Demailly investigated holomorphic contact structures which are hyperbolic in the sense that the Poincaré metrics on holomorphic Legendre disks yield a metric on the manifold. Reeb manifolds are those which admit a holomorphic vector Reeb vector field (i.e. transverse symmetry vector field) flowing freely. The quotient is a Kobayashi hyperbolic holomorphic symplectic manifold. This gives a classification of Reeb manifolds in low dimension.