmanned Aerial Vehicles, UAVs) are receiving increasing attention
during the past decade, due to their flexible, extensive, and dynamic space
sensing availability. The commercial UAV market report describes that the
global commercial drone market size will reach 501.4 billion by 2028 [1].
These drones can be efficiently used for autonomous facility maintenance,
scientific discoveries, or agricultural monitoring via the equipped cameras.
Consequently, high efficiency compression and low-cost storage scheme of
visual data collected from drones become highly demanding, which brings video
coding to UVAs even more closely.

Lossy video compression using neural networks has been popular research topic
in recent years, the objective of which is to achieve the compact
representation with minimal bit-rate (entropy) while preserving maximal video
signal quality, yielding the typical rate-distortion (R-D) optimization
problem. The video compression community has continued to gain coding
performance along with the local module advancement of hybrid coding framework
in a standardized fashion [2, 3, 4, 5, 6]. Recently, with the tremendous
success in artificial intelligence, neural video compression (NVC) attracts
increasing attention from both academic and industry realms for the capability
of end-to-end global optimization.

However, it is observed that fewer efforts have been dedicated to the field of
UAV video coding, leaving such research topic less studied. Existing UAV-
oriented video coding methods tend to build auxiliary modules or particular-
optimized algorithms on the top of existing video coding standards and their
extensions [7, 8] to adapt the specific content characteristics of UAV videos.
The motion model capacity between learned codecs and conventional codecs may
be quite different. Indeed, the pixel-wise fine-grained motion is more
flexible for drone video compression to describe aerial motion compared with
the commonly used hybrid coding techniques like block-based motion-
compensation prediction., the pixel-wise fine-grained motion is more flexible
for the drone video compression to describe aerial motion. Moreover,
considering that the UAV videos have completely different field-of-view,
motion dynamics, and viewpoint distance against compared to natural videos,
the compression of variable-size block-wise motion representation in most
existing hybrid coding approaches will consume a large number of bits for
compression. Therefore, it is nontrivial to develop a new framework for UAV
videos.

In this paper, we propose to build a novel task named learning based UAV video
coding. Furthermore, we extensively collect the UAV videos with different
content variations, including in-door and out-door scenes, object-scale
variations and viewpoint distance, different climate condition etc. Then we
encode those properly-selected videos using popular end-to-end optimized video
codecs and conventional hybrid codecs, to form a comprehensive benchmark for
learned drone video compression. We also provide a detailed analysis and
envision the challenge of such task for future research. The main
contributions of this paper are three folds. First, we construct a
comprehensive benchmark for the task of drone video compression which consist
of the rate-distortion (R-D) behavior of both hybrid and learned video codecs.
To our knowledge, it is the first attempt in end-to-end optimized solution to
compress drone videos. Second, we provide the review and analysis of the
learned drone video compression schemes and further discuss the challenges of
encoding UAV videos. Third, this benchmark and related research is
accomplished as a milestone MPAI111https://mpai.community/ End-to-end Video
(EEV) coding project. The proposed benchmark has constructed a solid baseline
for compressing UAV videos and facilitates the future research works for
related task.

##  2 Learned UAV Video Coding Benchmark

Video Sequences Characteristics. Prior UAV related research mainly focuses on
high level computer vision tasks for on-device cameras, such as
single/multiple object detection, image/video object tracking, etc. Notably,
there are plenty of workshops and grand challenges that aim to provide large-
scale drone video datasets, with thousands of images frames and still images.
Given those high quality UAV captured data, we collect a set of video
sequences to build the UAV video coding benchmark from those diverse contents,
considering the record device type (various model of drone-mounted cameras),
diverse in many aspects including location (in-door and out-door places),
environment (traffic workload, urban and rural regions), objects (e.g.,
pedestrian and vehicles), and scene object density (sparse and crowded
scenes). For better understanding of those videos, we provide comprehensive
summary of the prepared learned drone video coding benchmark in Table 1. The
corresponding thumbnail of each clip is depicted in Fig. 1 as supplementary
information. There are 14 video clips from multiple different source UAV video
datasets [1, 9, 10]. The resolutions and frame rate of them range from
$2720\times 1520$ down to $640\times 352$, 24 to 30 respectively. The frame
count of each test video is 100.

