The study of immersions of a smooth manifold M into holomorphic Riemannian manifolds suggests to define the notion of comple-valued) metrics, namely complex quadratic forms on ℂTM
with some condition of non-degeneracy.
The aim of the talk is to show some geometric properties of complex metrics which can be deduced from the study of this kind of immersions: in particular, we will present a correspondence between complex metrics with constant curvature and some pairs of developing maps for projective structures.
Time permitting, we will show a Uniformization Theorem in this setting and an application to the theory of immersions in ℍ3
This is joint work with Francesco Bonsante.