Gray s decomposition on doubly warped product manifolds and applications

TL;DR

This paper investigates conditions under which factor manifolds of doubly warped product manifolds inherit Gray's decomposition properties, leading to new insights into Einstein-like structures and applications in space-time models.

Contribution

It provides necessary and sufficient conditions on warping functions for factor manifolds to belong to the same Einstein-like class, extending Gray's decomposition to doubly warped products.

Findings

Inheritance of Einstein-like properties under specific warping conditions

Characterization of Einstein-like doubly warped space-times of types A, B, and P

Extension of Gray's decomposition to complex manifold structures

Abstract

A. Gray presented an interesting $O\left( n\right) $ invariant decomposition of the covariant derivative of the Ricci tensor. Manifolds whose Ricci tensor satisfies the defining property of each orthogonal class are called Einstein-like manifolds. In the present paper, we answered the following question: Under what condition(s), does a factor manifold $M_{i},i=1,2$ of a doubly warped product manifold $M=_{f_{2}}M_{1}\times _{f_{1}}M_{2}$ lie in the same Einstein-like class of $M$? By imposing sufficient and necessary conditions on the warping functions, an inheritance property of each class is proved. As an application, Einstein-like doubly warped product space-times of type $\mathcal{A},$ $\mathcal{B}$ or $\mathcal{P}$ are considered.

Equations

$${\mathcal{L}}_{ij}=R_{ij}-\frac{2R}{n+2}{g}_{ij}$$

$${\mathcal{H}}_{ij}=R_{ij}-\frac{R}{2\left(n-1\right)}{g}_{ij}$$

$$g={\left(f_2\circ {\pi }_2\right)}^2{\pi }_1^{\ast }\left(g_1\right)\oplus {\left(f_1\circ {\pi }_1\right)}^2{\pi }_2^{\ast }\left(g_2\right),$$

$${\mathcal{F}}^i\left(X_i,Y_i\right)=\frac{n_j}{f_i}{H}^{f_i}\left(X_i,Y_i\right),$$

$$D_{X_i}{X}_j$$

$$D_{X_i}{Y}_i$$

$$\mathrm{Ric}\left(X_i,Y_i\right)$$

$$\mathrm{Ric}\left(X_i,Y_j\right)$$

$$\left(D_X\mathrm{Ric}\right)\left(Y,Z\right)+\left(D_Y\mathrm{Ric}\right)\left(Z,X\right)+\left(D_Z\mathrm{Ric}\right)\left(X,Y\right)=0,$$

$$\left(D_X\mathrm{Ric}\right)\left(X,X\right)=0,$$

$$\left(D_{X_i}^i{\mathcal{F}}^i\right)\left(X_i,X_i\right)=\frac{2}{f_i^3}{X}_i\left(f_i\right)g_i\left(X_i,X_i\right)\left[f_j^{\diamond }+\left(n-2\right)\left({\nabla }^j{f}_j\right)\left(f_j\right)\right],$$

$$0$$

$$\left(D_X\mathrm{Ric}\right)\left(Y,Z\right)=\left(D_Y\mathrm{Ric}\right)\left(X,Z\right).$$

$$\left(D_{X_i}^i{\mathcal{F}}^i\right)\left(Y_i,Z_i\right)$$

$$B\left(X,Y\right)Z=\left(D_X\mathrm{Ric}\right)\left(Y,Z\right)-\left(D_Y\mathrm{Ric}\right)\left(X,Z\right).$$

$$B\left(X_i,Y_i,Z_i\right)=\left(D_{X_i}\mathrm{Ric}\right)\left(Y_i,Z_i\right)-\left(D_{Y_i}\mathrm{Ric}\right)\left(X_i,Z_i\right).$$

$$\left(D_{X_i}\mathrm{Ric}\right)\left(Y_i,Z_i\right)$$

$$\left(D_{X_1}^1{\mathcal{F}}^1\right)\left(Y_1,Z_1\right)=\left(D_{Y_1}^1{\mathcal{F}}^1\right)\left(X_1,Z_1\right),$$

