A Poincar\'e-Bendixson theorem for meromorphic connections on Riemann surfaces

TL;DR

This paper establishes a Poincaré-Bendixson theorem for the asymptotic behavior of geodesics on Riemann surfaces with meromorphic connections, extending classical results to complex geometric contexts.

Contribution

It introduces a Poincaré-Bendixson theorem for meromorphic connections on Riemann surfaces, including non-compact cases and geodesics on complex tori.

Findings

Describes asymptotic behavior of geodesics on compact Riemann surfaces

Analyzes geodesics for holomorphic connections on complex tori

Discusses extensions to non-compact Riemann surfaces

Abstract

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.