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# Killing's equations for invariant metrics on Lie groups

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

 Citation:

•  [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," 4th 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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