$$\left(D_X\mathrm{Ric}\right)\left(Y,Z\right)=0.$$

$$\left(D_{X_i}^i{\mathcal{F}}^i\right)\left(Y_i,Z_i\right)$$

$$0=\left(D_X\mathrm{Ric}\right)\left(Y,Z\right)$$

$$\left(D_{X_i}\mathrm{Ric}\right)\left(Y_i,Z_i\right)$$

$$\left(D_{X_i}^i{\mathrm{Ric}}^i\right)\left(Y_i,Z_i\right)$$

$$\left(D_{X_1}^1{\mathcal{F}}^1\right)\left(Y_1,Z_1\right)=0,$$

$${\nabla }_{\gamma }\left[R_{\alpha \beta }-\frac{R}{2\left(n-1\right)}{g}_{\alpha \beta }\right]={\nabla }_{\alpha }\left[R_{\gamma \beta }-\frac{R}{2\left(n-1\right)}{g}_{\gamma \beta }\right],$$

$${\nabla }_{\epsilon }{\mathcal{C}}_{\alpha \beta \gamma }^{\epsilon }=0,$$

$${\nabla }_{\epsilon }{\mathcal{C}}_{\alpha \beta \gamma }^{\epsilon }={\bar{\nabla }}_{\epsilon }{\bar{\mathcal{C}}}_{\alpha \beta \gamma }^{\epsilon }-\frac{n-3}{\phi }\left({\nabla }_{\epsilon }\phi \right){\bar{\mathcal{C}}}_{\alpha \beta \gamma }^{\epsilon }.$$

$$g$$

Full text

1. An introduction

Alfred Gray in [22] presented $O\left(n\right)$ invariant orthogonal irreducible decomposition of the space $W$ of all $\left(0,3\right)$ tensors satisfying only the identities of the gradient of the Ricci tensor $\nabla_{k}R_{ij}$. The space $W$ is decomposed into three orthogonal irreducible subspaces, that is, $W=\mathcal{A\oplus B\oplus I}$. This decomposition produces seven classes of Einstein-like manifolds, that is, manifolds whose Ricci tensor satisfies the defining identity of each subspace. They are the trivial class $\mathcal{P}$, the classes $\mathcal{A}$, $\mathcal{B}$, $\mathcal{I}$ and three composite classes $\mathcal{I\oplus A}$, $\mathcal{I\oplus B}$ and $\mathcal{A\oplus B}$.

In class $\mathcal{P}$, the Ricci tensor is parallel i.e. $\nabla_{k}R_{ij}=0$ whereas class $\mathcal{A}$ contains manifolds whose Ricci tensor is Killing. The Ricci tensor of manifolds in class $\mathcal{B}$ is a Codazzi tensor i.e. $\nabla_{k}R_{ij}=\nabla_{i}R_{kj}$. The traceless part of the Ricci tensor vanishes in class $\mathcal{I}$ i.e. class $\mathcal{I}$ contains Sinyukov manifolds[26]. The tensor

[TABLE]

is Killing in class $\mathcal{I\oplus A}$ whereas the tensor

[TABLE]

is a Codazzi tensor in class $\mathcal{I\oplus B}$. The class $\mathcal{A\oplus B}$ is identified by having constant scalar curvature. The same decomposition is discussed extensively in [3, Chapter 16] (see also [24, 26] and Section 3 for more details and equivalent conditions). Thereafter, Einstein-like manifolds have been studied by many authors such as G. Calvaruso in [7, 8, 9, 10] Mantica et al in [24, 25, 26] and many others [2, 5, 6, 30, 36]. An interesting study in [26] shows Einstein-like generalized Robertson-Walker space-times are perfect fluid space-times except those in class $\mathcal{I}$ which are not restricted. Sufficient conditions on generalized Robertson-Walker space-times in this class to be a perfect fluid are derived in [13].

Doubly warped products is a generalization of singly warped products introduced in [4]. The geometric properties of doubly warped product manifolds have been investigated by many authors such as pseudo-convexity in [1], harmonic Weyl conformal curvature tensor in [18], conformal flatness in [20, 21], geodesic completeness in [35], doubly warped product submanifolds in [17, 28, 29, 31] and conformal vector fields in [15]. Doubly warped space-times are widely used as exact solutions of Einstein’s field equations. Recently, the existence of compact Einstein doubly warped product manifolds is considered in [23].

