GEOMETRYISRN Geometry2090-63152090-6307International Scholarly Research Network21713210.5402/2012/217132217132Research ArticleSome Results on Super Quasi-Einstein ManifoldsHuiShyamal Kumar1LemenceRichard S.2, 3CoppensM.MorozovA.VisinescuM.1Nikhil Banga Sikshan MahavidyalayaBishnupur722122 West Bengal, BankuraIndianbsmahavidyalayabped.net2Institute of Mathematics College of ScienceUniversity of the Philippines DilimanQuezon City 1101Philippinesupd.edu.ph3Academic Production, Ochanomizu University2-1-1 OtsukaBunkyo-kuTokyo 112-8610Japanocha.ac.jp20121522012201207112011031220112012Copyright © 2012 Shyamal Kumar Hui and Richard S. Lemence.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper deals with the study of super quasi-Einstein manifolds admitting W2-curvature tensor. The totally umbilical hypersurfaces of S(QE)n are also studied. Among others, the existence of such a manifold is ensured by a nontrivial example.

1. Introduction

It is well known that a Riemannian manifold (Mn,g)  (n>2) is Einstein if its Ricci tensor S of type (0,2) is of the form S=ag, where a is a constant, which turns into S=(r/n)g,  r being the scalar curvature (constant) of the manifold. Let (Mn,g)  (n>2) be a Riemannian manifold. Let US={xM:S(r/n)g  at  x}, then the manifold (Mn,g) is said to be quasi-Einstein manifold  if on USM, we haveS-ag=bAA, where A is a 1-form on US and, a, b are some functions on US. It is clear that the 1-form A as well as the function b are nonzero at every point on US. From the above definition, it follows that every Einstein manifold is quasi-Einstein. In particular, every Ricci-flat manifold (e.g., Schwarzschild spacetime) is quasi-Einstein. The scalars a, b are known as the associated scalars of the manifold. Also, the 1-form A is called the associated 1-form of the manifold defined by g(X,ρ)=A(X) for any vector field X,  ρ being a unit vector field, called the generator of the manifold. Such an n-dimensional quasi-Einstein manifold is denoted by (QE)n. The quasi-Einstein manifolds have also been studied by De and Ghosh , Shaikh et al. , and Shaikh and Patra .

As a generalization of quasi-Einstein manifold, Chaki  introduced the notion of generalized quasi-Einstein manifolds. A Riemannian manifold (Mn,g)  (n>2) is said to be generalized quasi-Einstein manifold if its Ricci tensor S of type (0,2) is not identically zero and satisfies the following:S(X,Y)=ag(X,Y)+bA(X)A(Y)+c[A(X)B(Y)+A(Y)B(X)], where a, b, and c are scalars of which b0,  c0, A, B are nonzero 1-forms such that g(X,ρ)=A(X),  g(X,μ)=B(X) for all X and ρ,  μ are two unit vector fields mutually orthogonal to each other. In such a case, a,  b, and c are called the associated scalars, A,  B are called the associated 1-forms, and ρ,  μ are the generators of the manifold. Such an n-dimensional manifold is denoted by G(QE)n.

In , Chaki also introduced the notion of super quasi-Einstein manifold. A Riemannian manifold (Mn,g)  (n>2) is called super quasi-Einstein manifold if its Ricci tensor S of type (0,2) is not identically zero and satisfies the following:S(X,Y)=ag(X,Y)+bA(X)A(Y)+c[A(X)B(Y)+A(Y)B(X)]+dD(X,Y), where a, b, c, and d are nonzero scalars, A, B are two nonzero 1-forms such that g(X,ρ)=A(X), g(X,μ)=B(X) for all vector fields X, and ρ, μ are unit vectors such that ρ is perpendicular to μ and D is a symmetric (0,2) tensor with zero trace, which satisfies the condition D(X,ρ)=0 for all vector fields X. Here, a, b, c, and d are called the associated scalars, A, B are the associated 1-forms of the manifold, and D is called the structure tensor. Such an n-dimensional manifold is denoted by S(QE)n. The super quasi-Einstein manifolds have also been studied by Debnath and Konar , Özgür , and many others.

In 1970, Pokhariyal and Mishra  introduced new tensor fields, called W2 and E tensor fields, in a Riemannian manifold and studied their properties. According to them, a W2-curvature tensor on a manifold (Mn,g)  (n>2) is defined byW2(X,Y)Z=R(X,Y)Z+1n-1[g(X,Z)QY-g(Y,Z)QX], where Q is the Ricci operator, that is, g(QX,Y)=S(X,Y) for all X,Y. In this connection, it may be mentioned that Pokhariyal and Mishra [20, 21] and Pokhariyal  introduced some new curvature tensors defined on the line of Weyl projective curvature tensor.

