ISRN Geometry

VolumeΒ 2012, Article IDΒ 217132, 10 pages

http://dx.doi.org/10.5402/2012/217132

## Some Results on Super Quasi-Einstein Manifolds

^{1}Nikhil Banga Sikshan Mahavidyalaya, Bishnupur, 722122 West Bengal, Bankura, India^{2}Institute of Mathematics College of Science, University of the Philippines Diliman, Quezon City 1101, Philippines^{3}Academic 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 -curvature tensor. The totally umbilical hypersurfaces of 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 is Einstein if its Ricci tensor of type is of the form , where is a constant, which turns into being the scalar curvature (constant) of the manifold. Let be a Riemannian manifold. Let , then the manifold is said to be quasi-Einstein manifold [1β12] if on , we have where is a 1-form on and, , are some functions on . It is clear that the 1-form as well as the function are nonzero at every point on . 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 , are known as the associated scalars of the manifold. Also, the 1-form is called the associated 1-form of the manifold defined by for any vector field being a unit vector field, called the generator of the manifold. Such an -dimensional quasi-Einstein manifold is denoted by . 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 is said to be generalized quasi-Einstein manifold if its Ricci tensor of type is not identically zero and satisfies the following: where , , and are scalars of which , , are nonzero 1-forms such that for all and are two unit vector fields mutually orthogonal to each other. In such a case, , and are called the associated scalars, are called the associated 1-forms, and are the generators of the manifold. Such an -dimensional manifold is denoted by .

In [17], Chaki also introduced the notion of super quasi-Einstein manifold. A Riemannian manifold is called super quasi-Einstein manifold if its Ricci tensor of type is not identically zero and satisfies the following: where , , , and are nonzero scalars, , are two nonzero 1-forms such that , for all vector fields , and , are unit vectors such that is perpendicular to and is a symmetric tensor with zero trace, which satisfies the condition for all vector fields . Here, , , , and are called the associated scalars, , are the associated 1-forms of the manifold, and is called the structure tensor. Such an -dimensional manifold is denoted by . The super quasi-Einstein manifolds have also been studied by Debnath and Konar [18], ΓzgΓΌr [19], and many others.

In 1970, Pokhariyal and Mishra [20] introduced new tensor fields, called and tensor fields, in a Riemannian manifold and studied their properties. According to them, a -curvature tensor on a manifold is defined by where is the Ricci operator, that is, for all . 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 -curvature tensor was introduced on the line of Weyl projective curvature tensor, and by breaking into skew-symmetric parts, the tensor has been defined. Rainich conditions for the existence of the nonnull electrovariance can be obtained by and if we replace the matter tensor by the contracted part of these tensors. The tensor 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 -curvature tensor possesses the properties almost similar to the Weyl projective curvature tensor. Thus, we can very well use -curvature tensor in various physical and geometrical spheres in place of the Weyl projective curvature tensor.

The -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 -curvature tensor of a super quasi-Einstein manifold. It is proved that if in an the associated scalars are constants, the structure tensor is of Codazzi type and the generators and are vector fields with the associated 1-forms and not being the 1-forms of recurrences, then the manifold is -conservative.

Recently, Γzen and Altay [34] studied the totally umbilical hypersurfaces of weakly and pseudosymmetric spaces. Again, Γzen 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 . It is proved that the totally umbilical hypersurfaces of are 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 , which will be required in the sequel. Let be an orthonormal frame field at any point of the manifold, then setting in (1.3) and taking summation over , we obtain where is the scalar curvature of the manifold.

Also from (1.3), we have

#### 3. -Curvature Tensor of

Let a manifold be an , which is -flat, then from (1.4), we get Setting and in (3.1) and using (2.2) and (2.4), we obtain Again, plugging and in (3.1) and using (2.2) and (2.4), we get From (3.2) and (3.3), we have that is, where for all . From (3.5), we may conclude that the two vector fields and are codirectional, provided .

If , then we have which implies that is an eigenvalue of the tensor corresponding to the eigenvector . Thus, we have the following result.

Theorem 3.1. *Let a manifold be a -flat such that is not an eigenvalue of the tensor corresponding to the eigenvector defined by , then the vector fields and corresponding to the 1-forms and , respectively, are codirectional.*

From (1.4), we get that where βdivβ denotes the divergence.

