Projective Collineations in Warped Product Manifolds and (PRS)_n Manifolds

Abstract

The current work first explores projective collineations on pseudo-Riemannian manifolds. Projective collineations, curvature collineations, and Ricci curvature collineations are examined in relation to one another. On warped product manifolds, the projective collineations of the form $\zeta ={\zeta }_1+{\zeta }_2$ are investigated. We scrutinize various inheritance aspects in projective collineations from warped product manifolds to its factor manifolds. This provides, for example, a partially negative solution to Besse’s problem regarding the existence of Einstein warped product manifolds. Finally, Pseudo-Ricci symmetric space-times admitting projective collineations are investigated.

1. Introduction

The orbits of a vector field $\zeta$ on a Riemannian manifold M are defined by

$$\frac{d{u}^i}{d\theta }={\zeta }^i,i=1,2,3,\dots ,n$$

(1)

where $\left(u^i\right)$ represents a local coordinate system on M, $\theta \in J$, and J is an open interval of the real line. The above system of differential equations has a unique solution locally. These orbits generate a local flow $\psi$ of $\zeta$ on M. The flow lines $\psi$ of $\zeta$ are used to transform any object $\Omega$ from one point to another point. The Lie derivative of $\Omega$ in the direction of $\zeta$, denoted by ${\mathcal{L}}_{\zeta }\Omega$, measures the invariance of $\Omega$ under a change of position in the direction of $\zeta$. $\zeta$ is said to define a symmetry of $\Omega$ on M if ${\mathcal{L}}_{\zeta }\Omega$ vanishes. For example, if $\Omega$ is the metric tensor g (or curvature tensor $\mathcal{R}$, Ricci curvature tensor $Ric$, and matter tensor $\mathcal{T}$), that is, ${\mathcal{L}}_{\zeta }g=0$ (resp. ${\mathcal{L}}_{\zeta }\mathcal{R}=0,$${\mathcal{L}}_{\zeta }Ric=0,$ and ${\mathcal{L}}_{\zeta }\mathcal{T}=0$), then $\zeta$ is called a Killing vector field (or curvature collineation, Ricci curvature collineation, and matter collineation). The physics of a space-time is intimately related to these symmetries. Many generalizations of such symmetries are extensively investigated in the literature. One of these generalizations is the projective collineation $PC$. The local flow of a projective collineation $\zeta$ maps geodesics to geodesics. The generator of this group of local diffeomorphisms is called a projective collineation. The reader is directed to [1,2,3] for more information on such space-time symmetries.

The Ricci tensor $R_{ij}$ of an n-dimensional Einstein space-time M is proportional to the metric tensor $g_{ij}$ with constant of proportionality $\mu$, that is, $R_{ij}=\mu g_{ij}$ where $\mu =\frac{R}{n}$ and R is the scalar curvature of M [4]. One of the primary characteristics of Einstein space-times is the disappearance of the gradient of the Ricci tensor, so ${\nabla }_h{R}_{ij}$ is zero. Ricci symmetric space-times are space-times that make this property possible, where ∇ is the covariant derivative. As a result, Einstein manifolds are naturally classified as Ricci symmetric space-times. In a Ricci recurrent space-time, the gradient of the Ricci tensor is related to the Ricci tensor in the form ${\nabla }_h{R}_{ij}={\omega }_h{R}_{ij}$ [5]. The Ricci recurrent space-time class is undoubtedly a subclass of the class of all Ricci symmetric space-times. An n-dimensional space-time M is called a pseudo-Ricci symmetric space-time [6], (PRS)${}_n$, if the gradient of the Ricci tensor is given as

$${\nabla }_h{R}_{ij}=2{\omega }_h{R}_{ij}+{\omega }_i{R}_{jh}+{\omega }_j{R}_{ih}.$$

(2)

Weak Ricci symmetries of Riemannian manifolds were first explored by Tamássy and Binh [7]. The class of pseudo-Ricci symmetric manifolds is a subclass of weakly Ricci symmetric manifolds. There has been a great deal of emphasis on the topic of (PRS)${}_n$ manifolds; for instance, a sufficient condition of a (PRS)${}_n$ manifolds to be quasi-Einstein manifolds was introduced by De and Gazi [8]. (PRS)${}_n$ manifolds whose scalar curvature satisfies ${\nabla }_{k}R=0$ have zero scalar curvature [9]. A concrete example of pseudo-Ricci symmetric manifolds was given in [10].

Projective collineations have not been investigated for either warped product manifolds or (PRS)${}_n$ manifolds. With this work, we want to close the gap that has been identified. On warped product manifolds, a projective collineation’s factors must meet certain requirements on warped product manifolds to be a projective collineation on the factor manifolds. Many characterizations of projective collineations on warped product manifolds are given. We can find more motivations and future research directions of our work in several papers (see [11,12,13,14,15,16,17,18,19,20]).

2. Projective Collineations

A vector field $\zeta$ is called projective collineation if

$$\left({\mathcal{L}}_{\zeta }\nabla \right)\left(X,V\right)=p\left(X\right)V+p\left(V\right)X$$

(3)

for any vector fields $X,V$ where p is a $1-$form. In local coordinates, it is

$${\mathcal{L}}_{\zeta }{\Gamma }_{ij}^h={\delta }_i^h{p}_j+{\delta }_j^h{p}_i.$$

(4)

If p is constant, then $\zeta$ is an affine collineation $AC$. It is known that

$$\begin{array}{ccc}& & {\mathcal{L}}_{\zeta }{\Gamma }_{ij}^h\hfill \\ & =& \frac{1}{2}{g}^{hm}\left[{\nabla }_i\left({\mathcal{L}}_{\zeta }{g}_{mj}\right)+{\nabla }_j\left({\mathcal{L}}_{\zeta }{g}_{im}\right)-{\nabla }_m\left({\mathcal{L}}_{\zeta }{g}_{ij}\right)\right]\hfill \\ & =& \frac{1}{2}{g}^{hm}\left[{\nabla }_i\left({\nabla }_m{\zeta }_j+{\nabla }_j{\zeta }_m\right)+{\nabla }_j\left({\nabla }_m{\zeta }_i+{\nabla }_i{\zeta }_m\right)-{\nabla }_m\left({\nabla }_i{\zeta }_j+{\nabla }_j{\zeta }_i\right)\right]\hfill \\ & =& \frac{1}{2}{g}^{hm}\left[{\nabla }_i{\nabla }_m{\zeta }_j+{\nabla }_i{\nabla }_j{\zeta }_m+{\nabla }_j{\nabla }_m{\zeta }_i+{\nabla }_j{\nabla }_i{\zeta }_m-{\nabla }_m{\nabla }_i{\zeta }_j-{\nabla }_m{\nabla }_j{\zeta }_i\right]\hfill \end{array}$$

