# Homogeneous Riemann Surfaces

A Riemann surface $$X$$ is a connected complex manifold of complex dimension one. A homogeneous space is a manifold with a transitive smooth action of a Lie group. I guess there must be a classification of those Riemann surfaces which are also homogeneous spaces . . . however, I can't seem to find it.

• So you don't require the action to preserve the holomorphic structure? Eventually I think it doesn't matter at least in the closed case (where homogeneous $\Leftrightarrow$ genus $\le 1$). In the non-closed case, it matters: the quotient of the hyperbolic plane by a loxodromic isometry is homogeneous as smooth manifold, but not as Riemann surface. – YCor Feb 5 at 17:39
• What if I include compact? This is what I am really interested in. – Pierre Dubois Feb 5 at 18:07
• Ok, so I guess yes, I would like that the action preserves the holomorphic structure – Pierre Dubois Feb 5 at 19:46

If you only care about compact (oriented) surfaces, then it is easy to see that the only homogeneous examples are the sphere $$S^2=\mathbb C P^1$$ and the torus $$T^2=S^1\times S^1$$. Indeed, if $$M$$ is a compact homogeneous space, then its Euler characteristic is $$\chi(M)\geq0$$; so for an oriented surface this implies genus $$\leq1$$, as Pierre Dubois stated in his comment. Perhaps the references mentioned here can be helpful to you: How do you see that higher genus surfaces are not homogeneous?