arXiv:2104.03495v1 [gr-qc] 8 Apr 2021 nteapiainof application the On ic esrSo ye(0 type of S tensor Ricci Introduction 1 equation. Einstein field, vector forming Keywords: M.S.C.2010: asi htoaha* rna htahry* n Dipanka and Bhattacharyya** Arindam Chattopadhyay*, Kaushik nEnti aiodi imnino suoReana aiodwhos manifold pseudo-Riemannian or Riemannian a is manifold Einstein An o field relative tor isotropic spatially infinitesimally is spacetime htan that eismercqaiEnti pctm aifigadefi a N satisfying spacetime quasi-Einstein semi-symmetric eea hoyo eaiiyhsbe tde.Frty eh we Firstly, studied. been has M relativity of theory general field ic esri hstp fsaeieadas rvdta if that proved also of and square spacetime the of type out semi- this found becomes in have tensor also we Ricci spacetime Lastly, spac the conditions. such these that of curvature der theorem the the out proving found have by we then inflation, an nascae ymti esrfil,floe ytetheore the by vanishing followed with field, tensor symmetric associated an uvtr.Te ehv rvdta an that proved have we Then curvature. osat ihtrefrigvco field vector equati torse-forming field with Einstein constant, satisfying spacetime fluid perfect flat aallto parallel ( poetvl a us-isensaeiei faspecial a of is spacetime quasi-Einstein flat -projectively 2 nti ae h plcto fthe of application the paper this In l − ξ 6 m aifigEnti edeuto ihcsooia const cosmological with equation field Einstein satisfying ) M ξ M qaiEnti pctm.I h atscin ehv firstl have we section, last the In spacetime. Einstein -quasi ntenx eto ehv rvdta an that proved have we section next the In . ∇ poetvl a efc udsaeiewt torse-formi with spacetime fluid perfect flat -projectively poetv uvtr esr imnincrauetno,t tensor, curvature Riemannian tensor, curvature -projective rmr 32 eodr 31,53C44. 53D10, Secondary ; 53C25 Primary aapkrHg colHS,Nbdi,India Nabadwip, School(H.S), High Bamanpukur hneither then M aaprUiest,Klaa703,India Kolkata-700032, University, Jadavpur India Kolkata-700032, University, Jadavpur E-mail:[email protected] poetv uvtr esri pctm fquasi-const of spacetime a is tensor curvature -projective E-mail:[email protected] , E-mail:[email protected] ngnrlrelativity general in *Dprmn fMathematics, of ***Department )i o-eoadpootoa otemti esr Ein- tensor. metric the to proportional and non-zero is 2) *eateto Mathematics, of **Department Dprmn fMathematics, of *Department λ = k 2 M ( p Abstract − poetv uvtr tensor curvature -projective σ ) or M ξ α poetv uvtr esri the in tensor curvature -projective disasymmetric a admits M sacntn utpeof multiple constant a is poetvl a quasi-Einstein flat -projectively M oteui ieievec- timelike unit the to poetvl a Ricci flat -projectively nwt cosmological with on iecniini an is condition nite ls ihrsetto respect with class v rvdta an that proved ave htaspacetime a that m an h egho the of length the tm,followed etime, ymti un- symmetric M (0 n represents ant , -projectively 2) Debnath*** r gvector ng tensor proved y g . ant α orse- e 2 stein manifolds form a natural subclass of various classes of Riemannian or semi- Riemannian manifolds by a curvature condition imposed on their Ricci tensor [4]. Also in Riemannian geometry as well as in general relativity theory, the Einstein manifold plays a very important role.

Chaki and Maity [18] generalised the concept of Einstein manifold and intro- duced the notion of quasi-Einstein manifold. According to them, a Riemannian or semi-Riemannian manifold is said to be a quasi-Einstein manifold if its Ricci tensor S of type (0, 2) is non-zero and satisfies the condition

