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

Some Results on Super Quasi-Einstein Manifolds

Shyamal Kumar Hui1Β and Richard S. Lemence2,3

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‘Š 2 -curvature tensor. The totally umbilical hypersurfaces of 𝑆 ( 𝑄 𝐸 ) 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 ( 𝑀 𝑛 , 𝑔 ) ( 𝑛 > 2 ) is Einstein if its Ricci tensor 𝑆 of type ( 0 , 2 ) is of the form 𝑆 = π‘Ž 𝑔 , 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 ( 𝑀 𝑛 , 𝑔 ) ( 𝑛 > 2 ) be a Riemannian manifold. Let π‘ˆ 𝑆 = { π‘₯ ∈ 𝑀 ∢ 𝑆 β‰  ( π‘Ÿ / 𝑛 ) 𝑔 a t π‘₯ } , then the manifold ( 𝑀 𝑛 , 𝑔 ) is said to be quasi-Einstein manifold [1–12] if on π‘ˆ 𝑆 βŠ‚ 𝑀 , we have 𝑆 βˆ’ π‘Ž 𝑔 = 𝑏 𝐴 βŠ— 𝐴 , ( 1 . 1 ) where 𝐴 is a 1-form on π‘ˆ 𝑆 and, π‘Ž , 𝑏 are some functions on π‘ˆ 𝑆 . It is clear that the 1-form 𝐴 as well as the function 𝑏 are nonzero at every point on π‘ˆ 𝑆 . 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 ( 𝑄 𝐸 ) 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 ( 𝑀 𝑛 , 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‘Ž 𝑔 ( 𝑋 , π‘Œ ) + 𝑏 𝐴 ( 𝑋 ) 𝐴 ( π‘Œ ) + 𝑐 𝐴 ( 𝑋 ) 𝐡 ( π‘Œ ) + 𝐴 ( π‘Œ ) 𝐡 ( 𝑋 ) ( 1 . 2 ) where π‘Ž , 𝑏 , and 𝑐 are scalars of which 𝑏 β‰  0 , 𝑐 β‰  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 ( 𝑄 𝐸 ) 𝑛 .

In [17], Chaki also introduced the notion of super quasi-Einstein manifold. A Riemannian manifold ( 𝑀 𝑛 , 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‘Ž 𝑔 ( 𝑋 , π‘Œ ) + 𝑏 𝐴 ( 𝑋 ) 𝐴 ( π‘Œ ) + 𝑐 𝐴 ( 𝑋 ) 𝐡 ( π‘Œ ) + 𝐴 ( π‘Œ ) 𝐡 ( 𝑋 ) + 𝑑 𝐷 ( 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 ( 𝑄 𝐸 ) 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 π‘Š 2 and 𝐸 tensor fields, in a Riemannian manifold and studied their properties. According to them, a π‘Š 2 -curvature tensor on a manifold ( 𝑀 𝑛 , 𝑔 ) ( 𝑛 > 2 ) is defined by π‘Š 2 1 ( 𝑋 , π‘Œ ) 𝑍 = 𝑅 ( 𝑋 , π‘Œ ) 𝑍 + [ ] , 𝑛 βˆ’ 1 𝑔 ( 𝑋 , 𝑍 ) 𝑄 π‘Œ βˆ’ 𝑔 ( π‘Œ , 𝑍 ) 𝑄 𝑋 ( 1 . 4 ) where 𝑄 is the Ricci operator, that is, 𝑔 ( 𝑄 𝑋 , 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‘Š 2 -curvature tensor was introduced on the line of Weyl projective curvature tensor, and by breaking π‘Š 2 into skew-symmetric parts, the tensor 𝐸 has been defined. Rainich conditions for the existence of the nonnull electrovariance can be obtained by π‘Š 2 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‘Š 2 -curvature tensor possesses the properties almost similar to the Weyl projective curvature tensor. Thus, we can very well use π‘Š 2 -curvature tensor in various physical and geometrical spheres in place of the Weyl projective curvature tensor.

The π‘Š 2 -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 π‘Š 2 -curvature tensor of a super quasi-Einstein manifold. It is proved that if in an 𝑆 ( 𝑄 𝐸 ) 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‘Š 2 -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 ( 𝑄 𝐸 ) 𝑛 . It is proved that the totally umbilical hypersurfaces of 𝑆 ( 𝑄 𝐸 ) 𝑛 + 1 are 𝑆 ( 𝑄 𝐸 ) 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 ( 𝑄 𝐸 ) 𝑛 , which will be required in the sequel. Let { 𝑒 𝑖 ∢ 𝑖 = 1 , 2 , … , 𝑛 } be an orthonormal frame field at any point of the manifold, then setting 𝑋 = π‘Œ = 𝑒 𝑖 in (1.3) and taking summation over 𝑖 , 1 ≀ 𝑖 ≀ 𝑛 , we obtain π‘Ÿ = 𝑛 π‘Ž + 𝑏 , ( 2 . 1 ) where π‘Ÿ is the scalar curvature of the manifold.

