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={x∈M:S≠(r/n)gatx}, then the manifold (Mn,g) is said to be quasi-Einstein manifold [1–12] if on US⊂M, we haveS-ag=bA⊗A,
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 [13], Shaikh et al. [14], and Shaikh and Patra [15].

As a generalization of quasi-Einstein manifold, Chaki [16] 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 b≠0,c≠0, 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 [17], 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 [18], Özgür [19], and many others.

In 1970, Pokhariyal and Mishra [20] 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 [22] 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 [20] 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 [25], Matsumoto et al. [26], Pokhariyal [23, 24, 27], Shaikh et al. [28], Shaikh et al. [29], Taleshian and Hosseinzadeh [30], Tripathi and Gupta [31], Venkatesha et al. [32], and Yíldíz and De [33].

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 [34] studied the totally umbilical hypersurfaces of weakly and pseudosymmetric spaces. Again, Özen and Altay [35] also studied the totally umbilical hypersurfaces of weakly concircular and pseudoconcircular symmetric spaces. In this connection, it may be mentioned that Shaikh et al. [36] 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,1≤i≤n, 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.

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 E≠0.

If E(Y)=0, then we haveD(Y,μ)=bdB(Y)=bdg(Y,μ)sinceδ≠0,
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 [37], 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 [38], 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 [39]. 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=∂yα∂xi.
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 ℝ4 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, ℝ4 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)={12fori=1,1ex1fori=2,0otherwise,Bi(x)={-1ex3fori=1,1ex1fori=3,0otherwise,Dij(x)={ex1+x4fori=j=1,-4ex4fori=j=2,-ex3+x4fori=1,j=3,-2ex3+x4ex1fori=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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