Flat structure of meromorphic connections on Riemann surfaces

The possible omega limit sets of simple geodesics for meromorphic connections on compact Riemann surfaces have been studied by Abate, Tovena and Bianchi. In this paper, we study the same problem for infinite self-intersecting geodesics. In the first part of the paper we study relation among meromorphic $k$-differentials, singular flat metrics and meromorphic connections. Moreover, we prove a Poincar\'e-Bendixson theorem for infinite self-intersecting geodesics of meromorphic connections with monodromy in $G$, where $\arg G^k=\{0\}$ for some $k\in\mathbb{N}$.