Also from (1.3), we have 𝑆 ( 𝑋 , 𝜌 ) = ( π‘Ž + 𝑏 ) 𝐴 ( 𝑋 ) + 𝑐 𝐡 ( 𝑋 ) , ( 2 . 2 ) 𝑆 ( 𝜌 , 𝜌 ) = ( π‘Ž + 𝑏 ) , ( 2 . 3 ) 𝑆 ( 𝑋 , πœ‡ ) = π‘Ž 𝐡 ( 𝑋 ) + 𝑐 𝐴 ( 𝑋 ) + 𝑑 𝐷 ( 𝑋 , πœ‡ ) , ( 2 . 4 ) 𝑆 ( πœ‡ , πœ‡ ) = π‘Ž + 𝑑 𝐷 ( πœ‡ , πœ‡ ) , ( 2 . 5 ) 𝑆 ( 𝜌 , πœ‡ ) = 𝑐 . ( 2 . 6 )

3. π‘Š 2 -Curvature Tensor of 𝑆 ( 𝑄 𝐸 ) 𝑛

Let a manifold 𝑀 be an 𝑆 ( 𝑄 𝐸 ) 𝑛 ( 𝑛 > 2 ) , which is π‘Š 2 -flat, then from (1.4), we get 1 𝑅 ( π‘Œ , 𝑍 , π‘ˆ , 𝑉 ) = [ ] . 𝑛 βˆ’ 1 𝑔 ( 𝑍 , π‘ˆ ) 𝑆 ( π‘Œ , 𝑉 ) βˆ’ 𝑔 ( 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 obtain π‘Ž 𝑅 ( π‘Œ , 𝑍 , 𝜌 , πœ‡ ) = [ ] + 𝑑 𝑛 βˆ’ 1 𝐴 ( 𝑍 ) 𝐡 ( π‘Œ ) βˆ’ 𝐴 ( π‘Œ ) 𝐡 ( 𝑍 ) [ ] . 𝑛 βˆ’ 1 𝐴 ( 𝑍 ) 𝐷 ( π‘Œ , πœ‡ ) βˆ’ 𝐴 ( 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 get 𝑅 ( π‘Œ , 𝑍 , πœ‡ , 𝜌 ) = π‘Ž + 𝑏 [ ] . 𝑛 βˆ’ 1 𝐴 ( π‘Œ ) 𝐡 ( 𝑍 ) βˆ’ 𝐴 ( 𝑍 ) 𝐡 ( π‘Œ ) ( 3 . 3 ) From (3.2) and (3.3), we have [ ] [ ] , 𝐴 ( 𝑍 ) 𝑏 𝐡 ( π‘Œ ) βˆ’ 𝑑 𝐷 ( π‘Œ , πœ‡ ) = 𝐴 ( π‘Œ ) 𝑏 𝐡 ( 𝑍 ) βˆ’ 𝑑 𝐷 ( 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 ) = 𝑏 𝐡 ( π‘Œ ) βˆ’ 𝑑 𝐷 ( 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 β‰  0 .

If 𝐸 ( π‘Œ ) = 0 , then we have 𝑏 𝐷 ( π‘Œ , πœ‡ ) = 𝑑 𝑏 𝐡 ( π‘Œ ) = 𝑑 𝑔 ( π‘Œ , πœ‡ ) s i n c e 𝛿 β‰  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‘Š 2 -flat 𝑆 ( 𝑄 𝐸 ) 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 ) = 𝑏 𝐡 ( π‘Œ ) βˆ’ 𝑑 𝐷 ( 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 that ξ€· d i v π‘Š 2 ξ€Έ 1 ( π‘Œ , 𝑍 ) π‘ˆ = ( d i v 𝑅 ) ( π‘Œ , 𝑍 ) π‘ˆ + 2 [ ] , ( 𝑛 βˆ’ 1 ) 𝑑 π‘Ÿ ( 𝑍 ) 𝑔 ( π‘Œ , π‘ˆ ) βˆ’ 𝑑 π‘Ÿ ( π‘Œ ) 𝑔 ( 𝑍 , π‘ˆ ) ( 3 . 7 ) where “div” denotes the divergence.

