Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-10T12:07:45.614Z Has data issue: false hasContentIssue false
  • English
  • Français

Lagrangian foliations and Anosov symplectomorphisms on Kähler manifolds

Part of: Complex manifolds Symplectic geometry, contact geometry Dynamical systems with hyperbolic behavior Global differential geometry Local differential geometry

Published online by Cambridge University Press:  30 October 2020

M. J. D. HAMILTON  and
D. KOTSCHICK Open the ORCID record for D. KOTSCHICK [Opens in a new window]
M. J. D. HAMILTON
Affiliation:
Institut für Geometrie und Topologie, Universität Stuttgart, Pfaffenwaldring 57, 70569Stuttgart, Germany (e-mail: mark.hamilton@math.lmu.de)
D. KOTSCHICK*
Affiliation:
Mathematisches Institut, LMU München, Theresienstr. 39, 80333München, Germany
*
e-mail: dieter@math.lmu.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We investigate parallel Lagrangian foliations on Kähler manifolds. On the one hand, we show that a Kähler metric admitting a parallel Lagrangian foliation must be flat. On the other hand, we give many examples of parallel Lagrangian foliations on closed flat Kähler manifolds which are not tori. These examples arise from Anosov automorphisms preserving a Kähler form.

Keywords

Lagrangian foliationflat Kähler manifoldAnosov symplectomorphism

MSC classification

Primary: 32Q15: Kähler manifolds 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 53B35: Hermitian and Kählerian structures
Secondary: 53C12: Foliations (differential geometric aspects) 53C55: Hermitian and Kählerian manifolds 53D12: Lagrangian submanifolds; Maslov index
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 11 , November 2021 , pp. 3325 - 3335
Check for updates
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

References

Barth, W., Peters, C. and Van de Ven, A.. Compact Complex Surfaces. Springer, Berlin, 1984.CrossRefGoogle Scholar
Benoist, Y. and Labourie, F.. Sur les difféomorphismes d’Anosov affines à feuilletages stable et instable différentiables. Invent. Math. 111 (1993), 285–308.CrossRefGoogle Scholar
Catanese, F. and Demleitner, A.. Hyperelliptic threefolds with group ${D}_4$ , the dihedral group of order $8$ . Preprint, 2018, arXiv:1805.01835v1.Google Scholar
Cruceanu, V., Fortuny, P. and Gadea, P. M.. A survey on paracomplex geometry. Rocky Mountain J. Math. 26 (1996), 83–115.CrossRefGoogle Scholar
Dekimpe, K., Halenda, M. and Szczepanski, A.. Kähler flat manifolds. J. Math. Soc. Japan 61 (2009), 363–377.CrossRefGoogle Scholar
Duistermaat, J. J.. On global action-angle coordinates. Comm. Pure Appl. Math. 33 (1980), 687–706.CrossRefGoogle Scholar
Epstein, D. and Shub, M.. Expanding endomorphisms of flat manifolds. Topology 7 (1968), 139–141.CrossRefGoogle Scholar
Gordejuela, F. E. and Santamaría, R.. The canonical connection of a bi-Lagrangian manifold. J. Phys. A 34 (2001), 981–987.CrossRefGoogle Scholar
Hamilton, M. J. D. and Kotschick, D.. Künneth Geometry (London Mathematical Society Student Texts). Cambridge University Press, Cambridge (in press).Google Scholar
Johnson, F. E. A.. Flat algebraic manifolds. Geometry of Low-Dimensional Manifolds, Vol. 1 (London Mathematical Society Lecture Notes, 150). Cambridge University Press, Cambridge, 1990, pp. 73–91.Google Scholar
Johnson, F. E. A.. A flat projective variety with ${D}_8$ -holonomy. Tohoku Math. J. 71 (2019), 319–326.CrossRefGoogle Scholar
Johnson, F. E. A. and Rees, E. G.. Kähler groups and rigidity phenomena. Math. Proc. Cambridge Philos. Soc. 109 (1991), 31–44.CrossRefGoogle Scholar
Lange, H.. Hyperelliptic varieties. Tohoku Math. J. 53 (2001), 491–510.CrossRefGoogle Scholar
Porteous, H. L.. Anosov diffeomorphisms of flat manifolds. Topology 11 (1972), 307–315.CrossRefGoogle Scholar
Vaisman, I.. Basics of Lagrangian foliations. Publ. Mat. 33(3) (1989), 559–575.CrossRefGoogle Scholar
Wolf, J. A.. Spaces of Constant Curvature, 5th edn. Publish or Perish, Houston, TX, 1984.Google Scholar