Certain results on N(k)-quasi Einstein manifolds

Afrika Matematika https://doi.org/10.1007/s13370-018-0631-z Certain results on N(k)-quasi Einstein manifolds S. K. Chaubey1 Received: 27 February 2017 / Accepted: 8 September 2018 © African Mathematical Union and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2018 Abstract The object of the present paper is to study the properties of N (k)-quasi Einstein manifolds. The existence of some classes of such manifolds are proved by constructing physical and geometrical examples. It is also shown that the characteristic vector field of the manifold is a unit parallel vector field as well as Killing vector field. Keywords Quasi Einstein · k-nullity distribution · N (k)-quasi Einstein · Kenmotsu manifolds · Z -tensor · f (r , T )-gravity · Ricci soliton · Torse forming vector field · Curvature tensor Mathematics Subject Classification 53C25 · 53C35 · 53D10 1 Introduction In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian manifold whose Ricci tensor ‘S’ is proportional to the metric ‘g’. This statement is equivalent to the mathematical expression S = μg, where μ is a non-zero constant. The generalization or modification of Einstein manifold is a quasi Einstein manifold. The existence of quasi Einstein manifold is realized during study of exact solutions of Einstein field equations as well as during the consideration of quasi umbilical hypersurfaces. For instant, the Robertson–Walker space times in general relativity; toy universe in cosmology; models of gravitational collapse in astrophysics are quasi Einstein manifolds. El Naschie [1] turned the tables of the theory of elementary particles and showed that we could derive expectation number of elementary particles of the standard model using Einstein unified field equations. The Gödel’s classical solution of Einstein’s field equations and E-infinity have studied in [2,3]. A non-flat n-dimensional Riemannian manifold (Mn , g), (n > 2), is said to be quasi Einstein manifold if its Ricci tensor S of type (0, 2) is not identically zero and satisfies the tensorial expression S(X , Y ) = ag(X , Y ) + bη(X )η(Y ), ∀ X , Y ∈ T M B 1 (1) S. K. Chaubey [email protected] Section of Mathematics, Department of Information Technology, Shinas College of Technology, P.O. Box 77, 324 Shinas, Sultanate of Oman 123 S. K. Chaubey for smooth functions a and b( = 0), where η is a non-zero 1-form such that g(X , ξ ) = η(X ), g(ξ, ξ ) = η(ξ ) = 1 (2) for all vector field X and the associated unit vector field ξ [4]. The 1-form η is called the associated 1-form and the unit vector field ξ is called the generator of the manifold. It is found that a collection of non-interacting pressure less perfect fluid of general relativity is a four dimensional semi-Riemannian quasi Einstein manifold whose associated scalars are r2 and κρ, where κ is the gravitational constant, ρ and r are the energy density and scalar curvature, the generator of the manifold being the time like velocity vector field of the perfect fluid [5]. If the generator of a quasi Einstein manifold is parallel vector field, then the manifold is locally a product manifold of one-dimensional distribution U and (n − 1) dimensional distribution U ⊥ , where U ⊥ is involutive and integrable [6]. In an n-dimensional quasi Einstein manifold the Ricci tensor has precisely two distinct eigen values a and a + b, where the multiplicity of a is n − 1 and a + b is simple [4]. A proper η-Einstein contact metric manifold is a natural example of a quasi Einstein manifold [7,8]. The different geometrical properties of quasi Einstein manifolds have studied by Chaki [9], De and Gazi [10], Guha [11], De and Ghosh ([12–14]), Deszcz et al. [15], Mantica and Suh [16], De and De [17] and others. Let R denotes the