###### Abstract

In this paper we discuss when a quasi-conformally flat weakly Ricci symmetric
manifold (of dimension greater than 3) becomes a manifold of hyper quasi-
constant curvature, a quasi-Einstein manifold and a manifold of quasi-constant
curvature. Also we discuss when a pseudo projectively flat weakly Ricci
symmetric manifold (of dimension greater than 3) becomes pseudo-quasi constant
curvature and a quasi-Einstein manifold, and when a $W_{2}$-flat weakly Ricci
symmetric manifold (of dimension greater than 3) becomes a quasi-Einstein
manifold.

SOME TYPES OF WEAKLY RICCI SYMMETRIC RIEMANNIAN MANIFOLDS

Payel Karmakar1, Arindam Bhattacharyya2

1,2Department of Mathematics,

Jadavpur University, Kolkata-700032, India.

E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
Keywords: Weakly Ricci symmetric(WRS) manifold, quasi-conformal curvature
tensor, hyper quasi-constant curvature tensor, pseudo projective curvature
tensor, pseudo quasi-constant curvature, $W_{2}$-curvature tensor.

Mathematics Subject Classification[2010]: 53C20, 53C25.

##  1 Introduction

L.Tamassy and T.Q.Binh[11] introduced weakly symmetric Riemannian manifold. A
non-flat Riemannian manifold $(M^{n},g)(n>2)$ is called weakly symmetric if
the curvature tensor $\bar{R}$ of type (0,4) satisfies the condition
$(\nabla_{X}\bar{R})(Y,Z,U,V)=A(X)\bar{R}(Y,Z,U,V)+B(Y)\bar{R}(X,Z,U,V)+C(Z)\bar{R}(Y,X,U,V)\\\
+D(U)\bar{R}(Y,Z,X,V)+E(V)\bar{R}(Y,Z,U,X),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(1.1)
$\forall$vector fields $X,Y,Z,U,V\in\chi(M^{n})$, where $A,B,C,D$ and $E$ are
1-forms (non-zero simultaneously) and $\nabla$ is the operator of covariant
differentiation with respect to $g$. The 1-forms are called the associated
1-forms of the manifold and an n-dimensional manifold of this kind is denoted
by $(WS)_{n}$. If the associated 1-forms satisfy $B=C$ and $D=E$, the defining
condition of a $(WS)_{n}$ reduces to the following form[4]
$(\nabla_{X}\bar{R})(Y,Z,U,V)=A(X)\bar{R}(Y,Z,U,V)+B(Y)\bar{R}(X,Z,U,V)+B(Z)\bar{R}(Y,X,U,V)\\\
+D(U)\bar{R}(Y,Z,X,V)+D(V)\bar{R}(Y,Z,U,X).~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(1.2)

Let $\\{e_{i}\\}~{}~{}i=1,2,...,n$ be an orthonormal basis of the tangent
spaces in a neighbourhood of a point of the manifold. Then setting $Y=V=e_{i}$
in (1.7) and taking summation over $i,~{}~{}1\leq i\leq n$, we get
$(\nabla_{X}S)(Z,U)=A(X)S(Z,U)+B(Z)S(X,U)+D(U)S(X,Z)+B(R(X,Z)U)+D(R(X,U)Z),$
where $S$ is the Ricci tensor of type (0,2). (1.3)

A non-flat Riemannian manifold $(M^{n},g)(n>2)$ is called weakly Ricci-
symmetric if the Ricci tensor $S$ of type (0,2) satisfies the condition
$(\nabla_{X}S)(Z,U)=A(X)S(Z,U)+B(Z)S(X,U)+D(U)S(X,Z)~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(1.4)

From (1.3) it follows that a $(WS)_{n}(n>2)$ is weakly Ricci symmetric
(briefly $(WRS)_{n}(n>2))$ if [12]
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}B(R(X,Z)U)+D(R(X,U)Z)=0,~{}~{}\forall
X,U,Z\in\chi(M^{n}).~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(1.5)

Putting $Z=U=e_{i}$ in (1.4) we get,
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}dr(X)=rA(X)+B(QX)+D(QX),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(1.6)
where $r$ is the scalar curvature of the manifold.

If a $(WRS)_{n}(n>2)$ is of zero scalar curvature, then from (1.6),
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}B(QX)+D(QX)=0,~{}~{}\forall
X.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(1.7)

From (1.6) we have if a $(WRS)_{n}(n>2)$ is of non-zero constant scalar
curvature, then the 1-form $A$ can be expressed as
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}A(X)=-\frac{1}{r}[B(QX)+D(QX)],~{}~{}\forall
X.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(1.8)

From (1.4) we obtain,
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}T(QX)=rT(X),~{}~{}\forall
X.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(1.9)
where the vector field $\rho$ is defined by
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}T(X)=g(X,\rho)=B(X)-D(X),~{}~{}\forall
X.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(1.10)

Also from (1.6) we obtain,
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}T(Z)S(X,U)-T(U)S(X,Z)=0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(1.11)
$\forall$vector fields $X,Z,U$ and $T$ is a 1-form.

B.Das and A.Bhattacharyya studied some types of weakly symmetric Riemannian
manifolds in 2011[3]. In 2000, U.C.De et al. discussed about weakly symmetric
and weakly Ricci symmetric K-contact manifolds[5] and in 2012, A.A.Shaikh and
H.Kundu discussed about weakly symmetric and weakly Ricci symmetric warped
product manifolds[10]. Various types of works on weakly Ricci symmetric
manifolds is done by S.Jana, A.Shaikh[7] and U.C.De and G.C.Ghosh[6].
Motivated from their work we have established some new results on weakly Ricci
symmetric manifolds. In this paper we study quasi-conformally flat, pseudo
projectively flat and $W_{2}$-flat weakly Ricci symmetric manifolds. Here we
prove a quasi-conformally flat $(WRS)_{n}(n>3)$ of non-zero constant scalar
curvature is a manifold of hyper quasi-constant curvature and this manifold of
non-vanishing scalar curvature is a quasi-Einstein manifold and a manifold of
quasi-constant curvature with respect to the 1-form $T$ defined by
$T(X)=B(X)-D(X)\neq 0~{}~{}\forall X$, where $B,D$ are 1-forms(non-zero
simultaneously). Also we obtain that a pseudo projectively flat
$(WRS)_{n}(n>3)$ with non-zero constant scalar curvature is a manifold of
pseudo quasi-constant curvature and with non-vanishing scalar curvature is a
quasi-Einstein manifold with respect to $T$. In the last section we prove that
a $W_{2}$-flat $(WRS)_{n}(n>3)$ of non-vanishing scalar curvature is a quasi-
Einstein manifold with respect to $T$.

