Can every compact connected 2 manifold be expressed as $$ \Gamma\backslash G/H $$ where $ G $ is a Lie group, $H $ a subgroup of $ G $, $G/H $ a symmetric space, and $ \Gamma $ a discrete subgroup of $ G $? I know this is true for the orientable case. In fact the manifold $ G/H $ is holomorphic as is the covering map with fiber $ \Gamma $. I know that $$ RP^2 \cong O(3)/O(2)\times O(1) $$ so it is in fact homogeneous. But what about the Klein bottle (connected sun of two projective planes)? Or even the connected sum of $ n $ projective planes? Are those surfaces locally symmetric?
In your answer please provide an example of a triple $ (\Gamma, G,H) $. Preferably with G simple