Inspired by the above studies of Einstein-like metrics and doubly warped product manifolds, we studied doubly warped product manifolds equipped with Einstein-like metrics. The inheritance properties of the Einstein-like class type $\mathcal{P}$, $\mathcal{A},$ $\mathcal{B}$, $\mathcal{I\oplus A}$, $\mathcal{I\oplus B}$ or $\mathcal{A\oplus B}$ are investigated. To assure that factor manifolds of a doubly warped product manifold inherits the Einstein-like class type, sufficient and necessary conditions are derived on the warping functions. Finally, we apply the results to doubly warped space-times.

2. Preliminaries

A doubly warped product manifold is the (pseudo-)Riemannian product manifold $M=M_{1}\times M_{2}$ of two (pseudo-)Riemannian manifolds $(M_{i},g_{i},D_{i}),i=1,2,$ furnished with the metric tensor

[TABLE]

where the functions $f_{i}:M_{i}\rightarrow\left(0,\infty\right),$ $i=1,2$ are the warping functions of $M$. $M$ is denoted by ${}_{f_{2}}M_{1}\times_{f_{1}}M_{2}$. The maps $\pi_{i}:M_{1}\times M_{2}\rightarrow M_{i}$ are the natural projections $M$ onto $M_{i}$ whereas ∗ denotes the pull-back operator on tensors. In particular, if for example $f_{2}=1$, then $M=M_{1}\times_{f_{1}}M_{2}$ is called a singly warped product manifold (see [15, 35] for doubly warped products and [4, 14, 16, 27, 33, 34] for singly warped products).

Notation 1**.**

Throughout this work, we use the following notations

1. (1)

All tensor fields on $M_{i}$ are identified with their lifts to $M$. For example, we use $f_{i}$ for a function on $M_{i}$ and for its lift $\left(f_{i}\circ\pi_{i}\right)$ on $M$. 2. (2)

The manifolds $M_{i}$ has dimensions $n_{i}$ where $n=n_{1}+n_{2}$. 3. (3)

$\mathrm{Ric}$* is the Ricci curvature tensor on $M$ and $\mathrm{Ric}$i is the Ricci tensor on $M^{i}$.* 4. (4)

The gradient of $f_{i}$ on $M_{i}$ is denoted by $\nabla^{i}f_{i}$ and the Laplacian by $\bigtriangleup^{i}f_{i}$ whereas $f_{i}^{\diamond}=f_{i}\bigtriangleup^{i}f_{i}+\left(n_{j}-1\right)g_{i}\left(\nabla^{i}f_{i},\nabla^{i}f_{i}\right),$ $i\neq j$. 5. (5)

The indices $i$ and $j$ to denote the geometric objects of the factor manifolds $M_{i}$ and $M_{j}$. 6. (6)

The $\left(0,2\right)$ tensors $\mathcal{F}^{i}$ is defined as

[TABLE]

for $X_{i},Y_{i}\in\mathfrak{X}\left(M_{i}\right)$ and $i,j=1,2,i\neq j$.

The Levi-Civita connection $D$ on $M=_{f_{2}}M_{1}\times_{f_{1}}M_{2}$ is given by

[TABLE]

where $i\neq j$ and $X_{i},Y_{i}\in\mathfrak{X}\left(M_{i}\right)$. Then the Ricci curvature tensor $\mathrm{Ric}$ on $M$ is given by

[TABLE]

where $i\neq j$ and $X_{i},Y_{i}\in\mathfrak{X}\left(M_{i}\right)$. The reader is referred to [12, 11, 19] for some studies of curvature conditions on warped product manifolds.

3. Einstein-like doubly warped product manifolds

The Einstein-like doubly warped product manifolds $M=_{f_{2}}M_{1}\times_{f_{1}}M_{2}$ are investigated in this section. Every subsection is devoted to the study of a class of Einstein-like doubly warped product manifolds. Sufficient and necessary conditions are derived on the warping functions $f_{i}$ for factor manifolds $M_{i}$ to acquire the same Einstein-like class type.

3.1. Class $\mathcal{A}$

A doubly warped product manifold $\left(M,g\right)$ whose Ricci tensor is Killing, that is,

[TABLE]

for any vector fields $X,Y,Z\in\mathfrak{X}\left(M\right)$ is called Einstein-like doubly warped product manifold of class $\mathcal{A}$. This condition equivalent to

[TABLE]

for any vector field $X\in\mathfrak{X}\left(M\right)$ and the Ricci tensor is also called cyclic parallel. The legacy of factor manifolds of $M$ in class $\mathcal{A}$ is as follows.