The W2-curvature tensor was introduced on the line of Weyl projective curvature tensor, and by breaking W2 into skew-symmetric parts, the tensor E has been defined. Rainich conditions for the existence of the nonnull electrovariance can be obtained by W2 and E if we replace the matter tensor by the contracted part of these tensors. The tensor E enables to extend Pirani formulation of gravitational waves to Einstein space [23, 24]. It is shown that  except the vanishing of complexion vector and property of being identical in two spaces which are in geodesic correspondence, the W2-curvature tensor possesses the properties almost similar to the Weyl projective curvature tensor. Thus, we can very well use W2-curvature tensor in various physical and geometrical spheres in place of the Weyl projective curvature tensor.

The W2-curvature tensor has also been studied by various authors in different structures such as De and Sarkar , Matsumoto et al. , Pokhariyal [23, 24, 27], Shaikh et al. , Shaikh et al. , Taleshian and Hosseinzadeh , Tripathi and Gupta , Venkatesha et al. , and Yíldíz and De .

Motivated by the above studies, in Section 3, we study W2-curvature tensor of a super quasi-Einstein manifold. It is proved that if in an S(QE)n  (n>2) the associated scalars are constants, the structure tensor is of Codazzi type and the generators ρ and μ are vector fields with the associated 1-forms A and B not being the 1-forms of recurrences, then the manifold is W2-conservative.

Recently, Özen and Altay  studied the totally umbilical hypersurfaces of weakly and pseudosymmetric spaces. Again, Özen and Altay  also studied the totally umbilical hypersurfaces of weakly concircular and pseudoconcircular symmetric spaces. In this connection, it may be mentioned that Shaikh et al.  studied the totally umbilical hypersurfaces of weakly conharmonically symmetric spaces. Section 4 deals with the study of totally umbilical hypersurfaces of S(QE)n. It is proved that the totally umbilical hypersurfaces of S(QE)n+1 are S(QE)n if and only if the hypersurface is a totally geodesic hypersurface.

Finally, in the last section, the existence of super quasi-Einstein manifold is ensured by a nontrivial example.

2. Preliminaries

In this section, we will obtain some formulas of S(QE)n, which will be required in the sequel. Let {ei:i=1,2,,n} be an orthonormal frame field at any point of the manifold, then setting X=Y=ei in (1.3) and taking summation over i,  1in, we obtainr=na+b, where r is the scalar curvature of the manifold.

Also from (1.3), we haveS(X,ρ)=(a+b)A(X)+cB(X),S(ρ,ρ)=(a+b),S(X,μ)=aB(X)+cA(X)+dD(X,μ),S(μ,μ)=a+dD(μ,μ),S(ρ,μ)=c.

3. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M116"><mml:mrow><mml:msub><mml:mrow><mml:mi>W</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:mrow></mml:math></inline-formula>-Curvature Tensor of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M117"><mml:mi>S</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>Q</mml:mi><mml:mi>E</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>

Let a manifold M be an S(QE)n  (n>2), which is W2-flat, then from (1.4), we getR(Y,Z,U,V)=1n-1[g(Z,U)S(Y,V)-g(Y,U)S(Z,V)]. Setting U=ρ and V=μ in (3.1) and using (2.2) and (2.4), we obtainR(Y,Z,ρ,μ)=an-1[A(Z)B(Y)-A(Y)B(Z)]+dn-1[A(Z)D(Y,μ)-A(Y)D(Z,μ)]. Again, plugging U=μ and V=ρ in (3.1) and using (2.2) and (2.4), we getR(Y,Z,μ,ρ)=a+bn-1[A(Y)B(Z)-A(Z)B(Y)]. From (3.2) and (3.3), we haveA(Z)[bB(Y)-dD(Y,μ)]=A(Y)[bB(Z)-dD(Z,μ)], that is,A(Z)E(Y)=A(Y)E(Z), where E(Y)=g(Y,σ)=bB(Y)-dD(Y,μ) for all Y. From (3.5), we may conclude that the two vector fields ρ and σ are codirectional, provided E0.

If E(Y)=0, then we haveD(Y,μ)=bdB(Y)=bdg(Y,μ)since  δ0, which implies that b/d is an eigenvalue of the tensor D corresponding to the eigenvector σ. Thus, we have the following result.

Theorem 3.1.

Let a manifold M be a W2-flat S(QE)n  (n>2) such that b/d is not an eigenvalue of the tensor D corresponding to the eigenvector σ defined by E(Y)=g(Y,σ)=bB(Y)-dD(Y,μ), then the vector fields ρ and σ corresponding to the 1-forms A and E, respectively, are codirectional.