(5)

$$\begin{array}{ccc}& =& \frac{1}{2}\left[{\nabla }_i{\nabla }^h{\zeta }_j+{\nabla }_i{\nabla }_j{\zeta }^h+{\nabla }_j{\nabla }^h{\zeta }_i+{\nabla }_j{\nabla }_i{\zeta }^h-{\nabla }^h{\nabla }_i{\zeta }_j-{\nabla }^h{\nabla }_j{\zeta }_i\right].\hfill \end{array}$$

(6)

Therefore,

$$\begin{array}{ccc}& & \frac{1}{2}\left[{\nabla }_i{\nabla }^h{\zeta }_j+{\nabla }_i{\nabla }_j{\zeta }^h+{\nabla }_j{\nabla }^h{\zeta }_i+{\nabla }_j{\nabla }_i{\zeta }^h-{\nabla }^h{\nabla }_i{\zeta }_j-{\nabla }^h{\nabla }_j{\zeta }_i\right]\hfill \\ & =& {\delta }_i^h{p}_j+{\delta }_j^h{p}_i.\hfill \end{array}$$

A simple contraction yields

$$\begin{array}{ccc}\hfill \frac{1}{2}\left[{\nabla }_i{\nabla }^i{\zeta }_j+{\nabla }_i{\nabla }_j{\zeta }^i+{\nabla }_j{\nabla }^i{\zeta }_i+{\nabla }_j{\nabla }_i{\zeta }^i-{\nabla }^i{\nabla }_i{\zeta }_j-{\nabla }^i{\nabla }_j{\zeta }_i\right]& =& \left(n+1\right)p_j,\hfill \\ \hfill {\nabla }_j{\nabla }_i{\zeta }^i& =& \left(n+1\right)p_j.\hfill \end{array}$$

Now, we get the form of p as

$$p_i=\frac{1}{n+1}{\nabla }_j{\nabla }_i{\zeta }^i.$$

(7)

That is, $p_i={\nabla }_{i}p$ where $p=\frac{1}{n+1}\mathrm{div}\zeta$. In addition, Equation (6) implies

$${\mathcal{L}}_{\zeta }{\Gamma }_{ij}^h={\nabla }_j{\nabla }_i{\zeta }^h+R_{imj}^h{\zeta }^m.$$

(8)

Thus,

$${\nabla }_j{\nabla }_i{\zeta }^h+R_{imj}^h{\zeta }^m={\delta }_i^h{p}_j+{\delta }_j^h{p}_i.$$

(9)

A contraction of the above equation yields

$${\nabla }_j{\nabla }_i{\zeta }^j-R_{im}{\zeta }^m=\left(n+1\right)p_i.$$

Theorem 1.

A projective collineation ζ on a pseudo-Riemannian manifold M satisfies the following conditions:

$$\begin{array}{ccc}\hfill {\nabla }_j{\nabla }_i{\zeta }^j-R_{im}{\zeta }^m& =& \left(n+1\right)p_i.\hfill \\ \hfill p_i& =& {\nabla }_{i}p\hfill \\ \hfill p& =& \frac{1}{n+1}\mathrm{div}\zeta ,\hfill \end{array}$$

Be aware of the fact that a PC satisfies

$${\mathcal{L}}_{\zeta }{\mathcal{W}}_{ijk}^h=0,$$

where

$${\mathcal{W}}_{ijk}^h=R_{ijk}^h-\frac{1}{\left(n-1\right)}\left({\delta }_k^h{R}_{ij}-{\delta }_j^h{R}_{ki}\right)$$

(10)

is the Weyl projective curvature tensor. Consequently, it is simple to obtain

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\zeta }{R}_{ijk}^h& =& \frac{1}{\left(n-1\right)}\left({\mathcal{L}}_{\zeta }\left({\delta }_k^h{R}_{ij}\right)-{\mathcal{L}}_{\zeta }\left({\delta }_j^h{R}_{ki}\right)\right)\hfill \\ & =& \frac{1}{\left(n-1\right)}\left({\delta }_k^h{\mathcal{L}}_{\zeta }{R}_{ij}+R_{ij}{\mathcal{L}}_{\zeta }{\delta }_k^h-{\delta }_j^h{\mathcal{L}}_{\zeta }{R}_{ki}-R_{ki}{\mathcal{L}}_{\zeta }{\delta }_j^h\right).\hfill \end{array}$$

Theorem 2.

A projective collineation ζ on a pseudo-Riemannian manifold M satisfies the following condition:

$${\mathcal{L}}_{\zeta }{R}_{ijk}^h=\frac{1}{\left(n-1\right)}\left({\delta }_k^h{\mathcal{L}}_{\zeta }{R}_{ij}+R_{ij}{\mathcal{L}}_{\zeta }{\delta }_k^h-{\delta }_j^h{\mathcal{L}}_{\zeta }{R}_{ki}-R_{ki}{\mathcal{L}}_{\zeta }{\delta }_j^h\right).$$

According to the covariant derivative of ${\delta }_k^h$, the Lie derivative of ${\delta }_k^h$ is given by

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\zeta }{\delta }_k^h& =& {\zeta }^m{\nabla }_m{\delta }_k^h-{\delta }_k^m{\nabla }_m{\zeta }^h+{\delta }_l^h{\nabla }_k{\zeta }^l\hfill \\ & =& 0.\hfill \end{array}$$

Similarly, ${\mathcal{L}}_{\zeta }{\delta }_j^h$. Now, the covariant derivative of the Ricci tensor is given by

$${\mathcal{L}}_{\zeta }{R}_{ij}={\zeta }^m{\nabla }_m{R}_{ij}+R_{ir}{\nabla }_j{\zeta }^r+R_{il}{\nabla }_j{\zeta }^l.$$

Likewise, it is

$${\mathcal{L}}_{\zeta }{R}_{ki}={\zeta }^m{\nabla }_m{R}_{ki}+R_{il}{\nabla }_k{\zeta }^l+R_{ir}{\nabla }_k{\zeta }^r.$$

The Lie derivative of the Riemann tensor may be simplified as

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\zeta }{R}_{ijk}^h& =& \frac{1}{n-1}{\delta }_k^h\left({\zeta }^m{\nabla }_m{R}_{ij}+R_{ir}{\nabla }_j{\zeta }^r+R_{il}{\nabla }_j{\zeta }^l\right)\hfill \\ & & -\frac{1}{n-1}{\delta }_j^h\left({\zeta }^m{\nabla }_m{R}_{ki}+R_{il}{\nabla }_k{\zeta }^l+R_{ir}{\nabla }_k{\zeta }^r\right).\hfill \end{array}$$

Theorem 3.

