Painlevé III: A Case Study in the Geometry of Meromorphic Connections

The purpose of this monograph is two-fold:  it introduces a conceptual language for the geometrical objects underlying Painlevé equations,  and it offers new results on a particular Painlevé III equation of type  PIII (D6), called PIII (0, 0, 4, −4), describing its relation to isomonodromic families of vector bundles on P1  with meromorphic connections.  This equation is equivalent to the radial sine (or sinh) Gordon equation and, as such, it appears