Again, it is known that in a Riemannian manifold, we have ξ€· βˆ‡ ( d i v 𝑅 ) ( π‘Œ , 𝑍 ) π‘ˆ = π‘Œ 𝑆 ξ€Έ ξ€· βˆ‡ ( 𝑍 , π‘ˆ ) βˆ’ 𝑍 𝑆 ξ€Έ ( π‘Œ , π‘ˆ ) . ( 3 . 8 ) Consequently, by virtue of the above relation, (3.7) takes the form ξ€· d i v π‘Š 2 ξ€Έ ξ€· βˆ‡ ( π‘Œ , 𝑍 ) π‘ˆ = π‘Œ 𝑆 ξ€Έ ξ€· βˆ‡ ( 𝑍 , π‘ˆ ) βˆ’ 𝑍 𝑆 ξ€Έ 1 ( π‘Œ , π‘ˆ ) + 2 ( 𝑛 βˆ’ 1 ) { 𝑑 π‘Ÿ ( 𝑍 ) 𝑔 ( π‘Œ , π‘ˆ ) βˆ’ 𝑑 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 𝑑 π‘Ÿ ( 𝑋 ) = 0 for all 𝑋 . Consequently, (3.9) yields ξ€· d i v π‘Š 2 ξ€Έ ξ€· βˆ‡ ( π‘Œ , 𝑍 ) π‘ˆ = π‘Œ 𝑆 ξ€Έ ξ€· βˆ‡ ( 𝑍 , π‘ˆ ) βˆ’ 𝑍 𝑆 ξ€Έ ( π‘Œ , π‘ˆ ) . ( 3 . 1 0 ) Since π‘Ž , 𝑏 , 𝑐 , and 𝑑 are constants, we have from (1.3) that ξ€· βˆ‡ π‘Œ 𝑆 ξ€Έ βˆ‡ ( 𝑍 , π‘ˆ ) = 𝑏 ξ€Ί ξ€· π‘Œ 𝐴 ξ€Έ ξ€· βˆ‡ ( 𝑍 ) 𝐴 ( π‘ˆ ) + 𝐴 ( 𝑍 ) π‘Œ 𝐴 ξ€Έ ξ€» βˆ‡ ( π‘ˆ ) + 𝑐 ξ€Ί ξ€· π‘Œ 𝐴 ξ€Έ ( ξ€· βˆ‡ 𝑍 ) 𝐡 ( π‘ˆ ) + 𝐴 ( 𝑍 ) π‘Œ 𝐡 ξ€Έ ( + ξ€· βˆ‡ π‘ˆ ) π‘Œ 𝐴 ξ€Έ ξ€· βˆ‡ ( π‘ˆ ) 𝐡 ( 𝑍 ) + 𝐴 ( π‘ˆ ) π‘Œ 𝐡 ξ€Έ ξ€» ξ€· βˆ‡ ( 𝑍 ) + 𝑑 π‘Œ 𝐷 ξ€Έ ( 𝑍 , π‘ˆ ) . ( 3 . 1 1 ) We now assume that the structure tensor 𝐷 of such as 𝑆 ( 𝑄 𝐸 ) 𝑛 is of Codazzi type [37], then for all vector fields π‘Œ , 𝑍 , and π‘ˆ , we have ξ€· βˆ‡ π‘Œ 𝐷 ξ€Έ ξ€· βˆ‡ ( 𝑍 , π‘ˆ ) = 𝑍 𝐷 ξ€Έ ( π‘Œ , π‘ˆ ) . ( 3 . 