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ISRN Geometry
VolumeΒ 2012Β (2012), Article IDΒ 217132, 10 pages
http://dx.doi.org/10.5402/2012/217132
Research Article

Some Results on Super Quasi-Einstein Manifolds

1Nikhil Banga Sikshan Mahavidyalaya, Bishnupur, 722122 West Bengal, Bankura, India
2Institute of Mathematics College of Science, University of the Philippines Diliman, Quezon City 1101, Philippines
3Academic Production, Ochanomizu University, 2-1-1 Otsuka, Bunkyo-ku, Tokyo 112-8610, Japan

Received 7 November 2011; Accepted 3 December 2011

Academic Editors: M.Β Coppens, A.Β Morozov, and M.Β Visinescu

Copyright Β© 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.

Abstract

This paper deals with the study of super quasi-Einstein manifolds admitting π‘Š2-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=π‘Žπ‘”, where π‘Ž is a constant, which turns into 𝑆=(π‘Ÿ/𝑛)𝑔,π‘Ÿ being the scalar curvature (constant) of the manifold. Let (𝑀𝑛,𝑔)(𝑛>2) be a Riemannian manifold. Let π‘ˆπ‘†={π‘₯βˆˆπ‘€βˆΆπ‘†β‰ (π‘Ÿ/𝑛)𝑔atπ‘₯}, then the manifold (𝑀𝑛,𝑔) is said to be quasi-Einstein manifold [1–12] if on π‘ˆπ‘†βŠ‚π‘€, we haveπ‘†βˆ’π‘Žπ‘”=π‘π΄βŠ—π΄,(1.1) where 𝐴 is a 1-form on π‘ˆπ‘† and, π‘Ž, 𝑏 are some functions on π‘ˆπ‘†. It is clear that the 1-form 𝐴 as well as the function 𝑏 are nonzero at every point on π‘ˆπ‘†. 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 π‘Ž, 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,π‘Œ)=π‘Žπ‘”(𝑋,π‘Œ)+𝑏𝐴(𝑋)𝐴(π‘Œ)+𝑐𝐴(𝑋)𝐡(π‘Œ)+𝐴(π‘Œ)𝐡(𝑋)(1.2) where π‘Ž, 𝑏, and 𝑐 are scalars of which 𝑏≠0,𝑐≠0, 𝐴, 𝐡 are nonzero 1-forms such that 𝑔(𝑋,𝜌)=𝐴(𝑋),𝑔(𝑋,πœ‡)=𝐡(𝑋) for all 𝑋 and 𝜌,πœ‡ are two unit vector fields mutually orthogonal to each other. In such a case, π‘Ž,𝑏, and 𝑐 are called the associated scalars, 𝐴,𝐡 are called the associated 1-forms, and 𝜌,πœ‡ 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,π‘Œ)=π‘Žπ‘”(𝑋,π‘Œ)+𝑏𝐴(𝑋)𝐴(π‘Œ)+𝑐𝐴(𝑋)𝐡(π‘Œ)+𝐴(π‘Œ)𝐡(𝑋)+𝑑𝐷(𝑋,π‘Œ),(1.3) where π‘Ž, 𝑏, 𝑐, and 𝑑 are nonzero scalars, 𝐴, 𝐡 are two nonzero 1-forms such that 𝑔(𝑋,𝜌)=𝐴(𝑋), 𝑔(𝑋,πœ‡)=𝐡(𝑋) for all vector fields 𝑋, and 𝜌, πœ‡ are unit vectors such that 𝜌 is perpendicular to πœ‡ 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, π‘Ž, 𝑏, 𝑐, and 𝑑 are called the associated scalars, 𝐴, 𝐡 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], Γ–zgΓΌr [19], and many others.

