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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.

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  • $\begingroup$ What do you mean by a "maximally noncompact subgroup"? Can you give an example of a homogeneous space you are interested in? $\endgroup$ – Igor Belegradek Dec 7 '18 at 22:06
  • $\begingroup$ 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. $\endgroup$ – Ovserger Dec 7 '18 at 22:45
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    $\begingroup$ 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. $\endgroup$ – Igor Belegradek Dec 7 '18 at 22:49
  • $\begingroup$ 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. $\endgroup$ – Igor Belegradek Dec 7 '18 at 22:59
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    $\begingroup$ No, here $G_2$ is that exceptional simple Lie group with two Cartan generators and total dimension of 14. $\endgroup$ – Ovserger Dec 7 '18 at 23:31
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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.

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  • $\begingroup$ Thank you very much, I will look through it by tomorrow morning $\endgroup$ – Ovserger Dec 7 '18 at 23:33

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