From (1.4), we get that(divW2)(Y,Z)U=(divR)(Y,Z)U+12(n-1)[dr(Z)g(Y,U)-dr(Y)g(Z,U)], where “div” denotes the divergence.

Again, it is known that in a Riemannian manifold, we have(divR)(Y,Z)U=(YS)(Z,U)-(ZS)(Y,U). Consequently, by virtue of the above relation, (3.7) takes the form(divW2)(Y,Z)U=(YS)(Z,U)-(ZS)(Y,U)+12(n-1){dr(Z)g(Y,U)-dr(Y)g(Z,U)}. We now consider the associated scalars a, b, c, and d as constants, then (2.1) yields that the scalar curvature r is constant, and hence dr(X)=0 for all X. Consequently, (3.9) yields(divW2)(Y,Z)U=(YS)(Z,U)-(ZS)(Y,U). Since a, b, c, and d are constants, we have from (1.3) that(YS)(Z,U)=b[(YA)(Z)A(U)+A(Z)(YA)(U)]+c[(YA)(Z)B(U)+A(Z)(YB)(U)+(YA)(U)B(Z)+A(U)(YB)(Z)]+d(YD)(Z,U). We now assume that the structure tensor D of such as S(QE)n is of Codazzi type , then for all vector fields Y, Z, and U, we have(YD)(Z,U)=(ZD)(Y,U). By virtue of (3.11) and (3.12), (3.10) yields  (divW2)(Y,Z)U=b[(YA)(Z)A(U)+A(Z)(YA)(U)-(ZA)(Y)A(U)-A(Y)(ZA)(U)]+c[(YA)(Z)B(U)+A(Z)(YB)(U)+(YA)(U)B(Z)+A(U)(YB)(Z)-(ZA)(Y)B(U)-A(Y)(ZB)(U)-(ZA)(U)B(Y)-A(U)(ZB)(Y)]. Now, if the generators ρ and μ of the manifold are recurrent vector fields , then we have Yρ=π1(Y)ρ and Yμ=π2(Y)μ, where π1 and π2 are called the 1-forms of recurrence such that π1 and π2 are different from A and B. Consequently, we get (YA)(Z)=g(Yρ,Z)=g(π1(Y)ρ,Z)=π1(Y)A(Z),(YB)(Z)=g(Yμ,Z)=g(π2(Y)ρ,Z)=π2(Y)B(Z). In view of (3.14), (3.13) turns into(divW2)(Y,Z)U=2bπ1(Y)A(Z)A(U)+c[{π1(Y)+π2(Y)}{A(Z)B(U)+A(U)B(Z)}-{π1(Z)+π2(Z)}{A(Y)B(U)+A(U)B(Y)}]. Since g(ρ,ρ)=g(μ,μ)=1, it follows that (YA)(ρ)=g(Yρ,ρ)=0, and hence (3.14) reduces to π1(Y)=0 for all Y. Similarly, we have π2(Y)=0. Hence, from (3.15), we have (divW2)(Y,Z)U=0, that is, the manifold under consideration is W2-conservative . Hence, we can state the following.

Theorem 3.2.

Suppose that a manifold M is an S(QE)n  (n>2) such that associated scalars are constants and the structure tensor is of Codazzi type. If the generators ρ and μ corresponding to the associated 1-forms A and B are not being the 1-forms of recurrences, then the manifold is W2-conservative.

4. Totally Umbilical Hypersurfaces of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M200"><mml:mi>S</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>Q</mml:mi><mml:mi>E</mml:mi><mml:msub><mml:mrow><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>

Let (V̅,g̅) be an (n+1)-dimensional Riemannian manifold covered by a system of coordinate neighbourhoods {U,yα}. Let (V,g) be a hypersurface of (V̅,g̅) defined in a locally coordinate system by means of a system of parametric equation yα=yα(xi), where Greek indices take values 1,2,,n+1 and Latin indices take values 1,2,,n. Let Nα be the components of a local unit normal to (V,g), then we havegij=g̅αβyiαyjβ,g̅αβNαyjβ=0,g̅αβNαNβ=e=1,yiαyjβgij=g̅αβ-NαNβ,yiα=yαxi. The hypersurface (V,g) is called a totally umbilical hypersurface [40, 41] of (V̅,g̅) if its second fundamental form Ωij satisfiesΩij=Hgij,yi,jα=gijHNα, where the scalar function H is called the mean curvature of (V,g) given by H=(1/n)gijΩij. If, in particular, H=0, that is,Ωij=0, then the totally umbilical hypersurface is called a totally geodesic hypersurface of (V̅,g̅).

