  We approach the study of totally real immersions of smooth manifolds into
holomorphic Riemannian space forms of constant sectional curvature -1. We
introduce a notion of first and second fundamental form, we prove that they
satisfy a similar version of the classic Gauss-Codazzi equations, and
conversely that solutions of Gauss-Codazzi equations are immersion data of some
equivariant map. This study has some interesting geometric consequences:
  1) it provides a formalism to study immersions of surfaces into SL(2,C) and
into the space of geodesics of H^3;
  2) it generalizes the classical theory of immersions into non-zero curvature
space forms, leading to a model for the transitioning of hypersurfaces among
H^n, AdS^n, dS^n and S^n;
  3) we prove that a holomorphic family of immersion data corresponds to a
holomorphic family of immersions, providing an effective way to construct
holomorphic maps into the SO(n,C)-character variety. In particular we will
point out a simpler proof of the holomorphicity of the complex landslide;
  4) we see how, under certain hypothesis, complex metrics on a surface (i.e.
complex bilinear forms of its complexified tangent bundle) of constant
curvature -1 correspond to pairs of projective surfaces with the same holonomy.
Applying Bers Double Uniformization Theorem to this construction we prove a
Uniformization Theorem for complex metrics on a surface.