In 1970, Pokhariyal and Mishra [20] introduced new tensor fields, called π‘Š2 and 𝐸 tensor fields, in a Riemannian manifold and studied their properties. According to them, a π‘Š2-curvature tensor on a manifold (𝑀𝑛,𝑔)(𝑛>2) is defined byπ‘Š21(𝑋,π‘Œ)𝑍=𝑅(𝑋,π‘Œ)𝑍+[],π‘›βˆ’1𝑔(𝑋,𝑍)π‘„π‘Œβˆ’π‘”(π‘Œ,𝑍)𝑄𝑋(1.4) where 𝑄 is the Ricci operator, that is, 𝑔(𝑄𝑋,π‘Œ)=𝑆(𝑋,π‘Œ) for all 𝑋,π‘Œ. 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 π‘Š2-curvature tensor was introduced on the line of Weyl projective curvature tensor, and by breaking π‘Š2 into skew-symmetric parts, the tensor 𝐸 has been defined. Rainich conditions for the existence of the nonnull electrovariance can be obtained by π‘Š2 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 π‘Š2-curvature tensor possesses the properties almost similar to the Weyl projective curvature tensor. Thus, we can very well use π‘Š2-curvature tensor in various physical and geometrical spheres in place of the Weyl projective curvature tensor.

The π‘Š2-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 YΓ­ldΓ­z and De [33].

Motivated by the above studies, in Section 3, we study π‘Š2-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 πœ‡ are vector fields with the associated 1-forms 𝐴 and 𝐡 not being the 1-forms of recurrences, then the manifold is π‘Š2-conservative.

Recently, Γ–zen and Altay [34] studied the totally umbilical hypersurfaces of weakly and pseudosymmetric spaces. Again, Γ–zen 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 {π‘’π‘–βˆΆπ‘–=1,2,…,𝑛} be an orthonormal frame field at any point of the manifold, then setting 𝑋=π‘Œ=𝑒𝑖 in (1.3) and taking summation over 𝑖,1≀𝑖≀𝑛, we obtainπ‘Ÿ=π‘›π‘Ž+𝑏,(2.1) where π‘Ÿ is the scalar curvature of the manifold.

Also from (1.3), we have𝑆(𝑋,𝜌)=(π‘Ž+𝑏)𝐴(𝑋)+𝑐𝐡(𝑋),(2.2)𝑆(𝜌,𝜌)=(π‘Ž+𝑏),(2.3)𝑆(𝑋,πœ‡)=π‘Žπ΅(𝑋)+𝑐𝐴(𝑋)+𝑑𝐷(𝑋,πœ‡),(2.4)𝑆(πœ‡,πœ‡)=π‘Ž+𝑑𝐷(πœ‡,πœ‡),(2.5)𝑆(𝜌,πœ‡)=𝑐.(2.6)

3. π‘Š2-Curvature Tensor of 𝑆(𝑄𝐸)𝑛

Let a manifold 𝑀 be an 𝑆(𝑄𝐸)𝑛(𝑛>2), which is π‘Š2-flat, then from (1.4), we get1𝑅(π‘Œ,𝑍,π‘ˆ,𝑉)=[].π‘›βˆ’1𝑔(𝑍,π‘ˆ)𝑆(π‘Œ,𝑉)βˆ’π‘”(π‘Œ,π‘ˆ)𝑆(𝑍,𝑉)(3.1) Setting π‘ˆ=𝜌 and 𝑉=πœ‡ in (3.1) and using (2.2) and (2.4), we obtainπ‘Žπ‘…(π‘Œ,𝑍,𝜌,πœ‡)=[]+π‘‘π‘›βˆ’1𝐴(𝑍)𝐡(π‘Œ)βˆ’π΄(π‘Œ)𝐡(𝑍)[].π‘›βˆ’1𝐴(𝑍)𝐷(π‘Œ,πœ‡)βˆ’π΄(π‘Œ)𝐷(𝑍,πœ‡)(3.2) Again, plugging π‘ˆ=πœ‡ and 𝑉=𝜌 in (3.1) and using (2.2) and (2.4), we get𝑅(π‘Œ,𝑍,πœ‡,𝜌)=π‘Ž+𝑏[].π‘›βˆ’1𝐴(π‘Œ)𝐡(𝑍)βˆ’π΄(𝑍)𝐡(π‘Œ)(3.3) From (3.2) and (3.3), we have[][],𝐴(𝑍)𝑏𝐡(π‘Œ)βˆ’π‘‘π·(π‘Œ,πœ‡)=𝐴(π‘Œ)𝑏𝐡(𝑍)βˆ’π‘‘π·(𝑍,πœ‡)(3.4) that is,𝐴(𝑍)𝐸(π‘Œ)=𝐴(π‘Œ)𝐸(𝑍),(3.5) where 𝐸(π‘Œ)=𝑔(π‘Œ,𝜎)=𝑏𝐡(π‘Œ)βˆ’π‘‘π·(π‘Œ,πœ‡) for all π‘Œ. From (3.5), we may conclude that the two vector fields 𝜌 and 𝜎 are codirectional, provided 𝐸≠0.

