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On the existence of permutations conditioned by certain rational functions
Ricci solitons of the $ \mathbb{H}^{2} \times \mathbb{R} $ Lie group
Department of Mathematics, Laboratory of Pure and Applied Mathematics, University of Mostaganem (U.M.A.B.), B.P.227, 27000, Mostaganem, Algeria |
In this work we consider the three-dimensional Lie group denoted by $ \mathbb{H}^{2} \times \mathbb{R} $, equipped with left-invariant Riemannian metric. The existence of non-trivial (i.e., not Einstein) Ricci solitons on three-dimensional Lie group $ \mathbb{H}^{2} \times \mathbb{R} $ is proved. Moreover, we show that there are not gradient Ricci solitons.
References:
[1] |
P. Baird and L. Danielo,
Three-dimensional Ricci solitons which project to surfaces, J. Reine Angew. Math., 608 (2007), 65-91.
doi: 10.1515/CRELLE.2007.053. |
[2] |
W. Batat,
Curvature properties and Ricci soliton of Lorentzian pr-waves manifolds, J. Geom. Phys., 75 (2014), 7-16.
doi: 10.1016/j.geomphys.2013.08.014. |
[3] |
W. Batat, M. Brozos-Vazquez, E. García-Río and S. Gavino-Fernández,
Ricci solitons on Lorentzian manifolds with large isometry groups, Bull. Lond. Math. Soc., 43 (2011), 1219-1227.
doi: 10.1112/blms/bdr057. |
[4] |
W. Batat and K. Onda,
Algebraic Ricci solitons of three-dimensional Lorentzian Lie groups, J. Geom. Phys., 114 (2017), 138-152.
doi: 10.1016/j.geomphys.2016.11.018. |
[5] |
L. Belarbi,
On the symmetries of the $Sol_{3}$ Lie group, J. Korean Math. Soc., 57 (2020), 523-537.
doi: 10.4134/JKMS.j190198. |
[6] |
M. Božek, Existence of generalized symmetric Riemannian spaces with solvable isometry group, Časopis Pěst. Mat., 105 (1980), 368–384. |
[7] |
M. Brozos-Vázquez, G. Calvaruso, E. García-Río and S. Gavino-Fernández,
Three-dimensional Lorentzian homogeneous Ricci solitons, Israel J. Math., 188 (2012), 385-403.
doi: 10.1007/s11856-011-0124-3. |
[8] |
G. Calvaruso and B. De Leo,
Ricci solitons on Lorentzian Walker three-manifolds, Acta Math. Hungar., 132 (2011), 269-293.
doi: 10.1007/s10474-010-0049-z. |
[9] |
G. Calvaruso and A. Fino,
Ricci solitons and geometry of four-dimensional non-reductive homogeneous spaces, Canad. J. Math., 64 (2012), 778-804.
doi: 10.4153/CJM-2011-091-1. |
[10] |
G. Calvaruso and A. Fino, Four-dimensional pseudo-Riemannian homogeneous Ricci solitons, Int. J. Geom. Methods Mod. Phys., 12 (2015), 21pp.
doi: 10.1142/S0219887815500565. |
[11] |
G. Calvaruso, O. Kowalski and A. Marinosci,
Homogeneous geodesics in solvable Lie groups, Acta. Math. Hungar., 101 (2003), 313-322.
doi: 10.1023/B:AMHU.0000004942.87374.0e. |
[12] |
H. D. Cao, Recent progress on Ricci solitons, in Recent Advances in Geometric Analysis, Adv. Lect. Math. (ALM), 11, Int. Press, Somerville, MA, 2010, 1-38. |
[13] |
H. D. Cao, Geometry of complete gradient shrinking Ricci solitons, in Geometry and Analysis, Adv. Lect. Math. (ALM), 11, Int. Press, Somerville, MA, 2011,227-246. |
[14] |
L. F. Cerbo,
Generic properties of homogeneous Ricci solitons, Adv. Geom., 14 (2014), 225-237.
doi: 10.1515/advgeom-2013-0031. |
[15] |
D. Friedan,
Nonlinear models in $2+$ $\varepsilon$ dimensions, Ann. Physics, 163 (1985), 318-419.
doi: 10.1016/0003-4916(85)90384-7. |
[16] |
R. S. Hamilton, The Ricci flow on surfaces, in Mathematics and General Relativity, Contemp. Math., 71, Amer. Math. Soc., Providence, RI, 1988,237–262.
doi: 10.1090/conm/071/954419. |
[17] |
R. S. Hamilton,
Three manifolds with positive Ricci curvature, J. Differential Geometry, 17 (1982), 255-306.
doi: 10.4310/jdg/1214436922. |
[18] |
S. Hervik,
Ricci nilsoliton black holes, J. Geom. Phys., 58 (2008), 1253-1264.
doi: 10.1016/j.geomphys.2008.05.001. |
[19] |
O. Kowalski, Generalized Symmetric Spaces, Lectures Notes in Mathematics, 805, Springer-Verlag, Berlin-New York, 1980.
doi: 10.1007/BFb0103324. |
[20] |
J. Lauret,
Ricci soliton solvmanifolds, J. Reine Angew. Math., 650 (2011), 1-21.
doi: 10.1515/CRELLE.2011.001. |
[21] |
A. Mostefaoui and L. Belarbi, On the symmetries of five-dimensional Solvable Lie group, J. Lie Theory, 30 (2020), 155-169. Google Scholar |
[22] |
A. Mostefaoui, L. Belarbi and W. Batat,
Ricci solitons of five-dimensional Solvable Lie group, PanAmer. Math J., 29 (2019), 1-16.
