UNIQUENESS OF AFFINE STRUCTURES ON RIEMANN SURFACES

Let $M$ be any compact Riemann surface of genus $g\ge 2$. It is first established that there do not exist on $M$ any generic low- degree simple polar variations of branched affine structures having fixed nonpolar and polar branch data and fixed induced character homomorphism $\tilde \psi$. Hence, these structures depend uniquely on the branch data and the homomorphism. A related result is also established concern­ing the nonexistence on $M$ of generic low-degree single-point variations of branched affine structures having fixed homomorphism $\tilde \psi$. These resuits depend on the Noether and Weierstrass gaps on $M$. Corollaries are derived concerning mappings induced by sections of vector bundles of affine structures and concerning structures on an arbitrary hyperelliptic or elliptic ($g =1$) surface $M$.