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
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.
is generated by a linear contraction, usually a diagonal matrix
Such manifold is called a classical Hopf manifold.
In fact, it is not even symplectic because the second cohomology group is zero.
Even-dimensional Hopf manifolds admit hypercomplex structure.
The Hopf surface is the only compact hypercomplex manifold of quaternionic dimension 1 which is not hyperkähler.