If 𝐸(π‘Œ)=0, then we have𝑏𝐷(π‘Œ,πœ‡)=𝑑𝑏𝐡(π‘Œ)=𝑑𝑔(π‘Œ,πœ‡)since𝛿≠0,(3.6) 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 π‘Š2-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(π‘Œ)=𝑔(π‘Œ,𝜎)=𝑏𝐡(π‘Œ)βˆ’π‘‘π·(π‘Œ,πœ‡), then the vector fields 𝜌 and 𝜎 corresponding to the 1-forms 𝐴 and 𝐸, respectively, are codirectional.

From (1.4), we get thatξ€·divπ‘Š2ξ€Έ1(π‘Œ,𝑍)π‘ˆ=(div𝑅)(π‘Œ,𝑍)π‘ˆ+2[],(π‘›βˆ’1)π‘‘π‘Ÿ(𝑍)𝑔(π‘Œ,π‘ˆ)βˆ’π‘‘π‘Ÿ(π‘Œ)𝑔(𝑍,π‘ˆ)(3.7) where β€œdiv” denotes the divergence.

Again, it is known that in a Riemannian manifold, we haveξ€·βˆ‡(div𝑅)(π‘Œ,𝑍)π‘ˆ=π‘Œπ‘†ξ€Έξ€·βˆ‡(𝑍,π‘ˆ)βˆ’π‘π‘†ξ€Έ(π‘Œ,π‘ˆ).(3.8) Consequently, by virtue of the above relation, (3.7) takes the formξ€·divπ‘Š2ξ€Έξ€·βˆ‡(π‘Œ,𝑍)π‘ˆ=π‘Œπ‘†ξ€Έξ€·βˆ‡(𝑍,π‘ˆ)βˆ’π‘π‘†ξ€Έ1(π‘Œ,π‘ˆ)+2(π‘›βˆ’1){π‘‘π‘Ÿ(𝑍)𝑔(π‘Œ,π‘ˆ)βˆ’π‘‘π‘Ÿ(π‘Œ)𝑔(𝑍,π‘ˆ)}.(3.9) We now consider the associated scalars π‘Ž, 𝑏, 𝑐, and 𝑑 as constants, then (2.1) yields that the scalar curvature π‘Ÿ is constant, and hence π‘‘π‘Ÿ(𝑋)=0 for all 𝑋. Consequently, (3.9) yieldsξ€·divπ‘Š2ξ€Έξ€·βˆ‡(π‘Œ,𝑍)π‘ˆ=π‘Œπ‘†ξ€Έξ€·βˆ‡(𝑍,π‘ˆ)βˆ’π‘π‘†ξ€Έ(π‘Œ,π‘ˆ).(3.10) Since π‘Ž, 𝑏, 𝑐, and 𝑑 are constants, we have from (1.3) thatξ€·βˆ‡π‘Œπ‘†ξ€Έβˆ‡(𝑍,π‘ˆ)=π‘ξ€Ίξ€·π‘Œπ΄ξ€Έξ€·βˆ‡(𝑍)𝐴(π‘ˆ)+𝐴(𝑍)π‘Œπ΄ξ€Έξ€»βˆ‡(π‘ˆ)+π‘ξ€Ίξ€·π‘Œπ΄ξ€Έ(ξ€·βˆ‡π‘)𝐡(π‘ˆ)+𝐴(𝑍)π‘Œπ΅ξ€Έ(+ξ€·βˆ‡π‘ˆ)π‘Œπ΄ξ€Έξ€·βˆ‡(π‘ˆ)𝐡(𝑍)+𝐴(π‘ˆ)π‘Œπ΅ξ€Έξ€»ξ€·βˆ‡(𝑍)+π‘‘π‘Œπ·ξ€Έ(𝑍,π‘ˆ).(3.11) We now assume that the structure tensor 𝐷 of such as 𝑆(𝑄𝐸)𝑛 is of Codazzi type [37], then for all vector fields π‘Œ, 𝑍, and π‘ˆ, we haveξ€·βˆ‡π‘Œπ·ξ€Έξ€·βˆ‡(𝑍,π‘ˆ)=𝑍𝐷(π‘Œ,π‘ˆ).