A Riemannian or a semi-Riemannian manifold (Mⁿ, g), n = dimM ≥ 2, is said to be an Einstein manifold if the following condition

holds on M, where S and r denote the Ricci tensor and the scalar curvature of (Mⁿ, g) respectively. According to Besse ([3], p. 432), (1.1) is called the Einstein metric condition. Einstein manifolds play an important role in Riemannian Geometry as well as in general theory of relativity. Also Einstein manifolds form a natural subclass of various classes of Riemannian or semi-Riemannian manifolds by a curvature condition imposed on their Ricci tensor ([3], p. 432–433). For instance, every Einstein manifold belongs to the class of Riemannian or semi-Riemannian manifolds (Mⁿ, g) realizing the following relation :

where a, b are smooth functions and η is a non-zero 1-form such that

for all vector fields X.

A non-flat Riemannian or semi-Riemannian manifold (Mⁿ, g) (n > 2) is defined to be a quasi Einstein manifold ([7]) if its Ricci tensor S of type (0, 2) is not identically zero and satisfies the condition (1.2). We shall call η the associated 1-form and the unit vector field ξ is called the generator of the manifold.

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. Several authors have studied Einstein’s field equations. For example, in [17], Naschie turned the tables on the theory of elementary particles and showed the expectation number of elementary particles of the standard model using Einstein’s unified field equation. He also discussed possible connections between Gödel’s classical solution of Einstein’s field equations and E-infinity in [16]. Also quasi-Einstein manifolds have some importance in the general theory of relativity. For instance, the Robertson-Walker spacetimes are quasi Einstein manifolds. Further, quasi Einstein manifold can be taken as a model of the perfect fluid spacetime in general relativity [10]. Perfect fluid spacetimes in n-dimensions subjected to the restriction ∇mCjklm=0, where C is the Weyl conformal curvature tensor, recently investigated in [27] by Mantica, Molinari and De (see also [28]).

The study of quasi Einstein manifolds was continued by Chaki ([8]), Guha ([18]), De and Ghosh ([11, 12]), Bejan ([1]), Debnath and Konar ([14]) and many others. The notion of quasi-Einstein manifolds have been generalized by several authors in several ways such as generalized Einstein manifolds ([2]), generalized quasi-Einstein manifolds ([4, 13]), mixed generalized quasi-Einstein manifolds[5] and many others. In recent papers Özgür studied super quasi-Einstein manifolds ([32]) and generalized quasi-Einstein manifolds ([33]). Also Nagaraja ([29]) studied N(k)-mixed quasi-Einstein manifolds.

Let R denote the Riemannian curvature tensor of a Riemannian manifold M. The k-nullity distribution N(k) of a Riemannian manifold M ([36]) is defined by

k being some smooth function. In a similar way we can define the k-nullity distribution N(k) of a Lorentzian manifold. In a quasi Einstein manifold M, if the generator ξ belongs to some k-nullity distribution N(k), then M is said to be an N(k)-quasi Einstein manifold ([30]). In fact k is not arbitrary as the following:

Lemma 1

([30]).In an n-dimensional N(k)-quasi Einstein manifold it follows that

Now, it is immediate to note that in an n-dimensional N(k)-quasi-Einstein manifold ([30])

which is equivalent to

From (1.5) we get

In [30] it was shown that an n-dimensional conformally flat quasi Einstein manifold is an N(a+bn-1)-quasi Einstein manifold and in particular a 3-dimensional quasi Einstein manifold is an N(a+b2)-quasi Einstein manifold. Also in [31] Özgür, cited some physical examples of N(k)-quasi Einstein manifolds. Recently, Taleshian and Hosseinzadeh ([20, 35]), De, De and Gazi ([9]) studied some curvature conditions on N(k)-quasi-Einstein manifold. All these motivated us to study such a manifold.

The conformal curvature tensor play an important role in differential geometry and also in general theory of relativity. The Weyl conformal curvature tensor C of a Riemannian manifold (Mⁿ, g) (n > 3) is defined by

where r is the scalar curvature and Q is the symmetric endomorphism of the tangent space at each point corresponding to the Ricci tensor S, that is, g(QX, Y) = S(X, Y ). If the dimension n = 3, then the conformal curvature tensor vanishes identically. The conformal curvature tensor have been studied by several authors in several ways such as ([15, 19, 21, 22]) and many others.

The projective curvature tensor P in a Riemannian manifold (Mⁿ, g) is defined by ([37])

In 2012, Mantica and Molinari ([26]) defined a generalized (0,2) symmetric tensor given by

where φ is an arbitrary scalar function. In Refs. [23], [26] and [24] various properties of the tensor were pointed out.

