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A remark on Liouville vector fields and a theorem of Manouchehri

Published online by Cambridge University Press:  02 April 2001

MARC CHAPERON
Affiliation:
Institut de mathématiques, UMR 7586 du CNRS, Université Paris 7, UFR de mathématiques, CASE 7012, 2, place Jussieu, F–75251 Paris Cedex 05, France (e-mail: chaperon@math.jussieu.fr)

Abstract

In a recent article, Manouchehri proved a ‘Sternberg theorem’ for Liouville vector fields and noticed that it provided normal forms for implicit differential equations and first-order partial differential equations. We establish local and global versions of Moser's celebrated result on volume and symplectic forms when they admit a non-trivial one parameter (pseudo-) group of homotheties—by definition, a Liouville field is the generator of such a flow. The local version implies that two germs of Liouville fields of a symplectic or volume form are conjugate by a diffeomorphism germ which preserves the form if and only if they are conjugate. This contains Manouchehri's theorem.

Type
Research Article
Copyright
1999 Cambridge University Press

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