The equation of Weingarten for (V,g) can be written as N,jα=-(H/n)yjα. The structure equations of Gauss and Codazzi [40, 41] for (V,g) and (V̅,g̅) are, respectively, given byRijkl=R̅αβγδFijklαβγδ+H2Gijkl,R̅αβγδFijkαβγNδ=H,igjk-H,jgik, where Rijkl and R̅αβγδ are curvature tensors of (V,g) and (V̅,g̅), respectively, andFijklαβγδ=FiαFjβFkγFlδ,Fiα=yiα,Gijkl=gilgjk-gikgjl. Also we have [40, 41]S̅αδFiαFjδ=Sij-(n-1)H2gij,S̅αδNαFiδ=(n-1)H,i,r̅=r-n(n-1)H2, where Sij and S̅αδ are the Ricci tensors of (V,g) and (V̅,g̅), respectively, and r and r̅ are the scalar curvatures of (V,g) and (V̅,g̅), respectively.

In terms of local coordinates, the relation (1.3) can be written asSij=agij+bAiAj+c[AiBj+AjBi]+dDij. Let (V̅,g̅) be an S(QE)n+1, then we getSαβ=agαβ+bAαAβ+c[AαBβ+AβBα]+dDαβ. Multiplying both sides of (4.10) by Fijαβ and then using (4.6) and (4.9), we obtain H=0, which implies that the hypersurface is a totally geodesic hypersurface.

Conversely, we now consider that the hypersurface (V,g) is totally geodesic hypersurface, that is,H=0. In view of (4.11), (4.6) yieldsS̅αδFiαFjδ=Sij. Using (4.12) in (4.10), we have the relation (4.9). Thus, we can state the following.

Theorem 4.1.

The totally umbilical hypersurface of an S(QE)n+1 is an S(QE)n if and only if the hypersurface is a totally geodesic hypersurface.

Note that the theorem is a statement on the hypersurface based on the restrictions of the associated scalars and 1-forms coming from the manifold.

5. Example of a Super Quasi-Einstein Manifold

This section deals with a nontrivial example of S(QE)4.

Example 5.1.

We define a Riemannian metric g on 4 by the formula ds2=gijdxidxj=(dx1)2+ex1[ex2(dx2)2+ex3(dx3)2+ex4(dx4)2],(i,j=1,2,3,4).

Then, the only nonvanishing components of the Christoffel symbols, the curvature tensor, the Ricci tensor, and the scalar curvature are given byΓ221=-12ex1+x2,Γ331=-12ex1+x3,Γ441=-12ex1+x4,Γ222=12=Γ333=Γ444=Γ122=Γ133=Γ144,R1221=14ex1+x2,R1331=14ex1+x3,R1441=12ex1+x4,R2332=14e2x1+x2+x3,R2442=14e2x1+x2+x4,R3443=14e2x1+x3+x4,S11=34,S22=34ex1+x2,S33=34ex1+x3,S44=34ex1+x4,r=3 and the components which can be obtained from these by the symmetry properties. Therefore, 4 is a Riemannian manifold (M4,g) of nonvanishing scalar curvature. We will now show that M4 is an S(QE)4, that is, it satisfies (1.3). Let us now consider the associated scalars as follows:a=34,b=2ex1,c=ex1+x3,d=12ex4. In terms of local coordinate system, let us consider the 1-forms A and B and the structure tensor D as follows:Ai(x)={12for  i=1,1ex1for  i=2,0otherwise,Bi(x)={-1ex3for  i=1,1ex1for  i=3,0otherwise,Dij(x)={ex1+x4for  i=j=1,-4ex4for  i=j=2,-ex3+x4for  i=1,  j=3,-2ex3+x4ex1for  i=2,  j=3,0otherwise. In terms of local coordinate system, the defining condition (1.3) of an S(QE)n can be written asSij=agij+bAiAj+c[AiBj+AjBi]+dDij,i,j=1,2,3,4. By virtue of (5.3) and (5.4), it can be easily shown that the relation (5.5) holds for i,j=1,2,3,4. Therefore, (M4,g) is an S(QE)4, which is neither quasi-Einstein nor generalized quasi-Einstein. Hence, we can state the following.

Theorem 5.2.

Let (M4,g) be a Riemannian manifold endowed with the metric given in (5.1), then (M4,g) is an S(QE)4 with nonvanishing scalar curvature which is neither quasi-Einstein nor generalized quasi-Einstein.

Acknowledgments

The authors wish to thank the referees and editors for their comments and suggestions. This work was funded by the Special Coordination Funds for Promoting Science and Technology, Japan.

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