arXiv subjects
Publications and source records attributed to Aleksy Tralle.
The paper contributes to the problem of finding non-positive invariant Weyl connections on Lie groups. The main result is that in the case of completely solvable 3-dimensional Lie groups only the group SOL admits such connections, confirming the conjecture of Wojtkowski. We conjecture that our method can be extended to the general case of completely solvable Lie groups.
We solve the long standing problem of classification of standard compact Clifford-Klein forms of homogeneous spaces of simple non-compact real Lie groups under the extra assumption that $G$, $H$, $L$ are simple and absolutely simple. Then the result is that standard compact Clifford-Klein forms always arise from triples $(\mathfrak{g},\mathfrak{h},\mathfrak{l})$ of real Lie algebras such that $\mathfrak{h}\subset\mathfrak{g},\mathfrak{l}\subset\mathfrak{g}$, $\mathfrak{g}$ is simple and absolutely simple, $\mathfrak{h},\mathfrak{l}$ are (non-compact) reductive, $\mathfrak{g}=\mathfrak{h}+\mathfrak{l}$, and the intersection $\mathfrak{h}\cap\mathfrak{l}$ is compact. The consequence of this is the following characterization of proper co-compact actions of reductive Lie subgroups $L\subset G$ on a homogeneous spaces $G/H$ determined by absolutely simple real Lie group $G$ and a closed reductive subgroup $H$: $L$ acts on $G/H$ properly and co-compactly if and only if $G=H\cdot L$ and $H\cap L$ is compact.
We develop algorithms and computer programs which verify criteria of properness of discrete group actions on semisimple homogeneous spaces. We apply these algorithms to find new examples of non-virtually abelian discontinuous group actions on homogeneous spaces which do not admit proper SL(2,R)-actions.
In the seminal book of Boyer and Galicki "Sasakian Geometry" the authors formulated a research program of studying topological properties and answering questions about the existence of Sasakian structures. We survey recent progress in this topic.
We find relations between real root decompositions of triples of Lie algebras corresponding to standard compact Clifford-Klein forms, under the assumption that these triples are not Lie algebra decompositions in the sense of Onishchik. This enables us to find new classes of homogeneous spaces of simple real Lie groups which do not admit standard compact Clifford-Klein forms. In particular, we show that proper R-regular subalgebras of simple real Lie algebras never generate homogeneous spaces which admit compact standard Cliffrod-Klein forms.
We determine the structure of solvable Lie groups endowed with invariant stretched non-positive Weyl connections and find classes of solvable Lie groups admitting and not admitting such connections. In dimension 4 we fully classify solvable Lie groups and compact solvmanifolds which admit invariant SNP connections.
Let G be a simple non-compact linear connected Lie group and H be a closed non-compact semisimple subgroup. We are interested in finding classes of homogeneous spaces G/H admitting proper actions of discrete non virtually abelian subgroups of G. We develop an algorithm for finding such homogeneous spaces. As a testing example we obtain a list of all non-compact homogeneous spaces G/H admitting proper action of a discrete and non virtually abelian subgroup of G in the case when G has rank at most 8, and H is a maximal proper semisimple subgroup.
We ask a general question: what are locally homogeneous compact pseudo-Riemannian Einstein manifolds? We show that any standard compact Clifford-Klein form of a simple non-compact Lie group admits at least one Einstein metric. We conjecture that these are basically the only possible locally homogeneous Einstein pseudo-Riemannian compact manifolds using T. Kobayashi's conjecture as a guiding principle.
Kollár has found subtle obstructions to the existence of Sasakian structures on 5-dimensional manifolds. In the present article we develop methods of using these obstructions to distinguish K-contact manifolds from Sasakian ones. In particular, we find the first example of a closed 5-manifold M with $H_1(M,Z)=0$ which is K-contact but which carries no semi-regular Sasakian structures.
Smale-Barden manifolds $M$ are classified by their second homology $H_2(M,{\mathbb Z})$ and the Barden invariant $i(M)$. It is an important and dificult question to decide when $M$ admits a Sasakian structure in terms of these data. In this work we show methods of doing this. In particular we realize all $M$ with $H_2(M)={\mathbb Z}^k\oplus(\oplus_{i=1}^r{\mathbb Z}_{m_i}^{2g_i})$ and $i=0,\infty$, provided that $k\geq 1$, $m_i\geq 2$, $g_i\geq 1$, $m_i$ are pairwise coprime. Using our methods we also contribute to the problem of the existence of definite Sasakian structures on rational homology spheres. Also, we give a complete solution to the problem of the existence of Sasakian structures on rational homology spheres in the class of semi-regular Sasakian structures.
This article continues a line of research aimed at solving an important problem of T. Kobayashi of the existence of compact Clifford-Klein forms of reductive homogeneous spaces. We contribute to this topic by showing that almost all symmetric spaces and 3-symmetric spaces do not admit solvable compact CliffordfKlein forms (with several possible exceptions). Our basic tool is a combination of the Hirzebruch-Kobayashi-Ono proportionality principle with the theory of syndetic hulls. Using this, we prove a general theorem which yields a sufficient condition for the non-existence of compact solvable CliffordKlein forms.
In this article we prove that under certain assumptions, a reductive homogeneous space G/H does not admit a solvable compact Clifford-Klein form. This generalizes the well known non-existence theorem of Benoist for nilpotent Clifford-Klein forms. This generalization works for a particular class of homogeneous spaces determined by "very regular" embeddings of H into G.
We use a computer-aided approach to prove that there are no standard compact Clifford-Klein forms of homogeneous spaces of exceptional Lie groups. This yields further support for Kobayashi's conjecture about possible compact Clifford-Klein forms. On one hand, our approach is based on the algorithms developed in this work which eliminate the majority of possibilities. On the other hand, we complete the proof using the algorithmic methods of classifying semisimple subalgebras in simple real Lie algebras developed by Faccin and de Graaf, as well as by calculating invariants like a-hyperbolic rank.
In this paper we develop algorithms of finding homogeneous spaces of semisimple non-compact Lie groups which do not admit compact Clifford-Klein forms. We propose a computer program which checks if the given homogeneous space has a non-vanishing cohomological obstruction (found by Tholozan) to compact Clifford-Klein forms. By a numerical experiment we show that there is a large class of homogeneous spaces satisfying Tholozan's condition.
We prove that 3-symmetric spaces of simple linear real Lie groups do not admit amenable compact Clifford-Klein forms. Our basic tool are totally non cohomologous to zero fibrations.
We give a comprehensive account of Chern's Theorem that S^6 admits no omega-compatible almost complex structures. No claim to originality is being made, as the paper is mostly an expanded version of material already in the literature. This article extends the talks that both authors gave in Marburg during the conference "(Non)existence of complex structures on S^6" in April 2017.
We prove the formality and the evenness of odd-degree Betti numbers for compact Kähler orbifolds, by adapting the classical proofs for Kähler manifolds. As a consequence, we obtain examples of symplectic orbifolds not admitting any Kähler orbifold structure. We also review the known examples of non-formal simply connected Sasakian manifolds, and produce an example of a non-formal quasi-regular Sasakian manifold with Betti numbers $b_1=0$ and $b_2\,> 1$.
Let $M$ be a closed K-contact $(2n+1)$-manifold equipped with a quasi-regular K-contact structure. Rukimbira proved that the Reeb vector field $ξ$ of this structure has at least $n+1$ closed characteristics. We note that $ξ$ has at least $2n+1$ closed characteristics provided that the space of leaves of the foliation determined by $ξ$ is symplectically aspherical.