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### Open Access Journals

JGM

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.

JGM

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.

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