# Conformally flat homogeneous spaces

Let’s say we have a homogeneous space $$H∖GH\backslash G$$.

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 $$H∖GH\backslash G$$ is maximally-noncompact, i.e. $$HH$$ is a maximally compact subgroup of $$GG$$.

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.

In the simply-connected case the list consists of the Euclidean space $$Rn\mathbb R^n$$, the hyperbolic space $$HnH^n$$, the round sphere $$SnS^n$$, and the products $$Hn−1×RH^{n-1}\times \mathbb R$$ and $$Hn−k×SkH^{n-k}\times S^k$$. They also give a longer list in the non-simply-connected case, which includes e.g. any flat torus.