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 [1–12] 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], Özgür [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 Yíldíz 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, 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 . 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 “div” 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.