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Integrability of singular distributions on Banach manifolds

  • David Chillingworth (a1) and Peter Stefan (a2)

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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.

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(1)Bourbaki, N.Variétés différentielles et analytiques, fascicule de résultats (Hermann; Paris, 1967 (§§1–7), 1971 (§§8–15)).
(2)Dieudonné, J.Foundations of modern analysis (Academic Press; New York and London, 1960).
(3)Hermann, R.The differential geometry of foliations, II. J. Math. Mech. 11 (1962), 303–15.
(4)Lang, S.Introduction to differentiable manifolds (Wiley (Interscience); New York, 1962).
(5)Narasimhan, R.Analysis on real and complex manifolds (Masson; Paris, 1968).
(6)Stefan, P. Accessibility and singular foliations, Thesis, University of Warwick, 1973.
(7)Stefan, P.Accessible sets, orbits and foliations with singularities. Proc. London Math. Soc. (3) 29 (1974), 699713.
(8)Stefan, P. Integrability of systems of vectorfields. (To appear.)

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