Given an oriented immersed hypersurface in hyperbolic space ${\mathbb{H}}^{n+1}$${\mathbb{H}}^{n+1}$, its Gauss map is defined with values in the space of oriented geodesics of ${\mathbb{H}}^{n+1}$${\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$$G$ of the universal cover of an $n$$n$-manifold $M$$M$, equivariant for some group representation of ${\pi }_1(M)$${\pi }_1(M)$ in $\mathrm{Isom}({\mathbb{H}}^{n+1})$$\mathrm{Isom}({\mathbb{H}}^{n+1})$, is the Gauss map of an equivariant immersion in ${\mathbb{H}}^{n+1}$${\mathbb{H}}^{n+1}$. We fully answer this question for immersions with principal curvatures in $(-1,1)$$(-1,1)$: while the only local obstructions are the conditions that $G$$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$$M$ compact) in terms of the action of the group of compactly supported Hamiltonian symplectomorphisms.

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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${\mathbb{H}}^{n+1}$${\mathbb{H}}^{n+1}$，它的高斯图是用定向测地线空间中的值定义的${\mathbb{H}}^{n+1}$${\mathbb{H}}^{n+1}$，它具有天然的对 Kähler 结构。在本文中，我们解决了是否沉浸式的问题$G$$G$的普遍覆盖$n$$n$-歧管$米$$M$, 等变量的某些组表示${\pi }_1(米)$${\pi }_1(M)$在$\mathrm{异构体}({\mathbb{H}}^{n+1})$$\mathrm{Isom}({\mathbb{H}}^{n+1})$, 是等变浸入的高斯图${\mathbb{H}}^{n+1}$${\mathbb{H}}^{n+1}$. 对于主曲率浸入式，我们完全回答了这个问题$(-1,1)$$(-1,1)$：虽然唯一的局部障碍是$G$$G$是拉格朗日和黎曼，全局障碍更微妙，我们提供了两个表征，第一个是根据马斯洛夫类，第二个（对于$米$$M$紧）根据紧支持的哈密顿辛同胚群的作用。