An elliptic surface is a surface equipped with an elliptic fibration ( a surjective holomorphic map to a curve " B " such that all but finitely many fibers are smooth irreducible curves of genus 1 ).
12.
This is a smooth diffeomorphism of } } onto the closed annulus " z " d " 1 } }, restricting to a holomorphic map in the interior and a smooth diffeomorphism on both boundary curves.
13.
In fact, M yields a duality ( contravariant equivalence ) between the category of compact connected Riemann surfaces ( with non-constant holomorphic maps as morphisms ) and function fields of one variable over "'C " '.
14.
Let ? be a bounded region in "'C "'with smooth boundary " ? and let ? be a univalent holomorphic map of the unit disk " D " onto ? extending to a smooth diffeomorphism of the circle onto " ?.
15.
If equality holds throughout in one of the two inequalities above ( which is equivalent to saying that the holomorphic map preserves the distance in the Poincar?metric ), then " f " must be an analytic automorphism of the unit disc, given by a M�bius transformation mapping the unit disc to itself.
16.
This geometry, and associated symmetry group, was studied by Felix Klein as the monodromy groups of a Belyi surface a Riemann surface with a holomorphic map to the Riemann sphere, ramified only at 0, 1, and infinity ( a Belyi function ) the cusps are the points lying over infinity, while the vertices and the centers of each edge lie over 0 and 1; the degree of the covering ( number of sheets ) equals 5.
17.
The Klein surfaces form a category; a morphism from the Klein surface " X " to the Klein surface " Y " is a differentiable map " f " : " X " ?! " Y " which on each coordinate patch is either holomorphic or the complex conjugate of a holomorphic map and furthermore maps the boundary of " X " to the boundary of " Y ".
