# Conformally flat homogeneous spaces

Let's say we have a homogeneous space $$H\backslash G$$.

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 $$H\backslash G$$ 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.

• What do you mean by a "maximally noncompact subgroup"? Can you give an example of a homogeneous space you are interested in? Dec 7, 2018 at 22:06
• Igor, I am sorry, I meant that $H$ is maximally compact, my mistake. To provide an example, $(SU(1,1)\times SU(1,1) \setminus G_{2(2)}$ where $G_{2(2)}$ is a group corresponding to $\mathscr{g}_2$ algebra with all (two) of its Cartan generators made noncompact. Dec 7, 2018 at 22:45
• The classification is in Alekseevskiĭ, D. V.; Kimelʹfelʹd, B. N. Classification of homogeneous conformally flat Riemannian manifolds. Mat. Zametki 24 (1978), no. 1, 103–110, 143. Dec 7, 2018 at 22:49
• Is $G_{2(2)}=U(1)\times U(1)$ where $U(1)$ is identified with the diagonal unitary matrices of determinant one? If so, then I think the homogeneous space is the product of two hyperbolic planes, which is not conformally flat. Dec 7, 2018 at 22:59
• No, here $G_2$ is that exceptional simple Lie group with two Cartan generators and total dimension of 14. Dec 7, 2018 at 23:31

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 $$\mathbb R^n$$, the hyperbolic space $$H^n$$, the round sphere $$S^n$$, and the products $$H^{n-1}\times \mathbb R$$ and $$H^{n-k}\times S^k$$. They also give a longer list in the non-simply-connected case, which includes e.g. any flat torus.

• Thank you very much, I will look through it by tomorrow morning Dec 7, 2018 at 23:33