Table 1: Video sequence characteristics of the proposed learned UAV video coding benchmark Source |  | Sequence
---
Name
| Spatial
---
Resolution
| Frame
---
Count
| Frame
---
Rate
| Bit
---
Depth
| Scene
---
Feature
Class A VisDrone-SOT TPAMI2021 [1] | BasketballGround | 960x528 | 100 | 24 | 8 | Outdoor
GrassLand | 1344x752 | 100 | 24 | 8 | Outdoor
Intersection | 1360x752 | 100 | 24 | 8 | Outdoor
NightMall | 1920x1072 | 100 | 30 | 8 | Outdoor
SoccerGround | 1904x1056 | 100 | 30 | 8 | Outdoor
Class B VisDrone-MOT TPAMI2021 [1] | Circle | 1360x752 | 100 | 24 | 8 | Outdoor
CrossBridge | 2720x1520 | 100 | 30 | 8 | Outdoor
Highway | 1344x752 | 100 | 24 | 8 | Outdoor
Class C Corridor IROS2018 [9] | Classroom | 640x352 | 100 | 24 | 8 | Indoor
Elevator | 640x352 | 100 | 24 | 8 | Indoor
Hall | 640x352 | 100 | 24 | 8 | Indoor
Class D UAVDT_S ECCV2018 [10] | Campus | 1024x528 | 100 | 24 | 8 | Outdoor
RoadByTheSea | 1024x528 | 100 | 24 | 8 | Outdoor
Theater | 1024x528 | 100 | 24 | 8 | Outdoor

The dataset contains various UAV videos captured under different conditions,
including environments, flight altitudes, and camera views, e.g., basketball
ground, highway, classroom and campus. These video clips are selected from
several categories of real-life objects in different scene object densities
and lighting conditions, representing diverse scenarios in our daily life.
Compared to natural videos, UAV-captured videos are generally recorded by
drone-mounted cameras in motion and at different viewpoints and altitudes.
These features bring several new challenges, such as motion blur, scale
changes and complex background. Heavy occlusion, non-rigid deformation and
tiny scales of objects might be of great challenge to drone video compression.

Different from natural videos, the UAV videos is not viewpoint and scale
invariant. Thus the predictive coding efficiency might be restricted due to
large or global motion. Moreover, the distortion characteristic (bird/fish eye
view) of the drone-equipped camera also makes UAV video distinct from natural
ones. Therefore, conventional hybrid video coding with block-based motion
compensation might not be ideal choice when compressing them. It is observed
from Fig. 1 that the in-door drone captured videos are highly distorted around
the view boundary and the deformation of objects also degrades the inter-
prediction efficacy, resulting more intra coded blocks after rate-distortion
optimization based mode decision during encoding procedure.

Figure 1: The snapshot of the UAV sequences with different content
characteristics.

Pre-processing. The drone videos usually have slight compression effect by the
on-device hardware codec. To overcome this issue and produce near pristine
video clips, we conduct a set of pre-processing procedures to those drone
videos including down-sampling, boundary cropping, and color space alignment
(in RGB domain). We further particular center-cropped the resolutions to be
multiple of 16 to be compatible with convolution operations in learned codecs.
Eventually we employ the RGB planar data structure to store the raw data. It
should also be pointed out that the processing color space is RGB in this
benchmark, including encoding, decoding, and quality assessment.

Codecs selection. To comprehensively reveal R-D efficiency of UAV video using
both conventional and learned codecs, we encode the above collected drone
video sequences using HEVC/H.265 reference software with screen content coding
(HEVC-SCC) extension reference software (HM-16.20-SCM-8.8) [11] and the
emerging learned video coding framework OpenDVC [12, 13]. Moreover, the
reference model of MPAI End-to-end Video (EEV) is also employed to compress
the UAV videos. As such, the baseline coding results are composed of three
different codecs.

