ON THE EXISTENCE OF NEARLY QUASI-EINSTEIN MANIFOLDS. 1. Introduction. Novi Sad J. Math. Vol. 39, No. 2, 2009,

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1 Novi Sad J. Math. Vol. 39, No. 2, 2009, ON THE EXISTENCE OF NEARLY QUASI-EINSTEIN MANIFOLDS Abul Kalam Gazi 1, Uday Chand De 2 Abstact. The objective of the pesent pape is to establish the existence of nealy quasi-einstein manifolds. AMS Mathematics Subject Classification (2000): 53C25 Key wods and phases: quasi-einstein manifolds, nealy quasi-einstein manifolds, quasi-constant cuvatue, nealy quasi-constant cuvatue 1. Intoduction A Riemannian o a semi-riemannian manifold (M n, g), n = dimm 2, is said to be an Einstein manifold if the following condition (1.1) S = n g holds on M, whee S and denote the Ricci tenso and the scala cuvatue of (M n, g) espectively. Accoding to ([1], p. 432), (1.1) is called the Einstein metic condition. Einstein manifolds play an impotant ole in Riemannian Geomety as well as in geneal theoy of elativity. Also, Einstein manifolds fom a natual subclass of vaious classes of Riemannian o semi-riemannian manifolds by a cuvatue condition imposed on thei Ricci tenso ([1], p ). Fo instance, evey Einstein manifold belongs to the class of Riemannian manifolds (M n, g) ealizing the following elation : (1.2) S(X, Y ) = ag(x, Y ) + ba(x)a(y ), whee a, b R and A is a non-zeo 1-fom such that (1.3) g(x, U) = A(X), fo all vecto fields X. Moeove, diffeent stuctues on Einstein manifolds have also been studied by seveal authos. In 1993 Tamassy and Binh [2] studied weakly symmetic stuctues on Einstein manifolds. A non-flat Riemannian manifold (M n, g) (n > 2) is defined to be a quasi- Einstein manifold [3] if its Ricci tenso S of type (0, 2) is not identically zeo 1 Moynagodi E.B.A.U. High Madasah, P.O.- Noapaa, Kolkata , West Bengal, India, abulkalamgazi@yahoo.com 2 Depatment of Pue Mathematics, Univesity of Calcutta, 35, Ballygaunge Cicula Road, Kolkata , West Bengal, India, uc_de@yahoo.com

2 112 A.K. Gazi, U.C. De and satisfies the condition (1.2). We shall call A the associated 1-fom and U is called the geneato of the manifold. Quasi-Einstein manifolds aose duing the study of exact solutions of the Einstein field equations as well as duing consideations of quasi-umbilical hypesufaces of semi-euclidean spaces. Fo instance, the Robetson-Walke spacetime is quasi-einstein manifold [4]. Also quasi-einstein manifold can be taken as a model of the pefect fluid spacetime in geneal elativity [5]. So quasi-einstein manifolds have some impotance in the geneal theoy of elativity. It is to be noted that M. C. Chaki and R. K. Maity [6] also intoduced the notion of quasi-einstein manifolds which is diffeent fom that of R. Deszcz [3]. They took a and b as scalas and the geneato U of the manifold as a unit vecto field. The notion of quasi-einstein manifolds have been genealized by many authos in seveal ways such as genealized quasi-einstein manifolds [7], [8]. In a ecent pape [9], the authos intoduced the notion of nealy quasi- Einstein manifolds. A non-flat Riemannian manifold (M n, g) (n > 2) is called a nealy quasi-einstein manifold if its Ricci tenso S of type (0, 2) is not identically zeo and satisfies the condition (1.4) S(X, Y ) = ag(x, Y ) + be(x, Y ), whee a and b ae non-zeo scalas and E is a non-zeo symmetic tenso of type (0, 2). An n-dimensional nealy quasi-einstein manifold was denoted by N(QE) n. We shall call E the associated (0, 2) tenso and a and b as associated scalas. Remak 1. It is known ([10], p.39) that the oute poduct of two covaiant vectos is a covaiant tenso of type (0,2) but the convese is not tue, in geneal. Hence the manifolds which ae quasi-einstein ae also nealy quasi-einstein, but the convese is not tue, in geneal. Fo this the name, nealy quasi-einstein, was chosen. A concete example of a nealy quasi-einstein manifold was also given in [9] by the following theoem: Theoem A. Let (R 4, g) be a Riemannian manifold endowed with the metic given by ds 2 = g ij dx i dx j = (x 4 ) 4 3 [(dx 1 ) 2 + (dx 2 ) 2 + (dx 3 ) 2 ] + (dx 4 ) 2, (i, j = 1, 2, 3, 4). Then (R 4, g) is a N(QE) 4 with non-zeo and non-constant scala cuvatue which is not a quasi-einstein manifold. In this pape we like to intoduce anothe notion which genealizes the notion of a manifold of quasi-constant cuvatue [11]. A Riemannian manifold is called a manifold of quasi-constant cuvatue, if it is confomally flat and the cuvatue tenso R of type (0, 4) satisfies the condition (1.5) R(X, Y, Z, W ) = p[g(y, Z)g(X, W ) g(x, Z)g(Y, W )] + q[g(x, W )T (Y )T (Z) g(x, Z)T (Y )T (W ) + g(y, Z)T (X)T (W ) g(y, W )T (X)T (Z)],

