From Surf Wiki (app.surf) — the open knowledge base
Hopf manifold
In complex geometry, a Hopf manifold is obtained as a quotient of the complex vector space (with zero deleted) ({\mathbb C}^n\backslash 0) by a free action of the group \Gamma \cong {\mathbb Z} of integers, with the generator \gamma of \Gamma acting by holomorphic contractions. Here, a holomorphic contraction is a map \gamma:; {\mathbb C}^n \to {\mathbb C}^n such that a sufficiently big iteration ;\gamma^N maps any given compact subset of {\mathbb C}^n onto an arbitrarily small neighbourhood of 0.
Two-dimensional Hopf manifolds are called Hopf surfaces.
Examples
In a typical situation, \Gamma is generated by a linear contraction, usually a diagonal matrix q\cdot Id, with q\in {\mathbb C} a complex number, 0. Such manifold is called a classical Hopf manifold.
Properties
A Hopf manifold H:=({\mathbb C}^n\backslash 0)/{\mathbb Z} is diffeomorphic to S^{2n-1}\times S^1. For n\geq 2, it is non-Kähler. In fact, it is not even symplectic because the second cohomology group is zero.
Hypercomplex structure
Even-dimensional Hopf manifolds admit hypercomplex structure. The Hopf surface is the only compact hypercomplex manifold of quaternionic dimension 1 which is not hyperkähler.
References
This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.
Ask Mako anything about Hopf manifold — get instant answers, deeper analysis, and related topics.
Research with MakoFree with your Surf account
Create a free account to save articles, ask Mako questions, and organize your research.
Sign up freeThis content may have been generated or modified by AI. CloudSurf Software LLC is not responsible for the accuracy, completeness, or reliability of AI-generated content. Always verify important information from primary sources.
Report