A quasi-conformal curvature tensor $C^{*}$ is defined by[14]
$C^{*}(X,Y)Z=aR(X,Y)Z+b[S(Y,Z)X-S(X,Z)Y+g(Y,Z)QX-g(X,Z)QY]\\\
-\frac{\gamma}{n}[\frac{a}{n-1}+2b][g(Y,Z)X-g(X,Z)Y],~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(1.12)
where $a,b$ are constants, $g$ is the Riemannian metric and $R,Q,\gamma$ are
the Riemannian curvature tensor of type (1,3), the Ricci operator defined by
$g(QX,Y)=S(X,Y)$ and the scalar curvature respectively.

Chen and Yano[2] introduced a Riemannian manifold $(M^{n},g)(n>3)$ of quasi-
constant curvature which is conformally flat and its curvature tensor
$\bar{R}$ of type (0,4) is of the form
$\bar{R}(X,Y,Z,W)=a[g(Y,Z)g(X,W)-g(X,Z)g(Y,W)]+b[g(X,W)A(Y)A(Z)-g(X,Z)A(Y)A(W)\\\
+g(Y,Z)A(X)A(W)-g(Y,W)A(X)A(Z)],~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(1.13)
where $a,b$ are non-zero scalars.

A Riemannian manifold $(M^{n},g)(n>3)$ is said to be of hyper quasi-constant
curvature if it is conformally flat and its curvature tensor $\bar{R}$ of type
(0,4) satisfies the condition[9]
$\bar{R}(X,Y,Z,W)=a[g(Y,Z)g(X,W)-g(X,Z)g(Y,W)]+g(X,W)P(Y,Z)-g(X,Z)P(Y,W)\\\
+g(Y,Z)P(X,W)-g(Y,W)P(X,Z),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(1.14)
where $a$ is a non-zero scalar and $P$ is a tensor of type (0,2).

A pseudo projective curvature tensor $\bar{P}$ is defined by[13]
$\bar{P}(X,Y)Z=aR(X,Y)Z+b[S(Y,Z)X-S(X,Z)Y]-\frac{r}{n}[\frac{a}{n-1}+b][g(Y,Z)X-g(X,Z)Y],~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(1.15)
where $a,b$ are constants such that $a,b\neq 0$; $R,S,r$ are the Riemannian
curvature tensor, the Ricci tensor and scalar curvature respectively.

A Riemannian manifold $(M^{n},g)(n>3)$ is said to be of pseudo quasi-constant
curvature if it is pseudo projectively flat and its curvature tensor $\bar{R}$
of type (0,4) satisfies[3]
$\bar{R}(X,Y,Z,W)=a[g(Y,Z)g(X,W)-g(X,Z)g(Y,W)]+P(Y,Z)g(X,W)-P(X,Z)g(Y,W),~{}~{}~{}~{}~{}$(1.16)
where $a$ is a constant and $P$ is a tensor of type (0,2).

Pokhariyal and Mishra have defined the $W_{2}$-curvature tensor on a
differential manifold of dimension $n$ is given by[8]
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}W_{2}(X,Y)Z=R(X,Y)Z+\frac{1}{n-1}\\{g(X,Z)QY-g(Y,Z)QX\\}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(1.17)

All these notions will be required in the next sections.

##  2 Quasi-conformally flat $(WRS)_{n}(n>3)$

In this section we prove that a quasi-conformally flat weakly Ricci symmetric
manifold $(WRS)_{n}\\\ (n>3)$ of non-zero constant scalar curvature is a
manifold of hyper quasi constant curvature and this manifold of non-vanishing
scalar curvature is a quasi-Einstein manifold and a manifold of quasi-constant
curvature with respect to the 1-form $T$ defined by $T(X)=B(X)-D(X)\neq
0~{}~{}\forall X$, where $B,D$ are 1-forms(non-zero simultaneously).

Let $(M^{n},g)(n>3)$ be a quasi-conformally flat $(WRS)_{n}$. Then from (1.12)
we obtain,
$\bar{R}(X,Y,Z,U)=-\frac{b}{a}[S(Y,Z)g(X,U)-S(X,Z)g(Y,U)+g(Y,Z)S(X,U)-g(X,Z)S(Y,U)]\\\
+\frac{\gamma}{n}[\frac{1}{n-1}+\frac{2b}{a}][g(Y,Z)g(X,U)-g(X,Z)g(Y,U)],~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(2.1)
where $g(R(X,Y)Z,U)=\bar{R}(X,Y,Z,U)$.

Putting $X=U=e_{i}$ in (2.1) where $\\{e_{i}\\}$ is an orthonormal basis of
the tangent space at each point of the manifold and taking the summation over
$i$, where $1\leq i\leq n$, we get
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}S(Y,Z)=\alpha
g(Y,Z),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(2.2)
where
$\alpha=\frac{\gamma}{1+\frac{b}{a}(n-2)}[-\frac{b}{a}+\frac{1}{n}(1+\frac{2b(n-1)}{a})].$

From (2.2) we get
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\nabla_{X}S)(Y,Z)=\alpha^{\prime}d\gamma(X)g(Y,Z),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(2.3)
where
$\alpha^{\prime}=\frac{1}{1+\frac{b}{a}(n-2)}[-\frac{b}{a}+\frac{1}{n}(1+\frac{2b(n-1)}{a})].$

Using (2.3) and (1.4) we get,
$(\nabla_{X}S)(Y,Z)-(\nabla_{Z}S)(Y,X)\\\
=S(Y,Z)[A(X)-D(X)]-S(X,Y)[A(Z)-D(Z)]\\\
=\alpha^{\prime}[d\gamma(X)g(Y,Z)-d\gamma(Z)g(Y,X)].~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(2.4)

Let $\rho_{1},\rho_{2},\rho_{3}$ be the associated vector fields corresponding
to the 1-forms $A,B,D$ respectively, i.e.,
$g(X,\rho_{1})=A(X),g(X,\rho_{2})=B(X),g(X,\rho_{3})=D(X).$

