  In this paper we investigate the properties of the real and complex
projective structures associated to Hitchin and quasi-Hitchin representations
that were originally constructed using Guichard-Wienhard's theory of domains of
discontinuity. We determine the topology of the underlying manifolds and we
prove that some of these geometric structures are fibered in a special standard
way. In order to prove these results, we give two new ways to construct these
geometric structures: we construct them using gauge theory, flat bundles and
Higgs bundles, and we also give a new geometric way to construct them.