The derivation conditions R(ξ,X) · R = 0 and R(ξ,X) · S = 0 have been studied in [36], where R and S denote the curvature and Ricci tensor respectively. In [30], the derivation conditions C̄ (ξ,X) · R = 0 and C̄(ξ,X) · C̄ = 0 on N(k)-quasi-Einstein manifold were studied, where C̄ is the concircular curvature tensor. Moreover in [30], for an N(k)-quasi-Einstein manifold it was proved that k=a+bn-1. Özgür in [31] studied the condition R · P = 0, P · S = 0 and P · P = 0 for an N(k)-quasi-Einstein manifolds, where P denotes the projective curvature tensor and some physical examples of N(k)-quasi-Einstein manifolds are given. Again, in 2008, Özgür and Sular ([34]) studied N(k)-quasi-Einstein manifold satisfying R · C = 0 and R · C̃ = 0, where C and C̃ represent the conformal curvature tensor and the quasi-conformal curvature tensor, respectively. Recently, Yildiz, De and Cetinkaya ([38]) studied quasi-conformally recurrent N(k)-quasi-Einstein manifold. This paper is a continuation of the previous studies.

The paper is organized as follows:

After preliminaries in Section 3, we study an N(k)-quasi-Einstein manifold satisfying the curvature condition R(ξ,X) · = 0. In the next two sections, we study an N(k)-quasi-Einstein manifold satisfying (X, ξ) ·R = 0, and P(ξ,X) · = 0 respectively. Section 6 deals with the nature of the curvature condition C · = 0 in an N(k)-quasi-Einstein manifold. We study -recurrent N(k)-quasi-Einstein manifolds in Section 7. Finally, we give some examples of N(k)-quasi-Einstein manifolds.

From (1.2) and (1.3) it follows that

where r is the scalar curvature and Q is the Ricci operator. In an n-dimensional N(k)-quasi-Einstein manifold M, the generalized tensor takes the form

and scalar takes the form

Also,

Also the projective curvature tensor P in an n-dimensional N(k)-quasi-Einstein manifold satisfies the following relations:

for all vector fields X, Y, U on M.

Again in an n-dimensional N(k)-quasi-Einstein manifold M, the conformal curvature tensor C satisfies

for all vector fields X, Y, Z on M.

Let us suppose that the manifold (Mⁿ, g) be an N(k)-quasi-Einstein manifold. Then the condition R(ξ,X) · = 0 gives

In view of (1.6) and (2.1) we get

Also, in view of (1.7) and (2.1) we obtain

Thus equations (3.1), (3.2) and (3.3) together give

Since in a quasi-Einstein manifold b = 0, therefore, either k = 0 or, g(X, Y) = η(X)η(Y ). In the second case, from (1.2) it follows that the manifold becomes an Einstein manifold. This is a contradiction. Thus we have k = 0, that is, a + b = 0. Conversely, if k = 0, then in view of (1.6) and (1.7) the manifold satisfies R(ξ,X) · = 0.

Thus we can state the following:

Theorem 3.1

An N(k)-quasi-Einstein manifold (Mⁿ, g) satisfies the condition R(ξ,X) · = 0 if and only if a + b = 0.

In this section we consider an n-dimensional N(k)-quasi-Einstein manifold (Mⁿ, g) satisfying the condition

Now we have

where the endomorphism ( U)V is defined by

Then

Then from (4.1) and (4.2) we have

Using (1.5), (1.6) and (2.1) in (4.3) and then taking inner product with ξ, we obtain

Putting V = ξ in (4.4) we get

Since in a quasi-Einstein manifold b = 0, the 1-form η is non-zero and g(X,U) – η(X)η(U) = 0, from equation (4.5) it follows that k = 0. Again, if we take k = 0, then the converse is trivial.

Thus we can state the following:

Theorem 4.1

An n-dimensional N(k)-quasi-Einstein manifold (Mⁿ, g) satisfies the condition(X, ξ) · R = 0 if and only if a + b = 0.

Therefore, by the Theorems 3.1 and 4.1 we can state the following corollary:

Corollary 4.1

Let (Mⁿ, g) be an n-dimensional N(k)-quasi-Einstein manifold. Then the following statements are equivalent:

• R(ξ,X) · = 0,

• (X, ξ) · R = 0,

• a + b = 0,

for every vector field X on (Mⁿ, g).

In this section we consider an n-dimensional N(k)-quasi-Einstein manifold (Mⁿ, g) satisfying the condition

From the condition P(ξ,X) · = 0, we get

which in view of (2.5) gives

Since b = 0, using (2.3) we obtain

which gives a + b + φ = 0.

