A criterion for the existence of a flat connection on a parabolic vector bundle

摘要

We define homomorphic connection on a parabolic vector bundle over a Riemann surface and prove that a parabolic vector bundle admits a homomorphic connection if and only if each direct summand of it is of parabolic degree zero. This is a generalization to the parabolic context of a well-known result of Weil which says that a holomorphic vector bundle on a Riemann surface admits a holomorphic connection if and only every direct summand of it is of degree zero.