Putting $Z=\rho_{2}$ in (2.4) we get,
$[A(X)-D(X)]B(QY)-S(X,Y)[A(\rho_{2})-D(\rho_{2})]\\\
=\alpha^{\prime}[d\gamma(X)B(Y)-d\gamma(\rho_{2})g(Y,X)]\\\
=\alpha^{\prime}[B(Y)\\{\gamma A(X)+B(QX)+D(QX)\\}-g(X,Y)\\{\gamma
A(\rho_{2})+B(Q\rho_{2})+D(Q\rho_{2})\\}].~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(2.5)

If the manifold has non-zero constant scalar curvature, then by virtue of
(1.8), (2.5) yields that
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}[A(X)-D(X)]B(QY)-S(X,Y)[A(\rho_{2})-D(\rho_{2})]=0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(2.6)
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\Rightarrow
S(X,Y)=\alpha_{1}A(X)\bar{B}(Y)+\alpha_{2}D(X)\bar{B}(Y),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(2.7)
where $\bar{B}(Y)=B(QY)$ and
$\alpha_{1}=\frac{1}{A(\rho_{2})-D(\rho_{2})},\alpha_{2}=-\frac{1}{A(\rho_{2})-D(\rho_{2})}.$

This leads to the following theorem:

Theorem 2.1: In a quasi-conformally flat $(WRS)_{n}(n>3)$ of non-zero constant
scalar curvature, the Ricci tensor $S$ has the form (2.7).

Again using (1.8) in (2.7) we get,
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}S(X,Z)=\alpha_{1}\bar{B}(Z).-\frac{1}{\gamma}[\bar{B}(X)+\bar{D}(X)]+\alpha_{2}D(X)\bar{B}(Z),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(2.8)
where $\bar{D}(X)=D(QX).$

Using (2.8) in (2.1) we obtain,
$\bar{R}(X,Y,Z,W)=\alpha[g(Y,Z)g(X,W)-g(X,Z)g(Y,W)]+g(X,W)P(Y,Z)-g(X,Z)P(Y,W)\\\
+g(Y,Z)P(X,W)-g(Y,W)P(X,Z),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(2.9)
where $\alpha$ is a non-zero scalar and
$P(Y,Z)=-\frac{b}{a}[-\frac{\alpha_{1}}{\gamma}\bar{B}(Y)\bar{B}(Z)+\alpha_{2}D(Y)\bar{B}(Z)-\frac{\alpha_{1}}{\gamma}\bar{B}(Z)\bar{D}(Y)].$

Hence comparing (2.9) with (1.14) we can state the following theorem:

Theorem 2.2: A quasi-conformally flat $(WRS)_{n}(n>3)$ of non-zero constant
scalar curvature is a manifold of hyper quasi-constant curvature.

Now putting $U=\rho$ in (1.11) and then using (1.19) we get,
$T(Z)S(X,\rho)-T(\rho)S(X,Z)=0$
$\Rightarrow T(Z)T(QX)-T(\rho)S(X,Z)=0$
$\Rightarrow\gamma
T(Z)T(X)-T(\rho)S(X,Z)=0.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(2.10)

Let us suppose that a $(WRS)_{n}(n>3)$ is quasi-conformally flat and of non-
zero scalar curvature.

From (2.10) we have,
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}S(X,Z)=\tilde{\alpha}T(X)T(Z),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(2.11)
where $\tilde{\alpha}=\frac{\gamma}{T(\rho)}.$

Again, a Riemannian manifold is called quasi-Einstein if its Ricci tensor is
of the form[1]$-$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}S=pg+q\omega\otimes\omega,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(2.12)
where $p,q$ are scalars of which $q\neq 0$ and $\omega$ is a 1-form.

Comparing (2.11) and (2.12) we can state the following theorem:

Theorem 2.3: A quasi-conformally flat $(WRS)_{n}(n>3)$ of non-vanishing scalar
curvature is a quasi-Einstein manifold with respect to the 1-form $T$ defined
by $T(X)=B(X)-D(X)\neq 0~{}~{}\forall X$.

Again using (2.11) in (2.1) we obtain,
$\bar{R}(X,Y,Z,W)=l[g(Y,Z)g(X,W)-g(X,Z)g(Y,W)]+\delta[g(X,W)T(Y)T(Z)-g(X,Z)T(Y)T(W)\\\
+g(Y,Z)T(X)T(W)-g(Y,W)T(X)T(Z)],~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(2.13)
where $l=\frac{\gamma}{n}[\frac{1}{n-1}+\frac{2b}{a}]$ and
$\delta=-\frac{b}{a}\tilde{\alpha.}$

Hence comparing (2.13) with (1.13) we have the following theorem:

Theorem 2.4: A quasi-conformally flat $(WRS)_{n}(n>3)$ of non-vanishing scalar
curvature is a manifold of quasi-constant curvature with respect to the 1-form
$T$ defined by $T(X)=B(X)-D(X)\neq 0~{}~{}\forall X$.

Using the expression of $T$ in (2.13) we have,
$\bar{R}(X,Y,Z,W)=l[g(Y,Z)g(X,W)-g(X,Z)g(Y,W)]+g(X,Z)\\{\delta
BD\\}(Y,Z)-g(X,Z)\\{\delta BD\\}(Y,W)\\\ +g(Y,Z)\\{\delta
BD\\}(X,W)-g(Y,W)\\{\delta BD\\}(X,Z),$
where $\\{\delta BD\\}=\delta(BB-BD-DB+DD).$

Comparing the above relation with (1.14) we can state:

Corollary 2.1: A quasi-conformally flat $(WRS)_{n}(n>3)$ of non-zero scalar
curvature is a manifold of hyper quasi-constant curvature.

##  3 Pseudo-projectively flat $(WRS)_{n}(n>3)$

In this section we obtain that a pseudo projectively flat $(WRS)_{n}(n>3)$
with non-zero constant scalar curvature is a manifold of pseudo-quasi constant
curvature and with non-vanishing scalar curvature is a quasi-Einstein manifold
with respect to the 1-form $T$ defined by $T(X)=B(X)-D(X)\neq 0~{}~{}\forall
X$, where $B,D$ are 1-forms(non-zero simultaneously).

