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Contact structures on 1-connected 5-manifolds

Part of: Global differential geometry

Published online by Cambridge University Press:  26 February 2010

Hansjörg Geiges
Hansjörg Geiges
Affiliation:
Peterhouse, Cambridge, CB2 1RD
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Extract

All manifolds in this paper are assumed to be closed, oriented and smooth.

A contact structure on a (2n + l)-dimensional manifold M is a maximally non-integrable hyperplane distribution D in the tangent bundle TM, i.e., D is locally denned as the kernel of a 1-form α satisfying α ۸ (da)n ۸ 0. A global form satisfying this condition is called a contact form. In the situations we are dealing with, every contact structure will be given by a contact form (see [5]). A manifold admitting a contact structure is called a contact manifold.

MSC classification

Secondary: 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
Type
Research Article
Information
Mathematika , Volume 38 , Issue 2 , December 1991 , pp. 303 - 311
Copyright
Copyright © University College London 1991

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References

1.Barden, D.. Simply-connected 5-manifolds. Ann. of Math., 82 (1965), 365–385.CrossRefGoogle Scholar
2.Boothby, W. M. and Wang, H. C.. On contact manifolds. Ann. of Math., 68 (1958), 721–734.Google Scholar
3.Duchamp, T.. The classification of Legendre immersions. Preprint (1984).Google Scholar
4.Geiges, H.. Contact structures on (n - l)-connected (2n + l)-manifolds. J. T. Knight Prize entry, Cambridge (1991).Google Scholar
5.Gray, J. W.. Some global properties of contact structures. Ann. of Math., 69 (1959), 421–450.CrossRefGoogle Scholar
6.Gromov, M.. Partial Differential Relations (Springer-Verlag, 1986).Google Scholar
7.Haefliger, A.. Plongements differentiables de variétés dans variétés. Comment. Math. Helv., 36 (1961), 47–82.CrossRefGoogle Scholar
8.Kervaire, M. A.. A note on obstructions and characteristic classes. Amer. J. Math., 81 (1959), 773–784.CrossRefGoogle Scholar
9.Lutz, R. and Meckert, C.. Structures de contact sur certaines sphàres exotiques. C. R. Acad. Sc. Paris, 282A (1976), 591–593.Google Scholar
10.Martinet, J.. Formes de contact sur les variétés de dimension 3. Proc. Liverpool Singularities Symp. II (Wall, C. T. C., ed.), LNM 209 (Springer-Verlag, 1971), 142–163.Google Scholar
11.Massey, W. S.. Obstructions to the existence of almost complex structures. Bull. Amer. Math. Soc, 67 (1961), 559–564.Google Scholar
12.McDuff, D.. The structure of rational and ruled symplectic 4-manifolds. J. Amer. Math. Soc, 3 (1990), 679–712.Google Scholar
13.Meckert, C.. Forme de contact sur la somme connexe de deux variétés de contact de dimension impaire. Ann. tnst. Fourier, 32, 3 (1982), 251–260.Google Scholar
14.Thomas, C. B.. Contact forms on 1-connected 5-manifolds. Mathematika, 24 (1977), 237–240.Google Scholar
15.Thomas, C. B.. Contact structures on (n - 1)-connected (2n + 1)-manifolds. Banach Center Publ, 18 (1986), 255–270.CrossRefGoogle Scholar
16.Wall, C. T. C.. Classification problems in differential topology VI-Classification of (s-l)-connected (2s + l)-manifolds. Topology, 6 (1967), 273–296.CrossRefGoogle Scholar
17.Weinstein, A.. Contact surgery and symplectic handlebodies (1990). Hokkaido Math. J., 20 (1991), 241–251.Google Scholar