|
[23] |
K. Onda,
Lorentz Ricci solitons on 3-dimensional Lie groups, Geom. Dedicata, 147 (2010), 313-322.
doi: 10.1007/s10711-009-9456-0. |
[24] |
T. L. Payne,
The existence of soliton metrics for nilpotent Lie groups, Geom. Dedicata, 145 (2010), 71-88.
doi: 10.1007/s10711-009-9404-z. |
show all references
References:
[1] |
P. Baird and L. Danielo,
Three-dimensional Ricci solitons which project to surfaces, J. Reine Angew. Math., 608 (2007), 65-91.
doi: 10.1515/CRELLE.2007.053. |
[2] |
W. Batat,
Curvature properties and Ricci soliton of Lorentzian pr-waves manifolds, J. Geom. Phys., 75 (2014), 7-16.
doi: 10.1016/j.geomphys.2013.08.014. |
[3] |
W. Batat, M. Brozos-Vazquez, E. García-Río and S. Gavino-Fernández,
Ricci solitons on Lorentzian manifolds with large isometry groups, Bull. Lond. Math. Soc., 43 (2011), 1219-1227.
doi: 10.1112/blms/bdr057. |
[4] |
W. Batat and K. Onda,
Algebraic Ricci solitons of three-dimensional Lorentzian Lie groups, J. Geom. Phys., 114 (2017), 138-152.
doi: 10.1016/j.geomphys.2016.11.018. |
[5] |
L. Belarbi,
On the symmetries of the $Sol_{3}$ Lie group, J. Korean Math. Soc., 57 (2020), 523-537.
doi: 10.4134/JKMS.j190198. |
[6] |
M. Božek, Existence of generalized symmetric Riemannian spaces with solvable isometry group, Časopis Pěst. Mat., 105 (1980), 368–384. |
[7] |
M. Brozos-Vázquez, G. Calvaruso, E. García-Río and S. Gavino-Fernández,
Three-dimensional Lorentzian homogeneous Ricci solitons, Israel J. Math., 188 (2012), 385-403.
doi: 10.1007/s11856-011-0124-3. |
[8] |
G. Calvaruso and B. De Leo,
Ricci solitons on Lorentzian Walker three-manifolds, Acta Math. Hungar., 132 (2011), 269-293.
doi: 10.1007/s10474-010-0049-z. |
[9] |
G. Calvaruso and A. Fino,
Ricci solitons and geometry of four-dimensional non-reductive homogeneous spaces, Canad. J. Math., 64 (2012), 778-804.
doi: 10.4153/CJM-2011-091-1. |
[10] |
G. Calvaruso and A. Fino, Four-dimensional pseudo-Riemannian homogeneous Ricci solitons, Int. J. Geom. Methods Mod. Phys., 12 (2015), 21pp.
doi: 10.1142/S0219887815500565. |
[11] |
G. Calvaruso, O. Kowalski and A. Marinosci,
Homogeneous geodesics in solvable Lie groups, Acta. Math. Hungar., 101 (2003), 313-322.
doi: 10.1023/B:AMHU.0000004942.87374.0e. |
[12] |
H. D. Cao, Recent progress on Ricci solitons, in Recent Advances in Geometric Analysis, Adv. Lect. Math. (ALM), 11, Int. Press, Somerville, MA, 2010, 1-38. |
[13] |
H. D. Cao, Geometry of complete gradient shrinking Ricci solitons, in Geometry and Analysis, Adv. Lect. Math. (ALM), 11, Int. Press, Somerville, MA, 2011,227-246. |
[14] |
L. F. Cerbo,
Generic properties of homogeneous Ricci solitons, Adv. Geom., 14 (2014), 225-237.
doi: 10.1515/advgeom-2013-0031. |
[15] |
D. Friedan,
Nonlinear models in $2+$ $\varepsilon$ dimensions, Ann. Physics, 163 (1985), 318-419.
doi: 10.1016/0003-4916(85)90384-7. |
[16] |
R. S. Hamilton, The Ricci flow on surfaces, in Mathematics and General Relativity, Contemp. Math., 71, Amer. Math. Soc., Providence, RI, 1988,237–262.
doi: 10.1090/conm/071/954419. |
[17] |
R. S. Hamilton,
Three manifolds with positive Ricci curvature, J. Differential Geometry, 17 (1982), 255-306.
doi: 10.4310/jdg/1214436922. |
[18] |
S. Hervik,
Ricci nilsoliton black holes, J. Geom. Phys., 58 (2008), 1253-1264.
doi: 10.1016/j.geomphys.2008.05.001. |
[19] |
O. Kowalski, Generalized Symmetric Spaces, Lectures Notes in Mathematics, 805, Springer-Verlag, Berlin-New York, 1980.
doi: 10.1007/BFb0103324. |
[20] |
J. Lauret,
Ricci soliton solvmanifolds, J. Reine Angew. Math., 650 (2011), 1-21.
doi: 10.1515/CRELLE.2011.001. |
[21] |
A. Mostefaoui and L. Belarbi, On the symmetries of five-dimensional Solvable Lie group, J. Lie Theory, 30 (2020), 155-169. Google Scholar |
[22] |
A. Mostefaoui, L. Belarbi and W. Batat,
Ricci solitons of five-dimensional Solvable Lie group, PanAmer. Math J., 29 (2019), 1-16.
|
[23] |
K. Onda,
Lorentz Ricci solitons on 3-dimensional Lie groups, Geom. Dedicata, 147 (2010), 313-322.
doi: 10.1007/s10711-009-9456-0. |
[24] |
T. L. Payne,
The existence of soliton metrics for nilpotent Lie groups, Geom. Dedicata, 145 (2010), 71-88.
doi: 10.1007/s10711-009-9404-z. |
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