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.


In a typical situation,   is generated by a linear contraction, usually a diagonal matrix  , with   a complex number,  . Such manifold is called a classical Hopf manifold.


A Hopf manifold   is diffeomorphic to  . For  , it is non-Kähler. In fact, it is not even symplectic because the second cohomology group is zero.

Hypercomplex structureEdit

Even-dimensional Hopf manifolds admit hypercomplex structure. The Hopf surface is the only compact hypercomplex manifold of quaternionic dimension 1 which is not hyperkähler.


  • Hopf, Heinz (1948), "Zur Topologie der komplexen Mannigfaltigkeiten", Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, Inc., New York, pp. 167–185, MR 0023054
  • Ornea, Liviu (2001) [1994], "H/h110270", in Hazewinkel, Michiel (ed.), Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4