Joint Number Theory/Logic Seminar: Two models for the hyperbolic plane and existence of the Poincare metric on compact Riemann surfaces

12 May 2016
Norbert A'Campo
An implicite definition for the hyperbolic plane $H=H_I$ is in:
${\rm Spec}(\mathbb{R}[X]) = H_I \setunion  \mathbb{R}$.
All geometric hyperbolic features will follow from this definition in an elementary way.
A second definition is 
$H=H_J=\{J \in {\rm End}(R^2) \mid J^2=-Id, dx \wedge dy(u,Ju) \geq 0 \}$.
Working with $H=H_J$ allows to prove rather directly main theorems about Riemann surfaces.