On Einstein-type contact metric manifolds
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 -dimensional smooth manifold (or M) is said to be contact if it has a global 1-form η such that 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 and . Polarization of dη on the contact sub-bundle (defined by ), 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 as Using this in the well known formula of curvature tensor: , by a direct calculation, we obtain
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.
References (40)
A note on critical point metrics of the total scalar curvature functional
J. Math. Anal. Appl.
(2015)- et al.
A note on static spaces and related problems
J. Geom. Phys.
(2013) On static three-manifolds with positive scalar curvature
J. Differ. Geom.
(2017)On critical point equation of compact manifolds with zero radial Weyl curvature
Geom. Dedic.
(2017)- et al.
On static manifolds and related critical spaces with zero radial Weyl curvature
Monatshefte Math.
(2020) - et al.
Remarks on critical metrics of the scalar curvature and volume functionals on compact manifolds with boundary
Pac. J. Math.
(2018) - et al.
Critical point equation on four-dimensional compact manifolds
Math. Nachr.
(2014) - et al.
Bach-flat critical metrics of the volume functional on 4-dimensional manifolds with boundary
J. Geom. Anal.
(2015) - et al.
Rigidity for critical metrics of the volume functional
Math. Nachr.
(2019) - et al.
Critical metrics of the volume functional on compact three-manifolds with smooth boundary
J. Geom. Anal.
(2017)
Einstein Manifolds
Riemannian Geometry of Contact and Symplectic Manifolds
When is the tangent sphere bundle conformally flat?
J. Geom.
Contact metric manifolds satisfying a nullity condition
Isr. J. Math.
Three dimensional locally symmetric contact metric manifolds
Boll. Unione Mat. Ital., A (7)
A full classification of contact metric -spaces
Ill. J. Math.
Uniqueness theorem for anti-de Sitter spacetime
Phys. Rev. D
Einstein manifolds and contact geometry
Proc. Am. Math. Soc.
Generalized quasi-Einstein manifold with harmonic Weyl tensor
Math. Z.
Killing fields generated by multiple solutions to the Fischer-Marsden equation
Int. J. Math.
Cited by (3)
On Einstein-type almost Kenmotsu manifolds
2023, Analysis (Germany)Einstein-Type Metrics on Almost Kenmotsu Manifolds
2023, Bulletin of the Malaysian Mathematical Sciences Society∗-Ricci tensor on (κ, µ)-contact manifolds
2022, AIMS Mathematics