Article (Scientific journals)
On the Gauss map of equivariant immersions in hyperbolic space
El Emam, Christian; Seppi, Andrea
2022In Journal of Topology, 15, p. 238-301
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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 :
no
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 :
15
Pages :
238-301
Peer reviewed :
Peer Reviewed verified by ORBi
Available on ORBilu :
since 22 January 2021

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