On the Gauss map of equivariant immersions in hyperbolic space
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866909362212044800 |
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| author | Emam, Christian El Seppi, Andrea |
| author_facet | Emam, Christian El Seppi, Andrea |
| contents | Given an oriented immersed hypersurface in hyperbolic space $\mathbb{H}^{n+1}$, its Gauss map is defined with values in the space of oriented geodesics of $\mathbb{H}^{n+1}$, 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 $π_1(M)$ in $\mathrm{Isom}(\mathbb{H}^{n+1})$, is the Gauss map of an equivariant immersion in $\mathbb{H}^{n+1}$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2008_07390 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On the Gauss map of equivariant immersions in hyperbolic space Emam, Christian El Seppi, Andrea Differential Geometry Geometric Topology Symplectic Geometry Given an oriented immersed hypersurface in hyperbolic space $\mathbb{H}^{n+1}$, its Gauss map is defined with values in the space of oriented geodesics of $\mathbb{H}^{n+1}$, 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 $π_1(M)$ in $\mathrm{Isom}(\mathbb{H}^{n+1})$, is the Gauss map of an equivariant immersion in $\mathbb{H}^{n+1}$. 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. |
| title | On the Gauss map of equivariant immersions in hyperbolic space |
| topic | Differential Geometry Geometric Topology Symplectic Geometry |
| url | https://arxiv.org/abs/2008.07390 |