| Abstrakt: |
We study immersions of smooth manifolds into holomorphic Riemannian space forms of constant sectional curvature -1, including |$SL(2,\mathbb{C})$| and the space of geodesics of |$\mathbb{H}^3$| , 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 |$\mathbb{H}^n$| , |${\mathbb{A}}\textrm{d}{\mathbb{S}}^n$| , |$\textrm{d}{\mathbb{S}}^n$| , and |${\mathbb{S}}^n$| ; (2) it provides an effective tool to construct holomorphic maps to the |$\textrm{SO}(n,\mathbb{C})$| -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. [ABSTRACT FROM AUTHOR] |