1 2 ) By virtue of (3.11) and (3.12), (3.10) yields ξ€· d i v π‘Š 2 ξ€Έ βˆ‡ ( π‘Œ , 𝑍 ) π‘ˆ = 𝑏 ξ€Ί ξ€· π‘Œ 𝐴 ξ€Έ ξ€· βˆ‡ ( 𝑍 ) 𝐴 ( π‘ˆ ) + 𝐴 ( 𝑍 ) π‘Œ 𝐴 ξ€Έ βˆ’ ξ€· βˆ‡ ( π‘ˆ ) 𝑍 𝐴 ξ€Έ ( ξ€· βˆ‡ π‘Œ ) 𝐴 ( π‘ˆ ) βˆ’ 𝐴 ( π‘Œ ) 𝑍 𝐴 ξ€Έ ( ξ€» βˆ‡ π‘ˆ ) + 𝑐 ξ€Ί ξ€· π‘Œ 𝐴 ξ€Έ ξ€· βˆ‡ ( 𝑍 ) 𝐡 ( π‘ˆ ) + 𝐴 ( 𝑍 ) π‘Œ 𝐡 ξ€Έ + ξ€· βˆ‡ ( π‘ˆ ) π‘Œ 𝐴 ξ€Έ ( ξ€· βˆ‡ π‘ˆ ) 𝐡 ( 𝑍 ) + 𝐴 ( π‘ˆ ) π‘Œ 𝐡 ξ€Έ ( βˆ’ ξ€· βˆ‡ 𝑍 ) 𝑍 𝐴 ξ€Έ ξ€· βˆ‡ ( π‘Œ ) 𝐡 ( π‘ˆ ) βˆ’ 𝐴 ( π‘Œ ) 𝑍 𝐡 ξ€Έ βˆ’ ξ€· βˆ‡ ( π‘ˆ ) 𝑍 𝐴 ξ€Έ ( ξ€· βˆ‡ π‘ˆ ) 𝐡 ( π‘Œ ) βˆ’ 𝐴 ( π‘ˆ ) 𝑍 𝐡 ξ€Έ ( ξ€» . π‘Œ ) ( 3 . 1 3 ) Now, if the generators 𝜌 and πœ‡ of the manifold are recurrent vector fields [38], then we have βˆ‡ π‘Œ 𝜌 = πœ‹ 1 ( π‘Œ ) 𝜌 and βˆ‡ π‘Œ π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‘Œ 𝐴 ξ€Έ ξ€· βˆ‡ ( 𝑍 ) = 𝑔 π‘Œ ξ€Έ ξ€· πœ‹ 𝜌 , 𝑍 = 𝑔 1 ξ€Έ ( π‘Œ ) 𝜌 , 𝑍 = πœ‹ 1 ξ€· βˆ‡ ( π‘Œ ) 𝐴 ( 𝑍 ) , π‘Œ 𝐡 ξ€Έ ( ξ€· βˆ‡ 𝑍 ) = 𝑔 π‘Œ ξ€Έ ξ€· πœ‹ πœ‡ , 𝑍 = 𝑔 2 ( ξ€Έ π‘Œ ) 𝜌 , 𝑍 = πœ‹ 2 ( π‘Œ ) 𝐡 ( 𝑍 ) . ( 3 . 1 4 ) In view of (3.14), (3.13) turns into ξ€· d i v π‘Š 2 ξ€Έ ( π‘Œ , 𝑍 ) π‘ˆ = 2 𝑏 πœ‹ 1 πœ‹ ( π‘Œ ) 𝐴 ( 𝑍 ) 𝐴 ( π‘ˆ ) + 𝑐 ξ€Ί ξ€½ 1 ( π‘Œ ) + πœ‹ 2 ( ξ€Ύ { βˆ’ ξ€½ πœ‹ π‘Œ ) 𝐴 ( 𝑍 ) 𝐡 ( π‘ˆ ) + 𝐴 ( π‘ˆ ) 𝐡 ( 𝑍 ) } 1 ( 𝑍 ) + πœ‹ 2 ξ€Ύ ξ€» . ( 𝑍 ) { 𝐴 ( π‘Œ ) 𝐡 ( π‘ˆ ) + 𝐴 ( π‘ˆ ) 𝐡 ( π‘Œ ) } ( 3 . 1 5 ) Since 𝑔 ( 𝜌 , 𝜌 ) = 𝑔 ( πœ‡ , πœ‡ ) = 1 , it follows that ( βˆ‡ π‘Œ 𝐴 ) ( 𝜌 ) = 𝑔 ( βˆ‡ π‘Œ 𝜌 , 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 ( d i v π‘Š 2 ) ( π‘Œ , 𝑍 ) π‘ˆ = 0 , that is, the manifold under consideration is π‘Š 2 -conservative [39]. Hence, we can state the following.

