Conformally flat homogeneous spaces

Let’s say we have a homogeneous space HG.

Is it possible to tell whether this homogeneous space admits a conformally flat metric just from its group structure?

I am particularly interested in a situation when HG is maximally-noncompact, i.e. H is a maximally compact subgroup of G.

I hope, my question does not sound too broad. Maybe this question has a trivial answer, but from a background of a theoretical physicist, it is not obvious.

Answer

A complete classification of homogeneous conformally flat Riemannian manifolds is given here, namely, [Alekseevskiĭ, D. V.; Kimelʹfelʹd, B. N. Classification of homogeneous conformally flat Riemannian manifolds. Mat. Zametki 24 (1978), no. 1, 103–110, 143].

In the simply-connected case the list consists of the Euclidean space Rn, the hyperbolic space Hn, the round sphere Sn, and the products Hn1×R and Hnk×Sk. They also give a longer list in the non-simply-connected case, which includes e.g. any flat torus.

Attribution
Source : Link , Question Author : Ovserger , Answer Author : Igor Belegradek

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