Theorem 1**.**

In a doubly warped product manifold $M=_{f_{2}}M_{1}\times_{f_{1}}M_{2}$ where $M$ is of class type $\mathcal{A}$, a factor manifold $\left(M_{i},g_{i}\right)$ is an Einstein-like manifold of class $\mathcal{A}$ if and only if

[TABLE]

where $i,j=1,2,i\neq j$ and $X_{i}\in\mathfrak{X}\left(M_{i}\right)$.

Proof.

In a doubly warped product manifold $M=_{f_{2}}M_{1}\times_{f_{1}}M_{2}$ of class $\mathcal{A}$, it is

[TABLE]

Thus, for a the special case where $X=X_{i}$ lands on one factor, one may get

[TABLE]

Thus, after lengthy computations, it is

[TABLE]

These equations complete the proof. ∎

It is now easy to recover a similar result on singly warped product manifolds.

Corollary 1**.**

In a singly warped product manifold $M=M_{1}\times_{f_{1}}M_{2}$ where $M$ is of class type $\mathcal{A}$, $\left(M_{1},g_{1}\right)$ is an Einstein-like manifold of class $\mathcal{A}$ if and only if $\mathcal{F}^{i}$ is Killing. In addition, $\left(M_{2},g_{2}\right)$ is of class type $\mathcal{A}$.

3.2. Class $\mathcal{B}$

Let $M$ be as Einstein-like doubly warped product manifold of class $\mathcal{B}$. Then, the Ricci tensor is a Codazzi tensor,that is,

[TABLE]

The above condition is equivalent to:

1. (1)

$M$ has a harmonic Riemann tensor, that is, $\nabla_{\varepsilon}\mathcal{R}_{\alpha\beta\gamma}^{\text{ \ \ }\varepsilon}=0$, or 2. (2)

$M$ admits a harmonic Weyl conformal tensor and the scalar curvature is constant, that is, $\nabla_{\varepsilon}\mathcal{C}_{\alpha\beta\gamma}^{\text{ \ \ }\varepsilon}=0$ and $\nabla_{\varepsilon}R=0$.

The base manifold and the fiber manifold gain the Einstein-like class type $\mathcal{B}$ according to.

Theorem 2**.**

In a doubly warped product manifold $M=_{f_{2}}M_{1}\times_{f_{1}}M_{2}$ where $M$ is of class type $\mathcal{B}$, the factor manifold $\left(M_{i},g_{i}\right)$ is an Einstein-like manifold of class $\mathcal{B}$ if and only if

[TABLE]

where $i,j=1,2,i\neq j$ and $X_{i},Y_{i},Z_{i}\in\mathfrak{X}\left(M_{i}\right)$.

Proof.

Let us define the deviation tensor $B\left(X,Y,Z\right)$ as follows

[TABLE]

There are three different cases. Let us consider the first case, that is,

[TABLE]

It is enough to find $\left(D_{X_{i}}\mathrm{Ric}\right)\left(Y_{i},Z_{i}\right)$ as

[TABLE]

Simplifying this expression, it is

[TABLE]

By exchanging $X_{i}$ and $Y_{i}$ in the last equation and substitution in Equation (3.1), one gets the deviation tensor. For Einstein-like manifolds of class $\mathcal{B}$, the deviation tensor vanishes from which the result hold. ∎

It is easy to retrieve a similar result on a singly warped product manifold.

Corollary 2**.**

In a singly warped product manifold $M=M_{1}\times_{f_{1}}M_{2}$ where $M$ is of class type $\mathcal{B}$, $\left(M_{1},g_{1}\right)$ is an Einstein-like manifold of class $\mathcal{B}$ if and only if

[TABLE]

where $X_{1},Y_{1},Z_{1}\in\mathfrak{X}\left(M_{1}\right)$. In addition, $\left(M_{2},g_{2}\right)$ is Einstein-like of class type $\mathcal{B}$.

3.3. Class $\mathcal{P}$

Let $M$ be an Einstein-like doubly warped product manifold of class $\mathcal{P}$. Thus, $M$ has a parallel Ricci tensor, that is,

[TABLE]

Manifolds in this class are usually called Ricci symmetric.