Theorem 3.2. Suppose that a manifold 𝑀 is an 𝑆 ( 𝑄 𝐸 ) 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‘Š 2 -conservative.

4. Totally Umbilical Hypersurfaces of 𝑆 ( 𝑄 𝐸 ) 𝑛

Let ( 𝑉 , 𝑔 ) be an ( 𝑛 + 1 ) -dimensional Riemannian manifold covered by a system of coordinate neighbourhoods { π‘ˆ , 𝑦 𝛼 } . Let ( 𝑉 , 𝑔 ) be a hypersurface of ( 𝑉 , 𝑔 ) defined in a locally coordinate system by means of a system of parametric equation 𝑦 𝛼 = 𝑦 𝛼 ( π‘₯ 𝑖 ) , where Greek indices take values 1 , 2 , … , 𝑛 + 1 and Latin indices take values 1 , 2 , … , 𝑛 . Let 𝑁 𝛼 be the components of a local unit normal to ( 𝑉 , 𝑔 ) , then we have 𝑔 𝑖 𝑗 = 𝑔 𝛼 𝛽 𝑦 𝛼 𝑖 𝑦 𝛽 𝑗 , 𝑔 𝛼 𝛽 𝑁 𝛼 𝑦 𝛽 𝑗 = 0 , 𝑔 𝛼 𝛽 𝑁 𝛼 𝑁 𝛽 𝑦 = 𝑒 = 1 , 𝛼 𝑖 𝑦 𝛽 𝑗 𝑔 𝑖 𝑗 = 𝑔 𝛼 𝛽 βˆ’ 𝑁 𝛼 𝑁 𝛽 , 𝑦 𝛼 𝑖 = πœ• 𝑦 𝛼 π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 Ξ© 𝑖 𝑗 satisfies Ξ© 𝑖 𝑗 = 𝐻 𝑔 𝑖 𝑗 , 𝑦 𝛼 𝑖 , 𝑗 = 𝑔 𝑖 𝑗 𝐻 𝑁 𝛼 , ( 4 . 2 ) where the scalar function 𝐻 is called the mean curvature of ( 𝑉 , 𝑔 ) given by βˆ‘ 𝑔 𝐻 = ( 1 / 𝑛 ) 𝑖 𝑗 Ξ© 𝑖 𝑗 . If, in particular, 𝐻 = 0 , that is, Ξ© 𝑖 𝑗 = 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 𝑁 𝛼 , 𝑗 = βˆ’ ( 𝐻 / 𝑛 ) 𝑦 𝛼 𝑗 . The structure equations of Gauss and Codazzi [40, 41] for ( 𝑉 , 𝑔 ) and ( 𝑉 , 𝑔 ) are, respectively, given by 𝑅 𝑖 𝑗 π‘˜ 𝑙 = 𝑅 𝛼 𝛽 𝛾 𝛿 𝐹 𝛼 𝛽 𝛾 𝛿 𝑖 𝑗 π‘˜ 𝑙 + 𝐻 2 𝐺 𝑖 𝑗 π‘˜ 𝑙 , 𝑅 𝛼 𝛽 𝛾 𝛿 𝐹 𝛼 𝛽 𝛾 𝑖 𝑗 π‘˜ 𝑁 𝛿 = 𝐻 , 𝑖 𝑔 𝑗 π‘˜ βˆ’ 𝐻 , 𝑗 𝑔 𝑖 π‘˜ , ( 4 . 4 ) where 𝑅 𝑖 𝑗 π‘˜ 𝑙 and 𝑅 𝛼 𝛽 𝛾 𝛿 are curvature tensors of ( 𝑉 , 𝑔 ) and ( 𝑉 , 𝑔 ) , respectively, and 𝐹 𝛼 𝛽 𝛾 𝛿 𝑖 𝑗 π‘˜ 𝑙 = 𝐹 𝛼 𝑖 𝐹 𝛽 𝑗 𝐹 𝛾 π‘˜ 𝐹 𝛿 𝑙 , 𝐹 𝛼 𝑖 = 𝑦 𝛼 𝑖 , 𝐺 𝑖 𝑗 π‘˜ 𝑙 = 𝑔 𝑖 𝑙 𝑔 𝑗 π‘˜ βˆ’ 𝑔 𝑖 π‘˜ 𝑔 𝑗 𝑙 . ( 4 . 5 ) Also we have [40, 41] 𝑆 𝛼 𝛿 𝐹 𝛼 𝑖 𝐹 𝛿 𝑗 = 𝑆 𝑖 𝑗 βˆ’ ( 𝑛 βˆ’ 1 ) 𝐻 2 𝑔 𝑖 𝑗 , ( 4 . 6 ) 𝑆 𝛼 𝛿 𝑁 𝛼 𝐹 𝛿 𝑖 = ( 𝑛 βˆ’ 1 ) 𝐻 , 𝑖 , ( 4 . 7 ) π‘Ÿ = π‘Ÿ βˆ’ 𝑛 ( 𝑛 βˆ’ 1 ) 𝐻 2 , ( 4 . 8 ) where 𝑆 𝑖 𝑗 and 𝑆 𝛼 𝛿 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 as 𝑆 𝑖 𝑗 = π‘Ž 𝑔 𝑖 𝑗 + 𝑏 𝐴 𝑖 𝐴 𝑗 ξ€Ί 𝐴 + 𝑐 𝑖 𝐡 𝑗 + 𝐴 𝑗 𝐡 𝑖 ξ€» + 𝑑 𝐷 𝑖 𝑗 . ( 4 . 9 ) Let ( 𝑉 , 𝑔 ) be an 𝑆 ( 𝑄 𝐸 ) 𝑛 + 1 , then we get 𝑆 𝛼 𝛽 = π‘Ž 𝑔 𝛼 𝛽 + 𝑏 𝐴 𝛼 𝐴 𝛽 ξ€Ί 𝐴 + 𝑐 𝛼 𝐡 𝛽 + 𝐴 𝛽 𝐡 𝛼 ξ€» + 𝑑 𝐷 𝛼 𝛽 . ( 4 . 1 0 ) Multiplying both sides of (4.10) by 𝐹 𝛼 𝛽 𝑖 𝑗 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 . 1 1 ) In view of (4.11), (4.6) yields 𝑆 𝛼 𝛿 𝐹 𝛼 𝑖 𝐹 𝛿 𝑗 = 𝑆 𝑖 𝑗 . ( 4 . 1 2 ) 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 ( 𝑄 𝐸 ) 𝑛 + 1 is an 𝑆 ( 𝑄 𝐸 ) 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 ( 𝑄 𝐸 ) 4 .

