Complex hyperbolic space

In mathematics, more precisely in differential geometry, complex hyperbolic space is the symmetric space associated with the Lie group ${\displaystyle \mathrm {SU} (n,1)}$. It can be explicitely constructed by giving a metric on the unit ball in ${\displaystyle \mathbb {C} ^{n}}$. It is characterised by being the only simply connected Hermitian manifold whose holomorphic sectional curvature which is constant equal to -1.