A projective collineation ζ on a pseudo-Riemannian manifold M satisfies the following condition:

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{\zeta }{R}_{ijk}^h& =& \frac{1}{n-1}{\delta }_k^h\left({\zeta }^m{\nabla }_m{R}_{ij}+R_{ir}{\nabla }_j{\zeta }^r+R_{il}{\nabla }_j{\zeta }^l\right)\hfill \\ & & -\frac{1}{n-1}{\delta }_j^h\left({\zeta }^m{\nabla }_m{R}_{ki}+R_{il}{\nabla }_k{\zeta }^l+R_{ir}{\nabla }_k{\zeta }^r\right).\hfill \end{array}$$

Corollary 1.

A projective collineation ζ on a pseudo-Riemannian manifold M is a curvature collineation if and only if it is a Ricci collineation.

3. Warped Product Manifolds

The Cartesian product manifold $M_1\times M_2$ of two pseudo-Riemannian manifolds $\left(M_1,g_1\right)$ and $\left(M_2,g_2\right)$ furnished with the metric tensor $g=g_1\oplus f^2{g}_2$ is called the warped product manifold $M_1{\times }_f{M}_2$, where f is a smooth positive real-valued function on $M_1$. Let the natural projection maps of the warped product manifold [21] be ${\pi }_1$ and ${\pi }_2$. It is clear that

$${∥V∥}^2={∥{\pi }_{\ast 1}\left(V\right)∥}^2+{\left(f\circ {\pi }_1\right)}^2{∥{\pi }_{\ast 2}\left(V\right)∥}^2$$

for any vector V tangent to the warped product manifold. Generalized Robertson–Walker space-time [22,23,24] $I{\times }_f{M}_2$ and standard static space-time [25,26,27] $M_1{\times }_{f}I$ are exceptional cases of Lorentzian warped product manifolds when a factor manifold is an open, connected interval of the real line. Warped product manifolds’ scalar curvatures, Ricci curvatures, and Riemann curvatures can all be described in terms of the lift of the relevant curvature tensors from the factor manifolds to the product manifold [28,29,30].

4. Projective Collineation on Warped Product Manifolds

It is clear that the defining property of projective collineations is

$$\left({\mathcal{L}}_{\zeta }\nabla \right)\left(U,V\right)=p\left(U\right)V+p\left(V\right)U.$$

Let us derive a form of the left-hand side on a warped product manifold $M=M_1{\times }_f{M}_2$. Let $U,V\in \mathfrak{X}\left(M\right)$, where $U_i,V_i\in \mathfrak{X}\left(M_i\right)$ and $U=U_1+U_2,V=V_1+V_2$

$$\begin{array}{ccc}\hfill \left({\mathcal{L}}_{\zeta }\nabla \right)\left(U,V\right)& =& {\mathcal{L}}_{\zeta }\left(\nabla \left(U,V\right)\right)-\nabla \left({\mathcal{L}}_{\zeta }U,V\right)-\nabla \left(U,{\mathcal{L}}_{\zeta }V\right)\hfill \\ & =& \left[\zeta ,\nabla \left(U,V\right)\right]-\nabla \left(\left[\zeta ,U\right],V\right)-\nabla \left(U,\left[\zeta ,V\right]\right)\hfill \\ & =& \left[\zeta ,{\nabla }_{U}V\right]-{\nabla }_{\left[\zeta ,U\right]}V-{\nabla }_U\left[\zeta ,V\right].\hfill \end{array}$$

Now, one can expand this formula to

$$\begin{array}{ccc}& & \left({\mathcal{L}}_{\zeta }\nabla \right)\left(U,V\right)\hfill \\ & =& \left[{\zeta }_1+{\zeta }_2,{\nabla }_{U_1}^1{V}_1+U_1\left(lnf\right)V_2+V_1\left(lnf\right)U_2+{\nabla }_{U_2}^2{V}_2-f{g}_2\left(U_2,V_2\right){\nabla }^{1}f\right]\hfill \\ & & -{\nabla }_{\left[{\zeta }_1,U_1\right]}^1{V}_1-\left[{\zeta }_1,U_1\right]\left(lnf\right)V_2-V_1\left(lnf\right)\left[{\zeta }_2,U_2\right]\hfill \\ & & -{\nabla }_{\left[{\zeta }_2,U_2\right]}^2{V}_2+f{g}_2\left(\left[{\zeta }_2,U_2\right],V_2\right){\nabla }^{1}f-{\nabla }_{U_1}^1\left[{\zeta }_1,V_1\right]-U_1\left(lnf\right)\left[{\zeta }_2,V_2\right]\hfill \\ & & -\left[{\zeta }_1,V_1\right]\left(lnf\right)U_2-{\nabla }_{U_2}^2\left[{\zeta }_2,V_2\right]+f{g}_2\left(U_2,\left[{\zeta }_2,V_2\right]\right){\nabla }^{1}f.\hfill \end{array}$$

Some lengthy computations will lead us to the following proposition.

Proposition 1.

