On the Gauss map of equivariant immersions in hyperbolic space

El Emam, Christian; Seppi, Andrea

doi:10.1112/topo.12225

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On the Gauss map of equivariant immersions in hyperbolic space

El Emam, Christian; Seppi, Andrea

2022 • In Journal of Topology, 15, p. 238-301

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https://hdl.handle.net/10993/45777

DOI
10.1112/topo.12225

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Keywords :

Differential geometry; Hyperbolic geometry; Space of geodesics; Lagrangian immersions

Abstract :

[en] Given an oriented immersed hypersurface in hyperbolic space H^{n+1}, its Gauss map is defined with values in the space of oriented geodesics of H^{n+1}, which is endowed with a natural para-Kähler structure. In this paper we address the question of whether an immersion G of the universal cover of an n-manifold M, equivariant for some group representation of π1(M) in Isom(H^{n+1}), is the Gauss map of an equivariant immersion in H^{n+1}. We fully answer this question for immersions with principal curvatures in (−1,1): while the only local obstructions are the conditions that G is Lagrangian and Riemannian, the global obstruction is more subtle, and we provide two characterizations, the first in terms of the Maslov class, and the second (for M compact) in terms of the action of the group of compactly supported Hamiltonian symplectomorphisms

Disciplines :

Mathematics

Author, co-author :

El Emam, Christian ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)

Seppi, Andrea; Centre National de la Recherche Scientifique - CNRS > Institut Fourier, Université Grenoble Alpes

External co-authors :

Language :

English

Title :

On the Gauss map of equivariant immersions in hyperbolic space

Publication date :

2022

Journal title :

Journal of Topology

ISSN :

1753-8416

eISSN :

1753-8424

Publisher :

Wiley

Volume :

Pages :

238-301

Peer reviewed :

Peer Reviewed verified by ORBi

Available on ORBilu :

since 22 January 2021

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