Reference : Immersions of surfaces into SL(2,C) and into the space of geodesics of Hyperbolic space
Dissertations and theses : Doctoral thesis
Physical, chemical, mathematical & earth Sciences : Mathematics
Immersions of surfaces into SL(2,C) and into the space of geodesics of Hyperbolic space
El Emam, Christian mailto [Università degli Studi di Pavia > Dipartimento di Matematica "Felice Casorati"]
University of Pavia, ​Pavia, ​​Italy
Docteur en Mathématiques
Bonsante, Francesco mailto
Segatti, Antonio mailto
[en] Differential geometry ; Holomorphic Riemannian manifolds ; Submanifolds ; Transition geometry ; Para-Kahler manifolds
[en] 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-Kahler structure of this space of geodesics, of the Riemannian immersions which turn out to be Gauss maps of equivariant immersions into the hyperbolic space.

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