3 On the existence of nealy quasi-einstein manifolds 113 whee R(X, Y, Z, W ) = g(r(x, Y )Z, W ), R is the cuvatue tenso of type (1, 3), p, q ae scala functions and λ is a unit vecto field defined by g(x, λ) = T (X). It can be easily seen that if the cuvatue tenso R is of the fom (1.5), then the manifold is confomally flat. On the othe hand, Gh. Vanceanu [12] defined the notion of almost constant cuvatue by the same expession (1.5). Late A. L. Mocanu [13] pointed out that the manifold intoduced by Chen and Yano and the manifold intoduced by Gh. Vanceanu ae the same. Hence a Riemannian manifold is said to be of quasi-constant cuvatue if the cuvatue tenso R satisfies the elation (1.5). If in (1.5) q = 0, then the manifold educes to a manifold of constant cuvatue. A Riemannian manifold is said to be a manifold of nealy quasi-constant cuvatue, if the cuvatue tenso R of type (0, 4) satisfies the condition (1.6) R(X, Y, Z, W ) = p[g(y, Z)g(X, W ) g(x, Z)g(Y, W )] + q[g(x, W )B(Y, Z) g(x, Z)B(Y, W ) + g(y, Z)B(X, W ) g(y, W )B(X, Z)], whee R(X, Y, Z, W ) = g(r(x, Y )Z, W ), R is the cuvatue tenso of type (1, 3), p, q ae scala functions and B is a non-zeo symmetic tenso of type (0, 2). An n-dimensional Riemannian manifold of nealy quasi-constant cuvatue shall be denoted by N(QC) n. The name nealy quasi-constant cuvatue is chosen fo the same eason as in Remak 1. In 1956 S. S. Chen [14] studied a type of Riemannian manifold whose cuvatue tenso R of type (0, 4) satisfies the condition (1.7) R(X, Y, Z, W ) = F (X, Z)F (Y, W ) F (Y, Z)F (X, W ), whee F is a non-zeo symmetic tenso of type (0, 2). Such an n-dimensional manifold was called a special manifold with the associated symmetic tenso F and was denoted by ψ(f ) n. Such a manifold is impotant fo the following easons: Fistly, fo possessing some emakable popeties elating to cuvatue and chaactestic classes and secondly, fo containing a manifold of quasi-constant cuvatue [11] as a subclass. The pape is oganized as follows: Section 2 contains the poof of the theoem fo the existence of a N(QE) n. In section 3 we pove that a nealy quasi-umbilical hypesuface of a special manifold, ψ(f ) n, is a N(QC) n. Finally, we have studied the elations between a N(QC) n and a N(QE) n. 2. Existence Theoem of a N(QE) n In this section we pove the following theoem:

4 114 A.K. Gazi, U.C. De Theoem 2.1. If the non-zeo Ricci tenso S of a Riemannian manifold with non-zeo scala cuvatue satisfies the elation (2.1) S(Y, Z)S(X, W ) S(X, Z)S(Y, W ) = µ[g(y, Z)g(X, W ) g(x, Z)g(Y, W )] whee µ is a non-zeo scala, then the manifold is a nealy quasi-eistein manifold. Poof. Contacting X and W in (2.1) we get (2.2) S(Y, Z) g(q 2 Y, Z) = µ(n 1)g(Y, Z) whee Q be the symmetic endomophism of the tangent space at each point coesponding to the Ricci tenso S, that is, g(qx, Y ) = S(X, Y ). Now, since, 0 we get fom (2.2) that (2.3) S(Y, Z) = µ(n 1) g(y, Z) + 1 E(Y, Z) whee E is a (0, 2) type non-zeo symmetic tenso defined E(Y, Z) = g(q 2 Y, Z), which shows that the manifold is a N(QE) n. 3. Existence of a Manifold of nealy Quasi-Constant Cuvatue In this section we pove the following. Theoem 3.1. A nealy quasi-umbilical hypesuface of a manifold of special cuvatue ψ(f ) n is a manifold of nealy quasi-constant cuvatue. Poof. Let (M n 1, g) be a hypesuface of (M n, g). If A is the (1,1) tenso coesponding to the nomal valued second fundamental tenso H, then we have ([15], p.41) (3.1) g(a ξ (X), Y ) = g(h(x, Y ), ξ) whee ξ is the unit nomal vecto field and X, Y ae tangent vecto fields. Let H ξ be the symmetic (0,2) tenso associated with A ξ in the hypesuface defined by (3.2) g(a ξ (X), Y ) = H ξ (X, Y ). A hypesuface of a Riemannian manifold (M n, g) shall be called nealy quasiumbilical if its second fundamental tenso has the fom (3.3) H ξ (X, Y ) = αg(x, Y ) + F (X, Y ) whee F is a symmetic (0,2) tenso and α is a scala. If α = 0 (esp. F = 0 o α = F = 0) holds, then it is called nealy cylindical (esp. umbilical o

