Complex hyperbolic space
In mathematics, the complex hyperbolic space is a Hermitian manifold which is the equivalent of the real hyperbolic space in the context of complex manifolds. The complex hyperbolic space is a Kähler manifold, and it is characterised by being the only simply connected complete Kähler manifold whose holomorphic sectional curvature is constant equal to -1. Its underlying Riemannian manifold has non-constant negative curvature, pinched between -1 and -1/4, or between -4 and -1, according to the choice of a normalization of the metric): in particular, it is a CAT(-1/4) space.
The complex hyperbolic space is also the symmetric space associated with the Lie group . These spaces constitute one of the three families of rank one symmetric spaces, together with real and quaternionic hyperbolic spaces, classification to which must be added one exceptional space, the Cayley plane. It is also the only family of Hermitian symmetric spaces having rank one.