S(U, V )= lg(U, V )+ mA(U)A(V ), (1.1) where l and m are two non-zero real-valued scalar functions and A is a non-zero 1-form equivalent to the unit vector field ξ, i.e. g(U, ξ) = A(U), g(ξ,ξ)= 1. If m = 0 then the manifold becomes Einstein. Quasi-Einstein manifolds are denoted by (QE)n, where n is the dimension of the manifold. The notion of quasi-Einstein manifolds arose during the study of exact solutions of the Einstein field equations as well as during the considerations of quasi-umbilical hypersurfaces of semi-Euclidean spaces. For instance, the Robertson-Walker spacetime is a quasi-Einstein manifold. Also, quasi-Einstein manifolds can be taken as a model of perfect fluid spacetime in general relativity. The importance of quasi-Einstein spacetimes lies in the fact that 4-dimensional semi-Riemannian manifolds are related to study of general relativistic fluid spacetimes, where the unit vector field ξ is taken as timelike velocity vector field, that is, g(ξ,ξ)= 1. − In the recent papers [1], [23], the application of quasi-Einstein spacetime and generalised quasi-Einstein spacetime in general relativity have been studied. Many more works have been done in the spacetime of general relativity [2], [15], [25], [26], [29], [30], [31].

∞ Let (Mn,g) be an n-dimensional differentiable manifold of class C with the metric tensor g and the Riemannian connection . In 1971 G. P. Pokhariyal and ∇ R. S. Mishra ([11]) defined the M-projective curvature tensor as follows 1 P˜(U, V )W = R(U, V )W [S(V, W )U S(U, W )V − 2(n 1) − − + g(V, W )QU g(U, W )QV ], (1.2) − where R and S are the curvature tensor and the Ricci tensor of Mn, respectively. Such a tensor field P˜ is known as the M-projective curvature tensor. Some authors studied the properties and applications of this tensor [10], [14], [20] and [21]. In 2010, S. K. Chaubey and R. H. Ojha investigated the M-projective curvature tensor of a Kenmotsu manifold [24].

The concept of perfect fluid spacetime arose while discussing the structure of 3 this universe. Perfect fluids are often used in the general relativity to model the idealised distribution of matter, such as the interior of a star or isotropic pressure. The energy-momentum tensor T˜ of a perfect fluid spacetime is given by the following equation T˜(U, V )= pg(U, V )+(σ + p)A(U)A(V ), (1.3) where σ is the energy-density and p is the isotropic pressure, A is defined earlier and the unit vector field ξ is timelike, i.e. g(ξ,ξ)= 1. Einstein field equation with cosmological constant([6]) is given by − r˜ S(U, V ) g(U, V )+ λg(U, V )= kT˜(U, V ), (1.4) − 2 where, S is the Ricci tensor,r ˜ is the scalar curvature of the spacetime while λ, k are the cosmological constant and the gravitational constant respectively. It’s used to describe the dark energy of this universe in modern cosmology, which is responsible for the possible acceleration of this universe. The equations (1.3) and (1.4) together give r˜ S(U, V )=( λ + pk)g(U, V )+ k(σ + p)A(U)A(V ). (1.5) 2 − Comparing to the equation (1.1) we can say the tensor of the equation (1.5) repre- sents the tensor of a quasi-Einstein manifold.

The k-nullity distribution N(k) of a Riemannian manifold M is defined by

N(k): p Np(k)= → W Tp(M): R(U, V )W = k[g(V, W )U g(U, W )V ] , (1.6) { ∈ − } for all U, V TpM, where k is a smooth function. For a quasi-Einstein manifold M, ∈ if the generator ξ belongs to some N(k), then M is said to be N(k)-quasi-Einstein manifold [19] . Ozg¨ur¨ and Tripathi proved that for an n-dimensional N(k)-quasi l+m Einstein manifold [8], k = n−1 , where l and m are the respective scalar functions and n is the dimension of the manifold.

In this paper we have first derived some theorems on M-projectively flat space- times. After that we have introduced the concept of Ricci semi-symmetric spacetime with vanishing M-projective curvature tensor. Lastly we introduced the concept of torse-forming vector field in this spacetime and derived some theorems on it, thereby finding the curvature of the spacetime and finding the square of the length of the Ricci tensor for this spacetime with torse-forming vector field.

