On Immersions of Surfaces into SL(2, ℂ) and Geometric Consequences

We study immersions of smooth manifolds into holomorphic Riemannian space forms of constant sectional curvature -1, including SL(2,C) and the space of geodesics of H³, and we prove a Gauss–Codazzi theorem in this setting. This approach has some interesting geometric consequences: (1) it provides a model for the transitioning of hypersurfaces among Hⁿ, AdSⁿ, dSⁿ, and Sⁿ; (2) it provides an effective tool to construct holomorphic maps to the SO(n,C)-character variety, bringing to a simpler proof of the holomorphicity of the complex landslide; and (3) it leads to a correspondence, under certain hypotheses, between complex metrics on a surface (i.e., complex bilinear forms of its complexified tangent bundle) and pairs of projective structures with the same holonomy. Through Bers theorem, we prove a uniformization theorem for complex metrics.