Hopf manifold

In complex geometry, a Hopf manifold (Hopf 1948) is obtained as a quotient of the complex vector space (with zero deleted) ${\displaystyle ({\mathbb {C} }^{n}\backslash 0)}$ by a free action of the group ${\displaystyle \Gamma \cong {\mathbb {Z} }}$ of integers, with the generator ${\displaystyle \gamma }$ of ${\displaystyle \Gamma }$ acting by holomorphic contractions. Here, a holomorphic contraction is a map ${\displaystyle \gamma$ :\;{\mathbb {C} }^{n}\to {\mathbb {C} }^{n}} such that a sufficiently big iteration ${\displaystyle \;\gamma ^{N}}$ maps any given compact subset of ${\displaystyle {\mathbb {C} }^{n}}$ onto an arbitrarily small neighbourhood of 0.

Two-dimensional Hopf manifolds are called Hopf surfaces.