Let’s say we have a homogeneous space H∖G.

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 H∖G 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 Hn−1×R and Hn−k×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*