Abstract
We study the spaces of left-invariant Riemannian metrics on a Lie group up to isometry, and up to isometry and scaling. In this paper, we see that such spaces can be identified with the orbit spaces of certain isometric actions on noncompact symmetric spaces. We also study some Lie groups whose spaces of left-invariant metrics up to isometry and scaling are small.
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Kodama, H., Takahara, A. & Tamaru, H. The space of left-invariant metrics on a Lie group up to isometry and scaling. manuscripta math. 135, 229–243 (2011). https://doi.org/10.1007/s00229-010-0419-4
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DOI: https://doi.org/10.1007/s00229-010-0419-4