  We discuss some topological aspects of the Riemann-Hilbert transmission
problem and Riemann-Hilbert monodromy problem on Riemann surfaces. In
particular, we describe the construction of a holomorphic vector bundle
starting from the given representation of the fundamental group and investigate
the local behaviour of connexions on this bundle. We give formulae for the
partial indices of the Riemann-Hilbert transmission problem in the
three-dimensional case in terms of the correspoding vector bundle on the
Riemann sphere.