Example 5.1. We define a Riemannian metric 𝑔 on ℝ 4 by the formula 𝑑 𝑠 2 = 𝑔 𝑖 𝑗 𝑑 π‘₯ 𝑖 𝑑 π‘₯ 𝑗 = ξ€· 𝑑 π‘₯ 1 ξ€Έ 2 + 𝑒 π‘₯ 1  𝑒 π‘₯ 2 ξ€· 𝑑 π‘₯ 2 ξ€Έ 2 + 𝑒 π‘₯ 3 ξ€· 𝑑 π‘₯ 3 ξ€Έ 2 + 𝑒 π‘₯ 4 ξ€· 𝑑 π‘₯ 4 ξ€Έ 2 ξ‚„ , ( 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 by Ξ“ 1 2 2 1 = βˆ’ 2 𝑒 π‘₯ 1 + π‘₯ 2 , Ξ“ 1 3 3 1 = βˆ’ 2 𝑒 π‘₯ 1 + π‘₯ 3 , Ξ“ 1 4 4 1 = βˆ’ 2 𝑒 π‘₯ 1 + π‘₯ 4 , Ξ“ 2 2 2 = 1 2 = Ξ“ 3 3 3 = Ξ“ 4 4 4 = Ξ“ 2 1 2 = Ξ“ 3 1 3 = Ξ“ 4 1 4 , 𝑅 1 2 2 1 = 1 4 𝑒 π‘₯ 1 + π‘₯ 2 , 𝑅 1 3 3 1 = 1 4 𝑒 π‘₯ 1 + π‘₯ 3 , 𝑅 1 4 4 1 = 1 2 𝑒 π‘₯ 1 + π‘₯ 4 , 𝑅 2 3 3 2 = 1 4 𝑒 2 π‘₯ 1 + π‘₯ 2 + π‘₯ 3 , 𝑅 2 4 4 2 = 1 4 𝑒 2 π‘₯ 1 + π‘₯ 2 + π‘₯ 4 , 𝑅 3 4 4 3 = 1 4 𝑒 2 π‘₯ 1 + π‘₯ 3 + π‘₯ 4 , 𝑆 1 1 = 3 4 , 𝑆 2 2 = 3 4 𝑒 π‘₯ 1 + π‘₯ 2 , 𝑆 3 3 = 3 4 𝑒 π‘₯ 1 + π‘₯ 3 , 𝑆 4 4 = 3 4 𝑒 π‘₯ 1 + π‘₯ 4 , π‘Ÿ = 3 ( 5 . 2 ) and the components which can be obtained from these by the symmetry properties. Therefore, ℝ 4 is a Riemannian manifold ( 𝑀 4 , 𝑔 ) of nonvanishing scalar curvature. We will now show that 𝑀 4 is an 𝑆 ( 𝑄 𝐸 ) 4 , that is, it satisfies (1.3). Let us now consider the associated scalars as follows: 3 π‘Ž = 4 , 𝑏 = 2 𝑒 π‘₯ 1 , 𝑐 = 𝑒 π‘₯ 1 + π‘₯ 3 1 , 𝑑 = 2 𝑒 π‘₯ 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: 𝐴 𝑖 ⎧ βŽͺ βŽͺ ⎨ βŽͺ βŽͺ ⎩ 1 ( π‘₯ ) = 2 1 f o r 𝑖 = 1 , √ 𝑒 π‘₯ 1 𝐡 f o r 𝑖 = 2 , 0 o t h e r w i s e , 𝑖 ⎧ βŽͺ βŽͺ ⎨ βŽͺ βŽͺ ⎩ βˆ’ 1 ( π‘₯ ) = 𝑒 π‘₯ 3 1 f o r 𝑖 = 1 , 𝑒 π‘₯ 1 𝐷 f o r 𝑖 = 3 , 0 o t h e r w i s e , 𝑖 𝑗 ⎧ βŽͺ βŽͺ βŽͺ ⎨ βŽͺ βŽͺ βŽͺ ⎩ 𝑒 ( π‘₯ ) = π‘₯ 1 + π‘₯ 4 f o r 𝑖 = 𝑗 = 1 , βˆ’ 4 𝑒 π‘₯ 4 f o r 𝑖 = 𝑗 = 2 , βˆ’ 𝑒 π‘₯ 3 + π‘₯ 4 βˆ’ f o r 𝑖 = 1 , 𝑗 = 3 , 2 𝑒 π‘₯ 3 + π‘₯ 4 √ 𝑒 π‘₯ 1 f o r 𝑖 = 2 , 𝑗 = 3 , 0 o t h e r w i s e . ( 5 . 4 ) In terms of local coordinate system, the defining condition (1.3) of an 𝑆 ( 𝑄 𝐸 ) 𝑛 can be written as 𝑆 𝑖 𝑗 = π‘Ž 𝑔 𝑖 𝑗 + 𝑏 𝐴 𝑖 𝐴 𝑗 ξ€Ί 𝐴 + 𝑐 𝑖 𝐡 𝑗 + 𝐴 𝑗 𝐡 𝑖 ξ€» + 𝑑 𝐷 𝑖 𝑗 , 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, ( 𝑀 4 , 𝑔 ) is an 𝑆 ( 𝑄 𝐸 ) 4 , which is neither quasi-Einstein nor generalized quasi-Einstein. Hence, we can state the following.