(3.12) By virtue of (3.11) and (3.12), (3.10) yieldsξ€·divπ‘Š2ξ€Έβˆ‡(π‘Œ,𝑍)π‘ˆ=π‘ξ€Ίξ€·π‘Œπ΄ξ€Έξ€·βˆ‡(𝑍)𝐴(π‘ˆ)+𝐴(𝑍)π‘Œπ΄ξ€Έβˆ’ξ€·βˆ‡(π‘ˆ)𝑍𝐴(ξ€·βˆ‡π‘Œ)𝐴(π‘ˆ)βˆ’π΄(π‘Œ)𝑍𝐴(ξ€»βˆ‡π‘ˆ)+π‘ξ€Ίξ€·π‘Œπ΄ξ€Έξ€·βˆ‡(𝑍)𝐡(π‘ˆ)+𝐴(𝑍)π‘Œπ΅ξ€Έ+ξ€·βˆ‡(π‘ˆ)π‘Œπ΄ξ€Έ(ξ€·βˆ‡π‘ˆ)𝐡(𝑍)+𝐴(π‘ˆ)π‘Œπ΅ξ€Έ(βˆ’ξ€·βˆ‡π‘)π‘π΄ξ€Έξ€·βˆ‡(π‘Œ)𝐡(π‘ˆ)βˆ’π΄(π‘Œ)π‘π΅ξ€Έβˆ’ξ€·βˆ‡(π‘ˆ)𝑍𝐴(ξ€·βˆ‡π‘ˆ)𝐡(π‘Œ)βˆ’π΄(π‘ˆ)𝑍𝐡(ξ€».π‘Œ)(3.13) Now, if the generators 𝜌 and πœ‡ of the manifold are recurrent vector fields [38], then we have βˆ‡π‘ŒπœŒ=πœ‹1(π‘Œ)𝜌 and βˆ‡π‘Œπœ‡=πœ‹2(π‘Œ)πœ‡, where πœ‹1 and πœ‹2 are called the 1-forms of recurrence such that πœ‹1 and πœ‹2 are different from 𝐴 and 𝐡. Consequently, we get ξ€·βˆ‡π‘Œπ΄ξ€Έξ€·βˆ‡(𝑍)=π‘”π‘Œξ€Έξ€·πœ‹πœŒ,𝑍=𝑔1ξ€Έ(π‘Œ)𝜌,𝑍=πœ‹1ξ€·βˆ‡(π‘Œ)𝐴(𝑍),π‘Œπ΅ξ€Έ(ξ€·βˆ‡π‘)=π‘”π‘Œξ€Έξ€·πœ‹πœ‡,𝑍=𝑔2(ξ€Έπ‘Œ)𝜌,𝑍=πœ‹2(π‘Œ)𝐡(𝑍).(3.14) In view of (3.14), (3.13) turns intoξ€·divπ‘Š2ξ€Έ(π‘Œ,𝑍)π‘ˆ=2π‘πœ‹1πœ‹(π‘Œ)𝐴(𝑍)𝐴(π‘ˆ)+𝑐1(π‘Œ)+πœ‹2(ξ€Ύ{βˆ’ξ€½πœ‹π‘Œ)𝐴(𝑍)𝐡(π‘ˆ)+𝐴(π‘ˆ)𝐡(𝑍)}1(𝑍)+πœ‹2ξ€Ύξ€».(𝑍){𝐴(π‘Œ)𝐡(π‘ˆ)+𝐴(π‘ˆ)𝐡(π‘Œ)}(3.15) Since 𝑔(𝜌,𝜌)=𝑔(πœ‡,πœ‡)=1, it follows that (βˆ‡π‘Œπ΄)(𝜌)=𝑔(βˆ‡π‘ŒπœŒ,𝜌)=0, and hence (3.14) reduces to πœ‹1(π‘Œ)=0 for all π‘Œ. Similarly, we have πœ‹2(π‘Œ)=0. Hence, from (3.15), we have (divπ‘Š2)(π‘Œ,𝑍)π‘ˆ=0, that is, the manifold under consideration is π‘Š2-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 πœ‡ corresponding to the associated 1-forms 𝐴 and 𝐡 are not being the 1-forms of recurrences, then the manifold is π‘Š2-conservative.

