Immersions of surfaces into SL(2,C) and into the space of geodesics of Hyperbolic space

TL;DR

This thesis explores advanced geometric immersions into complex and hyperbolic spaces, establishing new theorems and characterizations that connect hypersurface theory with holomorphic and pseudo-Riemannian geometry.

Contribution

It introduces a Gauss-Codazzi theorem for immersions into holomorphic Riemannian space forms and characterizes Riemannian immersions into the space of geodesics of hyperbolic space.

Findings

Gauss-Codazzi theorem for holomorphic Riemannian space forms

Holomorphic transition approach for pseudo-Riemannian immersions

Characterization of Riemannian immersions as Gauss maps in hyperbolic space

Abstract

This thesis mainly treats two developments of the classical theory of hypersurfaces inside pseudo-Riemannian space forms. The former - a joint work with Francesco Bonsante - consists in the study of immersions of smooth manifolds into holomorphic Riemannian space forms of constant curvature -1 (including SL(2,C) with a multiple of its Killing form): this leads to a Gauss-Codazzi theorem, it suggests an approach to holomorphic transitioning of immersions into pseudo-Riemannian space forms, a trick to construct holomorphic maps into the PSL(2,C)-character variety, and leads to a restatement of Bers theorem. The latter - a joint work with Andrea Seppi - consists in the study of immersions of n-manifolds inside the space of geodesics of the hyperbolic (n+1)-space. We give a characterization, in terms of the para-Sasaki structure of this space of geodesics, of the Riemannian immersions which turn out to be Gauss maps of equivariant immersions into the hyperbolic space.