2 Preliminaries

Consider a quasi-Einstein spacetime with associated scalars l, m and associated 1-form A. Then by (1.1), we have r =4l m, (2.1) − 4 where r is a scalar curvature of the spacetime. If ξ is a unit timelike vector field, then g(ξ,ξ)= 1. Again from the equation (1.1), we have − S(ξ,ξ)= m l, (2.2) − For all vector fields U and V we have the following equation, g(QU, V )= S(U, V ), (2.3) where Q is the symmetric endomorphism of the tangent space at each point of the manifold corresponding to the Ricci tensor S. From the equation (1.5) and (2.3) we can get r˜ QU =( λ + pk)U + k(σ + p)A(U)ξ. (2.4) 2 −

If the unit timelike vector field ξ is a torse-forming vector field ([5], [16]) then it satisfies the following equation,

U ξ = U + A(U)ξ. (2.5) ∇ In [28] Venkatesha and H. A. Kumara proved that:

Theorem 2.1: On a perfect fluid spacetime with torse-forming vector field ξ, the following relation holds

( U A)(V )= g(U, V )+ A(U)A(V ). (2.6) ∇

Considering a frame field and taking a contraction over U and V from the equation (1.5) we get, r˜ =4λ + k(σ 3p). (2.7) −

3 M-projectively flat quasi-Einstein spacetime

In this section we consider a quasi-Einstein spacetime with vanishing M-projective curvature tensor. If a spacetime with dimension n = 4 is M-projectively flat then from the equation (1.2) we have 1 R(U, V )W = [S(V, W )U S(U, W )V + g(V, W )QU g(U, W )QV ]. (3.1) 6 − − Using the equation (2.3) in the equation (1.1) we get QU = lU + mA(U)ξ. (3.2) 5

Using the equations (1.1), (3.2) and taking the inner product with T from (3.1) we get l R˜(U, V, W, T )= [g(V, W )g(U, T ) g(U, W )g(V, T )] 3 − m + [g(U, T )A(V )A(W )+ g(V, W )A(U)A(T ) 6 g(V, T )A(U)A(W ) g(U, W )A(V )A(T )]. (3.3) − − Now taking l m D(U, V )= g(U, V )+ A(U)A(V ), (3.4) r3 2√3l from the equation (3.3) we have R˜(U, V, W, T )= D(V, W )D(U, T ) D(U, W )D(V, T ). (3.5) −

It is known that an n-dimensional Riemannian or semi-Riemannian manifold whose curvature tensor R˜ of type (0, 4) satisfies the condition (3.5), is called a special man- ifold with the associated symmetric tensor D and is denoted by the symbol ψ(D)n, where D is a symmetric tensor field of type (0, 2). Recently, these types of manifolds are studied in [9] and [27]. With the use of the equations (3.4) and (3.5) we can state the following theorem:

Theorem 3.1: An M-projectively flat quasi-Einstein spacetime is ψ(D)4, where D is the associated symmetric tensor field.

In [7] B. Y. Chen and K. Yano introduced the concept of quasi-constant cur- vature. A manifold is said to be a manifold of quasi-constant curvature if it satisfies the following condition R˜(U, V, W, T )= p[g(V, W )g(U, T ) g(U, W )g(V, T )] − + q[g(U, T )η(V )η(W ) g(V, T )η(U)η(W ) − + g(V, W )η(U)η(T ) g(U, W )η(V η(T )], (3.6) − where R˜ is the scalar curvature of type (0, 4), p and q are scalar functions while g(U, ν)= η(U), ν is the unit vector field, η is the respective 1-form and g(ν, ν) = 1. Thus in the view of (3.3) and (3.6) we state the following theorem:

Theorem 3.2: A spacetime with vanishing M-projective curvature tensor is a space- time of quasi-constant curvature.

Now, let us consider the space ξ⊥ = X : g(X,ξ)=0, X χ(M) . Let ⊥ { ∀ ∈ } U, V , W ξ , then the equation (3.3) will imply ∈ l R(U, V )W = [g(V, W )U g(U, W )V ]. (3.7) 3 − 6

So, we can state the following theorem:

l Theorem 3.3: An M-projectively flat quasi-Einstein spacetime becomes an N( 3 )- quasi Einstein spacetime provided U, V , W ξ⊥, ξ is a unit timelike vector field ∈ and l is a non-zero real-valued scalar function.

We also derive the following corollary:

Corollary 3.1: An M-projectively flat quasi-Einstein spacetime satisfies the fol- lowing results, m 2l (i)R(U, ξ)V = − g(U, V )ξ, 6 m 2l (ii)R(U, ξ)ξ = − U, 6 (3.8)

where U, V ξ⊥, ξ is a unit timelike vector field and l, m are two non-zero real- valued scalar∈ functions.