Again, it is known that in a Riemannian manifold, we have Consequently, by virtue of the above relation, (3.7) takes the form We now consider the associated scalars , , , and as constants, then (2.1) yields that the scalar curvature is constant, and hence for all . Consequently, (3.9) yields Since , , , and are constants, we have from (1.3) that We now assume that the structure tensor of such as is of Codazzi type [37], then for all vector fields , , and , we have By virtue of (3.11) and (3.12), (3.10) yields Now, if the generators and of the manifold are recurrent vector fields [38], then we have and , where and are called the 1-forms of recurrence such that and are different from and . Consequently, we get In view of (3.14), (3.13) turns into Since , it follows that , and hence (3.14) reduces to for all . Similarly, we have . Hence, from (3.15), we have , that is, the manifold under consideration is -conservative [39]. Hence, we can state the following.

Theorem 3.2. *Suppose that a manifold is an such that associated scalars are constants and the structure tensor is of Codazzi type. If the generators and corresponding to the associated 1-forms and are not being the 1-forms of recurrences, then the manifold is -conservative.*

#### 4. Totally Umbilical Hypersurfaces of

Let be an -dimensional Riemannian manifold covered by a system of coordinate neighbourhoods . Let be a hypersurface of defined in a locally coordinate system by means of a system of parametric equation , where Greek indices take values and Latin indices take values . Let be the components of a local unit normal to , then we have The hypersurface is called a totally umbilical hypersurface [40, 41] of if its second fundamental form satisfies where the scalar function is called the mean curvature of given by . If, in particular, , that is, then the totally umbilical hypersurface is called a totally geodesic hypersurface of .

The equation of Weingarten for can be written as . The structure equations of Gauss and Codazzi [40, 41] for and are, respectively, given by where and are curvature tensors of and , respectively, and Also we have [40, 41] where and are the Ricci tensors of and , respectively, and and are the scalar curvatures of and , respectively.

In terms of local coordinates, the relation (1.3) can be written as Let be an , then we get Multiplying both sides of (4.10) by and then using (4.6) and (4.9), we obtain , which implies that the hypersurface is a totally geodesic hypersurface.

Conversely, we now consider that the hypersurface is totally geodesic hypersurface, that is, In view of (4.11), (4.6) yields 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 is an 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 .

*Example 5.1. *We define a Riemannian metric on by the formula

Then, the only nonvanishing components of the Christoffel symbols, the curvature tensor, the Ricci tensor, and the scalar curvature are given by and the components which can be obtained from these by the symmetry properties. Therefore, is a Riemannian manifold of nonvanishing scalar curvature. We will now show that is an , that is, it satisfies (1.3). Let us now consider the associated scalars as follows: In terms of local coordinate system, let us consider the 1-forms and and the structure tensor as follows: In terms of local coordinate system, the defining condition (1.3) of an can be written as By virtue of (5.3) and (5.4), it can be easily shown that the relation (5.5) holds for . Therefore, is an , which is neither quasi-Einstein nor generalized quasi-Einstein. Hence, we can state the following.

Theorem 5.2. *Let be a Riemannian manifold endowed with the metric given in (5.1), then is an 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.

