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9 - Nilmanifolds

Published online by Cambridge University Press:  07 December 2023

M. J. D. Hamilton
Affiliation:
Universität Stuttgart
D. Kotschick
Affiliation:
Ludwig-Maximilians-Universität München
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Summary

This chapter contains a brief introduction to nilmanifolds, and a discussion of Künneth and related structures on nilmanifolds. Nilmanifolds are homogeneous spaces for nilpotent Lie groups, and for them the discussions of geometric structures can often be reduced to the consideration of left-invariant structures. Left-invariant structures in turn arise from the corresponding linear structures on the Lie algebra, and these linear structures are usually much more tractable than arbitrary geometric structures on smooth manifolds. The nilmanifolds of abelian Lie groups are just tori, so that in some sense nilmanifolds are the simplest generalisations of tori.

We do not give a systematic treatment of nilmanifolds here, but focus on providing a few explicit examples of Künneth structures, of hypersymplectic structures, and of Anosov symplectomorphisms in this setting. For more information on topics from the theory of nilmanifolds that we treat rather breezily, we refer to the books by Gorbatsevich, Onishchik and Vinberg [GOV-97] and by Knapp [Kna-96].

Type
Chapter
Information
Künneth Geometry
Symplectic Manifolds and their Lagrangian Foliations
, pp. 123 - 150
Publisher: Cambridge University Press
Print publication year: 2023

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  • Nilmanifolds
  • M. J. D. Hamilton, Universität Stuttgart, D. Kotschick, Ludwig-Maximilians-Universität München
  • Book: Künneth Geometry
  • Online publication: 07 December 2023
  • Chapter DOI: https://doi.org/10.1017/9781108902977.009
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  • Nilmanifolds
  • M. J. D. Hamilton, Universität Stuttgart, D. Kotschick, Ludwig-Maximilians-Universität München
  • Book: Künneth Geometry
  • Online publication: 07 December 2023
  • Chapter DOI: https://doi.org/10.1017/9781108902977.009
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Nilmanifolds
  • M. J. D. Hamilton, Universität Stuttgart, D. Kotschick, Ludwig-Maximilians-Universität München
  • Book: Künneth Geometry
  • Online publication: 07 December 2023
  • Chapter DOI: https://doi.org/10.1017/9781108902977.009
Available formats
×