Reference : On immersions of surfaces into SL(2,C) and geometric consequences
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Physical, chemical, mathematical & earth Sciences : Mathematics
On immersions of surfaces into SL(2,C) and geometric consequences
Bonsante, Francesco mailto [Università degli Studi di Pavia > Mathematics]
El Emam, Christian mailto [University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH) >]
[en] Differential geometry ; Hyperbolic geometry ; Transition geometry
[en] 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.
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