Fifteen years of contact circles and contact spheres
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FIFTEEN YEARS OF CONTACT CIRCLES AND CONTACT SPHERES Hansjörg Geiges · Jesús Gonzalo Pérez
Received: 11 October 2012 / Published online: 28 February 2013 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2013
Abstract Contact circles and contact spheres are geometric structures on 3-manifolds introduced by the authors some fifteen years ago. In this survey we review the main results about these structures and we include two as yet unpublished proofs. Keywords Contact circle · Contact sphere · Contact structure · Thurston geometries · Moduli space · Spin structure Mathematics Subject Classification (2010) 53D35 · 53C15 · 53C26 · 57R15 1 Introduction This is a survey of the geometry and topology of contact circles and contact spheres. A contact sphere is a triple of contact forms satisfying a natural nondegeneracy condition and is the quaternionic analog of a contact structure. A contact circle is a pair with the analogous nondegeneracy property. Two very different classes of such structures have been studied so far: taut ones, which satisfy a natural system of PDEs, and general type ones. Section 2 contains the most basic definitions. Section 3 explains how to recognize a contact circle or contact sphere, and shows the relations with symplectic and complex geometry. Section 4 explains the very important notions of taut contact circle and taut contact sphere. We want to stress that structures of general type (i.e. possibly non-taut ones) are also very interesting, with their own set of natural questions. Sections 5, 6, and 7 are devoted to the topology and geometry of taut structures. The reader will notice that these structures are by now fairly well understood.
H. Geiges Mathematisches Institut, Universität zu Köln, Weyertal 86–90, 50931 Köln, Germany e-mail: [email protected]
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J. Gonzalo Pérez ( ) Universidad Autónoma of Madrid, Departamento de Matemáticas, 28049 Madrid, Spain e-mail: [email protected]
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H. GEIGES, J. GONZALO PÉREZ
A lot remains to be discovered about contact circles and contact structures of general type. They are treated in Sects. 8, 9, and 10. Section 8 is the longest one, because it is also the only section where original proofs, not previously published, are given. Section 11 briefly discusses the interaction of this theory with other areas of Mathematics.
2 Basic definitions and motivation We work on 3-dimensional manifolds. A contact form on M 3 is a Pfaff form α on M such that α ∧ dα is nowhere zero, hence a volume form on M. A contact structure on M is a 2-plane field on M which is the kernel of some contact form. The identity: (f α) ∧ d(f α) = f 2 α ∧ dα
(1)
shows that if α is a contact form then so is f α for any nowhere zero function f . In other words: if a 2-plane field ξ is a contact structure, then all Pfaff forms having ξ as kernel are contact forms. Each contact form α defines an orientation of the 3-manifold through the volume form α ∧ dα, and (1) implies that f α defines the same ori
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