In a warped product manifold $M=M_1{\times }_f{M}_2$, it is

$$\begin{array}{ccc}\hfill \left({\mathcal{L}}_{\zeta }\nabla \right)\left(U,V\right)& =& \left({\mathcal{L}}_{{\zeta }_1}^1{\nabla }^1\right)\left(U_1,V_1\right)+\left({\mathcal{L}}_{{\zeta }_2}^2{\nabla }^2\right)\left(U_2,V_2\right)\hfill \\ & & -f{g}_2\left(U_2,V_2\right)\left[{\zeta }_1,{\nabla }^{1}f\right]-\left[{\zeta }_1,U_1\right]\left(lnf\right)V_2-\left[{\zeta }_1,V_1\right]\left(lnf\right)U_2\hfill \\ & & +f\left[{\zeta }_2\left(g_2\left(U_2,V_2\right)\right)-\left({\mathcal{L}}_{{\zeta }_2}^2{g}_2\right)\left(U_2,V_2\right)\right]{\nabla }^{1}f,\hfill \end{array}$$

for every vector fields ζ,$U,V$.

Thus, we have the following cases:

$$\begin{array}{ccc}\hfill \left({\mathcal{L}}_{{\zeta }_1}\nabla \right)\left(U_1,V_1\right)& =& \left({\mathcal{L}}_{{\zeta }_1}^1{\nabla }^1\right)\left(U_1,V_1\right)\hfill \end{array}$$

(11)

$$\begin{array}{ccc}\hfill \left({\mathcal{L}}_{{\zeta }_1}\nabla \right)\left(U_1,V_2\right)& =& -\left[{\zeta }_1,U_1\right]\left(lnf\right)V_2\hfill \end{array}$$

(12)

$$\begin{array}{ccc}\hfill \left({\mathcal{L}}_{{\zeta }_1}\nabla \right)\left(U_2,V_2\right)& =& f\left[{\zeta }_2\left(g_2\left(U_2,V_2\right)\right)-\left({\mathcal{L}}_{{\zeta }_2}^2{g}_2\right)\left(U_2,V_2\right)\right]{\nabla }^{1}f\hfill \end{array}$$

(13)

$$\begin{array}{ccc}\hfill \left({\mathcal{L}}_{{\zeta }_2}\nabla \right)\left(U_1,V_1\right)& =& 0\hfill \end{array}$$

(14)

$$\begin{array}{ccc}\hfill \left({\mathcal{L}}_{{\zeta }_2}\nabla \right)\left(U_1,V_2\right)& =& 0\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill \left({\mathcal{L}}_{{\zeta }_2}\nabla \right)\left(U_2,V_2\right)& =& \left({\mathcal{L}}_{{\zeta }_2}^2{\nabla }^2\right)\left(U_2,V_2\right)\hfill \\ & & +f\left[{\zeta }_2\left(g_2\left(U_2,V_2\right)\right)-\left({\mathcal{L}}_{{\zeta }_2}^2{g}_2\right)\left(U_2,V_2\right)\right]{\nabla }^{1}f.\hfill \end{array}$$

(16)

Theorem 4.

In a warped product manifold $M=M_1{\times }_f{M}_2$, a vector field ζ is a affine collineation on M if and only if the following conditions hold:

1. 

$\left({\mathcal{L}}_{{\zeta }_1}^1{\nabla }^1\right)\left(U_1,V_1\right)=\left({\mathcal{L}}_{{\zeta }_2}^2{\nabla }^2\right)\left(U_2,V_2\right)=0$,

2. 

$-\left[{\zeta }_1,U_1\right]\left(lnf\right)=0$,

3. 

$\left[{\zeta }_2\left(g_2\left(U_2,V_2\right)\right)-\left({\mathcal{L}}_{{\zeta }_2}^2{g}_2\right)\left(U_2,V_2\right)\right]$ or f is constant.

A $1-$form $p\left(U\right)$ may be rewritten as

$$\begin{array}{ccc}\hfill p\left(U\right)& =& g\left(P,U\right)\hfill \\ & =& g_1\left(P_1,U_1\right)+f^2{g}_2\left(P_2,U_2\right)\hfill \\ & =& p_1\left(U_1\right)+f^2{p}_2\left(U_2\right),\hfill \end{array}$$

where $p_1\left(U_1\right)=g_1\left(P_1,U_1\right),$ and $p_2\left(U_2\right)=g_2\left(P_2,U_2\right)$. Thus, we have the following cases

$$\begin{array}{ccc}\hfill p_1\left(U_1\right)V_1+p_1\left(V_1\right)U_1& =& \left({\mathcal{L}}_{{\zeta }_1}^1{\nabla }^1\right)\left(U_1,V_1\right)\hfill \\ \hfill p_1\left(U_1\right)V_2+f^2{p}_2\left(V_2\right)U_1& =& -\left[{\zeta }_1,U_1\right]\left(lnf\right)V_2\hfill \\ \hfill f^2{p}_2\left(U_2\right)V_2+f^2{p}_2\left(V_2\right)U_2& =& f\left[{\zeta }_2\left(g_2\left(U_2,V_2\right)\right)-\left({\mathcal{L}}_{{\zeta }_2}^2{g}_2\right)\left(U_2,V_2\right)\right]{\nabla }^{1}f\hfill \\ \hfill p_1\left(U_1\right)V_1+p_1\left(V_1\right)U_1& =& p_1\left(U_1\right)V_2+f^2{p}_2\left(V_2\right)U_1=0\hfill \\ \hfill f^2{p}_2\left(U_2\right)V_2+f^2{p}_2\left(V_2\right)U_2& =& \left({\mathcal{L}}_{{\zeta }_2}^2{\nabla }^2\right)\left(U_2,V_2\right)\hfill \\ & & +f\left[{\zeta }_2\left(g_2\left(U_2,V_2\right)\right)-\left({\mathcal{L}}_{{\zeta }_2}^2{g}_2\right)\left(U_2,V_2\right)\right]{\nabla }^{1}f.\hfill \end{array}$$

The first case is

$$\left({\mathcal{L}}_{\zeta }\nabla \right)\left(U_1,V_1\right)=p\left(U_1\right)V_1+p\left(V_1\right)U_1.$$

From Equations (11) and (14), the above equation yields

$$\left({\mathcal{L}}_{{\zeta }_1}^1{\nabla }^1\right)\left(U_1,V_1\right)=p_1\left(U_1\right)V_1+p_1\left(V_1\right)U_1,$$

(17)

that is, the vector field ${\zeta }_1$ is a projective collineation on the base manifold $M_1$.

Theorem 5.

