Killing's equations for invariant metrics on Lie groups
Firas Hindeleh Gerard Thompson
Journal of Geometric Mechanics 2011, 3(3): 323-335 doi: 10.3934/jgm.2011.3.323
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
Invariant metrics on Lie groups
Gerard Thompson
Journal of Geometric Mechanics 2015, 7(4): 517-526 doi: 10.3934/jgm.2015.7.517
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

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