论文标题
在高线浸入双曲线空间的高斯图上
On the Gauss map of equivariant immersions in hyperbolic space
论文作者
论文摘要
给定在双曲空间中定向沉浸式超曲面$ \ mathbb {h}^{n+1} $，其高斯映射在$ \ mathbb {h}^{n+1} $的定向地理学空间中定义为具有天然para-para-kähller结构。在本文中，我们解决了一个问题：$ n $ - manifold $ m $的通用封面是否会浸入$π_1（m）$ in $ \ mathrm {isom {isom}（\ Mathbb { $ \ 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.
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.
