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Contact and Symplectic Topology pp 137–172Cite as

Geometric Decompositions of Almost Contact Manifolds

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Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 26))

Abstract

These notes are intended to be an introduction to the use of approximately holomorphic techniques in almost contact and contact geometry. We develop the setup of the approximately holomorphic geometry. Once done, we sketch the existence of the two main geometric decompositions available for an almost contact or contact manifold: open books and Lefschetz pencils. The use of the two decompositions for the problem of existence of contact structures is mentioned.

The author was supported by the Spanish National Research Project MTM2010-17389 and the ESF Network Contact and Symplectic Topology.

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Notes

  1. 1.

    Just once and for all it is important to mention that all the results in these notes can be easily adapted to the non-coorientable case. The essential point being that any non-coorientable contact manifold admits a coorientable double-cover. Therefore to study non-coorientable manifolds is reduced to study coorientable ones with free \(\mathbb {Z}/2\mathbb {Z}\)-actions. See [21] for details.

  2. 2.

    The normal bundle TM/ξ of ξ as a subbundle of TM is trivial.

  3. 3.

    The maximum angle between two subspaces \(U,V \subset \mathbb {R}^{m}\) of the Euclidean space is by definition ∠ M (U,V)=max u∈U {∠(u,V)}.

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Acknowledgements

I want to thank Roger Casals by his proof-reading that has greatly improved the final version of this article. Special thanks also to Emmanuel Giroux who patiently explained to me the results contained in Section 3. Most of the ideas in these notes come from him. J.P. Mohsen kindly offered a copy of his thesis [27] to me. Discussions with Vincent Colin, Dishant Pancholi, Klaus Niederkrüeger, Thomas Vogel and Eva Miranda have been really helpful to write down these notes.

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Authors and Affiliations

  1. Instituto de Ciencias Matemáticas CSIC–UAM–UC3M–UCM, C. Nicolás Cabrera, 13–15, 28049, Madrid, Spain

    Francisco Presas

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  1. Francisco Presas
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Correspondence to Francisco Presas .

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Editors and Affiliations

  1. Laboratoire de Mathématiques d'Orsay, Université Paris-Sud, Orsay, France

    Frédéric Bourgeois

  2. Lab. de Mathématiques Jean Leray UMR6629, University of Nantes, Nantes, France

    Vincent Colin

  3. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary

    András Stipsicz

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Presas, F. (2014). Geometric Decompositions of Almost Contact Manifolds. In: Bourgeois, F., Colin, V., Stipsicz, A. (eds) Contact and Symplectic Topology. Bolyai Society Mathematical Studies, vol 26. Springer, Cham. https://doi.org/10.1007/978-3-319-02036-5_4

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