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Zh. Mat. Fiz. Anal. Geom., 2013, Volume 9, Number 3, Pages 400–420 (Mi jmag572)  

This article is cited in 5 scientific papers (total in 5 papers)

Local and Global Stability of Compact Leaves and Foliations

N. I. Zhukova

Department of Mechanics and Mathematics Nizhny Novgorod State University, Nizhny Novgorod, Russia

Abstract: The equivalence of the local stability of a compact foliation to the completeness and the quasi analyticity of its pseudogroup is proved. It is also proved that a compact foliation is locally stable if and only if it has the Ehresmann connection and the quasianalytic holonomy pseudogroup. Applications of these criterions are considered. In particular, the local stability of the complete foliations with transverse rigid geometric structures including the Cartan foliations is shown. Without assumption of the existence of an Ehresmann connection, the theorems on the stability of the compact leaves of conformal foliations are proved. Our results agree with the results of other authors.

Key words and phrases: foliation, compact foliation, Ehresmann connection for a foliation, holonomy pseudogroup, local stability of leaves.

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Bibliographic databases:
MSC: 57R30, 53D22
Received: 17.01.2012
Revised: 18.03.2013
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Citation: N. I. Zhukova, “Local and Global Stability of Compact Leaves and Foliations”, Zh. Mat. Fiz. Anal. Geom., 9:3 (2013), 400–420

Citation in format AMSBIB
\Bibitem{Zhu13}
\by N.~I.~Zhukova
\paper Local and Global Stability of Compact Leaves and Foliations
\jour Zh. Mat. Fiz. Anal. Geom.
\yr 2013
\vol 9
\issue 3
\pages 400--420
\mathnet{http://mi.mathnet.ru/jmag572}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3155148}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000322697400008}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. M. Ciska-Niedziaomska, “On the extremal function of the modulus of a foliation”, Arch. Math., 107:1 (2016), 89–100  crossref  mathscinet  isi  scopus
    2. H. I. Zhukova, K. I. Sheina, “Kriterii psevdorimanovosti sloeniya s transversalnoi lineinoi svyaznostyu”, Zhurnal SVMO, 18:2 (2016), 30–40  mathnet  elib
    3. N. I. Zhukova, “Influence of stratification on the groups of conformal transformations of pseudo-Riemannian orbifolds”, Ufa Math. J., 10:2 (2018), 44–57  mathnet  crossref  mathscinet  isi  scopus
    4. A. Yu. Dolgonosova, N. I. Zhukova, “Pseudo-Riemannian foliations and their graphs”, Lobachevskii J. Math., 39:1, SI (2018), 54–64  crossref  mathscinet  zmath  isi  scopus
    5. I N. Zhukova, “Automorphism groups of elliptic $G$-structures on orbifolds”, J. Geom. Phys., 132 (2018), 146–154  crossref  mathscinet  zmath  isi  scopus
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