Complex hyperbolic geometry

Abstract

In complex hyperbolic geometry we consider an open set biholomorphic to an open ball in ℂn, and we equip it with a particular metric that makes it have constant negative holomorphic curvature. This is analogous to but different from the real hyperbolic space. In the complex case, the sectional curvature is constant on complex lines, but it changes when we consider real 2-planes which are not complex lines.