Theorem 3**.**

In a doubly warped product manifold $M=_{f_{2}}M_{1}\times_{f_{1}}M_{2}$ where $M$ is of class type $\mathcal{P}$, $\left(M_{i},g_{i}\right)$ is an Einstein-like manifold of class $\mathcal{P}$ if and only if

[TABLE]

where $i,j=1,2,i\neq j$ and $X_{i},Y_{i},Z_{i}\in\mathfrak{X}\left(M_{i}\right)$.

Proof.

Let $M=_{f_{2}}M_{1}\times_{f_{1}}M_{2}$ be a Ricci symmetric doubly warped product manifold, that is,

[TABLE]

Equation (3.2) infers

[TABLE]

Thus, having a parallel Ricci tensor implies

[TABLE]

This equation completes the proof. ∎

The corresponding result on singly warped product manifolds is as follows.

Corollary 3**.**

In a singly warped product manifold $M=M_{1}\times_{f_{1}}M_{2}$ where $M$ is of class type $\mathcal{P}$. Then $\left(M_{1},g_{1}\right)$ is an Einstein-like manifold of class $\mathcal{P}$ if and only if

[TABLE]

where $X_{1},Y_{1},Z_{1}\in\mathfrak{X}\left(M_{1}\right)$. Also, $\left(M_{2},g_{2}\right)$ is Einstein-like of class type $\mathcal{P}$.

3.4. Class $\mathcal{I\oplus B}$

A doubly warped product manifold $M$ is of class type $\mathcal{I\oplus B}$ if its Ricci tensor satisfies

[TABLE]

that is, the tensor $\mathcal{H}_{\alpha\beta}=R_{\alpha\beta}-\frac{R}{2\left(n-1\right)}g_{\alpha\beta}$ is a Codazzi tensor. This condition is equivalent to

[TABLE]

where $\mathcal{C}$ is the Weyl conformal curvature tensor and $n\geq 3$, i.e., $M$ has a harmonic Weyl tensor. Let $g_{\beta\gamma}=\varphi^{2}\bar{g}_{\beta\gamma}$ be a conformal change of on a manifold $M$. It is well known that the Weyl tensor $\mathcal{C}_{\alpha\beta\gamma}^{\text{ \ \ }\varepsilon}$ remains invariant, that is, $\mathcal{\bar{C}}_{\alpha\beta\gamma}^{\text{ \ \ }\varepsilon}=\mathcal{C}_{\alpha\beta\gamma}^{\text{ \ \ }\varepsilon}$ however $\mathcal{C}_{\alpha\beta\gamma\varepsilon}=\varphi^{2}\mathcal{\bar{C}}_{\alpha\beta\gamma\varepsilon}$. The divergence of the Weyl tensor is given by[3]

[TABLE]

The doubly warped product metric may be rewritten as follows

[TABLE]

where $g_{i}=f_{i}^{2}\bar{g}_{i}$ and $\bar{g}=\bar{g}_{1}+\bar{g}_{2}$. The doubly warped product manifold $\left(M,g\right)$ has harmonic Weyl tensor if and only

[TABLE]

where $\varphi=f_{1}f_{2}$. Assume that $\nabla_{\varepsilon}\left(f_{1}f_{2}\right)\mathcal{C}_{\alpha\beta\gamma}^{\text{ \ \ }\varepsilon}=0$, then

[TABLE]

having a harmonic Weyl tensor is equivalent to the condition

[TABLE]

where $\mathcal{\bar{T}}$ is the Cotton tensor. The metric $\bar{g}$ splits as $\bar{g}=\bar{g}_{1}+\bar{g}_{2}$ and consequently the divergence of the Cotton tensor $\mathcal{\bar{T}}$ splits on the factor manifolds $\left(M_{i},\bar{g}_{i}\right)$ as

[TABLE]

In this case, $\left(\bar{\nabla}_{\gamma}^{i}\bar{R}^{i}\right)\left(\bar{g}_{i}\right)_{\alpha\beta}-\left(\bar{\nabla}_{\gamma}^{i}\bar{R}^{i}\right)\left(\bar{g}_{i}\right)_{\alpha\beta}=0$, that is, $\bar{R}^{i}$ is constant if and only if the cotton tensor $\mathcal{\bar{T}}^{i}$ on the doubly warped factor manifolds $\left(M^{i},\bar{g}_{i}\right)$ vanishes i.e.

