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Жесткие геометрии на пространстве слоев слоений и группы их автоморфизмов
С. 48-51.
We introduce a category of rigid geometries on smooth singular spaces of leaves of foliations.
A special category $\mathfrak F_0$ containing orbifolds is allocated. Unlike orbifolds, objects
of $\mathfrak F_0$ can have non-Hausdorff topology and can even not satisfy the separability axiom $T_0$.
It is shown that the rigid geometry $(N,\zeta)$, where $N\in (\mathfrak F_0)$, allows a desingularization. For each such geometry $( N,\zeta)$ we prove the existence and uniqueness of the structure of a finite-dimensional Lie group in the group of all automorphisms $Aut (N},\zeta)$.
The applications to the orbifolds are considered.
Keywords: foliationслоениегруппа автоморфизмовorbifoldautomorphism groupleaf spaceleaf manifoldrigid geometryпространство слоевмногообразие слоевжесткая геометрия орбифолд
Publication based on the results of:
In book
Каз. : Издательство Казанского университета, 2017
Nina I. Zhukova, / Cornell University. Series arXiv "math". 2018. No. 1704.04220.
We introduce a category of rigid geometries on singular spaces which are leaf spaces of foliations and are considered as leaf manifold. We separate out a special category F_0 of leaf manifolds containing the orbifold category as a complete subcategory. Objects of F_0 may be non-Hausdorff unlike orbifolds. The topology of some objects of F_0 ...
Added: April 14, 2017
Zhukova N., Moscow Mathematical Journal 2018
We introduce a category of rigid geometries on singular spaces which
are leaf spaces of foliations and are considered as leaf manifolds. We
single out a special category F_0 of leaf manifolds containing the orbifold
category as a full subcategory. Objects of F_0 may have non-Hausdorff
topology unlike the orbifolds. The topology of some objects of F_0 does
not satisfy ...
Added: April 2, 2018
Zhukova N., , in : The Conference NOMA-2017. Book of Abstracts. : Nizhny Novgorod : Nizhny Novgorod State University, 2017. P. 67-68.
We present a new method of investigation of G-structures on orbifolds. This method is founded on the consideration of a G-structure on an n-dimensional orbifold as the corresponding transversal structure of an associated foliation. For a given orbifold, there are different associated foliations. We construct and apply a compact associated foliation (M,F) on a compact ...
Added: April 14, 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
Zhukova N., Журнал Средневолжского математического общества 2017 Т. 19 № 4 С. 33-44
For any smooth orbifold $\mathcal N$ is constructed a foliated model, which is a foliation
with an Ehresmann, the leaf space of which is the same as $\mathcal N$. We investigate
the relationship relationship between some properties of orbifold and its foliated model.
The article discusses the application to Cartan orbifolds, that is orbifolds endowed with Cartan geometry. ...
Added: February 20, 2018
Zhukova N.I., K. I. Sheina, / 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
Perepechko A., Функциональный анализ и его приложения 2013 Т. 47 № 4 С. 45-52
We prove that the action of the special automorphism group on affine cones over del Pezzo surfaces of degree 4 and 5 is infinitely transitive. ...
Added: September 26, 2019
Kochetkov Y., Фундаментальная и прикладная математика 2007 Т. 13 № 6 С. 197-205
The geometry of foliations generated by some differentials of Abelian type is considered. The case where all fibers are closed is studied. ...
Added: May 15, 2012
Khoroshkin A., Transformation Groups 2015 P. 1-40
We prove the conjecture by Feigin, Fuchs and Gelfand describing the Lie algebra cohomology of formal vector fields on an n-dimensional space with coefficients in symmetric powers of the coadjoint representation. We also compute the cohomology of the Lie algebra of formal vector fields that preserve a given flag at the origin. The latter encodes ...
Added: April 9, 2015
Perepechko A., Математические заметки 2021 Т. 110 № 5 С. 744-750
Affine algebraic surfaces of Markov type of the form
x^2 + y^2 + z^2 − xyz = c
are studied. Their automorphism groups are found. ...
Added: October 12, 2021
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, , in : Progress in Analysis. Proceedings of the 8th congress of the International Society for Analysis, its Applications, and Computation (ISAAC), Moscow, Russia, August 22--27, 2011. Vol. 2.: M. : RUDN, 2012. P. 238-247.
We investigated conformal foliations $(M,F)$ of
codimension $q\geq 3$ and proved a criterion for them to be
Riemannian. In particular, the application of this criterion allowed
us to proof the existence of an attractor that is a minimal set for
each non-Riemannian conformal foliation. Moreover, if foliated
manifold is compact then non-Riemannian conformal foliation $(M,F)$
is $(Conf(S^q),S^q)$-foliation with finitely many minimal ...
Added: October 14, 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
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
Arzhantsev I., Perepechko A., / Bulletin des sciences mathématiques. Series 22-00305 "BULSCI-D". 2023.
We consider complete toric varieties X such that a maximal unipotent subgroup U of the automorphism group Aut(X) acts on X with an open orbit. It turns out that such varieties can be characterized by several remarkable properties. We study the set of Demazure roots of the corresponding complete fan, describe the structure of a maximal unipotent subgroup U in Aut(X), and find all regular subgroups ...
Added: October 6, 2023
Kuyumzhiyan K., Proceedings of the American Mathematical Society 2020 No. 148 P. 3723-3731
We prove the conjecture of Berest-Eshmatov-Eshmatov by showing that the group of automorphisms of a product of Calogero-Moser spaces C_n_i, where the n_i are pairwise distinct, acts m-transitively for each m. ...
Added: August 18, 2020
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
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
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
Arzhantsev I., Zaitseva Y., Research in Mathematical Sciences 2024 Vol. 11 No. 2 Article 27
An algebraic variety X is called a homogeneous variety if the automorphism group Aut(X) acts on X transitively, and a homogeneous space if there exists a transitive action of an algebraic group on X. We prove a criterion of smoothness of a suspension to construct a wide class of homogeneous varieties. As an application, we ...
Added: March 23, 2024
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., 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
Ivan V. Arzhantsev, Yulia I. Zaitseva, Kirill V. Shakhmatov, Proceedings of the Steklov Institute of Mathematics 2022 Vol. 318 No. 1 P. 13-25
Let X be an algebraic variety such that the group Aut(X) acts on X transitively. We define the transitivity degree of X as the maximum number m such that the action of Aut(X) on X is m-transitive. If the action of Aut(X) is m-transitive for all m, the transitivity degree is infinite. We compute the transitivity degree for all quasi-affine toric varieties and for many homogeneous spaces of algebraic groups. We also discuss a conjecture and ...
Added: November 5, 2022
Glutsyuk A., Ergodic Theory and Dynamical Systems 2023
Reflection in a strictly convex bounded planar billiard acts on the space of oriented lines and preserves a standard area form. A caustic is a curve C whose tangent lines are reflected by the billiard to lines tangent to C. The famous Birkhoff conjecture states that the only strictly convex billiards with a foliation by ...
Added: December 29, 2023