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March  2020, 28(1): 157-163. doi: 10.3934/era.2020010

Ricci solitons of the $ \mathbb{H}^{2} \times \mathbb{R} $ Lie group

Lakehal Belarbi ,

Department of Mathematics, Laboratory of Pure and Applied Mathematics, University of Mostaganem (U.M.A.B.), B.P.227, 27000, Mostaganem, Algeria

* Corresponding author: Lakehal Belarbi

Received  October 2019 Revised  February 2020 Published  March 2020

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.

Keywords: $ \mathbb{H}^{2} \times \mathbb{R} $ Lie group, Ricci soliton, Riemannian metrics.
Mathematics Subject Classification: 53C50, 53B30.
Citation: Lakehal Belarbi. Ricci solitons of the $ \mathbb{H}^{2} \times \mathbb{R} $ Lie group. Electronic Research Archive, 2020, 28 (1) : 157-163. doi: 10.3934/era.2020010
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

M. Božek, Existence of generalized symmetric Riemannian spaces with solvable isometry group, Časopis Pěst. Mat., 105 (1980), 368–384.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

L. F. Cerbo, Generic properties of homogeneous Ricci solitons, Adv. Geom., 14 (2014), 225-237.  doi: 10.1515/advgeom-2013-0031.  Google Scholar

[15]

D. Friedan, Nonlinear models in $2+$ $\varepsilon$ dimensions, Ann. Physics, 163 (1985), 318-419.  doi: 10.1016/0003-4916(85)90384-7.  Google Scholar

[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.  Google Scholar

[17]

R. S. Hamilton, Three manifolds with positive Ricci curvature, J. Differential Geometry, 17 (1982), 255-306.  doi: 10.4310/jdg/1214436922.  Google Scholar

[18]

S. Hervik, Ricci nilsoliton black holes, J. Geom. Phys., 58 (2008), 1253-1264.  doi: 10.1016/j.geomphys.2008.05.001.  Google Scholar

[19]

O. Kowalski, Generalized Symmetric Spaces, Lectures Notes in Mathematics, 805, Springer-Verlag, Berlin-New York, 1980. doi: 10.1007/BFb0103324.  Google Scholar

[20]

J. Lauret, Ricci soliton solvmanifolds, J. Reine Angew. Math., 650 (2011), 1-21.  doi: 10.1515/CRELLE.2011.001.  Google Scholar

[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.   Google Scholar

[23]

K. Onda, Lorentz Ricci solitons on 3-dimensional Lie groups, Geom. Dedicata, 147 (2010), 313-322.  doi: 10.1007/s10711-009-9456-0.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

M. Božek, Existence of generalized symmetric Riemannian spaces with solvable isometry group, Časopis Pěst. Mat., 105 (1980), 368–384.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

L. F. Cerbo, Generic properties of homogeneous Ricci solitons, Adv. Geom., 14 (2014), 225-237.  doi: 10.1515/advgeom-2013-0031.  Google Scholar

[15]

D. Friedan, Nonlinear models in $2+$ $\varepsilon$ dimensions, Ann. Physics, 163 (1985), 318-419.  doi: 10.1016/0003-4916(85)90384-7.  Google Scholar

[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.  Google Scholar

[17]

R. S. Hamilton, Three manifolds with positive Ricci curvature, J. Differential Geometry, 17 (1982), 255-306.  doi: 10.4310/jdg/1214436922.  Google Scholar

[18]

S. Hervik, Ricci nilsoliton black holes, J. Geom. Phys., 58 (2008), 1253-1264.  doi: 10.1016/j.geomphys.2008.05.001.  Google Scholar

[19]

O. Kowalski, Generalized Symmetric Spaces, Lectures Notes in Mathematics, 805, Springer-Verlag, Berlin-New York, 1980. doi: 10.1007/BFb0103324.  Google Scholar

[20]

J. Lauret, Ricci soliton solvmanifolds, J. Reine Angew. Math., 650 (2011), 1-21.  doi: 10.1515/CRELLE.2011.001.  Google Scholar

[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.   Google Scholar

[23]

K. Onda, Lorentz Ricci solitons on 3-dimensional Lie groups, Geom. Dedicata, 147 (2010), 313-322.  doi: 10.1007/s10711-009-9456-0.  Google Scholar

[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.  Google Scholar

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