American Institute of Mathematical Sciences

Advanced
December  2015, 7(4): 517-526. doi: 10.3934/jgm.2015.7.517

Invariant metrics on Lie groups

Gerard Thompson 1,

1. 

The University of Toledo, 2801 W Bancroft St., Toledo, OH 43606

Received  January 2015 Revised  August 2015 Published  October 2015

Index formulas for the curvature tensors of an invariant metric on a Lie group are obtained. The results are applied to the problem of characterizing invariant metrics of zero and non-zero constant curvature. Killing vector fields for such metrics are constructed and play an important role in the case of flat metrics.
Keywords: invariant Riemannian metric, Killing vector field., Lie algebra, Lie group.
Mathematics Subject Classification: Primary: 22E60, 53B21, 22E27; Secondary: 57S9.
Citation: Gerard Thompson. Invariant metrics on Lie groups. Journal of Geometric Mechanics, 2015, 7 (4) : 517-526. doi: 10.3934/jgm.2015.7.517
References:
[1]

M. Anderson, A survey of Einstein Metrics on 4-Dimensional Manifolds,, Handbook of Geometric Analysis, (2010).   Google Scholar

[2]

T. Arias-Marco and O. Kowalski, Classification of 4-dimensional homogeneous weakly Einstein manifolds,, Czechoslovak Math. J., 65 (2015), 21.  doi: 10.1007/s10587-015-0159-4.  Google Scholar

[3]

A. Besse, Einstein Manifolds,, 1st ed., (1987).  doi: 10.1007/978-3-540-74311-8.  Google Scholar

[4]

S. Chen and K. Liang, Left-invariant pseudo-Einstein metrics on Lie groups,, J. Nonlinear Math. Phys., 19 (2012).  doi: 10.1142/S1402925112500155.  Google Scholar

[5]

Z. Chen, D. Hou and C. Bai, A left-symmetric algebraic approach to left invariant flat pseudo-metrics on Lie groups,, J. Geom. Phys., 62 (2012), 1600.  doi: 10.1016/j.geomphys.2012.03.003.  Google Scholar

[6]

P. Gadea, J. González-Dávila and J. Oubina, Cyclic metric Lie groups,, Monatsh. Math., 176 (2015), 219.  doi: 10.1007/s00605-014-0692-5.  Google Scholar

[7]

M. Guediri, Novikov algebras carrying an invariant Lorentzian symmetric bilinear form,, J. Geom. Phys., 82 (2014), 132.  doi: 10.1016/j.geomphys.2014.04.007.  Google Scholar

[8]

F. Hindeleh and G. Thompson, Killing's equations for invariant metrics on Lie groups,, Journal of Geometry and Mechanics, 3 (2011), 323.  doi: 10.3934/jgm.2011.3.323.  Google Scholar

[9]

H. Kodama, A. Takahara and H. Tamaru, The space of left-invariant metrics on a Lie group up to isometry and scaling,, Manuscripta Math., 135 (2011), 229.  doi: 10.1007/s00229-010-0419-4.  Google Scholar

[10]

J. Milnor, Curvatures of left invariant metrics on Lie groups,, Advances in Math., 21 (1976), 293.  doi: 10.1016/S0001-8708(76)80002-3.  Google Scholar

[11]

J. Patera, R. T. Sharp, P. Winternitz and H. Zassenhaus, Invariants of real low dimension Lie algebras,, J. Math. Phys., 17 (1976), 986.  doi: 10.1063/1.522992.  Google Scholar

[12]

H. Wang and S. Deng, Left invariant Einstein-Randers metrics on compact Lie groups,, Canad. Math. Bull., 55 (2012), 870.  doi: 10.4153/CMB-2011-145-6.  Google Scholar

show all references

References:
[1]

M. Anderson, A survey of Einstein Metrics on 4-Dimensional Manifolds,, Handbook of Geometric Analysis, (2010).   Google Scholar

[2]

T. Arias-Marco and O. Kowalski, Classification of 4-dimensional homogeneous weakly Einstein manifolds,, Czechoslovak Math. J., 65 (2015), 21.  doi: 10.1007/s10587-015-0159-4.  Google Scholar

[3]

A. Besse, Einstein Manifolds,, 1st ed., (1987).  doi: 10.1007/978-3-540-74311-8.  Google Scholar

[4]

S. Chen and K. Liang, Left-invariant pseudo-Einstein metrics on Lie groups,, J. Nonlinear Math. Phys., 19 (2012).  doi: 10.1142/S1402925112500155.  Google Scholar

[5]

Z. Chen, D. Hou and C. Bai, A left-symmetric algebraic approach to left invariant flat pseudo-metrics on Lie groups,, J. Geom. Phys., 62 (2012), 1600.  doi: 10.1016/j.geomphys.2012.03.003.  Google Scholar

[6]

P. Gadea, J. González-Dávila and J. Oubina, Cyclic metric Lie groups,, Monatsh. Math., 176 (2015), 219.  doi: 10.1007/s00605-014-0692-5.  Google Scholar