The schematic diagrams of them are shown in Fig. 2. The left panel represents
the classical hybrid codec. The remaining two are learned codecs, OpenDVC and
EEV respectively. It is easy to observe that the EEV software is an enhanced
version of OpenDVC codec by incorporating more advanced modules such as motion
compensation prediction improvement, two-stage residual modeling, and in-loop
restoration network. Specifically, ${x_{t}}$ and $\hat{x}_{t}$ denote the un-
coded and restored image in both Fig. 2(b) and Fig. 2(c) while $\tilde{x}_{t}$
is the predicted frame. Analogously, ${v_{t}}$ and $\hat{v}_{t}$ correspond to
the pristine and restored optical flow. ${r_{t}}$ and $\hat{r}_{t}$ are the
uncompressed and reconstructed residual. Moreover, ${{r}^{{}^{\prime}}_{t}}$
and $\hat{r}^{{}^{\prime}}_{t}$ are coarse-to-fine (C2F) residual proposed by
EEV. Therefore, the reconstruction of learned codecs for OpenDVC and EEV can
be represented as $\hat{x}_{t}$=$\tilde{x}_{t}$+$\hat{r}_{t}$ and
$\hat{x}_{t}$=$\tilde{x}_{t}$+$\hat{r}_{t}$+$\hat{r}^{{}^{\prime}}_{t}$
respectively.

Another important factor for learned codecs is train-and-test data
consistency. It is widely accepted in machine learning community that train
and test data should be independent identically distributed. However, both
OpenDVC and EEV are trained using natural video dataset vimeo-90k [14] with
mean-square-error (MSE) as distortion metrics. We employ those pre-trained
weights of learned codecs without fine-tuning them on drone video data to
guarantee the generalisation of the benchmark. By doing so, the effectiveness
and drawbacks of learned and conventional codecs could be directly observed
when comparing the coding performances.

Figure 2: Block diagram of different codecs. (a) Conventional hybrid codec
HEVC. (b) OpenDVC. (3) MPAI EEV. Zoom-in for better visualization.

Parameter Setting. We follow the common test condition (CTC) to utilize
lowdelay-P configuration for HEVC codec. The main-RExt profile is utilized. We
set the $InputColourSpaceConvert$ parameter as $RGBtoGBR$ when encoding. When
decoding, we set the $OutputInternalColourSpace$ as $GBRtoRGB$ to ensure the
consistency of reconstruction results and decoded results. The internal bit-
depth is set to be 8 because the drone-captured videos are all using standard-
dynamic-range (SDR). The quantization parameters (QPs) are set to be
{30,34,38,42} to realize four different bit-rate points. Regarding the OpenDVC
and EEV software, the variable-rate coding is realized by training multiple
models using the hyper-parameter $\lambda$={2048,1024,512,256}, whose
functionality is to balance the trade-off between coding bits and distortion.
It is worth noting that the intra period (IP) number is set to be 16 for all
codecs for reasonable comparison. Note that when evaluating learned codecs,
the $mode$ and $metric$ parameter are set to be PSNR. $seq\\_wid$ and
$seq\\_hgt$ indicate the resolution of each sequence. In summary, the command
line settings for HM-16.20-SCC-8.8, OpenDVC, and EEV are listed below.

  * •

TAppEncoder -c encoder_LDP.cfg -InputBitDepth 8 -InputChromaFormat 444 -Level
6.2 -wdt seq_wid -hgt seq_hgt -f 100 -fr fps -q QP -IntraPeriod 16
-InputColourSpaceConvert RGBtoGBR -SNRInternalColourSpace 1
-OutputColourSpaceConvert GBRtoRGB

  * •

python test_opendvc.py -path seqname -mode PSNR -IntraPeriod 16 -metric PSNR
-l $\lambda$

  * •

python test_eev.py -path seqname -mode PSNR -IntraPeriod 16 -metric PSNR -l
$\lambda$

Evaluation Method. Since all drone videos in our proposed benchmark are using
RGB color space, the quality assessment methods are also applied to the
reconstruction in RGB domain. For each frame, the peak-signal-noise-ratio
(PSNR) values of $x_{t}$ and $\hat{x}_{t}$ are calculated for each component
channel then the RGB averaged value is obtained to indicate its picture
quality. Regarding bitrate, we calculate bit-per-pixel (BPP) using the binary
files produced by codecs. We report the coding efficiency of different codecs
using the Bjøntegaard delta bit rate (BD-rate) measurement [15].