Theorem 5.2. Let ( 𝑀 4 , 𝑔 ) be a Riemannian manifold endowed with the metric given in (5.1), then ( 𝑀 4 , 𝑔 ) is an 𝑆 ( 𝑄 𝐸 ) 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.

References

  1. M. C. Chaki and R. K. Maity, β€œOn quasi Einstein manifolds,” Publicationes Mathematicae Debrecen, vol. 57, no. 3-4, pp. 297–306, 2000. View at Zentralblatt MATH
  2. F. Defever, R. Deszcz, M. Hotloś, M. Kucharski, and Z. Sentürk, β€œGeneralisations of Robertson-Walker spaces,” Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae. Sectio Mathematica, vol. 43, pp. 13–24, 2000. View at Zentralblatt MATH
  3. R. Deszcz, F. Dillen, L. Verstraelen, and L. Vrancken, β€œQuasi-Einstein totally real submanifolds of S6(1),” The Tohoku Mathematical Journal, vol. 51, no. 4, pp. 461–478, 1999. View at Publisher Β· View at Google Scholar Β· View at MathSciNet
  4. R. Deszcz and M. Głogowska, β€œExamples of nonsemisymmetric Ricci-semisymmetric hypersurfaces,” Colloquium Mathematicum, vol. 94, no. 1, pp. 87–101, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  5. R. Deszcz, M. Głogowska, M. Hotloś, and Z. Sentürk, β€œOn certain quasi-Einstein semisymmetric hypersurfaces,” Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae. Sectio Mathematica, vol. 41, pp. 151–164, 1998. View at Zentralblatt MATH
  6. R. Deszcz and M. Hotloś, β€œOn some pseudosymmetry type curvature condition,” Tsukuba Journal of Mathematics, vol. 27, no. 1, pp. 13–30, 2003. View at Zentralblatt MATH
  7. R. Deszcz and M. Hotloś, β€œOn hypersurfaces with type number two in space forms,” Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae. Sectio Mathematica, vol. 46, pp. 19–34, 2003. View at Zentralblatt MATH
  8. R. Deszcz, M. Hotloś, and Z. Sentürk, β€œQuasi-Einstein hypersurfaces in semi-Riemannian space forms,” Colloquium Mathematicum, vol. 89, no. 1, pp. 81–97, 2001. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  9. R. Deszcz, M. Hotloś, and Z. Sentürk, β€œOn curvature properties of quasi-Einstein hypersurfaces in semi-Euclidean spaces,” Soochow Journal of Mathematics, vol. 27, no. 4, pp. 375–389, 2001. View at Zentralblatt MATH
  10. R. Deszcz, P. Verheyen, and L. Verstraelen, β€œOn some generalized Einstein metric conditions,” Institut Mathématique. Publications. Nouvelle Série, vol. 60, pp. 108–120, 1996. View at Zentralblatt MATH
  11. M. Głogowska, β€œSemi-Riemannian manifolds whose Weyl tensor is a Kulkarni-Nomizu square,” Institut Mathématique. Publications. Nouvelle Série, vol. 72, pp. 95–106, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  12. M. Głogowska, β€œOn quasi-Einstein Cartan type hypersurfaces,” Journal of Geometry and Physics, vol. 58, no. 5, pp. 599–614, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  13. U. C. De and G. C. Ghosh, β€œOn quasi Einstein manifolds,” Periodica Mathematica Hungarica, vol. 48, no. 1-2, pp. 223–231, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH Β· View at MathSciNet
  14. A. A. Shaikh, D. W. Yoon, and S. K. Hui, β€œOn quasi-Einstein spacetimes,” Tsukuba Journal of Mathematics, vol. 33, no. 2, pp. 305–326, 2009. View at Zentralblatt MATH
  15. A. A. Shaikh and A. Patra, β€œOn quasi-conformally flat quasi-Einstein spaces,” Differential Geometry—Dynamical Systems, vol. 12, pp. 201–212, 2010. View at Zentralblatt MATH
  16. M. C. Chaki, β€œOn generalized quasi Einstein manifolds,” Publicationes Mathematicae Debrecen, vol. 58, no. 4, pp. 683–691, 2001. View at Zentralblatt MATH
  17. M. C. Chaki, β€œOn super quasi Einstein manifolds,” Publicationes Mathematicae Debrecen, vol. 64, no. 3-4, pp. 481–488, 2004. View at Zentralblatt MATH
  18. P. Debnath and A. Konar, β€œOn super quasi Einstein manifold,” Institut Mathématique. Publications. Nouvelle Série, vol. 89, no. 103, pp. 95–104, 2011. View at Publisher Β· View at Google Scholar