4. Totally Umbilical Hypersurfaces of 𝑆(𝑄𝐸)𝑛

Let (𝑉,𝑔) be an (𝑛+1)-dimensional Riemannian manifold covered by a system of coordinate neighbourhoods {π‘ˆ,𝑦𝛼}. Let (𝑉,𝑔) be a hypersurface of (𝑉,𝑔) defined in a locally coordinate system by means of a system of parametric equation 𝑦𝛼=𝑦𝛼(π‘₯𝑖), 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=𝑔𝛼𝛽𝑦𝛼𝑖𝑦𝛽𝑗,𝑔𝛼𝛽𝑁𝛼𝑦𝛽𝑗=0,𝑔𝛼𝛽𝑁𝛼𝑁𝛽𝑦=𝑒=1,𝛼𝑖𝑦𝛽𝑗𝑔𝑖𝑗=π‘”π›Όπ›½βˆ’π‘π›Όπ‘π›½,𝑦𝛼𝑖=πœ•π‘¦π›Όπœ•π‘₯𝑖.(4.1) The hypersurface (𝑉,𝑔) is called a totally umbilical hypersurface [40, 41] of (𝑉,𝑔) if its second fundamental form Ω𝑖𝑗 satisfiesΩ𝑖𝑗=𝐻𝑔𝑖𝑗,𝑦𝛼𝑖,𝑗=𝑔𝑖𝑗𝐻𝑁𝛼,(4.2) where the scalar function 𝐻 is called the mean curvature of (𝑉,𝑔) given by βˆ‘π‘”π»=(1/𝑛)𝑖𝑗Ω𝑖𝑗. If, in particular, 𝐻=0, that is,Ω𝑖𝑗=0,(4.3) then the totally umbilical hypersurface is called a totally geodesic hypersurface of (𝑉,𝑔).

The equation of Weingarten for (𝑉,𝑔) can be written as 𝑁𝛼,𝑗=βˆ’(𝐻/𝑛)𝑦𝛼𝑗. The structure equations of Gauss and Codazzi [40, 41] for (𝑉,𝑔) and (𝑉,𝑔) are, respectively, given byπ‘…π‘–π‘—π‘˜π‘™=π‘…π›Όπ›½π›Ύπ›ΏπΉπ›Όπ›½π›Ύπ›Ώπ‘–π‘—π‘˜π‘™+𝐻2πΊπ‘–π‘—π‘˜π‘™,π‘…π›Όπ›½π›Ύπ›ΏπΉπ›Όπ›½π›Ύπ‘–π‘—π‘˜π‘π›Ώ=𝐻,π‘–π‘”π‘—π‘˜βˆ’π»,π‘—π‘”π‘–π‘˜,(4.4) where π‘…π‘–π‘—π‘˜π‘™ and 𝑅𝛼𝛽𝛾𝛿 are curvature tensors of (𝑉,𝑔) and (𝑉,𝑔), respectively, andπΉπ›Όπ›½π›Ύπ›Ώπ‘–π‘—π‘˜π‘™=πΉπ›Όπ‘–πΉπ›½π‘—πΉπ›Ύπ‘˜πΉπ›Ώπ‘™,𝐹𝛼𝑖=𝑦𝛼𝑖,πΊπ‘–π‘—π‘˜π‘™=π‘”π‘–π‘™π‘”π‘—π‘˜βˆ’π‘”π‘–π‘˜π‘”π‘—π‘™.(4.5) Also we have [40, 41]𝑆𝛼𝛿𝐹𝛼𝑖𝐹𝛿𝑗=π‘†π‘–π‘—βˆ’(π‘›βˆ’1)𝐻2𝑔𝑖𝑗,(4.6)𝑆𝛼𝛿𝑁𝛼𝐹𝛿𝑖=(π‘›βˆ’1)𝐻,𝑖,(4.7)π‘Ÿ=π‘Ÿβˆ’π‘›(π‘›βˆ’1)𝐻2,(4.8) where 𝑆𝑖𝑗 and 𝑆𝛼𝛿 are the Ricci tensors of (𝑉,𝑔) and (𝑉,𝑔), respectively, and π‘Ÿ and π‘Ÿ are the scalar curvatures of (𝑉,𝑔) and (𝑉,𝑔), respectively.

