On some curvature functionals over homogeneous Siklos space-times

الموضوعات :

A. Zaeim ¹ , M. Jafari ² , R. Kafimoosavi ³

1 - Department of Mathematics, Payame Noor University (PNU), P.O. Box 19395-4697, Tehran, Iran
2 - Department of Mathematics, Payame Noor University (PNU), P.O. Box 19395-4697, Tehran, Iran
3 - Department of Mathematics, Payame Noor University (PNU), P.O. Box 19395-4697, Tehran, Iran

تاريخ الإرسال : 27 الأربعاء , شوال, 1444 تاريخ التأكيد : 22 الأحد , ذو القعدة, 1444 تاريخ الإصدار : 12 الخميس , ذو القعدة, 1444

الکلمات المفتاحية: Siklos space-times‎, ‎quadratic curvature functional‎, ‎critical metric‎, ‎homogeneous space,

ملخص المقالة :

‎Some curvature functionals which are defined according to the quadratic curvature invariants were studied on a special class of space-times‎. ‎We exactly determine metrics that are critical for those considering curvature functionals‎, ‎through homogeneous classes‎.

المصادر:

[1] M. Berger, Quelques formules de variation pour une structure riemannienne, Ann. Sci. École Norm. Sup. 3 (1970), 285-294.
[2] A. L. Besse, Einstein Manifolds, Springer-Verlag, Berlin, 2008.
[3] J. Bicák, J. Podolsky, Gravitational waves in vacuum spacetimes with cosmological constant. I. Classification and geometrical properties of nontwisting type N solutions, J. Math. Phys. 40 (1999), 4495-4505.

[4] J. Bicák, J. Podolsky, Gravitational waves in vacuum spacetimes with cosmological constant. II. Deviation of geodesics and interpretation of nontwisting type solutions, J. Math. Phys. 40 (1999), 4506-4517.

[5] M. Brozos-Vazquez, S. Caeiro-Oliveira, E. Garcia-Rio, Critical metrics for all quadratic curvature functionals, Bull. London. Math. Soc. 53 (2021), 680-685.

[6] G. Calvaruso, Siklos spacetimes as homogeneous Ricci solitons, Class. Quantum. Grav. 36 (2019), 36:095011.

[7] G. Calvaruso, The Ricci soliton equation for homogeneous Siklos spacetimes, Note. Mat. 41 (2021), 31-44.

[8] G. Calvaruso, M. Kaflou, A. Zaeim, On the symmetries of Siklos spacetimes, Gen. Rel. Grav. 54 (2022), 54:60.

[9] G. Calvaruso, A. Zaeim, Critical metrics for quadratic curvature functionals on some solvmanifolds, Rev. Mat. Complut. (2023), In press.

[10] G. Catino, Some rigidity results on critical metrics for quadratic functionals, Calc. Var. Part. Diff. Eq. 54 (2015), 2921-2937.

[11] L. Defrise, Groupes D’Isotropie et Groupes de Stabilité Conforme Dans les Espaces Lorentziens, Thesis, Université Libre de Bruxelles, 1969.

[12] S. Deser, B. Tekin, Gravitational energy in quadratic-curvature gravities, Phys. Rev. Lett. 89 (2002), 10:101101.

[13] J. Ehlers, W. Kundt, Exact Solutions of the Gravitational Field Equations, The Theory of Gravitation, John Wiley & Sons, 1962.

[14] M. J. Gursky, J. A. Viaclovsky, Rigidity and stability of Einstein metrics for quadratic curvature functionals, J. Reine. Angew. Math. 700 (2015), 37-91.

[15] V. R. Kaigorodov, Einstein spaces of maximum mobility, Sov. Phys. Dok. 7 (1963), 893-895.

[16] M. Mohseni, Vacuum polarization in Siklos spacetimes, Phys. Rev. 97 (2018), 97:024006.

[17] I. Ozsváth, Homogeneous solutions of the Einstein-Maxwell equations, J. Math. Phys. 6 (1965), 1255-65.

[18] I. Ozsváth, I Robinson, K. Rózga, Plane-fronted gravitational and electromagnetic waves in spaces with cosmological constant, J. Math. Phys. 26 (1985), 1755-1761.

[19] J. Podolsky, Exact non-singular waves in the anti-de Sitter Universe, Gen. Relative. Gravit. 33 (2001), 1093-1113.

[20] J. Podolsky, Interpretation of the Siklos solutions as exact gravitational waves in the anti-de Sitter universe, Class. Quantum. Grav. 15 (1998), 719-733.

[21] J. Podolsky, J. B Griffiths, Impulsive waves in de Sitter and anti-de Sitter space-times generated by null particles with an arbitrary multipole structure, Class. Quantum. Grav. 15 (1998), 453-463.

[22] W. Sheng, L. Wang, Bach-flat critical metrics for quadratic curvature functionals, Ann. Glob. Anal. Geom. 54 (2018), 365-375.

[23] S. T. C. Siklos, Lobatchevski Plane Gravitational Waves, Cambridge University Press, Cambridge, 1985.

[24] H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, E. Herlt, Exact Solutions of Einstein’s Field Equations, Cambridge University Press, Cambridge, 2003.

[25] J. Viaclovsky, Critical Metrics for Riemannian Curvature Functionals, Lectures on Riemannian Curvature Functionals, IAS/PCMI Program in Geometric Analysis, 2016.

[26] A. Zaeim, J. Cheshmavar, M. A. Mousavi, Generalized Ricci solitons on homogeneous Siklos space-times, Aut. Jour. Math. Comput. (2023), In press.

[27] A. Zaeim, M. Jafari, M. Yaghoubi, Harmonic metrics on Gödel-type spacetimes, Inter. J. Geometric. Methods. Modern. Physics. 17 (2020), 6:2050092.