Global smooth geodesic foliations of the hyperbolic space

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• 作者：Yamile Godoy ; Marcos Salvai
• 关键词：Geodesic foliation ; Hyperbolic space ; Space of oriented lines ; 53C12 ; 53C40 ; 53C50
• 刊名：Mathematische Zeitschrift
• 出版年：2015
• 出版时间：October 2015
• 年：2015
• 卷：281
• 期：1-2
• 页码：43-54
• 全文大小：453 KB
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• 作者单位：Yamile Godoy (1)
  Marcos Salvai (1)
  
  1. FaMAF - CIEM, Ciudad Universitaria, 5000, C贸rdoba, Argentina
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-1823

文摘

We consider foliations of the whole three dimensional hyperbolic space \({\mathbb {H}}^{3}\) by oriented geodesics. Let \({\mathcal {L}}\) be the space of all the oriented geodesics of \({\mathbb {H}}^{3}\), which is a four dimensional manifold carrying two canonical pseudo-Riemannian metrics of signature \(\left( 2,2\right) \). We characterize, in terms of these geometries of \({\mathcal {L}}\), the subsets \({\mathcal {M}}\) in \({\mathcal {L}}\) that determine foliations of \({\mathbb {H}}^{3}\). We describe in a similar way some distinguished types of geodesic foliations of \({\mathbb {H}}^{3}\), regarding to which extent they are in some sense trivial in some directions: On the one hand, foliations whose leaves do not lie in a totally geodesic surface, not even at the infinitesimal level. On the other hand, those for which the forward and backward Gauss maps \(\varphi ^{\pm }:{\mathcal {M}}\rightarrow {\mathbb {H}} ^{3}\left( \infty \right) \) are local diffeomorphisms. Besides, we prove that for this kind of foliations, \(\varphi ^{\pm }\) are global diffeomorphisms onto their images. The subject of this article is within the framework of foliations by congruent submanifolds, and follows the spirit of the paper by Gluck and Warner where they understand the infinite dimensional manifold of all the great circle foliations of the three sphere.