On the Gauss map of equivariant immersions in hyperbolic space

Given an oriented immersed hypersurface in hyperbolic space H n + 1 $\mathbb {H}^{n+1}$ , its Gauss map is defined with values the of geodesics which endowed a natural para-Kähler structure. In this paper, we address question whether immersion G $G$ universal cover $n$ -manifold M $M$ equivariant for some group representation π ( ) $\pi _1(M)$ Isom $\mathrm{Isom}(\mathbb {H}^{n+1})$ . We fully answer immersions principal curvatures − $(-1,1)$ : while only local obstructions are conditions that Lagrangian and Riemannian, global obstruction more subtle, provide two characterizations, first terms Maslov class, second (for compact) action compactly supported Hamiltonian symplectomorphisms.