Характеристика Эйлера-Сатаки компактных аффинных орбифолдов

Жукова Нина Ивановна; Багаев А. В.

Book chapter

Характеристика Эйлера-Сатаки компактных аффинных орбифолдов

С. 9-16.

Жукова Н. И., 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.

Language: Russian

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Keywords: orbifold affine orbifold Euler–Satake characteristic

Publication based on the results of: Numerical and topological methods for research of dynamical systems, approximation of evolution equation solutions and algebraic geometrical structures(2019)

In book

Международная конференция «Современная геометрия и её приложения - 2019»: сборник трудов

Каз.: Издательство Казанского университета, 2019.

Rigid geometries and their automorphism groups on leaf spaces of foliations

Nina I. Zhukova. math. arxive. Cornell University, 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 do not satisfies the separation axiom T_0. It is shown that for every object N of F_0 a rigid geometry on N admits a desingularization. Moreover, for every such N we prove the existence and the uniqueness of a finite-dimensional Lie group structure on the group of all automorphisms of the rigid geometry on N.

Added: Apr 14, 2017

Classification of compact Lorentzian 2-orbifolds with noncompact full isometry groups

N. I. Zhukova, E. A. Rogozhina .. Siberian Mathematical Journal. 2012. Vol. 53. No. 6. P. 1037-1050.

Among closed Lorentzian surfaces, only flat tori admit non-compact full isometry groups. Moreover, for every n > 2 the standard n-dimensional flat torus equipped with canonical metric has a non-compact full isometry Lie group. We show that this fails for n= 2 and classify the flat Lorentzian metrics on the torus with a non-compact full isometry Lie group. We also prove that every two dimensional Lorentzian orbifold is very good. This implies the existence of a unique smooth compact 2-orbifold, called the pillow, admitting Lorentzian metrics with a non-compact full isometry Lie group. We classify the metrics of this type and construct some examples.

Added: Oct 19, 2014

An analog of Chern's conjecture for the Euler-Satake characteristic of affine orbifolds

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 fact allowed us to extend to orbifolds the known results of B.~Klingler and also results of B.~Kostant and D.~Sullivan on sufficient conditions to fulfill Chern's conjecture. Thus we prove that if an $n$-dimensional compact affine orbifold $\mathcal N$ is complete or if its holonomy group belongs to the special linear group $SL(n,\mathbb R),$ then the Euler-Satake characteristic of $\mathcal N$ has to vanish. An application to pseudo-Riemannian orbifolds is considered. Examples of orbifolds belonging to the investigated class are given. In particular, we construct an example of a compact incomplete affine orbifold with the vanishing Euler characteristic, the holonomy group of which does not belong to $SL(n,\mathbb R).$

Added: Apr 26, 2019

Geometric structures on orbifolds and their automorphisms

Zhukova N. In bk.: 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 manifold M for a compact orbifold. If an orbifold admits a G-structure, we construct and use a foliated G-bundle for the compact associated foliation. Using our method we prove the following statement.

Theorem 1. On a compact orbifold N the group of all automorphisms of an elliptic G-structure is a Lie group, this group is equipped with the compact-open topology, and its Lie group structure is defined uniquely.

By the analogy with manifolds we define the notion of an almost complex structure on orbifolds and get the following statement.

Theorem 2. The automorphism group of an almost complex structure on a compact orbifold is a Lie group, its topology is compact-open and its Lie group structure is defined uniquely.

For manifolds, the statements of Theorems 1 – 2 are classical results. Theorem 1 for manifolds was proved by Ochiai. In particular, in the case of flat elliptic G-structures on manifolds, Theorem 1 was proved by Guillemin and Sternberg and also by Ruh. Theorem 2 for manifolds was proved by Boothby, Kobayashi, Wang.

Added: Apr 14, 2018

Слоеные модели для гладких орбифолдов и их применение

Жукова Н. И. 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: Feb 20, 2018

Жесткие геометрии на пространстве слоев слоений и группы их автоморфизмов

Жукова Н. И. 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. 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.

Added: Apr 1, 2018