[7]

M. Guediri, Novikov algebras carrying an invariant Lorentzian symmetric bilinear form,, J. Geom. Phys., 82 (2014), 132.  doi: 10.1016/j.geomphys.2014.04.007.  Google Scholar

[8]

F. Hindeleh and G. Thompson, Killing's equations for invariant metrics on Lie groups,, Journal of Geometry and Mechanics, 3 (2011), 323.  doi: 10.3934/jgm.2011.3.323.  Google Scholar

[9]

H. Kodama, A. Takahara and H. Tamaru, The space of left-invariant metrics on a Lie group up to isometry and scaling,, Manuscripta Math., 135 (2011), 229.  doi: 10.1007/s00229-010-0419-4.  Google Scholar

[10]

J. Milnor, Curvatures of left invariant metrics on Lie groups,, Advances in Math., 21 (1976), 293.  doi: 10.1016/S0001-8708(76)80002-3.  Google Scholar

[11]

J. Patera, R. T. Sharp, P. Winternitz and H. Zassenhaus, Invariants of real low dimension Lie algebras,, J. Math. Phys., 17 (1976), 986.  doi: 10.1063/1.522992.  Google Scholar

[12]

H. Wang and S. Deng, Left invariant Einstein-Randers metrics on compact Lie groups,, Canad. Math. Bull., 55 (2012), 870.  doi: 10.4153/CMB-2011-145-6.  Google Scholar

[1]

Ville Salo, Ilkka Törmä. Recoding Lie algebraic subshifts. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 1005-1021. doi: 10.3934/dcds.2020307
[2]

Hongliang Chang, Yin Chen, Runxuan Zhang. A generalization on derivations of Lie algebras. Electronic Research Archive, , () : -. doi: 10.3934/era.2020124
[3]

Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $ \mathcal{W}(a, b, r) $. Electronic Research Archive, , () : -. doi: 10.3934/era.2020123
[4]

Hongyan Guo. Automorphism group and twisted modules of the twisted Heisenberg-Virasoro vertex operator algebra. Electronic Research Archive, , () : -. doi: 10.3934/era.2021008
[5]

Yunping Jiang. Global graph of metric entropy on expanding Blaschke products. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1469-1482. doi: 10.3934/dcds.2020325
[6]

Buddhadev Pal, Pankaj Kumar. A family of multiply warped product semi-Riemannian Einstein metrics. Journal of Geometric Mechanics, 2020, 12 (4) : 553-562. doi: 10.3934/jgm.2020017
[7]

Manxue You, Shengjie Li. Perturbation of Image and conjugate duality for vector optimization. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020176
[8]

Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020464
[9]

Qiao Liu. Local rigidity of certain solvable group actions on tori. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 553-567. doi: 10.3934/dcds.2020269
[10]

Kien Trung Nguyen, Vo Nguyen Minh Hieu, Van Huy Pham. Inverse group 1-median problem on trees. Journal of Industrial & Management Optimization, 2021, 17 (1) : 221-232. doi: 10.3934/jimo.2019108
[11]

Meihua Dong, Keonhee Lee, Carlos Morales. Gromov-Hausdorff stability for group actions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1347-1357. doi: 10.3934/dcds.2020320
[12]

Awais Younus, Zoubia Dastgeer, Nudrat Ishaq, Abdul Ghaffar, Kottakkaran Sooppy Nisar, Devendra Kumar. On the observability of conformable linear time-invariant control systems. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020444
[13]

Giulia Luise, Giuseppe Savaré. Contraction and regularizing properties of heat flows in metric measure spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 273-297. doi: 10.3934/dcdss.2020327
[14]

Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002
[15]

Paul A. Glendinning, David J. W. Simpson. A constructive approach to robust chaos using invariant manifolds and expanding cones. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020409
[16]

Bixiang Wang. Mean-square random invariant manifolds for stochastic differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1449-1468. doi: 10.3934/dcds.2020324
[17]

Soonki Hong, Seonhee Lim. Martin boundary of brownian motion on gromov hyperbolic metric graphs. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021014
[18]

Simone Fiori. Error-based control systems on Riemannian state manifolds: Properties of the principal pushforward map associated to parallel transport. Mathematical Control & Related Fields, 2021, 11 (1) : 143-167. doi: 10.3934/mcrf.2020031
[19]

Ying Lin, Qi Ye. Support vector machine classifiers by non-Euclidean margins. Mathematical Foundations of Computing, 2020, 3 (4) : 279-300. doi: 10.3934/mfc.2020018
[20]

Wen Li, Wei-Hui Liu, Seak Weng Vong. Perron vector analysis for irreducible nonnegative tensors and its applications. Journal of Industrial & Management Optimization, 2021, 17 (1) : 29-50. doi: 10.3934/jimo.2019097

2019 Impact Factor: 0.649

Tools

Metrics

  • PDF downloads (306)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences