Riemann Surface, projective structures

Projective transformation. A change of local complex coordinates Z → Z′(Z) of a Riemann surface which has the form

(1)

is called a projective (or Möbius or fractional linear) transformation [1]. Such a transformation is characterized by the following property: the mapping Z → Z′(Z) is a projective transformation if and only if S(Z′, Z) = 0.

Projective structure. A projective structure on a Riemann surface Σ is an atlas of local complex coordinates for which all coordinate transformations are projective. Every compact Riemann surface admits such a structure. Moreover, there is a one-to-one correspondence between projective structures and projective connections R _zz , described by the relation R _zz (z) ≡ S(Z, z) where S denotes the Schwarzian derivative , and where the coordinates Z and z belong, respectively, to a projective and a generic atlas of Σ. Indeed, the previous expression has the correct transformation properties thanks to the chain rule for the...