Riemann curvature tensor of a Riemannian manifold Mn . The k-nullity distribution N (k) of a Riemannian manifold is defined by N (k) : p−→N p (k) = {Z ∈ T p M : R(X , Y )Z = k[g(Y , Z )X − g(X , Z )Y ]}, (3) where k is a smooth function [18]. If the generator ξ belongs to k-nullity distribution N (k), then the quasi Einstein manifold is called an N (k)-quasi Einstein manifold [19]. The derivation conditions R(ξ, X ) · R = 0, R(ξ, X ) · S = 0 have also been studied in [19], where R and S denote the curvature and Ricci tensors of the manifold respectively. In 2007, Özgür and Tripathi [20] studied the deviation conditions Ẑ (ξ, X ) · Ẑ = 0 and Ẑ (ξ, X ) · R = 0 on N (k)quasi Einstein manifolds, where Ẑ denotes the con-circular curvature tensor. Özgür and Sular [21] continued the study of N (k)-quasi Einstein manifolds with conditions R(ξ, X ) · C = 0 and R(ξ, X ) · C̃ = 0, where C and C̃ denote the Weyl conformal and quasi conformal curvature tensors respectively. Again, in 2008, Özgür [22] studied the deviation conditions R(ξ, X ) · P = 0, P(ξ, X ) · S = 0 and P(ξ, X ) · P = 0 for an N (k)-quasi Einstein manifold, where P denotes the projective curvature tensor and some physical examples of N (k)-quasi Einstein manifolds are given. Several geometrical properties of N (k)-quasi Einstein manifolds have studied by De et al. [23–25], Singh et al. [26], Taleshian and Hosseinzadeh [27], Hosseinzadeh and Taleshian [28], Yang and Xu [29], Chaubey et al. [5], Chaubey [30] and others. In Cosmology, the reason for studying various type of space-time models are mainly for the purpose of representing the different phases in the evolution of the Universe. The evolution of the universe to its present state can be divided into three phases: The initial phase just after the big bang when the effects of both viscosity and heat flux were quite pronounced. The intermediate phase when the effect of viscosity was no longer significant but the heat flux was still not negligible. The final phase, which extends to the present state of the Universe when both the effects of viscosity and the heat flux have became negligible and the matter content of the Universe may be assumed to be a perfect fluid. The importance of the study of the quasi Einstein manifold lies in the fact that these spacetime manifolds represent the second and the third phases respectively in the evolution of the Universe [11]. For instance, we refer [10] and the references therein. Motivated from the above studied, author continues the study of N(k)-quasi Einstein manifolds. The present paper is organized as follows: 123 Certain results on N(k)-quasi Einstein manifolds Section 2 is preliminaries which covers the basic known results of N (k)-quasi Einstein manifold, m-projective curvature tensor and Z -tensor. Section 3 is concerned with the physical and geometrical examples of certain classes of N (k)-quasi Einstein manifolds which support the existence of such manifolds. The properties of W ∗ -pseudosymmetric N (k)-quasi Einstein manifolds are studied in Sect. 4. In next section, it is showed that the generator of the manifold is Killing as well as parallel unit vector fields under certain restriction. 2 Preliminaries In consequence of (1)–(3), we get and S(X , ξ ) = (a + b)η(X ), a+b k= n−1 (4) r = na + b, (6) (5) where r denotes the scalar curvature of the Riemannian manifold (Mn , g). In an n-dimensional N (k)-quasi Einstein manifold (Mn , g), the following relations hold [19,20] R(X , Y )ξ = k[η(Y )X − η(X )Y ], (7) R(X , ξ )Y = k[η(Y )X − g(X , Y )ξ ] = −R(ξ, X )Y , (8) R(ξ, X )ξ = k[η(X )ξ − X ], (9) Qξ = k(n − 1)ξ, (10) η(R(X , Y )Z ) = k[η(X )g(Y , Z ) − η(Y )g(X , Z )] (11) for arbitrary vector fields X , Y and Z . The m-projective curvature tensor W ∗ [31] on the Riemannian