In a warped product manifold $M=M_1{\times }_f{M}_2$ admitting a projective collineation $\zeta ={\zeta }_1+{\zeta }_2,$ the component ${\zeta }_1$ is a projective collineation on the base manifold, that is,

$$\left({\mathcal{L}}_{{\zeta }_1}^1{\nabla }^1\right)\left(U_1,V_1\right)=p_1\left(U_1\right)V_1+p_1\left(V_1\right)U_1$$

where $p_1\left(U_1\right)=g_1\left(P_1,U_1\right)$.

The second case implies

$$\left({\mathcal{L}}_{\zeta }\nabla \right)\left(U_1,V_2\right)=p\left(U_1\right)V_2+p\left(V_2\right)U_1.$$

Now, Equations (12) and (14) imply

$$-\left[{\zeta }_1,U_1\right]\left(lnf\right)V_2=p_1\left(U_1\right)V_2+f^2{p}_2\left(V_2\right)U_1.$$

Therefore,

$$\begin{array}{ccc}\hfill -\left[{\zeta }_1,U_1\right]\left(lnf\right)& =& p_1\left(U_1\right)\hfill \end{array}$$

(18)

$$\begin{array}{ccc}\hfill 0& =& p_2\left(V_2\right).\hfill \end{array}$$

(19)

The tangential component of the associated vector field to $M_2$ vanishes, that is, $P=P_1$. Let a be a smooth function on $M_1$, then

$$\begin{array}{ccc}\hfill p_1\left(a{U}_1\right)& =& -\left[{\zeta }_1,a{U}_1\right]\left(lnf\right)\hfill \\ \hfill a{p}_1\left(U_1\right)& =& -a\left[{\zeta }_1,U_1\right]\left(lnf\right)-{\zeta }_1\left(a\right)U_1\left(lnf\right)\hfill \\ & =& a{p}_1\left(U_1\right)-{\zeta }_1\left(a\right)U_1\left(lnf\right)\hfill \end{array}$$

and hence, ${\zeta }_1\left(a\right)U_1\left(lnf\right)=0$, and consequently ${\zeta }_1=0$ or f is constant. This discussion results in a nonexistence result of warped product manifolds, which gives us a partially negative answer to the question posed by Besse in [4].

Theorem 6.

A warped product manifold $M=M_1{\times }_f{M}_2$ admitting a projective collineation $\zeta ={\zeta }_1+{\zeta }_2,$ where ${\zeta }_1\ne 0$ is a Riemannian product manifold.

The next case is

$$\left({\mathcal{L}}_{\zeta }\nabla \right)\left(U_2,V_2\right)=p\left(U_2\right)V_2+p\left(V_2\right)U_2.$$

(20)

Since $P_2=0,$ it is $p\left(U_2\right)=f^2{p}_2\left(U_2\right)=0$ and

$$\left({\mathcal{L}}_{{\zeta }_1}\nabla \right)\left(U_2,V_2\right)+\left({\mathcal{L}}_{{\zeta }_2}\nabla \right)\left(U_2,V_2\right)=0.$$

Thus,

$$0=2f\left[{\zeta }_2\left(g_2\left(U_2,V_2\right)\right)-\left({\mathcal{L}}_{{\zeta }_2}^2{g}_2\right)\left(U_2,V_2\right)\right]{\nabla }^{1}f+\left({\mathcal{L}}_{{\zeta }_2}^2{\nabla }^2\right)\left(U_2,V_2\right).$$

(21)

This equation implies

$$\begin{array}{ccc}\hfill \left({\mathcal{L}}_{{\zeta }_2}^2{\nabla }^2\right)\left(U_2,V_2\right)& =& 0\hfill \end{array}$$

(22)

$$\begin{array}{ccc}\hfill \left[{\zeta }_2\left(g_2\left(U_2,V_2\right)\right)-\left({\mathcal{L}}_{{\zeta }_2}^2{g}_2\right)\left(U_2,V_2\right)\right]{\nabla }^{1}f& =& 0.\hfill \end{array}$$

(23)

The vector field ${\zeta }_2$ is an affine collineation on the fiber manifold. In addition, the warping function is constant; otherwise,

$${\zeta }_2\left(g_2\left(U_2,V_2\right)\right)=\left({\mathcal{L}}_{{\zeta }_2}^2{g}_2\right)\left(U_2,V_2\right).$$

This leads us to a second non-existence result of warped product manifolds.

Theorem 7.

A warped product manifold $M=M_1{\times }_f{M}_2$ admitting a projective collineation $\zeta ={\zeta }_1+{\zeta }_2$ is a Riemannian product manifold; otherwise,

$$\left({\mathcal{L}}_{{\zeta }_2}^2{g}_2\right)\left(U_2,V_2\right)={\zeta }_2\left(g_2\left(U_2,V_2\right)\right).$$

Now, assume that f is not constant, then

$$\begin{array}{ccc}\hfill {\zeta }_2\left(g_2\left(U_2,V_2\right)\right)& =& \left({\mathcal{L}}_{{\zeta }_2}^2{g}_2\right)\left(U_2,V_2\right)\hfill \\ \hfill {\zeta }_2\left(g_2\left(U_2,V_2\right)\right)& =& {\zeta }_2\left(g_2\left(U_2,V_2\right)\right)-g_2\left({\mathcal{L}}_{{\zeta }_2}^2{U}_2,V_2\right)-g_2\left(U_2,{\mathcal{L}}_{{\zeta }_2}^2{V}_2\right).\hfill \end{array}$$

Hence,

$$0=g_2\left(\left[{\zeta }_2,U_2\right],V_2\right)+g_2\left(U_2,\left[{\zeta }_2,V_2\right]\right).$$

This equation leads to

$$0=g_2\left(\left[{\zeta }_2,U_2\right],U_2\right)$$

for every vector field $U_2$. The curvature tensor is given by

$$\begin{array}{ccc}& & R^2\left(X_2,Y_2,Z_2,W_2\right)\hfill \\ & =& g_2\left(R^2\left(X_2,Y_2\right)Z_2,W_2\right)\hfill \\ & =& g_2\left({\nabla }_{X_2}^2{\nabla }_{Y_2}^2{Z}_2-{\nabla }_{Y_2}^2{\nabla }_{X_2}^2{Z}_2-{\nabla }_{\left[X_2,Y_2\right]}^2{Z}_2,W_2\right)\hfill \\ & =& g_2\left({\nabla }_{X_2}^2{\nabla }_{Y_2}^2{Z}_2,W_2\right)-g_2\left({\nabla }_{Y_2}^2{\nabla }_{X_2}^2{Z}_2,W_2\right)-g_2\left({\nabla }_{\left[X_2,Y_2\right]}^2{Z}_2,W_2\right).\hfill \end{array}$$

