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Характеристика Эйлера-Сатаки компактных аффинных орбифолдов
С. 9–16.
Zhukova N., Багаев А. В.
According to Chern's conjecture, the Euler characteristic of a closed affine manifold
must be zero. We prove the equivalence of this Chern conjecture to the following conjecture
for orbifolds: the Euler--Sataki characteristic of a compact affine orbifold is zero.
We found the conditions under which the Euler--Sataki characteristic of a compact affine orbifold
vanishes. Examples are constructed.
In book
Каз.: Издательство Казанского университета, 2019.
Bagaev A., Zhukova N., Уфимский математический журнал 2025 Т. 17 № 4 С. 11–25
between orbifolds of same dimension. The definition of degree for the mentioned maps was introduced by Pasquoto and Rot (2020). We propose a new, simpler definition for the degree of proper maps between smooth oriented orbifolds of the same dimension and show that it is equivalent to the definition by Pasquotto and Rot. Using this ...
Added: January 24, 2025
Bogolepova E., Zhukova N., Известия высших учебных заведений. Поволжский регион. Физико-математические науки 2019 № 1 С. 14–28
Actuality and goals. Lorentzian geometry finds widespread application in physics and is radically different from Riemannian geometry. As it is known an every smooth orbifold admits a Riemannian metric. The existence of a Lorentzian metric on an orbifold imposes restrictions on its structure. The isometry group of a Lorentzian orbifold is called inessential if it acts properly, otherwise the ...
Added: December 1, 2022
Bogolepova E., Zhukova N., Lobachevskii Journal of Mathematics 2021 Vol. 42 No. 14 P. 3324–3335
We prove a criterion of an Anosov action of the isometry group for compact Lorentzian 2-orbifolds. It is proved also that a non-compact complete flat Lorentzian 2-orbifold has an Anosov isometry if and only if its isometry group Lie acts improperly. The existence of chaotic behavior of such Anosov actions is investigated. It is shown ...
Added: February 3, 2022
Gusein-Zade S., Алгебра и анализ 2021 Т. 33 № 3 С. 73–84
Indices of singular points of a vector field or of a 1-form on a smooth manifold are closely related with the Euler characteristic through the classical Poincar\'e--Hopf theorem. Generalized Euler characteristics (additive topological invariants of spaces with some additional structures) are sometimes related with corresponding analogues of indices of singular points. Earlier, there was defined ...
Added: May 2, 2021
Gusein-Zade S., Manuscripta Mathematica 2018 Vol. 155 No. 3-4 P. 335–353
For a germ of a quasihomogeneous function with an isolated critical point at the origin invariant with respect to an appropriate action of a finite abelian group, H. Fan, T. Jarvis, and Y. Ruan defined the so-called quantum cohomology group. It is considered as the main object of the quantum singularity theory (FJRW-theory). We define ...
Added: October 27, 2020
Gusein-Zade S., Функциональный анализ и его приложения 2018 Т. 52 № 4 С. 72–85
The Euler characteristic is the only additive topological invariant for spaces of certain sort, in particular, for manifolds with certain finiteness properties. A generalization of the notion of a manifold is the notion of a V-manifold. We discuss a universal additive topological invariant of V-manifolds, the universal Euler characteristic. It takes values in the ring ...
Added: October 27, 2020
Bagaev A. V., Zhukova N., Journal of Geometry and Physics 2019 Vol. 142 P. 80–91
S.S. Chern conjectured that the Euler characteristic of every closed affine
manifold has to vanish. We present an analog of this conjecture stating that
the Euler-Satake characteristic of any compact affine orbifold is equal to zero.
We prove that Chern's conjecture is equivalent to its analog for
the Euler-Satake characteristic of compact affine orbifolds, and
orbifolds may be ineffective. This ...
Added: April 26, 2019
Боголепова Е. В., Zhukova N., Известия высших учебных заведений. Поволжский регион. Физико-математические науки 2019 № 1 С. 14–28
Using the bundle of pseudo-orthogonal frames some canonical covering map for two-dimensional Lorentzian orbifolds is constructed and applied. The existence of such map shows that any two-dimensional Lorentzian orbifold is very good.
It is proved that there are only two (up to isomorphisms in the category of orbifolds) two-dimensional smooth noncompact orbifolds admitting complete flat Lorentzian ...
Added: April 10, 2019
Omelchenko A., Краско Е. С., Discrete Mathematics 2019 Vol. 342 No. 2 P. 600–614
The second part of the paper is devoted to enumeration of r-regular maps on the torus up to all its homeomorphisms (unsensed maps). We describe in detail the periodic orientation reversing homeomorphisms of the torus which turn out to be representable as glide reflections. We show that considering quotients of the torus with respect to ...
Added: September 21, 2018
Omelchenko A., Краско Е. С., Discrete Mathematics 2019 Vol. 342 No. 2 P. 584–599
The work that consists of two parts is devoted to the problem of enumerating unrooted r-regular maps on the torus up to all its symmetries. We begin with enumerating near-r- regular rooted maps on the torus, the projective plane and the Klein bottle, as well as some special kinds of maps on the sphere: near-r-regular ...
Added: September 21, 2018
Zhukova N., Ufa Mathematical Journal 2018 Vol. 10 No. 2 P. 44–57
We study the groups of conformal transformations of 𝑛-dimensional pseudoRiemannian orbifolds (𝒩, 𝑔) as 𝑛 > 3. We extend the Alekseevskii method for studying
conformal transformation groups of Riemannian manifolds to pseudo-Riemannian orbifolds.
We show that a conformal pseudo-Riemannian geometry is induced on each stratum of
such orbifold. Due to this, for 𝑘 ∈ {0, 1}∪{3, . . ...
Added: June 21, 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
Zhukova N., В кн.: Международная молодежная школа-семинар "Современная геометрия и ее приложения". Международная конференция "Современная геометрия и ее приложения". Материалы школы-семинара и конференции.: Каз.: Издательство Казанского университета, 2017. 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. ...
Added: April 1, 2018
Zhukova N., Уфимский математический журнал 2018 Т. 10 № 2 С. 43–56
The groups of conformal transformations of $n$-dimensional
pseudo-Riemannian orbifolds $({\mathcal N},g)$ are investigated for $n\geq 3$.
The Alekseevskii method of the investigation of the conformal transformation groups of
Riemannian manifolds is extended by us to psevdo-Riemannian orbifolds. It is shown that
a conformal pseudo-Riemannian geometry is induced on each stratum of that orbifold. Due to this,
for $k\in\{0,1\}\cup\{3,...,n-1\}$ exact estimates ...
Added: March 19, 2018