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Generalized Flow-Box property for singular foliations

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

We introduce a notion of generalized Flow-Box property valid for general singular distributions and sub-varieties (based on a dynamical interpretation). Just as in the usual Flow-Box Theorem, we characterize geometrical and algebraic conditions of (quasi) transversality in order for an analytic sub-variety X (not necessarily regular) to be a section of a line foliation. We also discuss the case of more general foliations. This study is originally motivated by a question of Jean-François Mattei (Invent Math 103(2):297–325, 1991, Theorem 3.2.1) about the existence of local slices for a (non-compact) Lie group action.

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Acknowledgements

We would like to thank Jean-François Mattei for bringing the problem to our attention and for the very useful discussions on the topic. Daniel Panazzolo was supported by ANR-16-CE40-0008. We would also like to thank the two anonymous referees for the many useful suggestions and the careful revision, in particular concerning Definition 4.10.

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  1. André Belotto da Silva

    Present address: Institut de Mathématiques de Marseille, UMR CNRS 7373, Université Aix-Marseille, 39, Rue F. Joliot Curie, 13013, Marseille, France

Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, UMR CNRS 5129, Université Paul Sabatier, 118 Route de Narbonne, 31062, Toulouse Cedex 9, France

    André Belotto da Silva

  2. Département de Mathématiques, IRIMAS, Université de Haute Alsace, 4 Rue des Frères Lumière, 68093, Mulhouse, France

    Daniel Panazzolo

  3. Université de Strasbourg, Strasbourg, France

    Daniel Panazzolo

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  1. André Belotto da Silva
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  2. Daniel Panazzolo
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Correspondence to André Belotto da Silva.

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Dedicado à Felipe Cano, em comemoração dos seus 60 anos.

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Belotto da Silva, A., Panazzolo, D. Generalized Flow-Box property for singular foliations. RACSAM 113, 3949–3965 (2019). https://doi.org/10.1007/s13398-018-0596-7

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  • DOI: https://doi.org/10.1007/s13398-018-0596-7

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