UNIQUENESS OF AFFINE STRUCTURES ON RIEMANN SURFACES
Main Article Content
Abstract
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 $M$.
Article Details

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
References
H. M. Farkas and I. Kra, "Riemann Surfaces", Springer-Verlag, New York, 1980.
R. C. Gunning, "Special Coordinate Coverings of Riemann Surfaces", Math. Ann., 170, 67-86 (1966).
T. Kra, "A Generalization of a Theorem of Poincare", Proc.. Am. Math. Soc., 27, 299-302 (1971).
R. Mandelbaum, "Branched Structures on Riemann Surfaces", Trans. Am. Math. Soc., 163, 261-275 (1972).
J. T. Masterson, "Branched Affine and Projective Structures on Compact Riemann Surfaces", Indag. Math., 91, 309-319 (1988).
J. T. Masterson, "Branched Structures Associated with Lame's Equation", Arkiv for Math., 28, 131-137 (1990).
H. Poincare, "Sur les groupes des equations lineaires", Acta Math., 4, 201-312 (1884).