#### References

- M. C. Chaki and R. K. Maity, βOn quasi Einstein manifolds,β
*Publicationes Mathematicae Debrecen*, vol. 57, no. 3-4, pp. 297β306, 2000. View at Google Scholar Β· View at Zentralblatt MATH - F. Defever, R. Deszcz, M. Hotloś, M. Kucharski, and Z. Sentürk, βGeneralisations of Robertson-Walker spaces,β
*Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae. Sectio Mathematica*, vol. 43, pp. 13β24, 2000. View at Google Scholar Β· View at Zentralblatt MATH - R. Deszcz, F. Dillen, L. Verstraelen, and L. Vrancken, βQuasi-Einstein totally real submanifolds of ${S}^{6}(1)$,β
*The Tohoku Mathematical Journal*, vol. 51, no. 4, pp. 461β478, 1999. View at Publisher Β· View at Google Scholar Β· View at MathSciNet - R. Deszcz and M. Głogowska, βExamples of nonsemisymmetric Ricci-semisymmetric hypersurfaces,β
*Colloquium Mathematicum*, vol. 94, no. 1, pp. 87β101, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - R. Deszcz, M. Głogowska, M. Hotloś, and Z. Sentürk, βOn certain quasi-Einstein semisymmetric hypersurfaces,β
*Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae. Sectio Mathematica*, vol. 41, pp. 151β164, 1998. View at Google Scholar Β· View at Zentralblatt MATH - R. Deszcz and M. Hotloś, βOn some pseudosymmetry type curvature condition,β
*Tsukuba Journal of Mathematics*, vol. 27, no. 1, pp. 13β30, 2003. View at Google Scholar Β· View at Zentralblatt MATH - R. Deszcz and M. Hotloś, βOn hypersurfaces with type number two in space forms,β
*Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae. Sectio Mathematica*, vol. 46, pp. 19β34, 2003. View at Google Scholar Β· View at Zentralblatt MATH - R. Deszcz, M. Hotloś, and Z. Sentürk, βQuasi-Einstein hypersurfaces in semi-Riemannian space forms,β
*Colloquium Mathematicum*, vol. 89, no. 1, pp. 81β97, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - R. Deszcz, M. Hotloś, and Z. Sentürk, βOn curvature properties of quasi-Einstein hypersurfaces in semi-Euclidean spaces,β
*Soochow Journal of Mathematics*, vol. 27, no. 4, pp. 375β389, 2001. View at Google Scholar Β· View at Zentralblatt MATH - R. Deszcz, P. Verheyen, and L. Verstraelen, βOn some generalized Einstein metric conditions,β
*Institut Mathématique. Publications. Nouvelle Série*, vol. 60, pp. 108β120, 1996. View at Google Scholar Β· View at Zentralblatt MATH - M. Głogowska, βSemi-Riemannian manifolds whose Weyl tensor is a Kulkarni-Nomizu square,β
*Institut Mathématique. Publications. Nouvelle Série*, vol. 72, pp. 95β106, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - M. Głogowska, βOn quasi-Einstein Cartan type hypersurfaces,β
*Journal of Geometry and Physics*, vol. 58, no. 5, pp. 599β614, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - U. C. De and G. C. Ghosh, βOn quasi Einstein manifolds,β
*Periodica Mathematica Hungarica*, vol. 48, no. 1-2, pp. 223β231, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet - A. A. Shaikh, D. W. Yoon, and S. K. Hui, βOn quasi-Einstein spacetimes,β
*Tsukuba Journal of Mathematics*, vol. 33, no. 2, pp. 305β326, 2009. View at Google Scholar Β· View at Zentralblatt MATH - A. A. Shaikh and A. Patra, βOn quasi-conformally flat quasi-Einstein spaces,β
*Differential Geometry—Dynamical Systems*, vol. 12, pp. 201β212, 2010. View at Google Scholar Β· View at Zentralblatt MATH - M. C. Chaki, βOn generalized quasi Einstein manifolds,β
*Publicationes Mathematicae Debrecen*, vol. 58, no. 4, pp. 683β691, 2001. View at Google Scholar Β· View at Zentralblatt MATH - M. C. Chaki, βOn super quasi Einstein manifolds,β
*Publicationes Mathematicae Debrecen*, vol. 64, no. 3-4, pp. 481β488, 2004. View at Google Scholar Β· View at Zentralblatt MATH - P. Debnath and A. Konar, βOn super quasi Einstein manifold,β
*Institut Mathématique. Publications. Nouvelle Série*, vol. 89, no. 103, pp. 95β104, 2011. View at Publisher Β· View at Google Scholar - C. Özgür, βOn some classes of super quasi-Einstein manifolds,β
*Chaos, Solitons and Fractals*, vol. 40, no. 3, pp. 1156β1161, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - G. P. Pokhariyal and R. S. Mishra, βCurvature tensors' and their relativistics significance,β
