### Article

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

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.

лоренцев орбифолд, лоренцева поверхность, группа изометрий, аносовский автоморфизм тора

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 $SL(n,\mathbb R).$

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.

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.

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.

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.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.