After Riemannian manifolds, Lorentzian manifolds form the most important sub- class of pseudo-Riemannian manifolds. They are important in applications of gen- eral relativity. A principal premise of general relativity is that spacetime can be modelled as a 4-dimensional Lorentzian manifold of signature (3, 1) or, equivalently, (1, 3). With a signature of (3, 1) or (1, 3), the manifold is also time-orientable. A Lorentzian manifold is called infinitesimally spatially isotropic ([12]) relative to a unit timelike vector field ξ if its curvature tensor R satisfies the relation

R(X,Y )Z = α[g(Y,Z)X g(X,Z)Y ], (3.9) − for all X, Y , Z ξ⊥ and R(X,ξ)ξ = βX for all X ξ⊥, α and β are two non-zero ∈ ∈ real-valued functions.

From the equation (3.7) and the result (ii) of corollary (3.1) it is obvious that the manifold is infinitesimally spatially isotropic. Thus we can state the following theorem:

Theorem 3.4: An M-projectively flat quasi-Einstein spacetime is infinitesimally spatially isotropic relative to the unit timelike vector field ξ. 7

4 M-projectively flat Ricci semi-symmetric quasi- Einstein spacetime

In this section we consider a quasi-Einstein spacetime which is Ricci semi- symmetric. An n-dimensional semi-Riemannian manifold is said to be Ricci semi- symmetric if the tensor R.S and the Tachibana tensor Q˜(g,S) are linearly dependent, i.e., R(U, V ) S(W, T )= FSQ˜(g,S)(W, T ; U, V ) (4.1) · r holds on US where US = x M : S = g at x and FS is a scalar function on { ∈ 6 n } US. Now we know that

R(U, V ) S(W, T )= S(R(U, V )W, T ) S(W, R(U, V )T ), (4.2) · − − using the equation (4.1) we have

FSQ˜(g,S)(W, T ; U, V )= S(R(U, V )W, T ) S(W, R(U, V )T ). (4.3) − − We also know that

(U g V )W = g(V, W )U g(U, W )V. (4.4) ∧ − Now if it is a Ricci semi-symmetric quasi-Einstein spacetime then using the equations (4.2), (4.3) and (4.4) we get

S(R(U, V )W, T )+ S(W, R(U, V )T )= FS[g(V, W )S(U, T ) g(U, W )S(V, T ) − + g(V, T )S(W, U) g(U, T )S(V, W )].(4.5) − Since we know R˜(U, V, W, T )= R˜(U,V,T,W ), thus using the equation (1.1) in the equation (4.5) we obtain −

A(R(U, V )W )A(T )+ A(W )A(R(U, V )T )

= FS[g(V, W )A(U)A(T ) g(U, W )A(V )A(T ) − +g(V, T )A(W )A(U) g(U, T )A(V )A(W )], (4.6) − putting T = ξ in the equation (4.6) and applying the result g(R(U, V )ξ,ξ) = g(R(ξ,ξ)U, V ) we get,

A(R(U, V )W )= FS[g(V, W )A(U) g(U, W )A(V )], (4.7) − applying the equation (3.3) from (4.7) we get, 2l m (FS − )[g(V, W )A(U) g(U, W )A(V )]=0. (4.8) − 6 − So, if g(V, W )A(U) g(U, W )A(V ) = 0 then − 6 2l m F = − , (4.9) S 6 8 thus using the equations (4.7) and (4.9) we get,

2l m R(U, V )W = − [g(V, W )U g(U, W )V ], (4.10) 6 − from the equations (1.6) and (4.10) we observe that the spacetime becomes an 2l−m N( 6 )-quasi Einstein spacetime provided g(V, W )A(U) g(U, W )A(V ) = 0. This leads us to the next theorem: − 6

Theorem 4.1: An M-projectively flat Ricci semi-symmetric quasi-Einstein space- time with g(V, W )A(U) g(U, W )A(V ) =0 is an N( 2l−m )-quasi Einstein spacetime, − 6 6 where l and m are two non-zero real valued scalar functions.

