  Let $(M,g)$ be a closed oriented negatively curved surface. A unitary
connection on a Hermitian vector bundle over $M$ is said to be transparent if
its parallel transport along the closed geodesics of $g$ is the identity. We
study the space of such connections modulo gauge and we prove a classification
result in terms of the solutions of certain PDE that arises naturally in the
problem. We also show a local uniqueness result for the trivial connection and
that there is a transparent SU(2)-connection associated to each meromorphic
function on $M$.