5 On the existence of nealy quasi-einstein manifolds 115 geodesic). The name nealy quasi-umbilical is chosen fo the same eason as in Remak 1. Now fom (3.1), (3.2) and (3.3) we obtain which implies that g(h(x, Y ), ξ) = αg(x, Y )g(ξ, ξ) + F (X, Y )g(ξ, ξ) (3.4) H(X, Y ) = αg(x, Y )ξ + F (X, Y )ξ, since ξ is the only unit nomal vecto field. We have the following equation of Gauss ([15], p.45) fo any vecto fields X, Y, Z, W tangent to the hypesuface (3.5) g(r(x, Y )Z, W ) = g( R(X, Y )Z, W ) g(h(x, W ), H(Y, Z)) + g(h(y, W ), H(X, Z)) whee R is the cuvatue tenso of the hypesuface. Let us assume that the hypesuface is nealy quasi-umbilical. (3.4) and (3.5) it follows that Then fom (3.6) g(r(x, Y )Z, W ) = g( R(X, Y )Z, W ) + α 2 [g(y, W )g(x, Z) g(x, W )g(y, Z)] + α[g(y, W )F (X, Z) + g(x, Z)F (Y, W ) g(x, W )F (Y, Z) g(y, Z)F (X, W )] +[F (Y, W )F (X, Z) F (X, W )F (Y, Z)]. Since g(r(x, Y )Z, W ) = R(X, Y, Z, W ), using (1.7) in (3.6) we have (3.7) g( R(X, Y )Z, W ) = α 2 [g(x, W )g(y, Z) g(y, W )g(x, Z)] +α[g(x, W )F (Y, Z) + g(y, Z)F (X, W ) g(y, W )F (X, Z) g(x, Z)F (Y, W )]. Hence the nealy quasi-umbilical hypesuface of a manifold of special cuvatue ψ(f ) n is a manifold of nealy quasi-constant cuvatue. 4. Relations Between N(QC) n and N(QE) n Theoem 4.1. A manifold of nealy quasi-constant cuvatue is a nealy quasi- Einstein manifold. Poof. Putting X = W = e i in (1.6) whee {e i } is an othonomal basis of the tangent space at each point of the manifold and taking summation ove i, 1 i n, we get (4.1) S(Y, Z) = [p(n 1) + q B]g(Y, Z) + q(n 2)B(Y, Z) whee B is the tace of B. Hence the manifold is a nealy quasi-einstein manifold.

6 116 A.K. Gazi, U.C. De Fom Theoem3.1 and Theoem4.1 we can state the following poposition: Poposition 1. A nealy quasi-umbilical hypesuface of a manifold of special cuvatue ψ(f ) n is a manifold of nealy quasi-einstein manifold. Now contacting (4.1) with espect to Y, and Z we get (4.2) = n(n 1)p + 2(n 1)q B. In a Riemannian manifold (M n, g) (n > 3) the confomal cuvatue tenso C of type (0, 4) has the following fom: (4.3) C(X, Y, Z, W ) = R(X, Y, Z, W ) 1 [S(Y, Z)g(X, W ) n 2 S(X, Z)g(Y, W ) + S(X, W )g(y, Z) S(Y, W )g(x, Z)] + [g(y, Z)g(X, W ) g(x, Z)g(Y, W )]. (n 1)(n 2) Using (1.6), (4.1) and (4.2) in (4.3) we see that C(X, Y, Z, W ) = 0, that is, the manifold unde consideation is confomally flat. Hence we can state the following: Remak 2. Evey N(QC) n (n > 3) is a confomally flat N(QE) n. In this section we have poved that evey N(QC) n (n > 3) is a confomally flat N(QE) n. Now we shall pove that the convese is also tue, that is, evey confomally flat N(QE) n (n > 3) is a N(QC) n. Theoem 4.2. Evey confomally flat N(QE) n (n > 3) is a manifold of nealy quasi-constant cuvatue. Poof. Since the manifold is confomally flat, we have (4.4), R(X, Y, Z, W ) = 1 [S(Y, Z)g(X, W ) S(X, Z)g(Y, W ) n 2 + S(X, W )g(y, Z) S(Y, W )g(x, Z)] + [g(x, Z)g(Y, W ) g(y, Z)g(X, W )]. (n 1)(n 2) Using (1.4) we have (4.5) R(X, Y, Z, W ) = p[g(y, Z)g(X, W ) g(x, Z)g(Y, W )] a+bẽ + q[g(x, W )E(Y, Z) g(x, Z)E(Y, W ) + g(y, Z)E(X, W ) g(y, W )E(X, Z)] whee p = (n 1)(n 2) and q = b n 2. Hee Ẽ is the tce of E. This shows that the manifold is a manifold of nealy quasi-constant cuvatue. Remak 3. If the dimension of a N(QE) n is thee, then the confomal cuvatue tenso vanishes identically and such a thee dimensional N(QE) 3 is a manifold of nealy quasi-constant cuvatue.

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