On Einstein-type contact metric manifolds

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Abstract

In this paper, we consider an Einstein-type equation on contact metric manifolds. First, we prove that any K-contact manifold admitting an Einstein-type metric is isometric to a unit sphere. Similar conclusion also holds for a contact metric manifold admitting an Einstein type metric with zero radial Weyl curvature and satisfying a commutativity condition. Finally, we study an Einstein-type equation on (k,μ)-contact manifold.

Section snippets

Introduction and main results

Recently, there has been a rising interest in the study of static spaces in Riemannian geometry and mathematical physics. They are specially important because of their relevance in general relativity. In particular, vacuum static spaces-time is closely related to the cosmic no-hair conjecture of general relativity (see Boucher et al. [15]). In this paper, we consider an extended version of static space considered in [27], [37] which include some well known critical point equations that arise as

Preliminaries and notations

In this section, we review some basic definitions and properties on contact metric manifolds, see details in [10], [12]. A (2n+1)-dimensional smooth manifold M2n+1 (or M) is said to be contact if it has a global 1-form η such that η(dη)n0 everywhere on M. The 1-form η is known as the contact form. Corresponding to this 1-form one can find a unit vector field ξ, called the Reeb vector field, such that dη(ξ,.)=0 and η(ξ)=1. Polarization of on the contact sub-bundle D (defined by η=0), yields

Proof of main results

In the present section we will prove Theorem 1.1, Theorem 1.3. The Einstein-type equation (1.1) can be written asfQX=XDf+σX,XX(M). Using this in the well known formula of curvature tensor: R(X,Y)=[X,Y][X,Y], by a direct calculation, we obtaing(R(X,Z)Df,Y)=X(f)Ricg(Y,Z)Z(f)Ricg(X,Y)+f{g((XQ)Z,Y)g((ZQ)X,Y)}X(σ)g(Y,Z)+Z(σ)g(X,Y).

Conclusion

In this paper, we use the methods of local Riemannian geometry to study solutions of the Einstein-type equation (1.1) and characterize Einstein metrics in such broader classes of metrics as generalization of the static vacuum Einstein, the static vacuum, the static perfect fluid, the critical point and the Maio-Tam equations and generalize some well known results. It is important not only for differential geometry, but also for theoretical physics. In particular, it is conjectured that the only

Acknowledgement

The authors are thankful to the referee for some valuable comments that improve the paper. Dr. D.S. Patra is grateful to Professor Vladimir Rovenski for his constant encouragement. He is financially supported by the Council for Higher Education, Planning and Budgeting Committee (PBC) and the University of Haifa, Israel.

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