Thus,

$$\begin{array}{ccc}& & g_2\left({\nabla }_{{\zeta }_2}^2{\nabla }_{X_2}^2{\zeta }_2,X_2\right)\hfill \\ & =& {\zeta }_2\left(g_2\left({\nabla }_{X_2}^2{\zeta }_2,X_2\right)\right)-g_2\left({\nabla }_{X_2}^2{\zeta }_2,{\nabla }_{{\zeta }_2}^2{X}_2\right)\hfill \\ & =& {\zeta }_2\left(g_2\left({\nabla }_{{\zeta }_2}^2{X}_2,X_2\right)\right)-g_2\left({\nabla }_{X_2}^2{\zeta }_2,{\nabla }_{{\zeta }_2}^2{X}_2\right)\hfill \\ & =& g_2\left({\nabla }_{{\zeta }_2}^2{\nabla }_{{\zeta }_2}^2{X}_2,X_2\right)+g_2\left({\nabla }_{{\zeta }_2}^2{X}_2,{\nabla }_{{\zeta }_2}^2{X}_2\right)-g_2\left({\nabla }_{X_2}^2{\zeta }_2,{\nabla }_{{\zeta }_2}^2{X}_2\right)\hfill \\ & =& g_2\left({\nabla }_{{\zeta }_2}^2{\nabla }_{{\zeta }_2}^2{X}_2,X_2\right)+g_2\left(\left[{\zeta }_2,X_2\right],{\nabla }_{{\zeta }_2}^2{X}_2\right).\hfill \end{array}$$

Theorem 8.

In a warped product manifold $M=M_1{\times }_f{M}_2$ admitting a projective collineation $\zeta ={\zeta }_1+{\zeta }_2,$ we have the following.

1. 

The component ${\zeta }_1$ is a projective collineation on the base manifold, that is,

$$\left({\mathcal{L}}_{{\zeta }_1}^1{\nabla }^1\right)\left(U_1,V_1\right)=p_1\left(U_1\right)V_1+p_1\left(V_1\right)U_1$$

where $p_1\left(U_1\right)=g_1\left(P_1,U_1\right)$.

2. 

The component ${\zeta }_2$ is an affine collineation on the fiber manifold $M_2$,

3. 

The warping function is constant; otherwise, ${\zeta }_2$ vanishes,

4. 

The component of P tangential to $M_2$ vanishes,

5. 

The component of P tangential to $M_1$ vanishes; otherwise, f is constant.

Theorem 9.

In a non-trivial warped product manifold $M=M_1{\times }_f{M}_2$ admitting a projective collineation $\zeta ={\zeta }_1+{\zeta }_2$, we have the following:

1. 

The associated vector field P vanishes, that is, ζ is an affine collineation,

2. 

The component ${\zeta }_1$ is an affine collineation on the factor manifold $M_1,$

3. 

The component ${\zeta }_2$ vanishes.

The maximum number of affine collineation vector fields in a non-flat manifold $\left(M,g\right)$ is

$$\frac{n\left(n+1\right)}{2}.$$

Theorem 10.

A warped product manifold $M=M_1{\times }_f{M}_2$ admitting independent affine collineations greater than $\frac{1}{2}{n}_1\left(n_1+1\right)$ is a Cartesian product manifold.

For an affine vector field $\zeta$, it is known that $\zeta$ is also a curvature collineation vector field, that is,

$$\left({\mathcal{L}}_{\zeta }R\right)\left(U,V,Z\right)=0.$$

Every curvature collineation is also a Ricci curvature collineation, that is,

$$\left({\mathcal{L}}_{\zeta }Ric\right)\left(U,V\right)=0.$$

In a non-trivial warped product manifold $M=M_1{\times }_f{M}_2$ admitting affine collineation $\zeta ={\zeta }_1+{\zeta }_2$, $\zeta$ is a curvature collineation and consequently is a Ricci curvature collineation. The component ${\zeta }_1$ is a curvature collineation and consequently is a Ricci collineation on the factor manifold $M_1$.

Every curvature collineation and every Ricci curvature collineation is an isometric symmetry in an Einstein manifold.

Theorem 11.

An Einstein warped product manifold $M=M_1{\times }_f{M}_2$ admitting an affine collineation $\zeta ={\zeta }_1+{\zeta }_2$ is a Cartesian product manifold if ζ is Killing.

The following identity gives a different perspective to understanding projective collineations on warped product manifolds. For vector fields $\zeta ,U,V\in \mathfrak{X}\left(M\right)$, it is

$$\left({\mathcal{L}}_{\zeta }\nabla \right)\left(U,V\right)={\nabla }_U{\nabla }_V\zeta -{\nabla }_{{\nabla }_{U}V}\zeta +R\left(\zeta ,U\right)V.$$

Thus,

$$\begin{array}{ccc}\hfill \left({\mathcal{L}}_{\zeta }\nabla \right)\left(U_1,V_1\right)& =& {\nabla }_{U_1}{\nabla }_{V_1}\zeta -{\nabla }_{{\nabla }_{U_1}{V}_1}\zeta +R\left(\zeta ,U_1\right)V_1\hfill \\ & =& {\nabla }_{U_1}{\nabla }_{V_1}{\zeta }_1-{\nabla }_{{\nabla }_{U_1}{V}_1}{\zeta }_1+R\left({\zeta }_1,U_1\right)V_1\hfill \\ & & +{\nabla }_{U_1}{\nabla }_{V_1}{\zeta }_1-{\nabla }_{{\nabla }_{U_1}{V}_1}{\zeta }_1+R\left({\zeta }_1,U_1\right)V_1\hfill \\ & =& \left({\mathcal{L}}_{{\zeta }_1}^1{\nabla }^1\right)\left(U_1,V_1\right)+{\nabla }_{U_1}\left(V_1\left(lnf\right){\zeta }_2\right)-\left({\nabla }_{U_1}^1{V}_1\right)\left(lnf\right){\zeta }_2\hfill \\ & & +R\left({\zeta }_2,U_1\right)V_1\hfill \\ & =& \left({\mathcal{L}}_{{\zeta }_1}^1{\nabla }^1\right)\left(U_1,V_1\right)+\left(U_1\left(V_1\left(lnf\right)\right)+V_1\left(lnf\right)U_1\left(lnf\right){\zeta }_2\right)\hfill \\ & & -\left({\nabla }_{U_1}^1{V}_1\right)\left(lnf\right){\zeta }_2+R\left({\zeta }_2,U_1\right)V_1.\hfill \end{array}$$

