Take the tour ×  This is maybe a dumb question.
$SL_2(\mathbb{R})$ has a natural action on the upper half plane
$\mathbb{H}$ which is transitive with stabilizer isomorphic to
$SO_2(\mathbb{R})$. For this reason, people sometimes write
$\mathbb{H}$ as the coset space
$SL_2(\mathbb{R})/SO_2(\mathbb{R})$.  Now, it's clear how this
description recovers the topology of $\mathbb{H}$: it's just the
quotient topology. But can you recover either the Riemann
surface structure or the hyperbolic metric on $\mathbb{H}$ from
this description? How much of the structure of
$SL_2(\mathbb{R})$ and $SO_2(\mathbb{R})$ do you need to do
this, if it's possible?  share|improve this question @Qiaochu:
that's a highly non-dumb question! It comes up in the theory of
moduli spaces. It's not hard to check that the points of SL_2(Z)
\ H parametrise elliptic curves over the complexes, but why
should such a set be a complex manifold? I think there's a very
fancy answer to do with variation of Hodge structures but I
don't think I ever under