5 M-projectively flat perfect fluid spacetime with torse-forming vector field

If a manifold is M-projectively flat then using the divergence to both the sides ∇ of the equation (3.1) we get

( U S)(V, W ) ( V S)(U, W )=0, (5.1) ∇ − ∇ using the equation (2.3) we get,

g(( U Q)V ( V Q)U, W )=0. (5.2) ∇ − ∇ From the equation (2.7) since we observer ˜ is a constant thus using the equation (2.4) we get

k(σ + p)[( U A)(V )ξ + A(V ) U ξ ( V A)(U)ξ A(U) V ξ]=0, (5.3) ∇ ∇ − ∇ − ∇ using the equations (2.5) and (2.6) we get

k(σ + p)[g(V,ξ)U g(U, ξ)V ]=0, (5.4) − since k is the gravitational constant hence k = 0. Thus g(V,ξ)U g(U, ξ)V = 0 implies 6 − 6 σ + p =0, (5.5) which means either σ = p = 0 (empty spacetime) or the perfect fluid satisfies the vacuum-like equation of state. This allows us to derive the following theorem:

Theorem 5.1: An M-projectively flat perfect fluid spacetime with torse-forming vector field ξ satisfying Einstein field equation with cosmological constant is ei- ther an empty spacetime or satisfies the vacuumlike equation of state, provided g(V,ξ)U g(U, ξ)V =0. − 6 9

Now σ + p = 0 means the fluid behaves as a cosmological constant [13]. This is also termed as Phantom Barrier [22]. Now in cosmology we know such a choice σ = p leads to rapid expansion of the spacetime which is now termed as inflation [17].− In 1981, to explain the conditions observed in the universe, astrophysicist Alan Guth proposed cosmic inflation [3]. The term inflation refers to the explosively rapid expansion of spacetime that occurred a tiny fraction of a second after the Big Bang. So, we obtain the following theorem:

Theorem 5.2: an M-projectively flat perfect fluid spacetime with torse-forming vector field ξ satisfying Einstein field equation with cosmological constant represents an inflation.

Now putting σ + p = 0 from the equation (1.5) we get, k S(U, V ) = [λ + (σ p)]g(U, V ), (5.6) 2 − thus the equation (2.4) becomes k QU = [λ + (σ p)]U. (5.7) 2 − Using the equations (5.6) and (5.7) in the equation (3.1) we get

2λ + k(σ p) R(U, V )W = − [g(V, W )U g(U, W )V ]. (5.8) { 6 } − Hence we can state the following theorem:

Theorem 5.3: An M-Projectively flat perfect fluid spacetime with torse-forming vector field ξ, satisfying Einstein field equation with cosmological constant is of con- 2λ+k(σ−p) stant curvature 6 .

Consequently we obtain the following theorem as:

Theorem 5.4: An M-projectively flat perfect fluid spacetime with torse-forming vector field ξ satisfying Einstein field equation with cosmological constant is an Ein- stein spacetime.

From the equation (5.8) we easily obtain

(R(U, V ) R˜)(X,Y,Z,W )= R˜(R(U, V )X,Y,Z,W ) R˜(X, R(U, V )Y,Z,W ) · − − R˜(X,Y,R(U, V )Z, W ) R˜(X,Y,Z,R(U, V )W )=0, (5.9) − − which implies the manifold is semi-symmetric. Hence we obtain the following theo- rem: 10

Theorem 5.5: An M-projectively flat perfect fluid spacetime with torse-forming vector field ξ satisfying Einstein field equation with cosmological constant is a semi- symmetric spacetime.

Replacing U by QU from the equation (5.6) we get k S(QU, V ) = [λ + (σ p)]g(QU, V ). (5.10) 2 − Using the equation (5.6)which becomes k k S(QU, V ) = [λ + (σ p)]S(U, V ) = [λ + (σ p)]2g(U, V ). (5.11) 2 − 2 − Considering a frame field and taking a contraction over U and V from the equation (5.11) we get k Q 2 = 4[λ + (σ p)]2 = [2λ + k(σ p)]2. (5.12) || || 2 − − Hence we can state the following theorem:

Theorem 5.6: The square of the length of the Ricci tensor of an M-projectively flat perfect fluid spacetime with torse-forming vector field ξ satisfying Einstein field equation with cosmological constant is [2λ + k(σ p)]2. − The Ricci identity is given by

2 α(X,Y ) 2 α(X,Y )= α(R(U, V )X,Y )+ α(X, R(U, V )Y ), (5.13) ∇U,V −∇V,U where α is a symmetric (0, 2) tensor. Now if α is parallel to then ∇ α =0, (5.14) ∇ which further implies 2α =0. (5.15) ∇ Thus from the equation (5.13) we get

