Abstract
The aim of this chapter is to study surfaces in three dimensional contact manifolds. This will give us a so-called oriented singular foliation on the surface which is an important invariant of the contact structure. Roughly speaking, the construction is as follows. The contact planes and the tangent planes of a given surface are either transverse or they coincide at some point. In the latter case we will speak of a singular point. In the first case the contact planes define some direction field on the surface. An oriented singular foliation will be a class of vector fields on the surface which are pointing into this direction and which are zero exactly at the singular points.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
R. Abraham, J. Marsden, Foundation of Mechanics (Benjamin/Cummings, Reading, 1978)
D. Bennequin, Entrelacements et équations de Pfaff. Astérisque 107–108, 83–161 (1983)
Y. Eliashberg, Filling by holomorphic discs and its applications, in Geometry of Low Dimensional Manifolds, ed. by S. K. Donaldson. London Mathematical Society Lecture Notes (1991), pp. 45–67. Series 151
Y. Eliashberg, Contact 3–manifolds, twenty year since J. Martinet’s work. Ann. Inst. Fourier 42, 165–192 (1992)
Y. Eliashberg, Legendrian and transversal knots in tight contact 3-manifolds, topological methods in modern mathematics. Publish or Perish (1993)
E. Giroux, Convexité en topologie de contact. Commun. Math. Helvetici 66, 637–677 (1991)
M.W. Hirsch, Differential Topology. Graduate Texts in Math. 33 (Springer, New York, 1976)
W. Meeks, S.-T. Yau, Topology of three dimensional manifolds and the embedding problems in minimal surface theory. Ann. Math. 112, 441–484 (1980)
J. Munkres, Elementary Differential Topology (Princeton University Press, Princeton, 1963)
J. Palis Jr., W. de Melo, Geometric Theory of Dynamical Systems an Introduction. Translated from the Portuguese by A. K. Manning, vol. Xii (Springer Verlag, New York, Berlin, 1982), p. 198
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Abbas, C., Hofer, H. (2019). Surfaces in Three Dimensional Contact Manifolds. In: Holomorphic Curves and Global Questions in Contact Geometry. Birkhäuser Advanced Texts Basler Lehrbücher. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-11803-7_3
Download citation
DOI: https://doi.org/10.1007/978-3-030-11803-7_3
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-11802-0
Online ISBN: 978-3-030-11803-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)