A Poincar,-Bendixson theorem for meromorphic connections on Riemann surfaces

摘要

We shall prove a Poincar,-Bendixson theorem describing the asymptotic behavior of geodesics for a meromorphic connection on a compact Riemann surface. We shall also briefly discuss the case of non-compact Riemann surfaces, and study in detail the geodesics for a holomorphic connection on a complex torus.