manifold (Mn , g) is defined as 1 W ∗ (X , Y )Z = R(X , Y )Z − [S(Y , Z )X 2(n − 1) − S(X , Z )Y + g(Y , Z )Q X − g(X , Z )QY ] (12) for arbitrary vector fields X , Y , Z ; where Q is the Ricci operator of the Riemannian manifold, that is S(X , Y ) = g(Q X , Y ). It bridges the gap between conformal curvature tensor, conharmonic curvature tensor and concircular curvature tensor on one side and H -projective curvature tensor on the other. The properties of such curvature tensor have noticed in [32–37]. In consequence of (1), (2), (4), (5) and (11), (12) becomes η(W ∗ (X , Y )Z ) = b {η(X )g(Y , Z ) − η(Y )g(X , Z )}, 2(n − 1) (13) for arbitrary vector fields X , Y and Z . Recently, Mantica and Molinari [38] defined a generalized (0, 2) type tensor Z as Z (X , Y ) = S(X , Y ) + f g(X , Y ), (14) for arbitrary vector fields X and Y , where f is a smooth function. In consequence of (1) and (14), we have Z (X , Y ) = (a + f )g(X , Y ) + bη(X )η(Y ). (15) 123 S. K. Chaubey Putting Y = ξ in (15) and then using (2), we get Z (X , ξ ) = (a + b + f )η(X ) (16) Z (ξ, ξ ) = a + b + f . (17) which is equivalent to 3 Examples of N(k)-quasi Einstein manifolds In 2011, Harko et al. [39] proposed the theory of f (r , T )-gravity which is the generalization or modification of general relativity. In this theory, gravitational Lagrangian is considered as an arbitrary function of r and T , where r is the trace of Ricci tensor Si j and T is the trace of stress energy tensor Ti j . The field equations for this theory are derived from the Hilbert–Einstein type variational by considering the action   1 [ f (r , T ) + L m ] ( − g)d 4 x, A= (18) 16π where L m is the matter Lagrangian density. The stress energy tensor of the matter is given by √ −2 δ( −g)L m . (19) Ti j = √ −g δi j Let us suppose that the matter Lagrangian density depends only on gi j , then the field equations of f (r , T ) gravity written as 1 f (r , T )gi j + (gi j ∇k ∇ k − ∇i ∇ j ) fr (r , T ) 2 = 8π Ti j − f T (r , T )Ti j − f T (r , T )i j , fr (r , T )Si j − (20) where fr and f T denote the partial derivatives of f with respect to r and T respectively and i j = −2Ti j + gi j L m − 2glk ∂2 Lm . ∂ g i j ∂ glk (21) Here ∇ i represents the covariant derivative. If we consider f (r , T ) = f (r ), then the Eqs. (18) and (19) give the field equations of f (r )-gravity [40]. Let us consider that the matter is a conformally flat perfect fluid of dimension four with energy density ρ, pressure p and four velocity u i . As we know that there is no unique definition of matter Lagrangian, thus we consider L m = − p and the stress energy tensor of the matter as Ti j = (ρ + p)u i u j − pgi j , (22) where u i ∇ j u i = 0, u i · u i = 1. (23) From Eq. (22), we can easily find that the variation of stress energy of perfect fluid assumes the form i j = −2Ti j − pgi j . (24) Generally the field equations depend on the physical nature of the matter field and therefore for each choice of f (r , T ), we get a theoretical model. For instant, we choose f (r , T ) = r + 2 f (T ), 123 (25) Certain results on N(k)-quasi Einstein manifolds where f (T ) is the arbitrary function of the trace of stress energy tensor of the matter. After considering Eq. (25), (20) assumes the form 1 Si j − rgi j = 8π Ti j − 2 f ′ (T )Ti j − 2 f ′ (T )i j + f (T )gi j . 2 In view of (22), (23) and (24), (26) becomes   1 Si j = r + f (T ) − 8 pπ gi j + {( p + ρ)(8π + 2 f ′ (T ))}u i u j . 2 (26) (27) Contracting Eq. (27), we get r = 2{12π p − ( p + ρ) f ′ (T ) − 2 f (T ) − 4πρ}. (28) It is obvious from (27) that the perfect fluid is a certain class of quasi Einstein manifold with associated scalars { 21 r + f (T ) − 8 pπ} and ( p + ρ)(8π + 2 f ′ (T )). It is well known that a conformally flat quasi Einstein manifold is an N (k)-quasi Einstein manifold [19]. From Eqs. (5) and (27), we have   1 1 ′ k= (29) r + f (T ) + 8πρ + 2( p + ρ) f (T ) . 