American Institute of Mathematical Sciences

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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, 3, International Press, Boston, 2010. Google Scholar

[2]

T. Arias-Marco and O. Kowalski, Classification of 4-dimensional homogeneous weakly Einstein manifolds, Czechoslovak Math. J., 65 (2015), 21-59. doi: 10.1007/s10587-015-0159-4.  Google Scholar

[3]

A. Besse, Einstein Manifolds, 1st ed., Springer, Berlin, Heidelberg, New York, 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), 1250015, 11pp. 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), 1600-1610. 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), 219-239. 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), 132-144. 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), 323-335. 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), 229-243. 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), 293-329. 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), 986-994. 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), 870-881. 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, 3, International Press, Boston, 2010. Google Scholar

[2]

T. Arias-Marco and O. Kowalski, Classification of 4-dimensional homogeneous weakly Einstein manifolds, Czechoslovak Math. J., 65 (2015), 21-59. doi: 10.1007/s10587-015-0159-4.  Google Scholar

[3]

A. Besse, Einstein Manifolds, 1st ed., Springer, Berlin, Heidelberg, New York, 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), 1250015, 11pp. 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), 1600-1610. 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), 219-239. 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), 132-144. 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), 323-335. 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), 229-243. 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), 293-329. 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), 986-994. 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), 870-881. doi: 10.4153/CMB-2011-145-6.  Google Scholar

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