(24)

However, the curvature of warped product manifolds

$$\begin{array}{ccc}\hfill R\left({\zeta }_2,U_1\right)V_1& =& \frac{1}{f}{H}^f\left(U_1,V_1\right){\zeta }_2\hfill \\ \hfill U_1\left(V_1\left(lnf\right)\right)+V_1\left(lnf\right)U_1\left(lnf\right)-\left({\nabla }_{U_1}^1{V}_1\right)\left(lnf\right)& =& \frac{1}{f}{U}_1\left(V_1\left(f\right)\right)-\left({\nabla }_{U_1}^1{V}_1\right)\left(f\right)\hfill \\ & =& \frac{1}{f}{H}^f\left(U_1,V_1\right).\hfill \end{array}$$

Then,

$$\left({\mathcal{L}}_{\zeta }\nabla \right)\left(U_1,V_1\right)=\left({\mathcal{L}}_{{\zeta }_1}^1{\nabla }^1\right)\left(U_1,V_1\right).$$

This identity leads us to the same results as those above.

5. Pseudo-Ricci Symmetric Manifolds

In (PRS)${}_n$ space-times, the gradient of the Ricci tensor $R_{ij}$ is given as

$${\nabla }_h{R}_{ij}=2{\omega }_h{R}_{ij}+{\omega }_i{R}_{jh}+{\omega }_j{R}_{ih}.$$

(25)

Interchanging i and h, we find

$${\nabla }_i{R}_{hj}=2{\omega }_i{R}_{hj}+{\omega }_h{R}_{ji}+{\omega }_j{R}_{ih}.$$

(26)

Subtracting (26) from (2), one gets

$${\nabla }_h{R}_{ij}-{\nabla }_i{R}_{hj}={\omega }_h{R}_{ij}-{\omega }_i{R}_{hj}.$$

(27)

Transvecting (2) with $g^{hj}$, we get

$${\nabla }_h{R}_i^h=3{\omega }^j{R}_{ij}+{\omega }_{i}R.$$

Utilizing ${\nabla }_k{R}_i^k=\frac{1}{2}{\nabla }_{i}R$ in the foregoing equation, one infers

$${\nabla }_{i}R=6{\omega }^j{R}_{ij}+2{\omega }_{i}R.$$

(28)

Contracting (2) with $g^{ij}$, one obtains

$${\nabla }_{h}R=2{\omega }_{h}R+2{\omega }^j{R}_{hj}.$$

(29)

The last two equations imply

$$\begin{array}{ccc}\hfill {\omega }^j{R}_{hj}& =& 0,\hfill \end{array}$$

(30)

$$\begin{array}{ccc}\hfill {\nabla }_{h}R& =& 2{\omega }_{h}R.\hfill \end{array}$$

(31)

In addition, it is noted that

$$\begin{array}{ccc}\hfill R^{ij}{\nabla }_h{R}_{ij}& =& 2{\omega }_h{R}^{ij}{R}_{ij}+{\omega }_i{R}^{ij}{R}_{jh}+{\omega }_j{R}^{ij}{R}_{ih}.\hfill \\ \hfill R^{ij}{\nabla }_h{R}_{ij}& =& 2{\omega }_h{R}^{ij}{R}_{ij}\hfill \\ \hfill {\nabla }_h\left(R^{ij}{R}_{ij}\right)& =& 4{\omega }_h{R}^{ij}{R}_{ij}.\hfill \end{array}$$

Thus, the trace of the squared Ricci tensor $\mathit{R}=R^{ij}{R}_{ij}$ satisfies

$${\nabla }_h\mathit{R}=4{\omega }_h\mathit{R}.$$

For a non-zero scalar curvature, it is

$$\begin{array}{ccc}\hfill R{\nabla }_h\mathit{R}& =& 2\left({\nabla }_{h}R\right)\mathit{R}\hfill \\ \hfill {\nabla }_h\left(R\mathit{R}\right)& =& 3\left({\nabla }_{h}R\right)\mathit{R}\hfill \\ \hfill {\nabla }_h\left(R\mathit{R}\right)& =& 6{\omega }_{h}R\mathit{R}.\hfill \end{array}$$

Theorem 12.

In a (PRS)${}_n$ space-time, the Ricci scalars R and $\mathit{R}$ satisfy

$$\begin{array}{ccc}\hfill {\nabla }_{h}R& =& 2{\omega }_{h}R,\hfill \\ \hfill {\nabla }_h\mathit{R}& =& 4{\omega }_h\mathit{R},\hfill \\ \hfill {\nabla }_h\left(R\mathit{R}\right)& =& 6{\omega }_{h}R\mathit{R}.\hfill \end{array}$$

It is clear that

$${\omega }_h={\nabla }_h\left(ln\sqrt{R}\right)={\nabla }_h\left(ln\sqrt[4]{\mathit{R}}\right)={\nabla }_h\left(ln\sqrt[6]{R\mathit{R}}\right),$$

whenever both R and $\mathit{R}$ are positive. Simple calculations imply

$$\mathit{R}={\tau }^{2}R$$

where $\tau$ is a constant.

