Einstein like manifolds which are not einstein

Such an -dimensional quasi-Einstein manifold is denoted by. The quasi-Einstein manifolds have also been studied in [ 8 — 11 ]. Generalizing the notion of quasi-Einstein manifold, in [ 12 ], De and Ghosh introduced the notion of generalized quasi-Einstein manifolds and studied its geometrical properties with the existence of such notion by several nontrivial examples.

The unit vectors and corresponding to 1-forms and are orthogonal to each other. Also and are the generators of the manifold. Such an -dimensional manifold is denoted by. The generalized quasi-Einstein manifolds have also been studied in [ 13 — 16 ]. In , De and Gazi [ 17 ] introduced the notion of nearly quasi-Einstein manifolds. An -dimensional nearly quasi-Einstein manifold was denoted by. The nearly quasi-Einstein manifolds have also been studied by Prakasha and Bagewadi [ 18 ].

The present paper is organized as follows. Section 2 deals with the preliminaries. Section 3 is concerned with conharmonic curvature tensor on. In this section, it is proved that a conharmonically flat is one of the manifold of generalized quasiconstant curvature. In the last section we study some geometrical properties of a.

Consider a with associated scalars , , and and associated -forms ,. Then from 2 we get where is the scalar curvature of the manifold. Since and are orthogonal unit vector fields, , , and. Again from 2 , we have. Let be the symmetric endomorphism of the tangent space at each point of the manifold corresponding to the Ricci tensor.

Then for all. The rank-four tensor that remains invariant under conharmonic transformation for an -dimensional Riemannian manifold is given by [ 19 ] where and denotes the Riemannian curvature tensor of type defined by , where is the Riemannian curvature tensor of type. A manifold of generalized quasiconstant curvature tensor is. But the converse is not true, in general. In this section, we enquire under what conditions the converse will be true. A manifold whose conharmonic curvature tensor vanishes at every point of the manifold is called conharmonically flat manifold.

Thus this tensor represents the deviation of the manifold from conharmonic flatness. It satisfies all the symmetric properties of the Riemannian curvature tensor.

There are many physical applications of the tensor. For example, in [ 20 ], Abdussattar showed that sufficient condition for a space-time to be conharmonic to a flat is either empty in which case it is flat or filled with a distribution represented by energy momentum tensor possessing the algebraic structure of an electromagnetic field and conformal to a flat space-time [ 20 ]. Let us consider that the manifold under consideration is conharmonically flat.

Then from 6 we have Using 2 in 7 , we obtain. In [ 12 ], De and Ghosh generalize the notion of quasiconstant curvature and prove the existence of such a manifold. A Riemannian manifold is said to be a manifold of generalized quasiconstant curvature, if the curvature tensor of type satisfies the condition: where , , and are scalars and and are nonzero 1-forms. We assume that the unit vector fields and defined by and are orthogonal; that is,.

Now the relation 8 can be written as where , , and. Comparing 10 and 11 , it follows that the manifold is of generalized quasiconstant curvature. Thus we have the following theorem. Theorem 1. A conharmonically flat is one of generalized quasiconstant curvature. Next, differentiating 6 covariantly and then contracting we obtain where denotes the divergence. Again, it is known that in a Riemannian manifold we have Consequently by virtue of 13 , the relation 12 takes the form: Now consider the associated scalars , , and as constants; then 4 yields that the scalar curvature is constant, and hence for all.

Consequently, 14 reduces to Since , , and are constants, we have from 2 that By virtue of 16 , we get from 15 that Next, if the generators and of the manifold under consideration are recurrent vector fields [ 22 ], then we have and , where and are the 1-forms of the recurrence such that and are different from and.

Consequently, we get In view of 18 , relation 17 reduces to Also, since , it follows that , and hence 18 reduces to for all. Similarly, we have. Hence, from 19 , we have ; that is, the manifold under consideration is conharmonically conservative. Hence, we can state the following theorem. Theorem 2. Then we have Setting and in 20 , we have By using 5 in 21 , we obtain Next, putting in 6 and then taking inner product with , we get where.

Again from 5 we obtain from 23 Again, plugging in 6 and then taking inner product with , we have where. From 24 and 25 , one can get By taking account of 26 in 22 , we obtain This implies either or. Now if , then, from 2 , we have where.

That is, the manifold is a. On the other hand, if , then 24 gives Thus we can state the following theorem. Theorem 3. If is a satisfying the condition , then either is or the curvature tensor of the manifold satisfies the property In [ 23 ], Gray introduced two classes of Riemannian manifolds determined by the covariant differentiation of Ricci tensor. Class A consists of all Riemannian manifolds whose Ricci tensor satisfies the condition: that is, Ricci tensor is a Codazzi type tensor.

Class B consists of all Riemannian manifolds whose Ricci tensor satisfies the condition: that is, Ricci tensor is cyclic parallel.

First suppose that the associated scalars are constants and the Ricci tensor is of Codazzi type. Then from 2 we obtain Interchanging and in 32 , we have Since is of Codazzi type, we have from 32 and 33 that Putting in 34 and using and , since is a unit vector, we get This shows that.

Again, putting in 34 and using and , since is a unit vector, we get This shows that. Theorem 4. In a , if the associated scalars are constants and the Ricci tensor is of Codazzi type, then the associated 1-forms and are closed.

Next, suppose that the generators and are Killing vector fields in a and the associated scalars are constants. Matsushima, Y. Nomizu, K. Yano , Kinokunuja, Tokyo, pp. Ryan, P.

Schouten, J. Simon, U. Sumitomo, T. Tani, M. Wegner, B. Dedicata to appear. Yano, K. Download references. You can also search for this author in PubMed Google Scholar. Reprints and Permissions. Einstein-like manifolds which are not Einstein. Geom Dedicata 7, — Download citation. Received : 04 November Revised : 04 February Issue Date : September Anyone you share the following link with will be able to read this content:.

Sorry, a shareable link is not currently available for this article. Provided by the Springer Nature SharedIt content-sharing initiative. Skip to main content.