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We approach the study of totally real immersions of smooth manifolds intoholomorphic Riemannian space forms of constant sectional curvature -1. Weintroduce a notion of first and second fundamental form, we prove that theysatisfy a similar version of the classic Gauss-Codazzi equations, andconversely that solutions of Gauss-Codazzi equations are immersion data of someequivariant map. This study has some interesting geometric consequences:  1) it provides a formalism to study immersions of surfaces into SL(2,C) andinto the space of geodesics of H^3;  2) it generalizes the classical theory of immersions into non-zero curvaturespace forms, leading to a model for the transitioning of hypersurfaces amongH^n, AdS^n, dS^n and S^n;  3) we prove that a holomorphic family of immersion data corresponds to aholomorphic family of immersions, providing an effective way to constructholomorphic maps into the SO(n,C)-character variety. In particular we willpoint 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 constantcurvature -1 correspond to pairs of projective surfaces with the same holonomy.Applying Bers Double Uniformization Theorem to this construction we prove aUniformization Theorem for complex metrics on a surface.

On immersions of surfaces into SL(2,C) and geometric consequences