American Institute of Mathematical Sciences

Advanced
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

[1]

Shengbing Deng. Construction solutions for Neumann problem with Hénon term in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 2233-2253. doi: 10.3934/dcds.2019094
[2]

Guoyuan Chen, Yong Liu, Juncheng Wei. Nondegeneracy of harmonic maps from $ {{\mathbb{R}}^{2}} $ to $ {{\mathbb{S}}^{2}} $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3215-3233. doi: 10.3934/dcds.2019228
[3]

Florin Diacu, Shuqiang Zhu. Almost all 3-body relative equilibria on $ \mathbb S^2 $ and $ \mathbb H^2 $ are inclined. Discrete & Continuous Dynamical Systems - S, 2020, 13 (4) : 1131-1143. doi: 10.3934/dcdss.2020067
[4]

Van Hoang Nguyen. A simple proof of the Adams type inequalities in $ {\mathbb R}^{2m} $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (10) : 5755-5764. doi: 10.3934/dcds.2020244
[5]

Imed Bachar, Habib Mâagli. Singular solutions of a nonlinear equation in a punctured domain of $\mathbb{R}^{2}$. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 171-188. doi: 10.3934/dcdss.2019012
[6]

Tingting Wu, Jian Gao, Yun Gao, Fang-Wei Fu. $ {{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{2}}{{\mathbb{Z}}_{4}}$-additive cyclic codes. Advances in Mathematics of Communications, 2018, 12 (4) : 641-657. doi: 10.3934/amc.2018038
[7]

Weiwei Ao, Chao Liu. Asymptotic behavior of sign-changing radial solutions of a semilinear elliptic equation in $ \mathbb{R}^2 $ when exponent approaches $ +\infty $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (8) : 5047-5077. doi: 10.3934/dcds.2020211
[8]

Vladimir Chepyzhov, Alexei Ilyin, Sergey Zelik. Strong trajectory and global $\mathbf{W^{1,p}}$-attractors for the damped-driven Euler system in $\mathbb R^2$. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1835-1855. doi: 10.3934/dcdsb.2017109
[9]

Kwangseok Choe, Hyungjin Huh. Chern-Simons gauged sigma model into $ \mathbb{H}^2 $ and its self-dual equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4613-4646. doi: 10.3934/dcds.2019189
[10]

Joaquim Borges, Cristina Fernández-Córdoba, Roger Ten-Valls. On ${{\mathbb{Z}}}_{p^r}{{\mathbb{Z}}}_{p^s}$-additive cyclic codes. Advances in Mathematics of Communications, 2018, 12 (1) : 169-179. doi: 10.3934/amc.2018011
[11]

Teresa Alberico, Costantino Capozzoli, Luigi D'Onofrio, Roberta Schiattarella. $G$-convergence for non-divergence elliptic operators with VMO coefficients in $\mathbb R^3$. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 129-137. doi: 10.3934/dcdss.2019009
[12]

James Tanis. Exponential multiple mixing for some partially hyperbolic flows on products of $ {\rm{PSL}}(2, \mathbb{R})$. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 989-1006. doi: 10.3934/dcds.2018042
[13]

Karim Samei, Arezoo Soufi. Quadratic residue codes over $\mathbb{F}_{p^r}+{u_1}\mathbb{F}_{p^r}+{u_2}\mathbb{F}_{p^r}+...+{u_t}\mathbb{F}_ {p^r}$. Advances in Mathematics of Communications, 2017, 11 (4) : 791-804. doi: 10.3934/amc.2017058
[14]

Yuan Cao, Yonglin Cao, Hai Q. Dinh, Ramakrishna Bandi, Fang-Wei Fu. An explicit representation and enumeration for negacyclic codes of length $ 2^kn $ over $ \mathbb{Z}_4+u\mathbb{Z}_4 $. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020067
[15]

Aleksander Denisiuk. On range condition of the tensor x-ray transform in $ \mathbb R^n $. Inverse Problems & Imaging, 2020, 14 (3) : 423-435. doi: 10.3934/ipi.2020020
[16]

Sanjiban Santra. On the positive solutions for a perturbed negative exponent problem on $\mathbb{R}^3$. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1441-1460. doi: 10.3934/dcds.2018059
[17]

Xiaopeng Zhao. Space-time decay estimates of solutions to liquid crystal system in $\mathbb{R}^3$. Communications on Pure & Applied Analysis, 2019, 18 (1) : 1-13. doi: 10.3934/cpaa.2019001
[18]

Edcarlos D. Silva, José Carlos de Albuquerque, Uberlandio Severo. On a class of linearly coupled systems on $ \mathbb{R}^N $ involving asymptotically linear terms. Communications on Pure & Applied Analysis, 2019, 18 (6) : 3089-3101. doi: 10.3934/cpaa.2019138
[19]

Fengshuang Gao, Yuxia Guo. Infinitely many solutions for quasilinear equations with critical exponent and Hardy potential in $ \mathbb{R}^N $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (9) : 5591-5616. doi: 10.3934/dcds.2020239
[20]

Holger R. Dullin, Jürgen Scheurle. Symmetry reduction of the 3-body problem in $ \mathbb{R}^4 $. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020011

2018 Impact Factor: 0.263

Tools

Article outline

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences