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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
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show all references

References:
[1]

Handbook of Geometric Analysis, 3, International Press, Boston, 2010. Google Scholar

[2]

Czechoslovak Math. J., 65 (2015), 21-59. doi: 10.1007/s10587-015-0159-4.  Google Scholar

[3]

1st ed., Springer, Berlin, Heidelberg, New York, 1987. doi: 10.1007/978-3-540-74311-8.  Google Scholar

[4]

J. Nonlinear Math. Phys., 19 (2012), 1250015, 11pp. doi: 10.1142/S1402925112500155.  Google Scholar

[5]

J. Geom. Phys., 62 (2012), 1600-1610. doi: 10.1016/j.geomphys.2012.03.003.  Google Scholar

[6]

Monatsh. Math., 176 (2015), 219-239. doi: 10.1007/s00605-014-0692-5.  Google Scholar

[7]

J. Geom. Phys., 82 (2014), 132-144. doi: 10.1016/j.geomphys.2014.04.007.  Google Scholar

[8]

Journal of Geometry and Mechanics, 3 (2011), 323-335. doi: 10.3934/jgm.2011.3.323.  Google Scholar

[9]

Manuscripta Math., 135 (2011), 229-243. doi: 10.1007/s00229-010-0419-4.  Google Scholar

[10]

Advances in Math., 21 (1976), 293-329. doi: 10.1016/S0001-8708(76)80002-3.  Google Scholar

[11]

J. Math. Phys., 17 (1976), 986-994. doi: 10.1063/1.522992.  Google Scholar

[12]

Canad. Math. Bull., 55 (2012), 870-881. doi: 10.4153/CMB-2011-145-6.  Google Scholar

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