6. Pseudo-Ricci Symmetric Spacetimes with RCI and PC Vector Field

A vector field $\zeta$ is said to be Ricci curvature inheritance RCI if, for a scalar field $\alpha$, it satisfies

$${\mathcal{L}}_{\zeta }{R}_{ij}=\alpha R_{ij}.$$

(32)

A projective collineation $\zeta$ satisfies

$${\mathcal{L}}_{\zeta }{R}_{ij}=\left(1-n\right){\nabla }_j{p}_i,$$

(33)

where $p_i={\partial }_{i}p$ for some scalar field $p=\mathrm{div}\zeta$. In a particular case, if p is constant, then $\zeta$ reduces to affine collineation (AC). As the vector field $\zeta$ is RCI and PC, hence Equations (32) and (33) together imply

$$\alpha R_{ij}=\left(1-n\right){\nabla }_j{p}_i.$$

(34)

A contraction with $g^{ij}$ gives

$$\alpha R=\left(1-n\right){\nabla }_i{p}^i.$$

(35)

It is known that a PC vector field satisfies

$$p_h{\mathcal{W}}_{ijk}^h=\frac{1}{n-1}{\mathcal{L}}_{\zeta }\left({\nabla }_k{R}_{ij}-{\nabla }_j{R}_{ik}\right).$$

The use of (26) implies

$$\begin{array}{ccc}\hfill p_h{\mathcal{W}}_{ijk}^h& =& \frac{1}{n-1}{\mathcal{L}}_{\zeta }\left({\omega }_k{R}_{ij}-{\omega }_j{R}_{ih}\right),\hfill \\ \hfill p_h{\mathcal{W}}_{ijk}^h& =& \frac{1}{n-1}\left({\lambda }_k{R}_{ij}-{\lambda }_j{R}_{ik}+\alpha {\omega }_k{R}_{ij}-\alpha {\omega }_j{R}_{ik}\right),\hfill \\ \hfill p_h{\mathcal{W}}_{ijk}^h& =& \frac{1}{n-1}\{\left({\lambda }_k{R}_{ij}-{\lambda }_j{R}_{ik}\right)+\alpha \left({\omega }_k{R}_{ij}-{\omega }_j{R}_{ik}\right)\},\hfill \end{array}$$

(36)

where ${\lambda }_k={\mathcal{L}}_{\zeta }{\omega }_k$. Differentiating (34) covariantly, we get

$${\alpha }_k{R}_{ij}+\alpha {\nabla }_k{R}_{ij}=\left(1-n\right){\nabla }_k{\nabla }_j{p}_i.$$

(37)

Interchanging k and $j,$ we find

$${\alpha }_j{R}_{ik}+\alpha {\nabla }_j{R}_{ik}=\left(1-n\right){\nabla }_j{\nabla }_k{p}_i.$$

(38)

Subtracting (37) from (38), one infers

$$p_h{R}_{ijk}^h=\frac{1}{\left(1-n\right)}\left[{\alpha }_j{R}_{ik}-{\alpha }_k{R}_{ij}+\alpha \left({\nabla }_j{R}_{ik}-{\nabla }_k{R}_{ij}\right)\right].$$

Using (25), we obtain

$$p_h{R}_{ijk}^h=\frac{1}{\left(1-n\right)}\left[{\alpha }_j{R}_{ik}-{\alpha }_k{R}_{ij}+\alpha \left({\omega }_j{R}_{ik}-{\omega }_k{R}_{ij}\right)\right].$$

(39)

Multiplying (10) by $p_h,$ one gets

$$p_h{\mathcal{W}}_{ijk}^h=p_h{R}_{ijk}^h-\frac{1}{\left(n-1\right)}\left(p_k{R}_{ij}-p_j{R}_{ki}\right).$$

(40)

Equations (40) and (36) together imply

$$\begin{array}{ccc}& & p_h{R}_{ijk}^h-\frac{1}{\left(n-1\right)}\left(p_k{R}_{ij}-p_j{R}_{ki}\right)\hfill \\ & =& \frac{1}{n-1}\{\left({\lambda }_k{R}_{ij}-{\lambda }_j{R}_{ik}\right)+\alpha \left({\omega }_k{R}_{ij}-{\omega }_j{R}_{ik}\right)\}.\hfill \end{array}$$

(41)

Subtracting (41) from (39), we infer

$$\begin{array}{ccc}\hfill \left({\alpha }_k+p_k-{\lambda }_k\right)R_{ij}& =& \left({\alpha }_j+p_j-{\lambda }_j\right)R_{ik},\hfill \\ \hfill {\eta }_k{R}_{ij}& =& {\eta }_j{R}_{ik},\hfill \end{array}$$

(42)

where ${\eta }_k=\left({\alpha }_k+p_k-{\lambda }_k\right)$.

Assume that ${\eta }_k$ is a unit time-like vector field. A contraction with ${\eta }^k$ yields

$$R_{ij}=-{\eta }^k{\eta }_j{R}_{ik}.$$

(43)

Contracting with $g^{ij}$, we infer

$${\eta }_{k}R={\eta }^j{R}_{jk}.$$

(44)

Using (44) in (43), one deduces

$$R_{ij}=-{\eta }_i{\eta }_{j}R,$$

(45)

which illustrates that the space-time is Ricci simple [31,32]. Hence, we can conclude the following result:

Theorem 13.

A (PRS)${}_n$ space-time with an RCI and PC vector field represents stiff matter fluid.

Now, multiplying (45) by ${\omega }^i$, one infers

$$\left({\omega }^i{\eta }_i\right){\eta }_{j}R=0.$$

A contraction with ${\eta }^j$ implies

$$\left({\omega }^i{\eta }_i\right)R=0,$$

If ${\omega }^i{\eta }_i\ne 0,$ then $R=0$, and hence Equation (45) implies

$$R_{ij}=0.$$

(46)

Theorem 14.

A (PRS)${}_n$ space-time with an RCI and PC vector field is a vacuum provided the associated $1-$form ω is not co-directional with η.

The divergence of the conformal curvature tensor $\mathcal{C}$ is given as

$${\nabla }_h{\mathcal{C}}_{ijk}^h=\frac{n-3}{n-2}\left[\left({\nabla }_k{R}_{ij}-{\nabla }_j{R}_{ik}\right)-\frac{1}{2\left(n-1\right)}\left(g_{ij}{\nabla }_{k}R-g_{ik}{\nabla }_{j}R\right)\right].$$

(47)

The use of (46) and its contraction in (47) give

$${\nabla }_h{\mathcal{C}}_{ijk}^h=0,$$

(48)

which demonstrates that the conformal curvature tensor is divergence-free. In [31], Mantica et al. proved that a Lorentzian manifold $M^n\left(n\succ 3\right)$ whose Ricci tensor is of the form $R_{ij}=-{\eta }_i{\eta }_{j}R$, and the conformal curvature tensor that is divergence-free is a GRW space-time. Thus, in view of (45) and (48), we can state the following result:

Theorem 15.

A (PRS)${}_n$ space-time with an RCI and PC vector field is a GRW space-time, provided ${\omega }^i{\eta }_i\ne 0$.