Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-25T01:36:02.388Z Has data issue: false hasContentIssue false
  • English
  • Français

Integrability of singular distributions on Banach manifolds

Published online by Cambridge University Press:  24 October 2008

David Chillingworth  and
Peter Stefan
David Chillingworth
Affiliation:
Department of Mathematics, University of Southampton, Southampton SO9 5NH, England
Peter Stefan
Affiliation:
School of Mathematics and Computer Science, University College of North Wales, Bangor LL57 2UW, G.B.
Get access Rights & Permissions [Opens in a new window]

Extract

One of the key results in the work of the second author ((7), (8)) on integrability of systems of vectorfields is the theorem which relates integrability of a distribution to the concept of homogeneity. In this paper, we show that the homogeneity theorem also applies in an infinite-dimensional context, and this allows us to derive infinite-dimensional versions of several further results in (7) and (8), formulated in terms of distributions. In particular, we are able to express necessary and sufficient conditions for homogeneity in terms of Lie brackets (Theorems 3 and 4) and to characterize integrable real-analytic distributions (Theorem 5). As a corollary to our Theorem 2, we recover the standard Frobenius theorem on the integrability of regular distributions. We also discuss briefly a basic problem which arises in infinite dimensions when we view an integral manifold of an integrable distribution as part of a singular foliation.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 79 , Issue 1 , January 1976 , pp. 117 - 128
Copyright
Copyright © Cambridge Philosophical Society 1976

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)Bourbaki, N.Variétés différentielles et analytiques, fascicule de résultats (Hermann; Paris, 1967 (§§1–7), 1971 (§§8–15)).Google Scholar
(2)Dieudonné, J.Foundations of modern analysis (Academic Press; New York and London, 1960).Google Scholar
(3)Hermann, R.The differential geometry of foliations, II. J. Math. Mech. 11 (1962), 303–15.Google Scholar
(4)Lang, S.Introduction to differentiable manifolds (Wiley (Interscience); New York, 1962).Google Scholar
(5)Narasimhan, R.Analysis on real and complex manifolds (Masson; Paris, 1968).Google Scholar
(6)Stefan, P. Accessibility and singular foliations, Thesis, University of Warwick, 1973.Google Scholar
(7)Stefan, P.Accessible sets, orbits and foliations with singularities. Proc. London Math. Soc. (3) 29 (1974), 699–713.CrossRefGoogle Scholar
(8)Stefan, P. Integrability of systems of vectorfields. (To appear.)Google Scholar