Killing's equations for invariant metrics on Lie groups

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September 2011, 3(3): 323-335. doi: 10.3934/jgm.2011.3.323

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

Received October 2010 Revised May 2011 Published November 2011

Abstract
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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.

Keywords: right-invariant Riemannian metric, Lie algebra, Killing vector field., Lie group.

Mathematics Subject Classification: Primary: 22E60, 53B21, 22E27; Secondary: 57S9.

Citation: Firas Hindeleh, Gerard Thompson. Killing's equations for invariant metrics on Lie groups. Journal of Geometric Mechanics, 2011, 3 (3) : 323-335. doi: 10.3934/jgm.2011.3.323

References:

[1]	R. Ghanam, I. Strugar and G. Thompson, Matrix representations for low dimensional Lie algebras, Extracta Mathematica, 20 (2005), 151-184.
[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.

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References:

[1]	R. Ghanam, I. Strugar and G. Thompson, Matrix representations for low dimensional Lie algebras, Extracta Mathematica, 20 (2005), 151-184.
[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.

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