The existence of gradient Yamabe solitons on spacetimes. (English) Zbl 1507.53048
The authors investigate the existence of the non-trivial gradient Yamabe soliton on generalized Robertson-Walker spacetimes, standard static spacetimes, Walker manifolds and pp-wave spacetimes. The most remarkable results concern gradient Yamabe solitons on pp-wave spacetimes (see Section 3.5). All results of the paper have physical relevance and the geometry described is very interesting.
Reviewer: Sunil Kumar Yadav (Gorakhpur)
MSC:
| 53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |
| 53C40 | Global submanifolds |
| 53Z05 | Applications of differential geometry to physics |
| 83C75 | Space-time singularities, cosmic censorship, etc. |
Keywords:
generalized quasi-gradient Yamabe solitons; warped product manifolds; generalized Robertson-Walker spacetimes; standard static spacetimes; Walker manifolds; pp-wave spacetimesReferences:
| [1] | Hamilton, R.S.: The Ricci flow on surfaces, in: Mathematics and General Relativity. Contemp. Math., 71, 237-262 (1988) · Zbl 0663.53031 |
| [2] | Brendle, S., Convergence of the Yamabe flow for arbitrary initial energy, J. Diff. Geom., 69, 2, 217-278 (2005) · Zbl 1085.53028 |
| [3] | Brendle, S., Convergence of the Yamabe flow in dimension 6 and higher, Invent. Math., 170, 3, 541-576 (2007) · Zbl 1130.53044 · doi:10.1007/s00222-007-0074-x |
| [4] | Daskalopoulosa, P.; Sesum, N., The classification of locally conformally flat Yamabe solitons, Adv. Math., 240, 346-369 (2013) · Zbl 1290.35217 · doi:10.1016/j.aim.2013.03.011 |
| [5] | Yang, F.; Zhang, L., Geometry of gradient Yamabe solitons, Ann. Glob. Anal. Geom., 50, 367-379 (2016) · Zbl 1377.53066 · doi:10.1007/s10455-016-9516-2 |
| [6] | Cao, H-D; Sun, X.; Zhang, Y., On the structure of gradient Yamabe solitons, Math. Res. Lett., 19, 767-774 (2012) · Zbl 1267.53066 · doi:10.4310/MRL.2012.v19.n4.a3 |
| [7] | Maeta, S., Three-dimensional complete gradient Yamabe solitons with divergence-free Cotton tensor, Ann. Glob. Anal. Geom., 58, 227-237 (2020) · Zbl 1446.53030 · doi:10.1007/s10455-020-09722-9 |
| [8] | Bidabad, B.; Yar Ahmadi, M., On complete Yamabe solitons, Adv. Geom., 18, 1, 101-104 (2018) · Zbl 1386.53036 · doi:10.1515/advgeom-2017-0018 |
| [9] | Bejan, C-L; Güler, S., Laplace, Einstein and related equations on \(D\)-General Warping, Mediter. J. Math., 16, 19 (2019) · Zbl 1416.53025 · doi:10.1007/s00009-018-1283-9 |
| [10] | Catino, G., Generalized quasi-Einstein manifolds with harmonic Weyl tensor, Math. Z., 271, 3, 751-756 (2012) · Zbl 1246.53040 · doi:10.1007/s00209-011-0888-5 |
| [11] | Altay Demirbağ, S.; Güler, S., Rigidity of (m, \( \rho )\)-quasi Einstein manifolds, Math. Nachr.,, 290, 14-15, 2100-2110 (2017) · Zbl 1377.53039 · doi:10.1002/mana.201600186 |
| [12] | Naik, D.M.: Ricci solitons on Riemannian manifolds admitting certain vector field. Ricerche Mat. doi:10.1007/s11587-021-00622-z (2021) |
| [13] | Bejan, C-L; Crasmareanu, M., Second order parallel tensors and Ricci solitons in 3-dimensional normal paracontact geometry, Ann. Glob. Anal. Geom., 46, 117-127 (2014) · Zbl 1312.53049 · doi:10.1007/s10455-014-9414-4 |
| [14] | Huang, G.; Li, H., On a classification of the quasi-Yamabe gradient solitons, Methods Appl. Anal., 21, 379-390 (2014) · Zbl 1304.53033 · doi:10.4310/MAA.2014.v21.n3.a7 |
| [15] | Leandro Neto, B.: A note on (anti-)self dual quasi-Yamabe gradient solitons, Results Math. 1-7 (2015) |
| [16] | Leandro Neto, B.; Oliveira, HP, Generalized quasi-Yamabe gradient solitons, Diff. Geo. Appl., 49, 167-175 (2016) · Zbl 1352.53040 · doi:10.1016/j.difgeo.2016.07.008 |