3 2 From above discussions, we can state the following example: Example 3.1 A four dimensional conformally flat perfect fluid (M4 , g) is an N ( 31 { 21 r + f (T ) + 8πρ + 2( p + ρ) f ′ (T )})-quasi Einstein manifold. As we know that the simplest cosmological model is the dust universe ( p = 0, T = ρ). Let us suppose that the matter space is conformally flat dust universe with f (T ) = δT , where δ is constant. Thus Eq. (27) gives   1 r + δT gi j + 2ρ(4π + δ)u i u j , Si j = (30) 2 where the scalar curvature is given by r = −2(2δT + 4πρ + δρ). (31) Equation (30) clearly reflects that the dust universe is a certain class of quasi Einstein manifold with associated scalars ( 21 r + δT ) and 2ρ(4π + δ). From (5) and (30), we can easily find that   1 1 r + T δ + 2ρ(4π + δ) . (32) k= 3 2 Hence we can state: Example 3.2 A four dimensional conformally flat dust universe is an N ( 31 { 21 r +T δ+2ρ(4π + δ)})-quasi Einstein manifold. In 1972, Kenmotsu defined and studied a class of almost contact metric manifold and later renamed as Kenmotsu manifold [41]. Janssens and Vanheche [42] generalized the Kenmotsu manifold and called β-Kenmotsu manifold. Let (Mn , g), (n = 2m + 1 > 2), be an ndimensional β-Kenmotsu manifold, where φ is a tensor field of type (1, 1), ξ is the structure vector field, η is a covariant vector field associated with the Riemannian metric g satisfies the relations φ 2 (X ) = −X + η(X )ξ, η(ξ ) = 1, g(X , ξ ) = η(X ), and φξ = 0, (33) 123 S. K. Chaubey and ∇ X ξ = β(X − η(X )ξ ), (∇ X η)(Y ) = β{g(X , Y ) − η(X )η(Y )} (34) for arbitrary vector fields X , Y and β  = 0 [42]. Here ∇ represents the operator of covariant differentiation with respect to the Riemannian metric g. If β = 1, then β-Kenmotsu manifold converted into well known Kenmotsu manifold [41]. It can be easily see from (33) and (34) that (L ξ g)(X , Y ) = 2β{g(X , Y ) − η(X )η(Y )}, (35) where L ξ g denotes the Lie derivative of Riemannian metric along the characteristic vector field ξ . A smooth vector field V on a Riemannian manifold is said to be Ricci soliton if it satisfies the condition 1 L V g + S = λg, (36) 2 where g is the Riemannian metric associated to the vector field V , S is a Ricci tensor and λ is a non-zero real constant [43]. A Ricci soliton is a natural generalization of Einstein manifold and it has many applications in physics. Here λ is a real constant, therefore it is characterized in three categories: (i) λ = 0, Ricci soliton is steady, (ii) λ < 0, Ricci soliton is expanding, (iii) λ > 0, Ricci soliton is shrinking. In first two cases, Ricci soliton to be Einstein but we are interested in non-Einstein manifold and therefore we are going to consider the third case. Let us suppose that V = ξ and therefore the Eqs. (35) and (36) give S(X , Y ) = (λ − β)g(X , Y ) + βη(X )η(Y ), (37) a = λ − β( = 0) and b = β, (38) where which shows that the β-Kenmotsu manifold equipped with Ricci soliton is a quasi-Einstein manifold with associated scalars given in (38). Now with the help of (5) and (38), we have λ k = a+b n−1 = n−1 and thus we can state the following example as: Example 3.3 An n-dimensional conformally flat β-Kenmotsu manifold (Mn , g), (n > 2), λ equipped with Ricci soliton is an N ( n−1 )-quasi Einstein manifold. If we consider that β = λ, then Eq. (37) becomes S(X , Y ) = βη(X )η(Y ), (39) which represents a special type of quasi Einstein manifold with a = 0 and b = β. From Eqs. (5) and (39), we obtain β . (40) k= n−1 Hence we can state: Example 3.4 Let (Mn , g), (n > 2), be an n-dimensional conformally flat β-Kenmotsu manβ ifold equipped with Ricci soliton. Then it is an N ( n−1 )-quasi Einstein manifold. If we