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On the Gauss map of equivariant immersions in hyperbolic space
Journal of Topology ( IF 0.8 ) Pub Date : 2022-04-15 , DOI: 10.1112/topo.12225 Christian El Emam 1 , Andrea Seppi 2
Journal of Topology ( IF 0.8 ) Pub Date : 2022-04-15 , DOI: 10.1112/topo.12225 Christian El Emam 1 , Andrea Seppi 2
Affiliation
Given an oriented immersed hypersurface in hyperbolic space , its Gauss map is defined with values in the space of oriented geodesics of , which is endowed with a natural para-Kähler structure. In this paper, we address the question of whether an immersion of the universal cover of an -manifold , equivariant for some group representation of in , is the Gauss map of an equivariant immersion in . We fully answer this question for immersions with principal curvatures in : while the only local obstructions are the conditions that 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 compact) in terms of the action of the group of compactly supported Hamiltonian symplectomorphisms.
中文翻译:
关于双曲空间等变浸入的高斯图
给定双曲空间中的定向浸入超曲面,它的高斯图是用定向测地线空间中的值定义的,它具有天然的对 Kähler 结构。在本文中,我们解决了是否沉浸式的问题的普遍覆盖-歧管, 等变量的某些组表示在, 是等变浸入的高斯图. 对于主曲率浸入式,我们完全回答了这个问题:虽然唯一的局部障碍是是拉格朗日和黎曼,全局障碍更微妙,我们提供了两个表征,第一个是根据马斯洛夫类,第二个(对于紧)根据紧支持的哈密顿辛同胚群的作用。
更新日期:2022-04-15
中文翻译:
关于双曲空间等变浸入的高斯图
给定双曲空间中的定向浸入超曲面,它的高斯图是用定向测地线空间中的值定义的,它具有天然的对 Kähler 结构。在本文中,我们解决了是否沉浸式的问题的普遍覆盖-歧管, 等变量的某些组表示在, 是等变浸入的高斯图. 对于主曲率浸入式,我们完全回答了这个问题:虽然唯一的局部障碍是是拉格朗日和黎曼,全局障碍更微妙,我们提供了两个表征,第一个是根据马斯洛夫类,第二个(对于紧)根据紧支持的哈密顿辛同胚群的作用。