| [17] | Wang, LF, On noncompact quasi-Yamabe gradient solitons, Differ. Geom. Appl., 31, 337-348 (2013) · Zbl 1279.53039 · doi:10.1016/j.difgeo.2013.03.005 |
| [18] | Bishop, RL; O’Neill, B., Manifolds of negative curvature, Trans. Amer. Math. Soc., 145, 1-49 (1969) · Zbl 0191.52002 · doi:10.1090/S0002-9947-1969-0251664-4 |
| [19] | O’Neill, B., Semi-Riemannian Geometry with Applications to Relativity (1983), London: Academic Press Limited, London · Zbl 0531.53051 |
| [20] | Shenawy, S.; Ünal, B., \(2-\) Killing vector fields on warped product manifolds, Int. J. Math., 26, 1550065 (2015) · Zbl 1334.53077 · doi:10.1142/S0129167X15500652 |
| [21] | Cheeger, J.; Colding, T., Lower bounds on Ricci curvature and the almost rigidity of warped products, Ann. Math., 144, 189-237 (1996) · Zbl 0865.53037 · doi:10.2307/2118589 |
| [22] | Sánchez, M., On the Geometry of Generalized Robertson-Walker Spacetimes: geodesics, Gen. Relat. Gravit., 30, 915-932 (1998) · Zbl 1047.83509 · doi:10.1023/A:1026664209847 |
| [23] | Sánchez, M., On the geometry of generalized Robertson-Walker spacetimes: curvature and killing fields, J. Geom. Phys., 31, 1-15 (1999) · Zbl 0964.53046 · doi:10.1016/S0393-0440(98)00061-8 |
| [24] | Mantica, CA; Molinari, LG; De, UC, A condition for a perfect fluid spacetime to be a generalized Robertson-Walker spacetime, J. Math. Phys., 57, 2 (2016) · Zbl 1336.83053 · doi:10.1063/1.4941942 |
| [25] | Mantica, CA; Suh, YJ; De, UC, A note on generalized Robertson-Walker spacetimes, Int. J. Geom. Meth. Mod. Phys., 13, 1650079 (2016) · Zbl 1348.53070 · doi:10.1142/S0219887816500791 |
| [26] | Chen, B-Y, A simple characterization of generalized Robertson-Walker spacetimes, Gen. Relativ. Gravit., 46, 1833 (2014) · Zbl 1308.83159 · doi:10.1007/s10714-014-1833-9 |
| [27] | Allison, DE, Energy conditions in standard static spacetimes, Gen. Relativ. Gravit., 20, 115-122 (1988) · Zbl 0638.53059 · doi:10.1007/BF00759321 |
| [28] | Allison, DE, Geodesic completeness in static spacetimes, Geom. Dedicata, 26, 85-97 (1988) · Zbl 0644.53063 · doi:10.1007/BF00148016 |
| [29] | Allison, DE; Ünal, B., Geodesic structure of standard static spacetimes, J. Geo. Phys., 46, 193-200 (2003) · Zbl 1034.53040 · doi:10.1016/S0393-0440(02)00154-7 |
| [30] | Besse, AL, Einstein Manifolds, Classics in Mathematics (2008), Berlin: Springer, Berlin |
| [31] | Beem, J. K., Ehrlich, P. E., Easley, K. L.: Global Lorentzian Geometry. Marcel Dekker, Second Edition, NewYork, (1996) · Zbl 0846.53001 |
| [32] | Dobarro, F.; Ünal, B., Special standard static spacetimes, Nonlinear Anal. Theory Methods Appl., 59, 5, 759-770 (2004) · Zbl 1073.53087 · doi:10.1016/j.na.2004.07.035 |
| [33] | Walker, AG, Canonical form for a Riemannian space with a parallel field of null planes, Quart. J. Math. Oxford, 21, 69-79 (1950) · Zbl 0036.38303 · doi:10.1093/qmath/1.1.69 |
| [34] | Chaichi, M.; Garcia-Rio, E.; Vazquez-Abal, ME, Three-dimensional Lorentz manifolds admitting a parallel null vector field, J. Phys. A Math. Gen., 38, 841-850 (2005) · Zbl 1068.53049 · doi:10.1088/0305-4470/38/4/005 |
| [35] | Chaichi, M.; Garcia-Rio, E.; Matsushita, Y., Curvature properties of four-dimensional Walker metrics, Class. Quant. Gravity, 22, 559-577 (2005) · Zbl 1086.83031 · doi:10.1088/0264-9381/22/3/008 |
| [36] | Sidhoumi, N.; Batat, W., Ricci solitons on four-dimensional Lorentzian Walker manifolds Adv, Geom., 17, 4, 397-406 (2017) · Zbl 1390.53076 |
| [37] | Brinkmann, HW, Einstein spaces which are mapped conformally on each other, Math. Ann., 18, 119-145 (1925) · JFM 51.0568.03 · doi:10.1007/BF01208647 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
