  Using fiber bundle theory and conformal mappings, we continuously select a
point from the interior of Jordan domains in Riemannian surfaces. This
selection can be made equivariant under isometries, and take on prescribed
values such as the center of mass when the domains are convex. Analogous
results for conformal transformations are obtained as well. It follows that the
space of Jordan domains in surfaces of constant curvature admits an
isometrically equivariant strong deformation retraction onto the space of round
disks. Finally we develop a canonical procedure for selecting points from
planar Jordan domains.