Let $(M^{n},g)(n>3)$ be a pesudo-projectively flat $(WRS)_{n}$. Then from
(1.15) we obtain,
$\bar{R}(X,Y,Z,W)=-\frac{b}{a}[S(Y,Z)g(X,U)-S(X,Z)g(Y,U)]+\frac{r}{an}[\frac{a}{n-1}+b][g(Y,Z)g(X,U)\\\
-g(X,Z)g(Y,U)].~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(3.1)

Putting $X=U=e_{i}$ in (3.1) where $\\{e_{i}\\}$ is an orthonormal basis of
the tangent space at each point of the manifold and taking summation over $i$,
$1\leq i\leq n$, we get,
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}S(Y,Z)=\alpha
g(Y,Z),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(3.2)
where $\alpha=\frac{r}{n}.$

(3.2) indicates that a pseudo-projectively flat manifold is an Einstein
manifold.

Now from (3.2) we get,
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\nabla_{X}S)(Y,Z)=\alpha_{1}dr(X)g(Y,Z),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(3.3)
where $\alpha_{1}=\frac{1}{n}.$

Similarly
$(\nabla_{Z}S)(Y,X)=\alpha_{1}dr(Z)g(Y,X).~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(3.4)

Subtracting (3.4) from (3.3) we have,
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\nabla_{X}S)(Y,Z)-(\nabla_{Z}S)(Y,X)=\alpha_{1}[dr(X)g(Y,Z)-dr(Z)g(Y,X)].~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(3.5)

Interchanging $X,U$ in (1.4) and then subtracting the resultant from (1.4) we
obtain by virtue of (3.5) we obtain,
$[A(X)-D(X)]S(U,Z)-[A(U)-D(U)]S(X,Z)=\alpha_{1}[dr(X)g(Z,U)-dr(U)g(Z,X)].~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(3.6)

Let $\rho_{1},\rho_{2},\rho_{3}$ be the associated vector fields corresponding
to the 1-forms $A,B,D$ respectively,
i.e., $g(X,\rho_{1})=A(X),g(X,\rho_{2})=B(X),g(X,\rho_{3})=D(X).$

Substituting $U$ by $\rho_{2}$ in (3.6) and then using (1.6) we get,
$[A(X)-D(X)]B(QZ)-[A(\rho_{2})-D(\rho_{2})]S(X,Z)=\alpha_{1}[B(Z)\\{rA(X)+B(QX)+D(QX)\\}\\\
-g(X,Z)\\{rA(\rho_{2})+B(Q\rho_{2})+D(Q\rho_{2})\\}].~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(3.7)

If the manifold has non-zero constant scalar curvature, then by virtue of
(1.8), (3.7) yields that
$[A(\rho_{2})-D(\rho_{2})]S(X,Z)-[A(X)-D(X)]B(QZ)=0~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(3.8)
$\Rightarrow
S(X,Z)=\alpha_{2}A(X)\bar{B}(Z)+\alpha_{3}D(X)\bar{B}(Z),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(3.9)
where
$\bar{B}(Z)=B(QZ),\alpha_{2}=\frac{1}{A(\rho_{2})-D(\rho_{2})},\alpha_{3}=-\frac{1}{A(\rho_{2})-D(\rho_{2})}.$

This leads to the following:

Theorem 3.1: In a pseudo-projectively flat $(WRS)_{n}(n>3)$ of non-zero
constant scalar curvature, the Ricci tensor $S$ has the form (3.9).

Again using (1.8) in (3.9) we obtain,
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}S(X,Z)=-\frac{\alpha_{2}}{r}\bar{B}(Z)[\bar{B}(X)+\bar{D}(X)]+\alpha_{3}D(X)\bar{B}(Z).~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(3.10)

Using (3.10) in (3.1) we have,
$\bar{R}(X,Y,Z,W)=\alpha[g(Y,Z)g(X,W)-g(X,Z)g(Y,W)]+g(X,W)P(Y,Z)-g(Y,W)P(X,Z),~{}~{}~{}$(3.11)
where $\alpha=\frac{r}{an}[\frac{a}{n-1}+b]$ and
$P(Y,Z)=-\frac{b}{a}[-\frac{\alpha_{2}}{r}\bar{B}(Z)\\{\bar{B}(Y)+\bar{D}(Y)\\}+\alpha_{3}D(Y)\bar{B}(Z)].$

Comparing (3.11) with (1.16) we can state the following theorem:

Theorem 3.2: A pseudo-projectively flat $(WRS)_{n}(n>3)$ of non-zero constant
scalar curvature is a manifold of pseudo quasi-constant curvature.

Now putting $U=\rho$ in (1.11) and then using (1.9) we get,
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}rT(X)T(Z)-T(\rho)S(X,Z)=0.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(3.12)

From (3.12) we have,
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}S(X,Z)=\frac{r}{T(\rho)}T(X)T(Z).~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(3.13)

Now $T(\rho)\neq 0$ for if $T(\rho)=0$, then (3.13) implies that $rT(X)T(Z)=0$
which implies that $r=0$ (as $T(X)\neq 0~{}~{}\forall X$), which is a
contradiction.

Hence
$S(X,Z)=\tilde{\alpha_{1}}T(X)T(Z),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(3.14)
where $\tilde{\alpha_{1}}$ is a non-zero scalar.

Hence comparing (3.14) with (2.12) we can state that:

Theorem 3.3: A pseudo-projectively flat $(WRS)_{n}(n>3)$ of non-vanishing
scalar curvature is a quasi-Einstein manifold with respect to the 1-form $T$
defined by $T(X)=B(X)-D(X)\neq 0~{}~{}\forall X.$

Again using (3.14) in (3.1) we obtain,
$\bar{R}(X,Y,Z,W)=\gamma_{1}[g(Y,Z)g(X,W)-g(X,Z)g(Y,W)]+\delta_{1}[T(Y)T(Z)g(X,W)\\\
-T(X)T(Z)g(Y,W)],~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(3.15)
where $\gamma_{1}=\frac{r}{an}[\frac{a}{n-1}+b]$ and
$\delta_{1}=-\frac{b}{a}\tilde{\alpha_{1}}.$

Comparing (3.15) with (1.16) we have the following theorem:

Theorem 3.4: A pseudo-projectively flat $(WRS)_{n}(n>3)$ of non-vanishing
scalar curvature is a manifold of pseudo quasi-constant curvature with respect
to 1-form $T$ defined by $T(X)=B(X)-D(X)\neq 0~{}~{}\forall X$.