α(R(U, V )X,Y )+ α(X, R(U, V )Y )=0. (5.16)

Thus from the equation (5.8) we get 2λ + k(σ p) − [g(V,X)α(U, Y ) g(U, X)α(V,Y ) { 6 } − +g(V,Y )α(U, X) g(U, Y )α(V,X)]=0. (5.17) − Putting X = Y = V = ξ in the equation (5.17) we get, 2λ + k(σ p) − [α(U, ξ)+ A(U)α(ξ,ξ)]=0, (5.18) −{ 3 } 11 which means either λ = k (p σ) or 2 − α(U, ξ)= A(U)α(ξ,ξ). (5.19) − Now taking the derivative of α(ξ,ξ) with respect to V and using the equations (2.5) and (5.19) we get V (α(ξ,ξ))=0. (5.20) Taking the derivative of the equation (5.19) with respect to V and using the equation (5.20) we get V (α(U, ξ)) = α(ξ,ξ)V (g(U, ξ)). (5.21) − Since α is parallel with respect to thus using the equation (2.5) from the equation ∇ (5.21) we get α(U, V )= α(ξ,ξ)g(U, V ). (5.22) − Therefore we obtain the following theorem as:

Theorem 5.7: If an M-projectively flat perfect fluid spacetime satisfying Einstein field equation with cosmological constant, with torse-forming vector field ξ admits a k symmetric (0, 2) tensor α parallel to then either λ = 2 (p σ) or α is a constant multiple of g. ∇ −

References

[1] A. A. Shaikh, D. Y. Yoon and S. K. Hui (2009). ”On quasi-Einstein spacetime”, Tsukuba J. Math. 33(2), 305-326, http://doi.org/10.21099/tkbjm/1267209423.

[2] A. Bhattacharyya, T. De and D. Debnath (2009). ”Space time with generalized covariant recurrent energy momentum tensor”, Tamsui Oxford Journal of Mathematical Sciences 25(3), 269-276. https://www.researchgate.net/publication/242597683.

[3] Alan H. Guth (1981). ”Inflationary universe: A possible solution to the horizon and flatness problems”, Carnegie Observatories Astrophysics Series, Vol. 2, Measuring and Modeling the Universe, Cambridge Univ. Press, Cambridge, https://doi.org/10.1103/PhysRevD.23.347.

[4] A. L. Besse (1987). ”Einstein manifolds”, Ergeb. Math. Grenzgeb., 3. Folge, Bd. 10. Berlin, Heidelberg, New York : Springer-Verlag, ISBN: 978-3-540- 74311-8. 12

[5] A. M. Blaga (2017). ”Solitons and geometrical structures in a perfect fluid spacetime”, http://doi.org/10.1216/rmj.2020.50.41.

[6] B. O’Neill (1983). ”Semi-Riemannian geometry and application to relativity”, Pure and applied mathematics, Academic press. Inc. New York, ISBN: 9780080570570.

[7]B. Y. Chen and K. Yano (1972). ”Hypersurfaces of a conformally flat space”, Tensor (N.S.) 26, 315-321, https://www.researchgate.net/publication/266853315.

[8] C. Ozg¨ur¨ and M. M. Tripathi (2007). ”On the concircular curvature tensor of an N(k)-Einstein manifold”, Math. Pannon. 18, 95-100, http://doi.org/10.4134/JKMS.2005.42.5.883.

[9] D. G. Prakasha and B. S. Hadimani (2017). ”η-Ricci soli- ton on para-Sasakian manifolds”, J. Geom. 108, 383-392, https://doi.org/10.1007/s00022-016-0345-z.

[10] F. O. Zengin (2012). ”M-projectively flat spacetimes”, Math. Rep. 4, no. 4, 363-370 https://www.researchgate.net/publication/267090824.

[11] G. P. Pokhariyal and R. S. Mishra (1971). ”Curvature tensors and their relativistic significance”, Yokohama Math. J. 19(2), 97-103, http://repository.ias.ac.in/31212/.

[12] H. Karcher (1982). ”Infinitesimale charakterisierung von Friedmann-Univeren”, Arch. Math. 38, 58-64, https://doi.org/10.1007/BF01304758.

[13] H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers and E. Herlt (2003). ”Exact Solutions of Einstein’s Field Equations Cambridge Monogr. on Math. Phys. Cambridge Univ. Press, 2nd edition, Cambridge, https://doi.org/10.1017/CBO9780511535185.