suppose that β = 1, then β-Kenmotsu manifold converts into Kenmotsu manifold. By considering this fact and Example 3.3, we can state the following example: 123 Certain results on N(k)-quasi Einstein manifolds Example 3.5 An n-dimensional conformally flat Kenmotsu manifold (Mn , g), (n > 2), λ equipped with Ricci soliton is an N ( n−1 )-quasi Einstein manifold. A vector field U on an n-dimensional Riemannian (semi-Riemannian) manifold (Mn , g) is said to be torse forming vector field if it satisfies the relation ∇ X U = α X + A(X )U , (41) for arbitrary vector field X [44], where α is a smooth function and A is a 1-form associated with the Riemannian metric g, i.e., g(X , U ) = A(X ). If α = 0, then torse forming vector field becomes recurrent vector field [45]. Let us suppose that U = ξ , then (41) gives ∇ X ξ = α X + η(X )ξ. (42) (L X g)(Y , Z ) = g(∇Y X , Z ) + g(Y , ∇ Z X ). (43) It is well known that Taking X = ξ in (43) and then using (2) and (42), we have (L ξ g)(Y , Z ) = 2{αg(Y , Z ) + η(X )η(Y )}. (44) In view of (36), (44) turns into the form S(X , Y ) = (λ − α)g(X , Y ) − η(X )η(Y ), (45) which shows that the manifold is quasi Einstein with smooth functions a = (λ − α) and b = −1. In consequence of (5) and (45), we can find the value of k as k= a+b λ−α−1 = . n−1 n−1 (46) If we suppose that the manifold (Mn , g), (n > 2), is conformally flat, then with the above discussion we can state the following examples: Example 3.6 A conformally flat Riemannian manifold of dimension n(> 2) equipped with Ricci soliton whose potential vector field is a torse forming vector field is an N ( λ−α−1 n−1 )-quasi Einstein manifold. Example 3.7 Let (Mn , g), (n > 2), be an n-dimensional conformally flat Riemannian manifold admitting Ricci soliton. If the characteristic vector field of the manifold is recurrent, then the manifold is an N ( λ−1 n−1 )-quasi Einstein manifold. If λ = α, then (45) assumes the form S(X , Y ) = −η(X )η(Y ), (47) which represents that the manifold is a certain class of quasi Einstein manifold with associated −1 scalars 0 and −1. From (5) and (47), we get k = n−1 . Thus we can state: Example 3.8 Let (Mn , g), (n > 2), be an n-dimensional conformally flat Riemannian manifold satisfies the condition λ = α. If the characteristic vector field of the manifold is recurrent −1 as well as Ricci soliton, then the manifold is an N ( n−1 )-quasi Einstein manifold. 123 S. K. Chaubey Example 3.9 Let (x 1 , x 2 , . . . , x n ) ∈ Rn , where Rn denotes n-dimensional real number space. We consider a Lorentzian metric g on R4 = (x 1 , x 2 , x 3 , x 4 ; x 1  = (1+2q)π , q ∈ Z), (Z is the 4 set of positive integer), by  ds 2 = gi j d x i d x j = {sin(x 1 ) − cos(x 1 )} (d x 1 )2 + (d x 2 )2 + (d x 3 )2 − (d x 4 )2 , (48) where (i, j = 1, 2, 3, 4). With the help of (48), we can see that the non-vanishing components of the Lorentzian metric are g11 = g22 = g33 = sin(x 1 ) − cos(x 1 ), g44 = −1 (49) and its associated components are g 11 = g 22 = g 33 = 1 , g 44 = −1. sin(x 1 ) − cos(x 1 ) (50) In consequence of (49) and (50), it can be find that the non-vanishing components of Christoffel symbols, curvature tensor, Ricci tensor and scalar curvature are given by sin(x 1 ) + cos(x 1 ) sin(x 1 ) + cos(x 1 ) 1 1 , Ŵ = Ŵ = , 22 33 2(sin(x 1 ) − cos(x 1 )) 2(cos(x 1 ) − sin(x 1 )) 1 + sin2(x 1 ) 1 + sin2(x 1 ) = R1331 = , S , 33 4(cos(x 1 ) − sin(x 1 )) 4(1 − sin(2x 1 )) 1 + sin2(x 1 ) ( = 0) (51) r= 4(sin(x 1 ) − cos(x 1 ))3 1 2 3 Ŵ11 = Ŵ12 = Ŵ13 = and the other components are obtained by the symmetric properties. From (51), it is clear that the manifold (R4 , g) is a Lorentzian manifold. Now, we are going to prove that the manifold (R4 , g) is a certain class of N (k)-quasi Einstein manifold. For this purpose we take the associated smooth functions a and b as follows: a = 0, b = 1 + sin2(x 1 ) ( = 0). 