Using the expression of $T$ in (3.15), it can be easily seen that
$\bar{R}(X,Y,Z,W)=\gamma_{1}[g(Y,Z)g(X,W)-g(X,Z)g(Y,W)]+\\{\delta_{1}BD\\}(Y,Z)g(X,W)\\\
-\\{\delta_{1}BD\\}(X,Z)g(Y,W),$
where $\\{\delta_{1}BD\\}=\delta_{1}(BB-BD-DB+DD)$.

Comparing the above relation with (1.16) we can state:

Corollary 3.1: A pseudo-projectively flat $(WRS)_{n}(n>3)$ of non-zero scalar
curvature is a manifold of pseudo quasi-constant curvature.

##  4 $W_{2}$-flat $(WRS)_{n}(n>3)$

In this section we prove that a $W_{2}$-flat $(WRS)_{n}(n>3)$ of non-vanishing
scalar curvature is a quasi-Einstein manifold with respect to the 1-form $T$
defined by $T(X)=B(X)-D(X)\neq 0~{}~{}\forall X$, where $B,D$ are 1-forms(non-
zero simultaneously).

Let $(M^{n},g)(n>3)$ be a $W_{2}$-flat $(WRS)_{n}$. Then from (1.17) we have,
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\bar{R}(X,Y,Z,W)=\frac{1}{n-1}[g(Y,Z)S(X,W)-g(X,Z)S(Y,W)].~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(4.1)

Putting $X=W=e_{i}$ in (4.1), where $\\{e_{i}\\}$ is an orthonormal basis of
the tangent space at each point of the manifold and taking summation over $i$,
where $1\leq i\leq n$, we obtain,
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}S(Y,Z)=\alpha
g(Y,Z),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(4.2)
where $\alpha=\frac{r}{n}.$

From (4.2) we get,
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(\nabla_{X}S)(Y,Z)=\alpha^{\prime}dr(X)g(Y,Z),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(4.3)
where $\alpha^{\prime}=\frac{1}{n}.$

Using (4.3) and (1.4) we get,
$(\nabla_{X}S)(Y,Z)-(\nabla_{Z}S)(Y,X)$
$=S(Y,Z)[A(X)-D(X)]-S(X,Y)[A(Z)-D(Z)]$
$=\alpha^{\prime}[dr(X)g(Y,Z)-dr(Z)g(Y,X)].~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(4.4)

Let $\rho_{1},\rho_{2},\rho_{3}$ be the associated vector fields corresponding
to the 1-forms $A,B,D$ respectively,
i.e., $g(X,\rho_{1})=A(X),g(X,\rho_{2})=B(X),g(X,\rho_{3})=D(X).$

Putting $Z=\rho_{2}$ in (4.4) we get,
$[A(X)-D(X)]B(QY)-[a(\rho_{2})-D(\rho_{2})]S(X,Y)$
$=\alpha^{\prime}[dr(X)B(Y)-dr(\rho_{2})g(Y,X)]$
$=\alpha^{\prime}[B(Y)\\{rA(X)+B(QX)+D(QX)\\}-g(X,Y)\\{rA(\rho_{2})+B(Q\rho_{2})+D(Q\rho_{2})\\}].~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(4.5)

If the manifold has non-zero constant scalar curvature, then by virtue of
(1.8), (4.5) yields that
$[A(X)-D(X)]B(QY)-[A(\rho_{2})-D(\rho_{2})]S(X,Y)=0$
$\Rightarrow
S(X,Y)=\alpha_{1}A(X)\bar{B}(Y)+\alpha_{2}D(X)\bar{B}(Y),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(4.6)
where
$\bar{B}(Y)=B(QY),\alpha_{1}=\frac{1}{A(\rho_{2})-D(\rho_{2})},\alpha_{2}=-\frac{1}{A(\rho_{2})-D(\rho_{2})}.$

This leads to the following:

Theorem 4.1: In a $W_{2}$-flat $(WRS)_{n}(n>3)$ of non-zero constant scalar
curvature, the Ricci tensor $S$ has the form (4.6).

Putting $U=\rho$ in (1.11) and then using (1.9) we get,
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}rT(X)T(Z)=T(\rho)S(X,Z).~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(4.7)

Let us now suppose that a $(WRS)_{n}(n>3)$ is $W_{2}$-flat and of non-zero
scalar curvature.

From (4.7) we have,
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}S(X,Z)=\frac{r}{T(\rho)}T(X)T(Z).~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(4.8)

Now $T(\rho)\neq 0$ as if $T(\rho)=0$ then (4.8) implies that
$rT(X)T(Z)=0\Rightarrow r=0$ (as $T(X)\neq 0~{}~{}\forall X$), which is a
contradiction.

Hence
$S(X,Z)=\tilde{\alpha_{1}}T(X)T(Z),~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$(4.9)
where $\tilde{\alpha_{1}}$ is a non-zero scalar.

Hence comparing (4.9) with (2.12) we can state that:

Theorem 4.2: A $W_{2}$-flat $(WRS)_{n}(n>3)$ of non-vanishing scalar curvature
is a quasi-Einstein manifold with respect to the 1-form $T$ defined by
$T(X)=B(X)-D(X)\neq 0~{}~{}\forall X$.

Acknowledgement: The first author has been sponsored by University Grants
Commission(UGC) Junior Research Fellowship, India. UGC-Ref. No.: 1139/(CSIR-
UGC NET JUNE 2018).

## References

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11institutetext:  University of Florence, Dept. of Mathematics and Computer
Science, Florence, Italy 22institutetext: University of Perugia, Dept. of
Mathematics and Computer Science, Perugia, Italy

# IoT to monitor people flow in areas of public interest

Damiano Perri ORCID:0000-0001-6815-6659
Marco Simonetti ORCID:0000-0003-2923-5519
Alex Bordini ORCID:0000-0002-4445-7876
Osvaldo Gervasi ORCID:0000-0003-4327-520X 112211222222

###### Abstract

The unexpected historical period we are living has abruptly pushed us to
loosen any sort of interaction between individuals, gradually forcing us to
deal with new ways to allow compliance with safety distances; indeed the
present situation has demonstrated more than ever how critical it is to be
able to properly organize our travel plans, put people in safe conditions, and
avoid harmful circumstances. The aim of this research is to set up a system to
monitor the flow of people inside public places and facilities of interest
(museums, theatres, cinemas, etc.) without collecting personal or sensitive
data. Weak monitoring of people flows (i.e. monitoring without personal
identification of the monitored subjects) through Internet of Things tools
might be a viable solution to minimize lineups and overcrowding. Our study,
which began as an experiment in the Umbria region of Italy, aims to be one of
several answers to automated planning of people’s flows in order to make our
land more liveable. We intend to show that the Internet of Things gives almost
unlimited tools and possibilities, from developing a basic information process
to implementing a true portal which enables business people to connect with
interested consumers.

