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## Compact leaves of structurally stable foliations

Proceedings of the Steklov Institute of Mathematics. 2012. Vol. 278. No. 1. P. 94-105.

We prove that any compact manifold whose fundamental group contains an abelian normal subgroup of positive rank can be represented as a leaf of a structurally stable suspended foliation on a compact manifold. In this case, the role of a transversal manifold can be played by an arbitrary manifold. We construct examples of structurally stable foliations that have a compact leaf with infinite solvable fundamental group which is not nilpotent. We also distinguish a class of structurally stable foliations each of whose leaves is compact and locally stable in sense of Ehresmann and Reeb.

N. I. Zhukova, Труды Математического института им. В.А. Стеклова РАН 2012 Т. 278 С. 102-113

We prove that any compact manifold whose fundamental group contains an abelian normal subgroup of positive rank can be represented as a leaf of a structurally stable suspended foliation on a compact manifold. In this case, the role of a transversal manifold can be played by an arbitrary manifold. We construct examples of structurally stable ...

Added: September 28, 2014

Zhukova N.I., K. I. Sheina, Basic automorphism groups of complete Cartan foliations covered by fibrations / Cornell University. Series math "arxiv.org". 2015. No. 1410.1144 .

We get sufficient conditions for the full basic automorphism group of a complete
Cartan foliation to admit a unique (finite-dimensional) Lie group structure in the category
of Cartan foliations. In particular, we obtain sufficient conditions for this group
to be discrete. Emphasize that the transverse Cartan geometry may be noneffective.
Some estimates of the dimension of this group depending ...

Added: November 10, 2014

Nina. I. Zhukova, Galaev A., Attractors of Cartan foliation / Cornell University. Series math "arxiv.org". 2017.

The paper is focused on the existence problem of attractors for foliations. Since the existence of an attractor is a transversal property of the foliation, it is natural to consider foliations admitting transversal geometric structures. As transversal structures are chosen Cartan geometries due to their universality. The existence problem of an attractor on a complete ...

Added: March 23, 2017

Zhukova N., Chebochko N., Известия высших учебных заведений. Математика 2020 № 11 С. 87-92

The aim of this work is to describe the structure of complete Lorentzian foliations $(M, F)$ of codimension two
on $n$-dimensional closed manifolds. It is proved that $(M, F)$ is either Riemannian or has a constant
transversal curvature and its structure is described. For such foliations $(M, F)$, the criterion is obtained,
reducing the chaos problem in $(M, ...

Added: October 6, 2020

N. I. Zhukova, Journal of Mathematical Sciences 2016 Vol. 219 No. 1 P. 112-124

We consider a Cartan foliation (M,F) of an arbitrary codimension q admitting an
Ehresmann connection such that all leaves of (M,F) are embedded submanifolds of M.
We prove that for any foliation (M,F) there exists an open, not necessarily connected,
saturated, and everywhere dense subset M0 of M and a manifold L0 such that the induced
foliation (M0, FM0) ...

Added: October 21, 2016

А.Ю. Долгоносова .., Н.И. Жукова, Труды Математического центра им. Н.И. Лобачевского 2013 Т. 47 С. 43-46

Different equivalent approaches to the notion of a foliation with transverse linear connection are
represented. ...

Added: October 18, 2014

Mangolte F., Kuyumzhiyan K., Journal of Pure and Applied Algebra 2012 Vol. 216 No. 10 P. 2106-2112

A group G acts infinitely transitively on a set Y if for every positive integer m, its action is m-transitive on Y. Given a real affine algebraic variety Y of dimension greater than or equal to 2, we show that, under a mild restriction, if the special automorphism group of Y (the group generated by ...

Added: September 10, 2012

L.I. Kuzmina, Osipov Y. V., Вестник Московского государственного строительного университета 2017 No. 11 P. 1278-1283

Subject: a groundwater filtration affects the strength and stability of underground and hydro-technical constructions. Research objectives: the study of one-dimensional problem of displacement of suspension by the flow of pure water in a porous medium. Materials and methods: when filtering a suspension some particles pass through the porous medium, and some of them are stuck ...

Added: January 6, 2018

Zhukova N., Труды Московского физико-технического института 2017 Т. 9 № 4 С. 132-141

Complete transversely affine foliations are studied. The strong transversal equivalence of
complete affine foliations is investigated, which is a more refined notion than the transverse
equivalence of foliations in the sense of Molino. A global holonomy group of a complete
affine foliations is determined and it is proved that this group is the complete invariant
of the foliation relatively ...

Added: November 28, 2017

N. I. Zhukova, Sbornik Mathematics 2012 Vol. 203 No. 2 P. 380-405

We prove that every complete foliation (M, F) of codimension q > 1 is either Riemannian or a (Conf (S^q), S^q)-foliation. We further prove that if (M, F) is not Riemannian, it has a global attractor which is either a nontrivial minimal set or a closed leaf or a union of two closed leaves. In ...

Added: October 19, 2014

Goncharuk N. B., Ilyashenko Y., Солодовников Н. А., Moscow Mathematical Journal 2019 Vol. 19 No. 4 P. 709-737

We classify global bifurcations in generic one-parameter local families of vector elds on S2 with a parabolic cycle. The classication is quite dierent from the classical results presented in monographs on the bifurcation theory. As a by product we prove that generic families described above are structurally stable. ...

