Smooth distributions are finitely generated

Abstract

A subbundle of variable dimension inside the tangent bundle of a smooth manifold is called a smooth distribution if it is the pointwise span of a family of smooth vector fields. We prove that all such distributions are finitely generated, meaning that the family may be taken to be a finite collection. Further, we show that the space of smooth sections of such distributions need not be finitely generated as a module over the smooth functions. Our results are valid in greater generality, where the tangent bundle may be replaced by an arbitrary vector bundle.

Singular foliations for M-theory compactification

“…This singular distribution is cosmooth (rather than smooth) in the sense of [11] (see appendix D). Notice that D is smooth iff ξ is everywhere non-chiral -i.e.…”

“…The geometric information along the non-chiral locus U is encoded [8] by a regular foliation F which carries a longitudinal G 2 structure and whose geometry is determined by the supersymmetry conditions in terms of the supergravity four-form field strength. When ∅ = W M , we show that F extends to a singular foliationF of the whole manifold M by adding leaves which are allowed to have singularities at points belonging to W. This singular foliation "integrates" a cosmooth 1 [11][12][13][14] singular distribution D (a.k.a. generalized sub-bundle of T M ), defined as the kernel distribution of a closed one-form ω which belongs to a cohomology class f ∈ H 1 (M, R) determined by the supergravity four-form field strength.…”

“…For each maximal component C max j , the diffeomorphism (5.11) can be chosen 11 such that the sign of the integral γ j ω is positive along any 11 The sign of γ j ω does not depend on the choice of γ since ω vanishes on the leaves of Fω. If the sign is negative, then it can be made positive by composing the diffeomorphism (5.11) with idL j × R, where 1)) is any orientation-reversing diffeomorphism of the interval (0, 1).…”

“…17 We say that F is algebraically locally finitely generated if every point of M has an open neighborhood U such that C ∞ (U, F ) is finitely generated as a C ∞ (U, R)-module. A generalized subbundle of E is called regular if it is an ordinary smooth subbundle of E. Some references for the theory of generalized subbundles are [11,12].…”

“…It was shown in [11] that a generalized subbundle F of E is smooth iff there exists a finite collection s 1 . .…”