In terms of local coordinates, the relation (1.3) can be written as𝑆𝑖𝑗=π‘Žπ‘”π‘–π‘—+𝑏𝐴𝑖𝐴𝑗𝐴+𝑐𝑖𝐡𝑗+𝐴𝑗𝐡𝑖+𝑑𝐷𝑖𝑗.(4.9) Let (𝑉,𝑔) be an 𝑆(𝑄𝐸)𝑛+1, then we get𝑆𝛼𝛽=π‘Žπ‘”π›Όπ›½+𝑏𝐴𝛼𝐴𝛽𝐴+𝑐𝛼𝐡𝛽+𝐴𝛽𝐡𝛼+𝑑𝐷𝛼𝛽.(4.10) Multiplying both sides of (4.10) by 𝐹𝛼𝛽𝑖𝑗 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.(4.11) In view of (4.11), (4.6) yields𝑆𝛼𝛿𝐹𝛼𝑖𝐹𝛿𝑗=𝑆𝑖𝑗.(4.12) 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=𝑔𝑖𝑗𝑑π‘₯𝑖𝑑π‘₯𝑗=𝑑π‘₯1ξ€Έ2+𝑒π‘₯1𝑒π‘₯2𝑑π‘₯2ξ€Έ2+𝑒π‘₯3𝑑π‘₯3ξ€Έ2+𝑒π‘₯4𝑑π‘₯4ξ€Έ2ξ‚„,(𝑖,𝑗=1,2,3,4).(5.1)

Then, the only nonvanishing components of the Christoffel symbols, the curvature tensor, the Ricci tensor, and the scalar curvature are given byΞ“1221=βˆ’2𝑒π‘₯1+π‘₯2,Ξ“1331=βˆ’2𝑒π‘₯1+π‘₯3,Ξ“1441=βˆ’2𝑒π‘₯1+π‘₯4,Ξ“222=12=Ξ“333=Ξ“444=Ξ“212=Ξ“313=Ξ“414,𝑅1221=14𝑒π‘₯1+π‘₯2,𝑅1331=14𝑒π‘₯1+π‘₯3,𝑅1441=12𝑒π‘₯1+π‘₯4,𝑅2332=14𝑒2π‘₯1+π‘₯2+π‘₯3,𝑅2442=14𝑒2π‘₯1+π‘₯2+π‘₯4,𝑅3443=14𝑒2π‘₯1+π‘₯3+π‘₯4,𝑆11=34,𝑆22=34𝑒π‘₯1+π‘₯2,𝑆33=34𝑒π‘₯1+π‘₯3,𝑆44=34𝑒π‘₯1+π‘₯4,π‘Ÿ=3(5.2) 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:3π‘Ž=4,𝑏=2𝑒π‘₯1,𝑐=𝑒π‘₯1+π‘₯31,𝑑=2𝑒π‘₯4.(5.3) In terms of local coordinate system, let us consider the 1-forms 𝐴 and 𝐡 and the structure tensor 𝐷 as follows:π΄π‘–βŽ§βŽͺβŽͺ⎨βŽͺβŽͺ⎩1(π‘₯)=21for𝑖=1,βˆšπ‘’π‘₯1𝐡for𝑖=2,0otherwise,π‘–βŽ§βŽͺβŽͺ⎨βŽͺβŽͺβŽ©βˆ’1(π‘₯)=𝑒π‘₯31for𝑖=1,𝑒π‘₯1𝐷for𝑖=3,0otherwise,π‘–π‘—βŽ§βŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽ©π‘’(π‘₯)=π‘₯1+π‘₯4for𝑖=𝑗=1,βˆ’4𝑒π‘₯4for𝑖=𝑗=2,βˆ’π‘’π‘₯3+π‘₯4βˆ’for𝑖=1,𝑗=3,2𝑒π‘₯3+π‘₯4βˆšπ‘’π‘₯1for𝑖=2,𝑗=3,0otherwise.(5.4) In terms of local coordinate system, the defining condition (1.3) of an 𝑆(𝑄𝐸)𝑛 can be written as𝑆𝑖𝑗=π‘Žπ‘”π‘–π‘—+𝑏𝐴𝑖𝐴𝑗𝐴+𝑐𝑖𝐡𝑗+𝐴𝑗𝐡𝑖+𝑑𝐷𝑖𝑗,𝑖,𝑗=1,2,3,4.(5.5) 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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