[TABLE]

The Weyl tensors $\mathcal{C}^{i}$ on doubly warped product factor manifolds $\left(M_{i},g_{i}\right)$ satisfy

[TABLE]

It is time now to write the following result.

Theorem 4**.**

In a doubly warped product manifold $M=_{f_{2}}M_{1}\times_{f_{1}}M_{2}$ where $M$ is of class type $\mathcal{I\oplus B}$. Assume that $\nabla_{\varepsilon}\left(f_{1}f_{2}\right)\mathcal{C}_{\alpha\beta\gamma}^{\text{ \ \ }\varepsilon}=0$ and the conformal change $\left(M_{i},f_{i}^{-2}g_{i}\right)$ has a constant scalar curvature. Then $\left(M_{i},g_{i}\right)$ is an Einstein-like manifold of class $\mathcal{I\oplus B}$ if and only if $\left(\nabla_{\varepsilon}^{i}f_{i}\right)\mathcal{C}_{\alpha\beta\gamma}^{i\text{ \ \ }\varepsilon}=0$ for each $i=1,2$.

A. Gebarowski proved an inheritance property of this class in [18, Theorem 2].

3.5. Class $\mathcal{I\oplus A}$

Doubly warped product manifolds where the tensor

[TABLE]

is Killing lies the class $\mathcal{I\oplus A}$. The above condition is equivalent to

[TABLE]

The following theorem draw the inheritance property of this class.

Theorem 5**.**

In a doubly warped product manifold $M=_{f_{2}}M_{1}\times_{f_{1}}M_{2}$ where $M$ is of class type $\mathcal{I\oplus A}$, the factor manifold $\left(M_{i},g_{i}\right)$ is of class type $\mathcal{I\oplus A}$ if and only if

[TABLE]

Proof.

Assume that $M=_{f_{2}}M_{1}\times_{f_{1}}M_{2}$ be a doubly warped product manifold of class type $\mathcal{I\oplus A}$. Then

[TABLE]

Using equation (3.2), it is

[TABLE]

and consequently, one has

[TABLE]

which completes the proof. ∎

3.6. Class $\mathcal{A\oplus B}$

This class is identified by having a constant scalar curvature. Let $M=_{f_{2}}M_{1}\times_{f_{1}}M_{2}$ be a doubly warped product manifold of class type $\mathcal{A\oplus B}$, that is, the scalar curvature $R$ of $M$ is constant, say $c$. The use of Equation 7 in [18] implies that $M_{i}$ is of class $\mathcal{A\oplus B}$ if there are two constants $c_{i}$ and $c_{j}$ such that

[TABLE]

where $F_{i}=g_{i}^{\alpha\beta}\nabla_{\alpha}^{i}\nabla_{\beta}^{i}f_{i}$.

4. Einstein-like doubly warped Relativistic space-times

Let $(M,g)$ be a Riemannian manifold, $f:M\rightarrow\left(0,\infty\right)$ and $\sigma:I\rightarrow\left(0,\infty\right)$ are smooth functions. The manifold $\bar{M}=_{f}I\times_{\sigma}M$ furnished with the metric tensor $\bar{g}=-f^{2}dt^{2}\oplus\sigma^{2}g$ is called a doubly warped space-time. For $U,V\in\mathfrak{X}\left(M\right),$ the covariant derivative $\bar{D}$ on $\bar{M}$ is given by

[TABLE]

whereas the Ricci tensor $\mathrm{\bar{R}ic}$ on $\bar{M}$ is given by

[TABLE]

For the definition and relativistic significance of doubly warped space-times, the reader is referred to [15, 32] and references therein.

Theorem 6**.**

In a doubly warped space-time $\bar{M}=_{f}I\times_{\sigma}M$ of class type $\mathcal{A}$, $M$ is an Einstein-like manifold of class type $\mathcal{A}$ if and only if

[TABLE]

Theorem 7**.**

In a doubly warped space-time $\bar{M}=_{f}I\times_{\sigma}M$ of class type $\mathcal{B}$, $M$ is an Einstein-like manifold of class type $\mathcal{B}$ if and only if

[TABLE]

Theorem 8**.**

In a doubly warped space-time $\bar{M}=_{f}I\times_{\sigma}M$ of class type $\mathcal{P}$, $M$ is an Einstein-like manifold of class type $\mathcal{P}$ if and only if

[TABLE]