###### Keywords:

IOT, Sensors, Image Processing, Augmented Reality, Virtual Reality

##  1 Introduction

One of the characteristics related to the pandemic of COVID19 as a result of
social distancing is that it has forced institutions to reduce the maximum
acceptable number of simultaneous visitors. This situation generated
difficulties from the organisational point of view of both the structures and
the tourists who wanted to visit certain areas of cultural and tourist
interest. The words ”Internet of Things” (IoT) refer to a collection of
objects that gain intelligence through the internet, exchanging data about
themselves and gaining access to information from other sources [1, 2].

The goal of this research is to maximize the potential of IoT devices to
collect particular data by combining privacy with useful information,
simplicity of use with usefulness, economy with power, and versatility with
accuracy. This project will aim at leveraging existing infrastructure to
develop the concept of indoor tracking, i.e. the ability to provide
information on the number of people presently inside a building, public or
private, and selectively providing the details publicly available.

The system we are proposing is made up of three major components: acquisition
devices, which deal with understanding how the flow varies over time based on
the impulses they receive, coordination devices, which deal with managing all
of the information received, and presentation devices, which make the
previously acquired information available to the public.

##  2 Related works

The effort to automate work and manufacturing processes that began in the
nineteenth century has never truly ended, due to the assistance of
increasingly efficient and performing technology. Today’s strategy is to make
machines increasingly autonomous and intelligent[3, 4, 5], rather than highly
reliant on human supervision[6, 7]. This has required interaction with a
complex world, which necessitates the development of numerous external inputs
on which to take decisions[8, 9]. As a result, the Internet of Things has
become a significant paradigm in a variety of fields, including
manufacturing[10, 11], remote control[12, 13] and monitoring[14, 15], home
automation[16, 17], gaming[18] and Virtual and Augmented Reality[19, 20]. Our
concept is based on the basic requirement to track the flow of individuals
moving through restricted settings for a variety of reasons, including real-
time monitoring and control to minimize congestion, and statistical assessment
for economic and logistical considerations. Although the goal is the same, the
methodologies used over time have been the most varied: RFID techniques[21],
ultrasound sensors[22], PIR sensors[23], closed-circuit TV (CCTV)[24], face
recognition by Artificial Intelligence [25, 26, 27], post-processed data local
memories or in Cloud Services[28].

The use of IoT has always proven to be a simple answer, given the low cost of
the components, which certainly favors the implementation of a do-it-yourself
solution. The method we presented is intended to be simple, both in terms of
design and implementation, as well as in terms of simplicity of use and with a
low learning curve.

##  3 The Architecture of the system

Our system is tasked with monitoring and tracing the movement of people
passing through a location, which is a confined environment such as a public
or private structure. This data is very useful in ensuring the safety and
security of people in the area, and it is controlled differently by those who
have the information, and decides what to publish, and those who utilize it,
i.e. the end user.

To ensure the generic collection of data, useful for guaranteeing service, and
to realize the anonymity of individual people, a scalable stack architecture,
as shown in Figure 1, is divided into three logical levels: data acquisition
layer, data coordination layer which has the function to improve a correct
exchange of information across devices, and presentation layer needed to allow
the flexible use of information in manageable formats.

Figure 1: System Architecture scheme

###  3.1 Data Acquisition Layer

This layer contains every object related to the collection and conversion of
sensor data into processable information from higher layers.

The first critical component of our research is a collection of flow
acquisition devices known as Flow-Meters. They can be heterogeneous or
homogeneous: in some situations, using different sensors may be convenient to
get a wide overview; in others, using sensors of the same kind may enhance
overall accuracy or area detection. The first scenario entails the employment
of numerous sensors of various sorts to improve detection possibilities,
utilizing the plurality of signals received to manage the complexity of the
surrounding environment. In the second case, the detection system is
implemented using a single type of sensor, like an emitter and a receiver
photo-diode; such a simple structure was initially considered but later
rejected because it is incapable of effectively describing contingent
situations, as the direction of the person.

To prevent issues with correctly recognizing the passing of one or more
persons, we ultimately choose a thermal camera sensor, best known as Grid-EYE
matrix sensor, shown in Figure 2a. This is one of the most common option in
the field of Indoor Localization. It is a tiny detector, less than a
centimeter in size, that creates a two-dimensional 8x8 matrix, able to detect
the average temperature for the section of the plane that is projected to the
ground using lasers similar to those used in thermoscanners - Figure 2c.

The size of the detectable plane sections is obviously determined by the
distance between the device and the ground: the higher the device’s
positioning height, the larger the effect on the single viewing angle
measurements, i.e. the incidence of each laser sensor on the ground. The area
covered by sensor and relative geometrical distortion is shown in Figure 2b.

(a) Thermal Camera Sensor

(b) Grid-EYE matrix sensor

(c) Thermoscanner

Figure 2: GRID-EYE matrix sensor

To manage the incoming flow from the different sensors, we utilize an ESP32
micro-controller, which is responsible for processing the raw data and
creating a version suitable for control devices. It continually interrogates
the sensor and extracts all of the information about the flow. The huge amount
of RAM memory required for the management of our matrices needed a
sophisticated micro-controller.

