On the Gauss map of equivariant immersions in hyperbolic space

English

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]

17-Aug-2020

1

55

No

[en] Differential geometry ; Hyperbolic geometry ; Space of geodesics ; Lagrangian immersions

[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