*Yokohama Mathematical Journal*, vol. 18, pp. 105β108, 1970. View at Google Scholar Β· View at Zentralblatt MATH - G. P. Pokhariyal and R. S. Mishra, βCurvature tensors and their relativistic significance. II,β
*Yokohama Mathematical Journal*, vol. 19, no. 2, pp. 97β103, 1971. View at Google Scholar Β· View at Zentralblatt MATH - G. P. Pokhariyal, βCurvature tensors and their relativistic significance. III,β
*Yokohama Mathematical Journal*, vol. 21, pp. 115β119, 1973. View at Google Scholar Β· View at Zentralblatt MATH - G. P. Pokhariyal, βRelativistic significance of curvature tensors,β
*International Journal of Mathematics and Mathematical Sciences*, vol. 5, no. 1, pp. 133β139, 1982. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH - G. P. Pokhariyal, βCurvature tensors on
*A*-Einstein Sasakian manifolds,β*Balkan Journal of Geometry and Its Applications*, vol. 6, no. 1, pp. 45β50, 2001. View at Google Scholar - U. C. De and A. Sarkar, βOn a type of
*P*-Sasakian manifolds,β*Mathematical Reports*, vol. 11, no. 2, pp. 139β144, 2009. View at Google Scholar Β· View at Zentralblatt MATH - K. Matsumoto, S. Ianuş, and I. Mihai, βOn
*P*-Sasakian manifolds which admit certain tensor fields,β*Publicationes Mathematicae Debrecen*, vol. 33, no. 3-4, pp. 199β204, 1986. View at Google Scholar - G. P. Pokhariyal, βStudy of a new curvature tensor in a Sasakian manifold,β
*The Tensor Society. Tensor. New Series*, vol. 36, no. 2, pp. 222β226, 1982. View at Google Scholar Β· View at Zentralblatt MATH - A. A. Shaikh, S. K. Jana, and S. Eyasmin, βOn weakly ${W}_{2}$-symmetric manifolds,β
*Sarajevo Journal of Mathematics*, vol. 3, no. 1, pp. 73β91, 2007. View at Google Scholar Β· View at Zentralblatt MATH - A. A. Shaikh, Y. Matsuyama, and S. K. Jana, βOn a type of general relativistic spacetime with ${W}_{2}$-curvature tensor,β
*Indian Journal of Mathematics*, vol. 50, no. 1, pp. 53β62, 2008. View at Google Scholar - A. Taleshian and A. A. Hosseinzadeh, βOn ${W}_{2}$-curvature tensor
*N(k)*-quasi Einstein manifolds,β*The Journal of Mathematics and Computer Science*, vol. 1, no. 1, pp. 28β32, 2010. View at Google Scholar - M. M. Tripathi and P. Gupta, βOn $\tau $-curvature tensor in
*K*-contact and Sasakian manifolds,β*International Electronic Journal of Geometry*, vol. 4, no. 1, pp. 32β47, 2011. View at Google Scholar - Venkatesha, C. S. Bagewadi, and K. T. Pradeep Kumar, βSome results on Lorentzian Para-Sasakian manifolds,β
*ISRN Geometry*, vol. 2011, Article ID 161523, 9 pages, 2011. View at Publisher Β· View at Google Scholar - A. Yíldíz and U. C. De, βOn a type of Kenmotsu manifolds,β
*Differential Geometry—Dynamical Systems*, vol. 12, pp. 289β298, 2010. View at Google Scholar Β· View at Zentralblatt MATH - F. Özen and S. Altay, βOn weakly and pseudo-symmetric Riemannian spaces,β
*Indian Journal of Pure and Applied Mathematics*, vol. 33, no. 10, pp. 1477β1488, 2002. View at Google Scholar Β· View at Zentralblatt MATH - F. Özen and S. Altay, βOn weakly and pseudo concircular symmetric structures on a Riemannian manifold,β
*Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica*, vol. 47, pp. 129β138, 2008. View at Google Scholar Β· View at Zentralblatt MATH - A. A. Shaikh, I. Roy, and S. K. Hui, βOn totally umbilical hypersurfaces of weakly conharmonically symmetric spaces,β
*Global Journal Science Frontier Research*, vol. 10, no. 4, pp. 28β30, 2010. View at Google Scholar - D. Ferus,
*A Remark on Codazzi Tensors on Constant Curvature Space*, vol. 838 of*Lecture Notes in Mathematics, Global Differential Geometry and Global Analysis*, Springer, New York, NY, USA, 1981. - J. A. Schouten,
*Ricci-Calculus. An Introduction to Tensor Analysis and Its Geometrical Applications*, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen mit Besonderer Berücksichtigung der Anwendungsgebiete, Bd X, Springer, Berlin, Germany, 1954. - N. J. Hicks,
*Notes on Differential Geometry*, Affiliated East West Press, 1969. - B.-Y. Chen,
*Geometry of Submanifolds*, Marcel Dekker, New York, NY, USA, 1973. - L. P. Eisenhart,
*Riemannian Geometry*, Princeton University Press, Princeton, NJ, USA, 1949.