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.
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3$\begingroup$ 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. $\endgroup$ – YCor Feb 5 at 17:39
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$\begingroup$ What if I include compact? This is what I am really interested in. $\endgroup$ – Pierre Dubois Feb 5 at 18:07
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$\begingroup$ You're not answering my question. In any case, I posted an answer covering both cases. $\endgroup$ – YCor Feb 5 at 18:09
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$\begingroup$ Ok, so I guess yes, I would like that the action preserves the holomorphic structure $\endgroup$ – Pierre Dubois Feb 5 at 19:46
