Abstract
The goal of this lecture is to give an introduction to existence problems of contact structures. So, in the first Section we define the notion of contact structure, as well as some specialized contact structures. We also study the rigidity and the local behavior of such a structure. Some basic problems concerning the geometry of contact manifolds are presented in Sect. 2. The existence of contact forms is studied in the next Section. Specially in the 3–dimensional case, some classical results and the new Geiges–Gonzalo theory of contact circles and contact spheres and the classification manifolds carrying such structures are presented. Some historical considerations pointing important steps in the development of contact geometry are finally presented.
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Notes
- 1.
The manifold M is called a CR manifold if there exists a complex distribution \(\mathcal T\) (i.e. a subbundle \(\mathcal T\) of the complexified tangent bundle \(T^{c}M\)) so that \([\mathcal T,\mathcal T] \subset \mathcal T\) and \(\mathcal T \cap \overline{\mathcal T}=0\).
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Pitiş, G. (2017). Contact Forms in Geometry and Topology. In: Haesen, S., Verstraelen, L. (eds) Topics in Modern Differential Geometry. Atlantis Transactions in Geometry, vol 1. Atlantis Press, Paris. https://doi.org/10.2991/978-94-6239-240-3_5
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