The object of the present paper is to study super quasi-Einstein manifolds. Some geometric properties of super quasi-Einstein manifolds have been studied. We also discuss
A Riemannian or semi-Riemannian manifold (
holds on M, where
where
for all vector fields
A non-flat Riemannian or semi-Riemannian manifold (
It is to be noted that Chaki and Maity [9] also introduced the notoin of quasi-Einstein manifolds in a different way. They have taken a, b are scalars and the vector field U metrically equivalent to the 1-form A as a unit vector field. Such an n-dimensional manifold is denoted by (
Quasi-Einstein manifolds arose during the study of exact solutions of the Einstein field equations as well as during considerations of quasi-umbilical hypersurfaces of semi-Euclidean spaces. For instance, the Robertson-Walker spacetimes are quasi-Einstein manifolds. So quasi-Einstein manifolds have some importance in the general theory of relativity.
Quasi-Einstein manifolds have been generalized by several authors in several ways such as generalized quasi-Einstein manifolds [7, 14, 40], super quasi-Einstein manifolds [8, 21, 34] and many others.
As a generalization of quasi-Einstein manifold Chaki introduced the notion of a super quasi-Einstein manifold in [8]. According to him a non-flat Riemannian or semi-Riemannian manifold (
where
for all
A Riemannian or semi-Riemannian manifold of quasi-constant curvature was given by Chen and Yano [10] as a conformally flat manifold with the curvature tensor
where
for all
It can be easily seen that if the curvature tensor
Subsequently Chaki [8] generalized the notion of (
A non-flat Riemannian or semi-Riemannian manifold (
where
The notion of quasi-conformal curvature tensor was given by Yano and Sawaki [43] and is defined as follows:
where
If
The spacetime of general relativity and cosmology is regarded as a connected 4-dimensional semi-Riemannian manifold (
The present paper is organized as follows:
After introduction in Section 2 we study
A
Using (
where
In virtue of (
Now we look for sufficient condition in order that a
In a
which implies that the scalar curvature
We know that (
since
Contracting (
Imposing the condition that the generators
and
Therefore from (
Thus we can state the following:
In this section let us consider the generators
and
where
From (
and
Since
and
for all
Similarly, we have
for all
We also assume that the associated scalars are constants. Then from (
Using (
By virtue of (
Thus we can state the following theorem:
A vector field
where
In this section we consider the vector fields
and
where
Now using (
Contracting (
where
Now contracting (
Thus,
Since
Thus the manifold reduces to a pseudo quasi-Einstein manifold. Hence we can state the following:
A vector fields
where
In this section we suppose that the generators
and
where λ and
A non-flat Riemannian or semi-Riemannian manifold (
where
Now, using (
We assume that the 1-forms λ and
for all Z. Then we obtain from (
Using (
where
Thus we can state the following:
In a smooth manifold (
where
for all
where
In this section we first study
If
The Einstein’s field equation without cosmological constant is given by [32, 33]
where
Now using (
In virtue of (
Thus we can state the following:
But it has been proved by Shaikh[39] that if a viscous fluid pseudo quasi-Einstein spacetime obeys Einstein’s field equation with a cosmological constant, then none of the energy density and isotropic pressure of the fluid can be a constant.
In view of Theorem 4.1 and the result of Shaikh leads to the following theorem:
This result has significant agreement with the recent day observational truth that energy density and isotropic pressure can not be a constant. Here we discuss the form of energy momentum tensor [27] as follows:
where
Hence we conclude that if the associated vector fields of a
In this section we prove the existence of a
We consider a Lorentzian manifold (
where
The only non-vanishing components of the Christoffel symbols, the curvature tensor and the Ricci tensor are
We take the scalars
We choose the 1-forms as follows:
and
at any point
We take the associated tensor as follows:
at any point
It can be easily prove that the
We shall now show that the 1-forms are unit and orthogonal. Here,
So, the manifold under consideration is a
The author is thankful to the referee for his/her critical remarks and constructive suggestions towards the improvement of the paper.