A Poincar\'e-Bendixson theorem for meromorphic connections on Riemann   surfaces
  We prove a Poincar\'e-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. 
Marco Abate, Fabrizio Bianchi
math.CV math.DS