4(cos(x 1 ) − sin(x 1 ))3 (52) i =3 other wise (53) Now we define the 1-forms Ai as follows:  cos(x 1 ) − sin(x 1 ), Ai = 0, Now, we have to prove the following: Si j = agi j + b Ai A j (54) for i, j = 1, 2, 3, 4. For instant, we have to show that S33 = ag33 + b A3 A3 . Left hand side of (55) = S33 = 1+sin2(x 1 ) (55) (from (51)). In view of (49), (52) and (53), right 4(1−sin(2x 1 )) 1+sin2(x 1 ) b A3 A3 = 4(1−sin(2x 1 )) . hand side of (55) = ag33 + In the similar way, we can verify for other components of Si j . From (1)–(3) and (52), it can be easily prove that k= 1 + sin2(x 1 ) a−b = n−1 12(sin(x 1 ) − cos(x 1 ))3 and r = 4a − b hold on (R4 , g) and therefore it is an N manifold. 123 1+sin2(x 1 ) 12(sin(x 1 )−cos(x 1 ))3 -quasi Einstein Certain results on N(k)-quasi Einstein manifolds √ Example 3.10 If R3 = (x 1 , x 2 , x 3 ; x 1  = (πq + ar ctan 2), q ∈ Z), (Z is the set of positive integer) be a three dimensional real number space, then the Lorentzian metric on R3 is defined as ds 2 = gi j d x i d x j = −(d x 1 )2 + cos 2 (x 1 )[(d x 2 )2 + (d x 3 )2 ], (56) where (i, j = 1, 2, 3). The non-vanishing components of the Lorentzian metric are g11 = −1, g22 = cos 2 (x 1 ) = g33 . (57) Also its associated components are g 11 = −1, g 22 = g 33 = sec2 (x 1 ). (58) From Eqs. (57) and (58), it can be easily calculate that the non-vanishing components of the Christoffel symbols, curvature tensors and the Ricci tensors are given by the following relations sin2(x 1 ) sin2(x 1 ) 2 3 = Ŵ13 = −tan(x 1 ), R2323 = , Ŵ12 , 2 4 = −(1 + sin 2 (x 1 )), S11 = −2, S22 = S33 = cos2(x 1 ) 1 1 = Ŵ33 =− Ŵ22 R1212 = R1313 (59) and the other components can be obtain from (59) by using the symmetric properties. Here Si j represents the components of the Ricci tensor. From Eq. (59), it is clear that the 3-dimensional space R3 with the Lorentzian metric g defined in (56) is a Lorentzian manifold of dimension 3. From (57)–(59), it can be easily see that the scalar curvature of the manifold (R3 , g) is non-zero, i.e., r = 2(2 − tan 2 (x 1 ))  = 0. (60) Now, we choose the scalars a and b as follows: a= cos2(x 1 ) , b = −sec2 (x 1 )  = 0. cos 2 (x 1 ) (61) We define the 1-form A as follows: Ai =  1, 0, i =1 other wise (62) Now, we have to prove that Si j = agi j + b Ai A j . (63) In consequence of (57), (59) and (61)–(63), we can verify that S11 = ag11 + b A11 and all the other components of S satisfy the relation (63). Hence (R3 , g) is a quasi-Einstein manifold. 3 3 Now, k = a−b n−1 = 1 and r = 3a − b hold on (R , g), therefore (R , g) is an N (1)-quasi Einstein manifold. Example 3.11 [22] A conformally flat perfect fluid space time (M 4 , g) satisfying Einstein’s equation with cosmological constant λ is an N ( λ3 + κ(3σ6+ p) )-quasi Einstein manifold. Example 3.12 [23] A special para-Sasakian manifold with vanishing D-concircular curvature tensor is an N (k)-quasi Einstein manifold. Example 3.13 [23] A perfect fluid pseudo Ricci symmetric space time is an N ( 2r9 )-quasi Einstein manifold. 123 S. K. Chaubey 4 W ∗ -pseudosymmetric N(k)-quasi Einstein manifolds An n-dimensional Riemannian (pseudo Riemannian) manifold (Mn , g) is said to be W ∗ pseudosymmetric if the tensors R · W ∗ and Q(g, W ∗ ) defined by (R(X , Y ) · W ∗ )(Z , W )U = R(X , Y )W ∗ (Z , W )U − W ∗ (R(X , Y )Z , W )U − W ∗ (Z , R(X , Y )W )U − W ∗ (Y , W )R(X , Y )U (64) and Q(g, W ∗ )(Z , W , U ; X , Y ) = −((X ∧W ∗ Y ) · W ∗ )(Z , W )U = (X ∧W ∗ Y )W ∗ (Z , W )U − W ∗ ((X ∧W ∗ Y )Z , W )U − W ∗ (Z , (X ∧W ∗ Y )W )U − W ∗ (Z , W )(X ∧W ∗ Y )U (65) are linearly dependent, i.e., (R(X , Y ) · W ∗ )(Z , W )U = L W ∗ Q(g, W ∗ )(Z , W , U ; X , Y ), (66) for arbitrary vector fields X , Y , Z , W and U . Here X ∧W ∗ Y denotes