In complex geometry, a Hopf manifold (Hopf 1948) is obtained as a quotient of the complex vector space (with zero deleted) by a free action of the group of integers, with the generator of acting by holomorphic contractions. Here, a holomorphic contraction is a map such that a sufficiently big iteration maps any given compact subset of onto an arbitrarily small neighbourhood of 0.
Two-dimensional Hopf manifolds are called Hopf surfaces.