If the uniformization theorem gives a 'first classification' of Riemann surfaces into hyperbolic, parabolic, and elliptic surfaces, then what would a 'second classification' look like? If it makes any sense to consider such a thing...
I know that there are (up to conformal equivalence) three simply connected surfaces, and that any riemann surface may be realised as the quotient space of the covering surface by the fundamental group. Within these classes, are there sub-classes, so to speak?