the endomorphism defined as (X ∧W ∗ Y )Z = g(Y , Z )X − g(X , Z )Y (67) and L W ∗ is a smooth function holds on UW ∗ = {x ∈ Mn : W ∗  = 0 at x}. If L W ∗ = 0, then the manifold (Mn , g) reduces to m-projective semisymmetric manifold (i.e., R · W ∗ = 0). In consequence of (64)–(67), we have R(X , Y )W ∗ (Z , W )U − W ∗ (R(X , Y )Z , W )U − W ∗ (Z , R(X , Y )W )U − W ∗ (Y , W )R(X , Y )U = L W ∗ {(X ∧W ∗ Y )W ∗ (Z , W )U − W ∗ ((X ∧W ∗ Y )Z , W )U − W ∗ (Z , (X ∧W ∗ Y )W )U − W ∗ (Z , W )(X ∧W ∗ Y )U }. (68) Putting X = ξ in (68) and then using (2), (8) and (67), we find that (k − L W ∗ ){′ W ∗ (Z , W , U , Y )ξ − η(W ∗ (Z , W )U )Y − g(Y , Z )W ∗ (ξ, W )U + η(Z )W ∗ (Y , W )U − g(Y , W )W ∗ (Z , ξ )U + η(W )W ∗ (Z , Y )U − g(Y , U )W ∗ (Z , W )ξ + η(U )W ∗ (Z , W )Y } = 0, (69) which is equivalent to (k − L W ∗ ){′ W ∗ (Z , W , U , Y ) − η(W ∗ (Z , W )U )η(Y ) − g(Y , Z )η(W ∗ (ξ, W )U ) + η(Z )η(W ∗ (Y , W )U ) − g(Y , W )η(W ∗ (Z , ξ )U ) + η(W )η(W ∗ (Z , Y )U ) − g(Y , U )η(W ∗ (Z , W )ξ ) + η(U )η(W ∗ (Z , W )Y )} = 0, (70) where ′ W ∗ (Z , W , U , Y ) = g(W ∗ (Z , W )U , Y ). From (70) it is clear that either k = L W ∗ or ′ W ∗ (Z , W , U , Y ) − η(W ∗ (Z , W )U )η(Y ) − g(Y , Z )η(W ∗ (ξ, W )U ) + η(Z )η(W ∗ (Y , W )U ) − g(Y , W )η(W ∗ (Z , ξ )U ) + η(W )η(W ∗ (Z , Y )U ) − g(Y , U )η(W ∗ (Z , W )ξ ) + η(U )η(W ∗ (Z , W )Y ) = 0. 123 (71) Certain results on N(k)-quasi Einstein manifolds Let us suppose that k  = L W ∗ , then with help of (1)–(3), (5) and (11)–(13), (71) gives ′ R(Z , W , U , Y ) = λ1 {g(Y , Z )g(U , W ) − g(Y , W )g(Z , U )} + λ2 {η(W )η(U )g(Y , Z ) − η(U )η(Z )g(Y , W ) + η(Y )η(Z )g(W , U ) − η(W )η(Y )g(Z , U )}, (72) 2a+b b where λ1 = 2(n−1) and λ2 = 2(n−1) . A Riemannian manifold (Mn , g) is said to possesses quasi-constant curvature if the curvature tensor R is not identically zero and satisfies the relation ′ R(X , Y , Z , U ) = λ1 {g(Y , Z )g(X , U ) − g(X , Z )g(Y , U )} + λ2 {η(Y )η(Z )g(X , U ) − η(X )η(Z )g(Y , U ) + η(X )η(U )g(Y , Z ) − η(Y )η(U )g(X , Z )} (73) for arbitrary vector fields X , Y , Z and U [46,47]. Here λ1 and λ2 are smooth functions on Mn . If λ2 = 0, then (Mn , g) reduces to a space of constant curvature λ1 . From (72) and (73), we can state: Theorem 4.1 If (Mn , g), (n > 2), be an n-dimensional W ∗ -pseudosymmetric N (k)-quasi Einstein manifold with k  = L W ∗ , then it is a manifold of quasi-constant curvature. Now we are going to prove the following theorem as: Theorem 4.2 An n-dimensional m-projectively pseudosymmetric N (k)-quasi Einstein manifold (Mn , g), (n > 2), satisfies the relation k = L W ∗ . Proof If possible, we suppose that k  = L W ∗ on an m-projectively pseudosymmetric N (k)-quasi Einstein manifold (Mn , g) and therefore Eq. (72) holds on (Mn , g). Let {ei , i = 1, 2, . . . , n} be an orthonormal basis of the tangent space at any point of the manifold (Mn , g). Then putting Z = Y = ei in (72) and taking summation over i, 1 ≤ i ≤ n, we get S(W , U ) = a1 g(W , U ) + b1 η(W )η(U ), (74) bn and b1 = − b(n−2) where a1 = a + 2(n−1) 2(n−1) . Since, on a quasi-Einstein manifold the smooth functions a and b are unique, as if S = a1 g + b1 η ⊗ η, then (a − a1 )g + (b − b1 )η ⊗ η=0 and thus g is of rank ≤ 1, a contradiction and therefore k = L W ∗ . Hence the statement of the theorem. ⊔ ⊓ In [5], authors proved the following theorems: Theorem 4.3 If an n-dimensional N (k)-quasi Einstein manifold (Mn , g) equipped with cyclic parallel Ricci tensor satisfies R(ξ, X ) · W ∗ = 0, then its generator ξ is a Killing vector field. Theorem 4.4 Let (Mn , g) be an n-dimensional N (k)-quasi Einstein manifold equipped with cyclic parallel Ricci tensor satisfying R(ξ, X ) · W ∗ = 0. Then the characteristic vector field on (Mn , g) is a parallel unit vector field. If we consider that L W ∗ = 0 ⇒ k = 0. Thus the manifold under consideration is an m-projective semisymmetric manifold. By considering this fact and Theorems 4.2–4.4, we can state the following lemmas: Lemma 4.1 If an n-dimensional m-projectively semisymmetric N (k)-quasi Einstein manifold (Mn , g), (n > 2), has a cyclic parallel Ricci tensor, then the generator of the manifold is a Killing vector field. 