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

Abstract

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 $\mathds{H}^3$, 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 $\mathds{H}^n$, ${\mathbb{A}}\textrm{d}{\mathbb{S}}^n$, $\textrm{d}{\mathbb{S}}^n$, 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.

On the Gauss map of equivariant immersions in hyperbolic space

“…By (10) and ( 14), the norm of vectors proportional to 𝜒 (𝑥,𝑣) is preserved. Together with (12) and (13), vectors of the form 𝑑𝜑 𝑡 (𝑤  ) and 𝑑𝜑 𝑡 (𝑤  ) are orthogonal to 𝑑𝜑 𝑡 (𝜒 (𝑥,𝑣) ) = 𝜒 𝜑 𝑡 (𝑥,𝑣) . This concludes the first part of the statement.…”

“…It is worth mentioning that in dimension 3, the pseudo‐Riemannian metric $$\mathbb {G}$$ of $$\mathcal {G}(\mathbb {H}^{3})$$ can be seen as the real part of a holomorphic Riemannian manifold of constant curvature $$-1$$ (see [13]).…”

On PSL(2,ℂ) and on the space of geodesics of ℍ 3 as holomorphic Riemannian manifolds

“…We examine some features of PSL(2, C) and G, the space of geodesics of H 3 , equipped with some natural holomorphic Riemannian structures. The main reference for this paper is [1].…”