The sensor’s output signal is a low-resolution thermal image. It is
accompanied with a lot of noise, which has been reduced by using interpolation
techniques and filtering matrices. Initially, we sought a method that
attempted to make these passages less apparent, incorporating filters to
tackle the problem; these filters are known as convolutive in the context of
digital image processing. A convolutive filter is a kernel matrix that is
applied to another matrix, in this case the one from the sensor, to produce an
effect based on the values it assumes. The initial working hypothesis was to
blur the image; however, after implementing the algorithm, it was discovered
that the benefits were minimal because the system was limited by the few
pixels available. As a result, the originally expected homogeneity would not
have been achieved, and some information would have been lost, eliminated by
the convolutive filter. This indicated to us that the primary requirement was
not to mix the pixels we already had, but to dilute them over a broader
region. Interpolation is a method that involves adding new pixels to existing
ones and calculating their value using a reconstruction algorithm. The
technique we have chosen is one of the most often used tools for upscaling in
photo editing, bicubic interpolation, whereas the technique to filter pixels
has been Background Subtraction [29].

Only when the noise has been adequately attenuated, the instrument’s
sensitivity can be modified using a threshold value. This allows you to
maintain the highest values of the matrix unchanged while attenuating the
minimums and clearly identifying the individual clusters as shown in Figure
3a. We called cluster a submatrix with variable shape and size, that
numerically represents the area which the person can be projected on.

(a)

(b)

Figure 3: Image Processing

The matrix is then scanned to determine the size of the clusters. The
objective is to minimize these regions to a single point. This point is their
center of gravity, and acquire one for each individual in order to work on
data distribution uniformity. To do this, a Flood-Fill algorithm is used to
extrapolate the individual clusters, and then their centers of gravity are
computed using a weighted average ah shown in Figure 3b.

To convey the results of this level’s computations to higher levels, the
device requires a communication logic, capable of being simple and reliable.
Physical communication in binary form is used to support this, with increments
and decrements communicated to the relevant pins through the electrical
potential difference. In this way a proper electrical diagram can help us to
quickly provide feedback on functioning and faults as shown in Figure 4.

Figure 4: Wiring diagram for the circuit

###  3.2 Coordination Layer

At this level, we can locate all of the equipment that deal with coordinating
the data from the Flow-Meters. They are classified into two kinds:

  * $\bullet$

within the infrastructure

  * $\bullet$

outside of the infrastructure

The former are microcontrollers that manage data synchronization from many
sorts of sensors. The latter are servers (called brokers) that communicate
with the former through the MQTT protocol and ensure the consistency of
infrastructure acquisitions. Microcontrollers have the responsibility to
manage this data in accordance with the standards of the linked devices and to
act as an interface with the MQTT broker as shown in Figure 5a.

Our infrastructure must be a closed system in order for the broker to be aware
of the full flow. This enables the broker to ensure that the incoming data is
consistent and to store it in a database. The broker and the coordinator use
the MQTT protocol to interact safely and reliably, and they take particular
actions based on the messages they receive as shown in Figure 5b.

(a) Interface among the MQTT broker and microcontrollers

(b) Data communication between MQTT broker and a coordinator

Figure 5: Coordination between microcontroller and broker

The connection initialization and initial configuration are performed during
the first start, and the device is resumed once done. During this phase, the
device runs in WIFI_STA111The possible working states are: WIFI_AP (Access
Point mode),WIFI_STA (Client only), WIFI_AP_STA (dual mode), WIFI_OFF
(disabled) modality and provides a setup web page via the Autoconnect library,
where the connection credentials may be entered. On the server side, there is
a validation and registration phase during the initial connection; from then
on, at each successive start, the device requests the server for its personal
data, publishing on the specific topic of the MQTT protocol. The server
responds with the device type, location information, and any setup constants,
which are collected by three distinct callbacks. If a single coordinator is
responsible for managing a large number of sensors, a first coordination is
necessary. The influx of data from these sensors must be coordinated based on
their characteristics; hence, the several Flow-Meters employ a common standard
to transmit the specifications and results of their processing. One of their
parameters is the region the sensor can cover: the objective is to broaden the
range of action of certain types of sensors while keeping the system scalable,
since signal processing and identification are delegated to a lower level, the
Flow-Meter. The coordinator then assigns a difference value to each of the
sensors and updates it at each state change with the increase or reduction.
Another thread is responsible for posting the updated value to the server.

At this level, coordination is based on an overview of the sites and we expect
that our sensors have mapped at least all of the entrance and exit areas. The
coordinators function as repeaters in the interaction with the Broker. As a
result, all coordinators within the location must be linked with it. The
broker must be aware of the whole flow, which occurs only if our site is a
closed system; at that point, it has visibility of what comes in and what
comes out.

Essentially, the protocol implemented provides us with channels to publish and
receive messages among the various clients via callbacks. As a result, this
paradigm does not provide intellectual ability for those that manage the
messages, the broker, which is only limited to coordinating communication. It
was necessary to build an additional layer on top of an existing broker that
would allow intercepting messages passing through the network and triggering
server-side actions. These activities include retrieving data, storing and
maintaining information, and performing tasks based on the kind of message
provided to the subjects, which are used as API endpoints. At this stage, the
server relays the message exchange, which does not just send the message but
also applies predefined logic to it: an example might be the phase of adding a
new device, which merely registers on the server. Following the authentication
and authorization stages, the server verifies and tells the device about the
place which it is allocated to, and then informs the other devices connected.

One of the challenges we faced throughout the project’s execution was how to
maintain the quantity of information arriving from the coordinators synced and
the quality on a high level, because these devices, as previously said, are
not so reliable when they perform this kind of tasks. To address this issue,
it was therefore decided to adopt the MQTT protocol, which has a very high
degree of reliability in handling the transmission of information across
various devices and equipment.

Security deserves a separate mention, as there are two ”by design”
vulnerabilities of the MQTT protocol: protection against unwanted
subscription/publication and encryption of packets exchanged between client
and broker. Unwanted subscription protection can be achieved through a
whitelist. Each device is provided with a unique key and the subscription is
made possible only if the key is valid. In the event that the key is
compromised, a new one can be generated, as the device is capable of receiving
updates via OTA (Over-The-Air). By default, even to keep the protocol light,
no encryption is applied to the packets.

There are several methods for securing message exchange and each of them
should be chosen based on the device’s power. According to RFC 7228, these
devices are classified into three types: our microcontroller belongs to the
class 2 devices, capable of managing the overhead of the SSL/TLS protocol;
additionally, because it is dual-core, it is possible to delegate the server
update to a separate process that manages the queue, without slowing down the
data acquisition process.