123 S. K. Chaubey Lemma 4.2 Let (Mn , g), (n > 2), be an n-dimensional m-projectively semisymmetric N (k)quasi Einstein manifold equipped with cyclic parallel Ricci tensor, then the generator of the manifold is a parallel unit vector field. 5 Z -recurrent N(k)-quasi Einstein manifolds An n-dimensional Riemannian (semi-Riemannian) manifold (Mn , g) is said to be Z -recurrent if the non-vanishing Z -tensor of type (0, 2) satisfies the condition (∇ X Z )(Y , Z ) = η(X )Z (Y , Z ), (75) for arbitrary vector fields X , Y and Z [48]. Here we suppose that the 1-form η associated with the Riemannian metric g is non-zero. If η = 0, then manifold (Mn , g) converted into Z -symmetric manifold. It is obvious from (2) and (15) that (∇ X Z )(Y , Z ) = {da(X ) + d f (X )}g(Y , Z ) + db(X )η(Y )η(Z ) + b{(∇ X η)(Y )η(Z ) + (∇ X η)(Z )η(Y )}. (76) In view of (75) and (76), we have η(X )Z (Y , Z ) = {da(X ) + d f (X )}g(Y , Z ) + db(X )η(Y )η(Z ) + b{(∇ X η)(Y )η(Z ) + (∇ X η)(Z )η(Y )}. (77) Replacing Y and Z by ξ in (77) and then using (2) and (17), we can find da(X ) + db(X ) + d f (X ) = (a + b + f )η(X ). (78) Let {ei , i = 1, 2, . . . , n}, be an orthonormal basis of the tangent space at any point of the manifold (Mn , g). Then putting Y = Z = ei in (77) and taking summation over i, 1 ≤ i ≤ n, we get {(a + f )n + b}η(X ) = {da(X ) + d f (X )}n + db(X ). (79) It is obvious that (∇ X Z )(Y , Z ) = X Z (Y , Z ) − Z (∇ X Y , Z ) − Z (Y , ∇ X Z ). (80) From (75) and (80), we have η(X )Z (Y , Z ) = X Z (Y , Z ) − Z (∇ X Y , Z ) − Z (Y , ∇ X Z ). (81) Putting Y = Z = ξ in (81) and the using (2), (15) and (17), we find that (a + b + f )η(X ) = X (a + b + f ). (82) With the help of (82), (78) becomes da(X ) + db(X ) + d f (X ) = X (a + b + f ). (83) If we consider that a + b + f is constant, then from (83) it is clear that da(X ) + db(X ) + d f (X ) = 0. (84) In consequence of (76) and (84), we get (∇ X Z )(Y , Z ) = db(X ){η(Y )η(Z )−g(Y , Z )}+b{(∇ X η)(Y )η(Z )+(∇ X η)(Z )η(Y )}. (85) 123 Certain results on N(k)-quasi Einstein manifolds Put Z = ξ in (85) and then using (2), we have (∇ X Z )(Y , ξ ) = b(∇ X η)(Y ). (86) The covariant derivative of (16) with respect to X gives (∇ X Z )(Y , ξ ) = (a + b + f )(∇ X η)(Y ) − (a + f )g(Y , ∇ X ξ ) − bη(Y )η(∇ X ξ ). (87) From Eqs. (86) and (87), we have (a + f )(∇ X η)(Y ) = (a + f )g(Y , ∇ X ξ ) + bη(Y )η(∇ X ξ ). (88) Putting Y = ξ in (88) and then using Eq. (2), we can find ∇ X ξ = 0. (89) From the above discussions, we can state the following theorem: Theorem 5.1 If a + b + f is constant on an n-dimensional Z -recurrent N (k)-quasi Einstein manifold, then the generator of the manifold is a parallel unit vector field. From Eqs. (88) and (89), we have (∇ X η)(Y ) + (∇Y η)(X ) = 0. (90) Hence we can state: Theorem 5.2 Let (Mn , g), (n > 2), be an n-dimensional Z -recurrent N (k)-quasi Einstein manifold with a + b + f = constant, then the generator of the manifold is a Killing vector field. 6 Conclusion Quasi Einstein manifolds play an important role in mathematics and physics. The existence of certain classes of N (k)-quasi-Einstein manifolds have been derived by proving several geometrical and physical examples. We considered m-projectively pseudosymmetric, semisymmetric and Z -recurrent N (k)-quasi Einstein manifolds and proved many interesting results. 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