  19. C. Özgür, β€œOn some classes of super quasi-Einstein manifolds,” Chaos, Solitons and Fractals, vol. 40, no. 3, pp. 1156–1161, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  20. G. P. Pokhariyal and R. S. Mishra, β€œCurvature tensors' and their relativistics significance,” Yokohama Mathematical Journal, vol. 18, pp. 105–108, 1970. View at Zentralblatt MATH
  21. G. P. Pokhariyal and R. S. Mishra, β€œCurvature tensors and their relativistic significance. II,” Yokohama Mathematical Journal, vol. 19, no. 2, pp. 97–103, 1971. View at Zentralblatt MATH
  22. G. P. Pokhariyal, β€œCurvature tensors and their relativistic significance. III,” Yokohama Mathematical Journal, vol. 21, pp. 115–119, 1973. View at Zentralblatt MATH
  23. G. P. Pokhariyal, β€œRelativistic significance of curvature tensors,” International Journal of Mathematics and Mathematical Sciences, vol. 5, no. 1, pp. 133–139, 1982. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  24. G. P. Pokhariyal, β€œCurvature tensors on A-Einstein Sasakian manifolds,” Balkan Journal of Geometry and Its Applications, vol. 6, no. 1, pp. 45–50, 2001.
  25. U. C. De and A. Sarkar, β€œOn a type of P-Sasakian manifolds,” Mathematical Reports, vol. 11, no. 2, pp. 139–144, 2009. View at Zentralblatt MATH
  26. K. Matsumoto, S. Ianuş, and I. Mihai, β€œOn P-Sasakian manifolds which admit certain tensor fields,” Publicationes Mathematicae Debrecen, vol. 33, no. 3-4, pp. 199–204, 1986.
  27. G. P. Pokhariyal, β€œStudy of a new curvature tensor in a Sasakian manifold,” The Tensor Society. Tensor. New Series, vol. 36, no. 2, pp. 222–226, 1982. View at Zentralblatt MATH
  28. A. A. Shaikh, S. K. Jana, and S. Eyasmin, β€œOn weakly W2-symmetric manifolds,” Sarajevo Journal of Mathematics, vol. 3, no. 1, pp. 73–91, 2007. View at Zentralblatt MATH
  29. A. A. Shaikh, Y. Matsuyama, and S. K. Jana, β€œOn a type of general relativistic spacetime with W2-curvature tensor,” Indian Journal of Mathematics, vol. 50, no. 1, pp. 53–62, 2008.
  30. A. Taleshian and A. A. Hosseinzadeh, β€œOn W2-curvature tensor N(k)-quasi Einstein manifolds,” The Journal of Mathematics and Computer Science, vol. 1, no. 1, pp. 28–32, 2010.
  31. M. M. Tripathi and P. Gupta, β€œOn τ-curvature tensor in K-contact and Sasakian manifolds,” International Electronic Journal of Geometry, vol. 4, no. 1, pp. 32–47, 2011.
  32. Venkatesha, C. S. Bagewadi, and K. T. Pradeep Kumar, β€œSome results on Lorentzian Para-Sasakian manifolds,” ISRN Geometry, vol. 2011, Article ID 161523, 9 pages, 2011. View at Publisher Β· View at Google Scholar
  33. A. Yíldíz and U. C. De, β€œOn a type of Kenmotsu manifolds,” Differential Geometry—Dynamical Systems, vol. 12, pp. 289–298, 2010. View at Zentralblatt MATH
  34. F. Özen and S. Altay, β€œOn weakly and pseudo-symmetric Riemannian spaces,” Indian Journal of Pure and Applied Mathematics, vol. 33, no. 10, pp. 1477–1488, 2002. View at Zentralblatt MATH
  35. F. Özen and S. Altay, β€œOn weakly and pseudo concircular symmetric structures on a Riemannian manifold,” Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica, vol. 47, pp. 129–138, 2008. View at Zentralblatt MATH
  36. A. A. Shaikh, I. Roy, and S. K. Hui, β€œOn totally umbilical hypersurfaces of weakly conharmonically symmetric spaces,” Global Journal Science Frontier Research, vol. 10, no. 4, pp. 28–30, 2010.
  37. 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.
  38. 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.
  39. N. J. Hicks, Notes on Differential Geometry, Affiliated East West Press, 1969.
  40. B.-Y. Chen, Geometry of Submanifolds, Marcel Dekker, New York, NY, USA, 1973.
  41. L. P. Eisenhart, Riemannian Geometry, Princeton University Press, Princeton, NJ, USA, 1949.