Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-25T14:21:45.163Z Has data issue: false hasContentIssue false
  • English
  • Français

Integrable almost cotangent structures and Legendrian bundles

Published online by Cambridge University Press:  24 October 2008

G. Thompson
G. Thompson
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ont. N2L 3G1, Canada
Get access Rights & Permissions [Opens in a new window]

Extract

Recently, the present author together with M. Crampin proved a structure theorem for a certain subclass of geometric objects known as almost tangent structures (Crampin and Thompson [8]). As the name suggests, an almost tangent structure is obtained by abstracting one of the tangent bundle's most important geometrical ingredients, namely its canonical 1–1 tensor, and using it to define a certain class of G-structures. Roughly speaking, the structure theorem referred to above may be paraphrased by saying that, if an almost tangent structure is integrable as a G-structure and satisfies some natural global hypotheses, then it is essentially the tangent bundle of some differentiable manifold. (I shall have a further remark to make about the conclusion of that theorem at the end of Section 2.)

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 101 , Issue 1 , January 1987 , pp. 61 - 78
Copyright
Copyright © Cambridge Philosophical Society 1987

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Abraham, R. and Marsden., J. E.Foundations of Mechanics (Benjamin, 1978).Google Scholar
[2]Arnold., V. I.Mathematical Methods in Classical Mechanics (Springer, 1978).CrossRefGoogle Scholar
[3]Arnold, V. I., Gusein-Zade, S. M. and Varchenko., A. N.Singularities of Differentiable Maps, vol. i (Birkhäuser, 1985).CrossRefGoogle Scholar
[4]Blair., D. E.Contact Manifolds in Riemmanian Geometry (Springer, 1976).CrossRefGoogle Scholar
[5]Boothby, W. and Wang., H. C.On contact manifolds. Ann. of Math. (2) 68 (1958), 721–734.CrossRefGoogle Scholar
[6]Bruckheimer., M. R. Thesis, University of Southampton (1960).Google Scholar
[7]Clark, R. S. and Goel., D. S.Almost cotangent manifolds. J. Differential Geom. 9 (1974), 109–122.CrossRefGoogle Scholar
[8]Crampin, M. and Thompson., G.Affine bundles and integrable almost tangent structures. Math. Proc. Cambridge Philos. Soc. 98 (1985), 61–71.CrossRefGoogle Scholar
[9]Goldschmidt., H.Integrability criteria for systems of nonlinear partial differential equations. J. Differential Geom. 1 (1967), 269–307.CrossRefGoogle Scholar
[10]Kobayashi., S.Transformation Groups in Differential Geometry (Springer, 1972).CrossRefGoogle Scholar
[11]Martinet., J. Formes de contact sur les variétés de dimension 3. Proc. of Liverpool Singularities Symposium II, Lectures Notes in Math. vol. 209 (Springer, 1971), 142–163.CrossRefGoogle Scholar
[12]Palais., R. S.A Global Formulation of the Lie Theory of Transformation Groups. A.M.S. Memoirs 22 (1957).Google Scholar
[13]Plante., J. F.Anosov flows. Amer. J. Math. 94 (1972), 729–754.CrossRefGoogle Scholar
[14]Reeb., G.Sur certaines propriétés topologiques des trajectoires des systémes dynamiques. Mémoires des L'Acad. Roy. de Belgique, Sci. Ser. 2, 27 (1952), 1–62.Google Scholar
[15]Simms, D. and Woodhouse., N. M. J.Lectures on Geometric Quantization (Springer, 1976).Google Scholar
[16]Stong., R. S.Contact manifolds. J. Differential Geom. 9 (1974), 219–238.CrossRefGoogle Scholar
[17]Thomas., C. B.Almost regular contact manifolds. J. Differential Geom. 11 (1976), 521–533.CrossRefGoogle Scholar
[18]Weil., A.Collected Papers (Springer, 1979).Google Scholar
[19]Weinstein., A.Symplectic manifolds and their Lagrangian submanifolds. Adv. in Math. 6 (1971), 329–346.CrossRefGoogle Scholar
[20]Weinstein., A.Lectures on Symplectic Manifolds, CBMS Conf. Series no. 29 (American Mathematical Society, 1977).CrossRefGoogle Scholar
[21]Yano, K. and Ishihara., S.Tangent and Cotangent Bundles (Marcel Dekker, 1973).Google Scholar