On the inner automorphisms of a singular foliation

  • Alfonso Garmendia
  • Ori Yudilevich


A singular foliation in the sense of Androulidakis and Skandalis is an involutive and locally finitely generated module of compactly supported vector fields on a manifold. An automorphism of a singular foliation is a diffeomorphism that preserves the module. In this note, we give a proof of the (surprisingly non-trivial) fundamental fact that the time-one flow of an element of a singular foliation (i.e. its exponential) is an automorphism of the singular foliation. This fact was previously proven in Androulidakis and Skandalis (J Reine Angew Math 626:1–37, 2009) using an infinite dimensional argument (involving differential operators), and the purpose of this note is to complement that proof with a finite dimensional proof in which the problem is reduced to solving an elementary ODE.


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    Androulidakis, I., Skandalis, Georges: The holonomy groupoid of a singular foliation. J. Reine Angew. Math. 626, 1–37 (2009)MathSciNetCrossRefGoogle Scholar
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    Rudolph, G., Schmidt, M.: Differential geometry and mathematical physics. Part I. Manifolds, Lie groups and Hamiltonian systems. Theoretical and Mathematical Physics. Springer Netherlands (2013)Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsKU LeuvenLeuvenBelgium

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