Killing's equations for invariant metrics on Lie groups

Firas Hindeleh; Gerard Thompson

doi:10.3934/jgm.2011.3.323

Article Contents

2011, Volume 3, Issue 3: 323-335. Doi: 10.3934/jgm.2011.3.323

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Killing's equations for invariant metrics on Lie groups

Firas Hindeleh^1, and
Gerard Thompson^2,

1.
Grand Valley State University, 1 Campus Dr., Allendale, MI 49401
2.
The University of Toledo, 2801 W Bancroft St., Toledo, OH 43606

Received: October 2010

Revised: May 2011

Published: September 2011

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Abstract

This article is the first in a series that will investigate symmetry and curvature properties of a right-invariant metric on a Lie group. This paper will consider Lie groups in dimension two and three and will focus on the solutions of Killing's equations. A striking result is that several of the three-dimensional Lie groups turn out to be spaces of constant curvature.

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Mathematics Subject Classification: Primary: 22E60, 53B21, 22E27; Secondary: 57S99.

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References

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[2]	W. H. Greub, "Linear Algebra," 4^th edition, Gradaute Texts in Mathematics, No. 23, Springer-Verlag, New York-Berlin, 1975.
[3]	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.
[4]	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.