##  3 R-D Results and Performance Analysis

Coding Performance The corresponding PSNR based R-D performances of the three
different codecs are shown in Table 2. Regarding the simulation results, it is
observed that around 20% bit-rate reduction could be achieved when comparing
EEV and OpenDVC codec. This shows promising performances for the learned
codecs and its improvement made by EEV software. By comparatively studying the
coding gain of different UAV videos, it is observed that the R-D efficiency is
consistent among diverse kinds of contents. This point corresponds to our
previous assumption that learning based codecs have partial generalisation
ability even without fine-tuning on drone video datasets. Some minor
turbulence is perceived in Class C, which confirms that in-door UAV content
has different texture characteristics with out-door ones.

Table 2: The BD-rate performance of different codecs (OpenDVC, EEV, and HM-16.20-SCM-8.8) on drone video compression. The distortion metric is RGB-PSNR. Category |  | Sequence
---
Name
| BD-Rate Reduction
---
EEV vs OpenDVC
| BD-Rate Reduction
---
EEV vs HEVC
Class A VisDrone-SOT | BasketballGround | -23.84% | 9.57%
GrassLand | -16.42% | -38.64%
Intersection | -18.62% | -28.52%
NightMall | -21.94% | -6.51%
SoccerGround | -21.61% | -10.76%
Class B VisDrone-MOT | Circle | -20.17% | -25.67%
CrossBridge | -23.96% | 26.66%
Highway | -20.30% | -12.57%
Class C Corridor | Classroom | -8.39% | 178.49%
Elevator | -19.47% | 109.54%
Hall | -15.37% | 58.66%
Class D UAVDT_S | Campus | -26.94% | -25.68%
RoadByTheSea | -20.98% | -24.40%
Theater | -19.79% | 2.98%
Class A | -20.49% | -14.97%
Class B | -21.48% | -3.86%
Class C | -14.41% | 115.56%
Class D | -22.57% | -15.70%
Average | -19.84% | 15.23%

When we directly compare the coding performance of EEV and HEVC, obvious
performance gap between the in-door and out-door sequences could be observed.
Generally speaking, the HEVC SCC codec outperforms the learned codec by 15.23%
over all videos. Regarding Class C, EEV is significantly inferior to HEVC by
clear margin, especially for the $Classroom$ and $elevator$ sequences. The
reason for such results might be domain shift of the train-test data, such as
content difference (natural scene for training and in-door drone video for
test) and imaging type change (in-door drone videos are highly distorted
around the boundary). However, the EEV codec has better coding efficiency than
HEVC in Class A, Class B and Class D by 14.97%, 3.86% and 15.70% respectively,
which shows great potentials for the learning based UAV video codec. It is
additionally observed that the coding performances are not consistency in
different drone sequences within a certain class. EEV outperforms HEVC by over
38% in $GrassLand$ but less performed in $BasketballGround$ by 9.5%. Such R-D
statistics reveal that learned codecs are more sensitive to the video content
variations than conventional hybrid codecs if we directly apply natural-video-
trained codec to UAV video coding. For future research, this point could be
resolved and modeled as an out-of-distribution problem and extensive knowledge
could be borrowed from the machine learning community.

R-D Curves and Analysis To further dive into the R-D efficiency interpretation
of different codecs, we plot the R-D curves of different methods in Fig. 3.
Specifically, we select one test video clip from each class to illustrate the
diversity and R-D behavior. The R-D curves of $Camplus$, $Highway$,
$NightMall$, $Elevator$, $SoccerGround$, and $Intersection$ are depicted. The
blue-violet, peach-puff, and steel-blue curves denote EEV, HEVC and OpenDVC
codec respectively. Given the BD-rate performances and R-D curves, interesting
observations and phenomenon could be perceived. It is observed from those
curves that the EEV consistently outperforms the OpenDVC for all bit-rate
points and all sequences. However, when comparing with HM-16.20-SCM-8.8 codec,
the EEV codec might be less effective is some cases (especially for bpp less
than 0.1 scenarios). Regarding the in-door test video, the conventional codec
performs much better than the remaining two learned codecs. As aforementioned,
the videos capture by in-door-drone-equipped cameras are highly distorted at
boundary area due to fish-eye imaging effect. The learned codecs are optimized
using natural videos, resulting domain gap. Therefore, the content
characteristic of UAV videos and its distance to the natural videos shall be
modeled and investigated in future research.

