Fórmulas de distância no espaço hiperbólico complexo

Abstract

This work consists of presenting formulas for the distance between points, real geodesics lines, complex geodesics, bisectors and any combination of these objects in complex hyperbolic space. Let us consider the two-dimension complex hyperbolic space, becausedemonstrated formulas generalize naturally to the hyperbolic space of any dimension, changing only the concept of geodesic complex by the concept of hyperplane complex. For most cases studied, the formulas presented provide explicit expressions for the desired distances. However, in the case of two real geodesic line two bisectors and in thecase of distance between a real geodesic line and a bisector, the distance depends on of a root of a polynomial of sixth degree and n-invariant. We will present examples in which it is soluble the polynomial and an example in which the polynomial is not soluble radical. Our study was based on the Hanna Sandler article .Distance formulas in complex hyperbolic space. [2], but used as main references the book .Complex Hyperbolic Geometry . of William M. Goldman [1] and John R. Parker notes .Notes on Complex Hyperbolic Geometry. [5].