Added: October 23, 2020

Ludmila I.Kuzmina, Osipov Y., Galaguz Y., International Journal for Computational Civil and Structural Engineering 2017 Vol. 13 No. 3 P. 70-76

The study of filtration as one of the problems of underground hydromechanics is necessary for the design and construction of tunnels, underground and hydraulic structures. Deep bed filtration of suspension in a porous medium with variable porosity and permeability and with an initial deposit is considered. An asymptotic solution to a model with small limit ...

Added: January 6, 2018

Н.И. Жукова, Шеина К. И., Труды Математического центра им. Н.И. Лобачевского 2014 Т. 50 С. 74-76

We investigate Cartan foliations covered by fibrations. We obtain a sufficient condition for the full
basic automorphism group of a complete Cartan foliation covered by fibration to admit a
unique (finite-dimensional) Lie group structure in the category of
Cartan foliations. The explicit new formula for determining its basic automorphism
Lie group is given. Examples of computing the full basic ...

Added: November 12, 2014

Nina I. Zhukova, Anna Yu. Dolgonosova .., Central European Journal of Mathematics 2013 Vol. 11 No. 12 P. 2076-2088

The category of foliations is considered. In this category
morphisms are differentiable mappings transforming leaves of one
foliation into leaves of the other foliation.
We proved that the automorphism group of the foliations
admitting a transverse linear connection is an infinite-dimensional
Lie group modeled on $LF$-spaces. This result extends the corresponding
result of Macias-Virgos and Sanmartin for Riemannian foliations.
In particular, our ...

Added: September 28, 2014

Zhukova N., Журнал Средневолжского математического общества 2018 Т. 20 № 4 С. 395-407

It is shown that the structural theory of Molino for Riemannian foliations on compact
manifolds and complete Riemannian manifolds is generalized to Riemannian foliations with
Ehresmann connection. There are no restrictions on the codimension of the foliation
and the dimension of the foliated manifold.
For a Riemannian foliation $(M, F)$ with Ehresmann connection
it is proved that the closure of ...

Added: December 27, 2019

Кузьмина Л.И., Осипов Ю. В., Вестник Московского государственного строительного университета 2016 № 2 С. 49-61

The problem of filtering a suspension of tiny solid particles in a porous medium is con-sidered. The suspension with constant concentration of suspended particles at the filter inlet moves through the empty filter at a constant speed. There are no particles before the front; behind the front of the fluid flow solid particles interact with ...

Added: February 26, 2016

Grines V., Левченко Ю. А., Доклады Российской академии наук. Математика, информатика, процессы управления (ранее - Доклады Академии Наук. Математика) 2012 Т. 447 № 2 С. 127-129

The paper is devoted to topological classifiication of cascades on 3-manifolds whose nonwandering set consists of surface 2-dimensional basic sets. ...

Added: February 25, 2015

Zhukova N., Математический сборник 2012 Т. 203 № 3 С. 79-106

Доказано, что любое полное конформное слоение (M,F) коразмерности q> 2 является либо римановым, либо (Conf(S^q),S^q)-слоением. Если (M,F) не является римановым слоением, то оно имеет глобальный аттрактор, представляющий собой либо нетривиальное минимальное множество, либо один замкнутый слой или объединение двух замкнутых слоев. При этом компактность многообразия M не предполагается. В частности, каждое собственное полное конформное не риманово ...

Added: September 28, 2014

Zhukova N., Journal of Mathematical Sciences 2015 Vol. 208 No. 1 P. 115-130

We study the problem of classification of complete non-Riemannian conformal foliations
of codimension q > 2 with respect to transverse equivalence. It is proved that two
such foliations are transversally equivalent if and only if their global holonomy groups
are conjugate in the group of conformal transformations of the q-dimensional sphere
Conf (Sq). Moreover, any countable essential subgroup of ...

Added: December 11, 2017

Zhukova N., Journal of Physics: Conference Series 2018 Vol. 990 No. 1 P. 1-15

A foliation that admits a Weyl structure arising from a pseudo-Riemannian metric of any signature as its transverse structure is called a pseudo-Riemannian Weyl foliation or (for short) a Weyl foliation. We investigate codimension q ≥ 2 Weyl foliations on (not necessarily compact) manifolds. Different interpretations of their holonomy groups are given. We prove a ...

Added: April 1, 2018

Н.И. Жукова, Mathematical Notes (Rusian Federation) 2013 Т. 93 № 6 С. 994-996

In this paper a unified method for studying foliations with transversal psrsbolic geometry of rank one is presented.
Ideas of Fraces' paper on parabolic geometry of rank one and of works of the author on conformal foliations
are developed. ...

Added: September 28, 2014

Zhukova N. I., Mathematical notes 2013 Vol. 93 No. 5-6 P. 928-931

In this paper a unified method for studying foliations with transversal parabolic geometry of rank one is presented.
Ideas of Fraces' paper on parabolic geometry of rank one and of works of the author on conformal foliations
are developed. ...

Added: October 19, 2014

Ilyashenko Y., Solodovnikov N., Goncharuk N. B., Ergodic Theory and Dynamical Systems 2018

We prove that generic one-parameter families of vector fields on S2 with a parabolic cycle (families of class PC) are structurally stable. The bifurcations in these families are classified. ...

Added: December 15, 2017