  It is shown that abelian Higgs vortices on a hyperbolic surface $M$ can be
constructed geometrically from holomorphic maps $f:M \to N$, where $N$ is also
a hyperbolic surface. The fields depend on $f$ and on the metrics of $M$ and
$N$. The vortex centres are the ramification points, where the derivative of
$f$ vanishes. The magnitude of the Higgs field measures the extent to which $f$
is locally an isometry.
  Witten's construction of vortices on the hyperbolic plane is rederived, and
new examples of vortices on compact surfaces and on hyperbolic surfaces of
revolution are obtained. The interpretation of these solutions as
SO(3)-invariant, self-dual SU(2) Yang--Mills fields on $\R^4$ is also given.