###  3.3 Presentation Layer

This level has the purpose of making the information stored to the user
accessible through the devices located at the underlying levels, and to manage
the access mode for single users, guaranteeing the separation of
responsibilities for any of them. So, main actions for this layer are:

  * $\bullet$

make the information acquired by the devices located at the underlying levels
available to the user

  * $\bullet$

manage access methods

  * $\bullet$

separate the responsibilities relating to the user profile

The first evident distinction is between the external user, called standard
user, who reads the information, and the internal user, called business, who
owns the information and determines how to display it. The latter, unlike the
former, must be registered in the system, in order to manage an activity.

This level’s structure has been extensively tested in various environments;
there is a Docker container which virtualizes the data archived in the MySQL
database in a secure environment, and the server, which runs on a virtual
machine and externally exposing only the endpoints of the APIs that the
application calls as shown in Figure 6.

Figure 6: Presentation Layer and docker container

The presentation system is made up of two parts: the web server and the
client. Infrastructure services manage main database activities (CRUD) using
APIs accessible to the application client. These APIs are classified into
three categories (authentication, task management, and device management),
corresponding to a different set of entities in the database.

The development scheme we have chosen provides the Model View Controller (MVC)
pattern is an approach that involves categorizing the elements of an
application into three distinct parts in order to make their modification and
management autonomous, thereby increasing maintainability and portability.
These three categories are the Model, the View, and the Controller, in that
order as shown in Figure 7.

  * $\bullet$

Model: contains the data access methods

  * $\bullet$

View: takes care of viewing the data to the user and manages the interaction
between the latter and the underlying infrastructure

  * $\bullet$

Controller: receives user commands through the View and reacts by performing
operations that may affect the Model and which generally lead to a change in
the View state

Figure 7: Scheme of MVC pattern

Therefore, the Model level is responsible for managing the data that the
application must manipulate. The term Model implies its nature that is a
representation of the object, which will subsequently be mapped to the
database according to rules set by the programmer, abstracting the structure
of the data within the database.

In the MVC framework, the Controller level is the most crucial. The Controller
level contains all of the classes that allow you to validate user requests and
process them in order to return a response. It is able of intercepting user
requests, choosing the components capable of processing them, mapping the
result into an object, and returning it to the caller.

To allow the user to interact with the system, we created a Progressive Web
Application that can run within the browser and so adapt to any operating
system or device, with a responsive UI. The application’s Home screen greets
the user with a map that shows nearby areas of interest as well as associated
information. By picking the point of interest, you will have access to extra
information that the management has decided to make available for the specific
user as shown in Figure 8.

Figure 8: App responsive graphic interface

The user must be registered and enabled in order to utilize the application’s
functions as a business role. Every time the user takes an action, a request
is sent to the server, and the request is successful if the formal check is
accurate. After an user has been verified and approved, he/she may manage
their corporate profiles as shown in Figure 9.

Figure 9: Registration, login and modification authorization phases

To publish updates on their activity, the user has to enter them and associate
the devices with them too. The localization information about the site is
retrieved through a Geocoding service. When the insertion is finished, the
server changes the tables and notifies the other clients via RPC and so the
device is completely associated with the support of an OTP Key.

The application has also been enriched with an additional module that allows
the visitor to obtain information on artifacts where appropriate (museums,
exhibitions, etc.). With the help of virtual reality techniques, the user can
take a guided tour of the facility and view the main works of art.
Furthermore, a series of QR codes have been created inside the structure that
allow the user to view the artifact through augmented reality technologies.

##  4 Discussion of Results

We proceeded with the verification of the overall correctness of the system
after completing all of the steps mentioned in the previous chapters. All
testing were carried out on a precised room within the University area with a
single entrance and exit door, with a setup consisting of a single Grid-EYE
sensor, a Flow-Meter, and a Coordinator. We tested the system in its entirety
after performing all of the tests required to ensure the system’s correct
operation and, later, those required to refine the functionality of the
individual components. The following operations were then performed:

  1. 1.

Positioning of the Grid-EYE sensor on the door jamb, at a height of about two
meters

  2. 2.

Connection of the sensor to the acquisition microcontroller, which in turn is
connected to the coordinator

  3. 3.

Connecting the microcontrollers to the network

  4. 4.

Configuration of Coordination Layer (Broker MQTT) and Presentation Layer
(database and web server) on two separate machines

  5. 5.

Installing the application on an Android device

  6. 6.

Registration via email and password

  7. 7.

Authentication and authorization within the system

  8. 8.

Insertion of a fictitious activity so that it can appear on the map and be
selected from the context menu

  9. 9.

Association of the coordinator via the OTP Key. The correct name for a
business role appeared on the microcontroller display and the device appeared
on the proper page.

At this point, the activity and the device had been properly setup, and the
fifteen-day testing period could begin. We turned on the detector at the start
of the first day, when all staff (professors, students, and so on) members
were not yet in the room. We found that every time a person went in, the
counter was incremented and when he/she went out, the number was decremented.
At the conclusion of the test period, late in the last day’s evening, the
counter was correctly returned to zero.

##  5 Conclusions and Future Works

The testing revealed that our system has a strong basic detection capacity, as
we discovered a result of zero for eight days, a value of one for four days,
and a value of two for the remaining three days on 42 average daily passes.
The detections were monitored using the program, which checked that the inputs
and outputs were appropriately synchronized whenever a state change happened.
It was also able to examine the logs, which revealed that the mistakes were
caused by erroneous assessments by the acquisition tools. The different test
phases have demonstrated how, although being in an experimental phase, our
technology seems to be adequate for creating a reliable infrastructure. It has
been shown that the Internet of Things provides nearly limitless tools and
potential, both in terms of opportunities and future extensions, ranging from
the development of a simple information process to the implementation of a
real portal capable of allowing those who do business to communicate with
interested customers. An exciting future use might be to make available, in
addition to the actual survey summary, booking systems capable of sending the
flow of people, which, when combined with the information status signals,
allows customers to have direct contact with who provides the service in real
time.

## Acronyms

The following acronyms are used in this manuscript:

OTP | One Time Password
---|---
RPC | Remote Procedure Call
CRUD | Create, Read, Update, Delete
MVC | Model View Controller
UI | User Interface
CCTV | closed-circuit TV
PIR | Passive InfraRed
IOT | Internet of Things
RAM | Read Only Memory
CPU | Central Processing Unit
OTA | Over The Air
API | Application Program Interface
MQTT | MQ Telemetry Transport or Message Queue Telemetry Transport

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