(a) Campus 1024$\times$528 (b) Highway 1344$\times$752 (c) NightMall
1920$\times$1072 (d) Elevator 640$\times$352

(e) SoccerGround 1904$\times$1056 (f) Intersection 1360$\times$752

Figure 3: R-D curves of different codecs (OpenDVC [13], EEV, and
HM-16.20-SCM-8.8 [11]) for difference drone video sequences. The blue-violet,
peach-puff, and steel-blue curves denote EEV, HEVC and OpenDVC codec
respectively.

Classical codecs are designed to be content independent by using crafted rules
and non-parametric models, such that different contents are treated equal
during deployment, which guarantees the robustness of them. For data-trained
codecs, the generalisation ability relies on both of the model-designation
principles and the training data as well as their optimization techniques,
which hints the versatility of learned codecs. For practical utility, on-line
model updating might be a critical factor for learned codecs to overcome the
unpredictable domain shift between the distribution of training data and
deployment environment.

Visual Quality Evaluation The subjective quality of reconstructed images of
$RoadByTheSea$ and $NightMall$ sequences for different codecs
(HM-16.20-SCM-8.8 [11], OpenDVC [13], and EEV) are shown in Fig. 4. The first
row represents the reconstructed frames of the 61st frame of $RoadByTheSea$
sequence. The test parameter setting for this set of visual comparison (Fig.
4(a)-(c)) is QP=42 in HEVC-SCC, $\lambda$=256 for the two learned codecs. It
is observed that the subjective quality of EEV codec is much better than the
remaining two codecs, which shows the superiority of the learned codec.
Specifically, textures of the EEV-coded picture are more visual pleasant, in
which the road-lanes and other salient objects remain crystal clear while the
conventional codec fails to restore such information. The blurring artifact
also degrades the visual quality HEVC-SCC-coded picture. In this regard, the
learned codec shows promising potentials for the autonomous drone video
analysis or other downstream drone-vision tasks. The second row depicts the
decoded pictures of the 23nd frame of $NightMall$ sequence. The test parameter
setting for this set of comparison (Fig. 4(d)-(f)) is QP=38 in HEVC-SCC,
$\lambda$=512 for the two learned codecs. Surprisingly, the learned codec EEV
obtains around 1dB PSNR improvement over HEVC-SCC when using similar bit-rate
cost. The textures of pedestrians and bicycles can be kept after EEV
compression. Given the subjective quality comparisons of all images in Fig. 4,
one can further conclude that the learned codecs are able to handle the UAV
videos under different flying conditions (height, day/night time) even without
fine-tuning on related datasets.

(a) HEVC 0.065bpp 25.20dB

(b) OpenDVC 0.078bpp 25.99dB

(c) EEV 0.077bpp 26.44dB

(d) HEVC 0.055bpp 32.30dB

(e) OpenDVC 0.063bpp 32.84dB

(f) EEV 0.058bpp 33.27dB

Figure 4: Subjective quality comparison of reconstructed images of
$RoadByTheSea$ and $NightMall$ for different codecs (HM-16.20-SCM-8.8 [11],
OpenDVC [13], and EEV). Zoom-in for better visualization.

##  4 Working Mechanism of MPAI-EEV

This work was accomplished in the MPAI-EEV coding project, which is an MPAI
standard project seeking to compress video by exploiting AI-based data coding
technologies1. Within this workgroup, experts around the globe gather and
review the progress, and plan new efforts every two weeks. In its current
phase, attendance at MPAI-EEV meetings is open to interested
experts222http://eev.mpai.community. Since its formal establishment in Nov.
2021, the MPAI EEV has released three major versions of it reference models.
MPAI-EEV plans on being an asset for AI-based end-to-end video coding by
continuing to contribute new development in the end-to-end video coding field.

##  5 Conclusion

In this paper, we build the first benchmark for the task of learning based UAV
video coding, which consists of diverse UAV video contents associated with the
R-D characteristics of classical and emerging learning based video codecs. It
is observed that learned codecs outperform the conventional hybrid codecs for
drone video compression since learned codecs have better model capacity in
capturing the motion blur recorded by the drone-mounted cameras. The proposed
benchmark has constructed a solid baseline for compressing UAV videos and
facilitates the future research works for related topics such as downstream
drone video analysis tasks.

##  6 References

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