论文标题
在Quatatrnionic双曲空间中
On bisectors in quaternionic hyperbolic space
论文作者
论文摘要
在本文中，我们研究了与季学双曲线几何形状几何形状有关的问题。我们开发了Quaternion双曲线空间中的一些基本理论$ h^n_q $。特别是，我们表明Quaternionic BiseTors享受$ h^n_q $的完全测量的Submanifolds的各种分解。与复杂的双曲线几何形状相反，两种分型仅接受两种类型的分解（由Mostow和Goldman描述），我们表明，在Quaternionic案例的几何形状中，分离剂的几何形状更丰富。本文的主要目的是描述一个由$ h^n_q $组成的无限分解的家族，该家族完全是$ h^n_q $的，该家族完全是$ h^n_q $ submanifolds of $ h^n_q $ simetrict to Complex Proffer -ybolic Space $ H^n_c $，我们称之为风扇分解。另外，我们将正交投影的公式推导到$ h^n_q $ simetric中的完全测量的子手机上，至$ h^n_c $。使用此功能，我们在$ h^n_q $中引入了一类新的Hypersurfaces，我们将其称为$ H^n_q $中的复杂双曲机包。我们希望复杂的双曲线包将有助于为Quaternionic双曲线空间的离散组构造基本多面体。
In this paper, we study a problem related to geometry of bisectors in quaternionic hyperbolic geometry. We develop some of the basic theory of bisectors in quaternionic hyperbolic space $H^n_Q$. In particular, we show that quaternionic bisectors enjoy various decompositions by totally geodesic submanifolds of $H^n_Q$. In contrast to complex hyperbolic geometry, where bisectors admit only two types of decomposition (described by Mostow and Goldman), we show that in the quaternionic case geometry of bisectors is more rich. The main purpose of the paper is to describe an infinite family of different decompositions of bisectors in $H^n_Q$ by totally geodesic submanifolds of $H^n_Q$ isometric to complex hyperbolic space $H^n_C$ which we call the fan decompositions. Also, we derive a formula for the orthogonal projection onto totally geodesic submanifolds in $H^n_Q$ isometric to $H^n_C$. Using this, we introduce a new class of hypersurfaces in $H^n_Q$, which we call complex hyperbolic packs in $H^n_Q$. We hope that the complex hyperbolic packs will be useful for constructing fundamental polyhedra for discrete groups of isometries of quaternionic hyperbolic space.