Thus we can state the following:

Theorem 5.1

Let M be an n-dimensional N(k)-quasi-Einstein manifold. Then M satifies the condition P(ξ,X) · = 0 if and only if a + b + φ = 0.

In this section we consider an n-dimensional N(k)-quasi-Einstein manifold (Mⁿ, g) satisfying the condition

Using (1.10) and (1.2) in (6.1) we get

from which we obtain

Using (2.9) in (6.3) we get

Hence we can state the following:

Theorem 6.1

In an n-dimensional N(k)-quasi-Einstein manifold (Mⁿ, g), the relation C · = 0 holds for all vector fields X, Y, U, V on (Mⁿ, g).

A non-flat Riemannian or semi-Riemannian manifold (Mⁿ, g) is said to be a -recurrent manifold ([25]) if its tensor satisfies the condition

where η is a non-zero 1-form. We have

Using (7.1) and (7.2) we get

Putting U = V = ξ in (7.3) we obtain

Thus we can state the following:

Theorem 7.1

If (Mⁿ, g) is a-recurrent N(k)-qusi-Einstein manifold, then

for all X ∈ TM.

A -recurrent manifold is -symmetric if and only if the 1-form η is zero. Thus we have the following corollaries:

Corollary 7.2

If (Mⁿ, g) is a-symmetric N(k)-quasi-Einstein manifold, then a + b + φ is constant.

Corollary 7.3

If (Mⁿ, g) is a-recurrent N(k)-quasi-Einstein manifold and if a+b+φ is constant, then either a+b+φ = 0 or (Mⁿ, g) reduces to a-symmetric N(k)-quasi-Einstein manifold.

Let us consider a Riemannian metric g on ℝ⁴ by

where q=ex1k2 and k is a non-zero constant, (i, j = 1, 2, 3, 4). Then the only non-vanishing components of the Christoffel symbols and the curvature tensors are:

and the components obtained by the symmetry properties. The non-vanishing components of the Ricci tensors are:

It can be easily shown that the scalar curvature r of the resulting space (ℝ⁴, g) is r=6q(1+q)(1+2q)3, which is non-vanishing and non-constant. Now we shall show that this (ℝ⁴, g) is an N(k)-quasi-Einstein manifold.

To show that the manifold under consideration is an N(k)-quasi-Einstein manifold, let us choose the scalar functions a, b and the 1-form η as follows:

at any point x ∈ ℝ⁴. Now the equation (1.2) reduces to the equations

and

since, for the other cases (1.2) holds trivially. By (8.3) and (8.4) we get

By similar argument it can be shown that (8.6), (8.7) and (8.8) are also true. So, (ℝ⁴, g) is an N (q(1+2q)3)-quasi-Einstein manifold.

Example 9.1

This example is concerned with example of an N(k)-quasi-Einstein manifold in general relativity by the coordinate free method of differential geometry. In this method of study the spacetime of general relativity is regarded as a connected four-dimensional semi-Riemannian manifold (M⁴, g) with Lorentzian metric g with signature (−,+,+, +). The geometry of the Lorentzian manifold begins with the study of causal character of vectors of the manifold. It is due to this causality that the Lorentzian manifold becomes a convenient choice for the study of general relativity.

An n-dimemsional Riemannian or semi-Riemannian manifold (Mⁿ, g), (n > 3) is said to be pseudo Ricci symmetric manifold[6] if its Ricci tensor S of type (0,2) is not identically zero and satisfies the condition

where A is a non-zero 1-form defined by

for all X. Such a manifold is denoted by (PRS)_n.

Here we consider a perfect fluid (PRS)₄ spacetime with non-zero scalar curvature and having the basic vector field U as the timelike vector field of the fluid, that is, g(U,U) = −1. In a recent paper[9] De et al. prove the following:

A perfect fluid pseudo Ricci symmetric spacetime with non-zero scalar curvature is an N(2r9)-quasi-Einstein manifold.

Example 9.2([31])

A conformally flat perfect fluid spacetime (M⁴, g) satisfying Einstein’s equation without cosmological constant is an N(k(3σ+p)6)-quasi-Einstein manifold.

Example 9.3([31])

A conformally flat perfect fluid spacetime (M⁴, g) satisfying Einstein’s equation with cosmological constant is an N(λ3+k(3σ+p)6)-quasi-Einstein manifold, where k is the gravitational constant, σ is the energy density and p is the isotropic pressure of the fluid.

The authors are thankful to the reviewer for careful reading of the manuscript and his/her thoughtful comments for the improvement of the paper.