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.

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

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.