[14] K. Chattopadhyay, A. Bhattacharyya and D. Debnath (2020). ”A Study of spacetimes with vanishing M-projective curvature tensor”, Accepted, Journal of the Tensor Society of India. 13

[15] K. Chattopadhyay, N. Bhunia and A. Bhattacharyya (2018). ”On Ricci- symmetric mixed generalized quasi Einstein spacetime”, Bull. Cal. Math. Soc. 110(6), 513-524, https://www.researchgate.net/publication/343136926.

[16] K. Yano (1994). ”On torse-forming direction in Riemannian spaces”, Proc. Imp. Acad. Tokyo. 20, 340-345, http:/doi.org/10.3792/pia/1195572958.

[17] L. Amendola and S. Tsujikawa (2010). ”Dark Energy: Theory and Observa- tions”, Cambridge Univ. Press, Cambridge, ISBN: 9780521516006.

[18]M. C. Chaki and R. K. Maity (2000). ”On quasi- Einstein manifolds”, Publ. Math. Debrecen 57, 297-306, http://publi.math.unideb.hu/load_pdf.php?p=641.

[19] M. M. Tripathi and J. S. Kim (2007). ”On N(k)-Einstein manifolds”, Commun. Korean Math. Soc. 22(3), 411-417, https://doi.org/10.4134/CKMS.2007.22.3.411.

[20] R. H. Ojha (1975). ”A note on the M-projective curva- ture tensor”, Indian J. Pure Appl. Math. 8(12), 1531-1534, https://insa.nic.in/writereaddata/UpLoadedFiles/IJPAM/20005a78_1531.pdf.

[21] R. H. Ojha (1986). ”M-projectively flat Sasakian man- ifolds”, Indian J. Pure Appl. Math. 17(4), 481-484, https://insa.nic.in/writereaddata/UpLoadedFiles/IJPAM/20005a4e_481.pdf.

[22] S. Chakraborty, N. Mazumder and R. Biswas (2011). ”Cosmological evolution across phantom crossing and the nature of the horizon”, Astrophys. Space Sci. Libr. 334, 183-186, http://doi.org/10.1007/s10509-011-0704-z.

[23] S. G¨uler and S. A. Demirbaˇg(2016). ”A study of generalized quasi-Einstein spacetime with application in general relativity”, International Journal of The- oretical Physics 55, 548-562, http://doi.org/10.1007/s10773-015-2692-1.

[24] S. K. Chaubey and R. H. Ojha (2010). ”On the M-projective curva- ture tensor of a Kenmotsu manifold”, Diff. Geom. Dyn. Syst. 12, 52-60, https://www.researchgate.net/publication/265403632. 14

[25] S. Mallick, Y. J. Suh and U. C. De (2016). ”A spacetime with pseudo- projective curvature tensor”, J. Math. Phys. 57, Paper ID 062501, https://doi.org/10.1063/1.4952699.

[26] S. Ray-Guha (2006). ”On perfect fluid pseudo Ricci symmetric spacetime”, Tensor (N.S.) 67, 101-107, https://www.researchgate.net/publication/263047376.

[27] Venkatesha and D. M. Naik (2017). ”Certain results on K-para contact and para Sasakian manifold”, J. Geom. 108, 939-952, https://doi.org/10.1007/s00022-017-0387-x.

[28] Venkatesha and H. A. Kumara (2019). ”Ricci soliton and geometrical struc- ture in a perfect fluid spacetime with torse-forming vector field”, Afr. Mat., http://doi.org/10.1007/s13370-019-00679-y.

[29] Z. Ahsan and M. Ali (2017). ”Curvature tensor for the spacetime of general relativity”, Int. J. of Geom. Methods Mod. Phys. 14(5), 1750078(13 pages), https://doi.org/10.1142/S0219887817500785.

[30] Z. Ahsan and M. Ali (2015). ”Quasi-conformal curvature tensor for the spacetime of general relativity”, Palestine Journal of Mathematics, ISSN- 2219-5688, 4(1), 233-240.

[31] Z. Ahsan and S. A. Siddiqui (2009). ”Concircular curvature ten- sor and fluid spacetimes”, Internat. J. Theoret. Phys. 48, 3202-3212, http://doi.org/10.1007/s10773-009-0121-z.