Clifford-Klein Forms Research Articles

Four-dimensional compact Clifford–Klein forms of pseudo-Riemannian symmetric spaces with signature (2, 2)

We give a classification of reducible and indecomposable pseudo-Riemannian symmetric spaces with signature [Formula: see text] which admit compact Clifford–Klein forms. For this, we develop a method that applies to particular 1-connected solvable symmetric spaces. We also examine a “solvable analogue” of Kobayashi’s conjecture for reductive type and find an evidence that the assumption “reductive type” in the conjecture is crucial.

A criterion of proper action on some compact extensions of ℝn and applications

Let K be a compact subgroup of [Formula: see text] and [Formula: see text] the semi-direct product group. Let H be a closed subgroup of G and [Formula: see text] a discontinuous group for the homogeneous space [Formula: see text]. We establish a geometrical criterion of the proper action of [Formula: see text] on [Formula: see text], which requires an accurate description of the structure of closed connected subgroups of Euclidean motion groups. As a consequence, we establish a criterion for the existence of a compact Clifford–Klein form [Formula: see text] and study the Calabi–Markus phenomenon.

Cartan Projections of Some Non-reductive Subgroups and Proper Actions on Homogeneous Spaces

Kobayashi (Duke Math. J. 67:653–664, 1992) gave a necessary condition for the existence of compact Clifford–Klein forms in terms of Cartan projections and non-compact dimensions of reductive subgroups. We extend his method to non-reductive subgroups and give some examples of homogeneous spaces of reductive type that do not admit compact Clifford–Klein forms by comparing Cartan projections and non-compact dimensions of reductive subgroups with those of non-reductive subgroups.

Obstructions to the existence of compact Clifford–Klein forms for tangential symmetric spaces

For a homogeneous space [Formula: see text] of reductive type, we consider the tangential homogeneous space [Formula: see text]. In this paper, we give obstructions to the existence of compact Clifford–Klein forms for such tangential symmetric spaces and obtain new tangential symmetric spaces which do not admit compact Clifford–Klein forms. As a result, in the class of irreducible classical semisimple symmetric spaces, we have only two types of symmetric spaces which are not proved not to admit compact Clifford–Klein forms. The existence problem of compact Clifford–Klein forms for homogeneous spaces of reductive type, which was initiated by Kobayashi in 1980s, has been studied by various methods but is not completely solved yet. On the other hand, the one for tangential homogeneous spaces has been studied since 2000s and an analogous criterion was proved by Kobayashi and Yoshino. In concrete examples, further works are needed to verify Kobayashi–Yoshino’s condition by direct calculations. In this paper, some easy-to-check necessary conditions ([Formula: see text][Formula: see text]obstructions) for the existence of compact quotients in the tangential setting are given, and they are applied to the case of symmetric spaces. The conditions are related to various fields of mathematics such as associated pair of symmetric space, Calabi–Markus phenomenon, trivializability of vector bundle (parallelizability, Pontrjagin class), Hurwitz–Radon number and Pfister’s theorem (the existence problem of common zero points of polynomials of odd degree).

A splitting of local rigidity of Clifford–Klein forms of homogeneous spaces of completely solvable Lie groups

In this paper, we discuss local rigidity of Clifford–Klein forms of homogeneous spaces of 1-connected completely solvable Lie groups. We split the property of local rigidity into two conditions: vertical rigidity and horizontal rigidity. By this separation, we discuss local rigidity, in particular, Baklouti’s conjecture.

On locally homogeneous pseudo-Riemannian compact Einstein manifolds

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 Riemannian compact manifolds using T. Kobayashi’s conjecture as a guiding principle.

Erratum to “On solvable compact Clifford-Klein forms”

Erratum to “On solvable compact Clifford-Klein forms”

On the Hirzebruch–Kobayashi–Ono proportionality principle and the non-existence of compact solvable Clifford–Klein forms of certain homogeneous spaces

Abstract 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 Clifford–Klein 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 Clifford–Klein forms.

Nonexistence of standard compact Clifford–Klein forms of homogeneous spaces of exceptional Lie groups

Nonexistence of standard compact Clifford–Klein forms of homogeneous spaces of exceptional Lie groups

Proof of Kobayashi's rank conjecture on Clifford–Klein forms

T. Kobayashi conjectured in the 36th Geometry Symposium in Japan (1989) that a homogeneous space G/H of reductive type does not admit a compact Clifford-Klein form if rank G - rank K < rank H - rank K_H. We solve this conjecture affirmatively. We apply a cohomological obstruction to the existence of compact Clifford-Klein forms proved previously by the author, and use the Sullivan model for a reductive pair due to Cartan-Chevalley-Koszul-Weil.

A local rigidity theorem for finite actions on Lie groups and application to compact extensions of Rn

Let G be a Lie group, and let Γ be a finite group. We show in this article that the space Hom(Γ,G)/G is discrete and—in addition—finite if G has finitely many connected components. This means that in the case in which Γ is a discontinuous group for the homogeneous space G/H, where H is a closed subgroup of G, all the elements of Kobayashi’s parameter space are locally rigid. Equivalently, any Clifford–Klein form of finite fundamental group does not admit nontrivial continuous deformations. As an application, we provide a criterion of local rigidity in the context of compact extensions of Rn.

Deforming discontinuous groups for Heisenberg motion groups

We study in this paper the local rigidity proprieties of deformation parameters of the natural action of a discontinuous group [Formula: see text] acting on a homogeneous space [Formula: see text], where [Formula: see text] stands for a closed subgroup of the Heisenberg motion group [Formula: see text]. That is, the parameter space admits a locally rigid (equivalently a strongly locally rigid) point if and only if [Formula: see text] is finite. Moreover, Calabi–Markus’s phenomenon and the question of existence of compact Clifford–Klein forms are also studied.

Classification of irreducible symmetric spaces which admit standard compact Clifford–Klein forms

We give a classification of irreducible symmetric spaces which admit standard compact Clifford–Klein forms. The method uses representation theory over the real number field and the criterion for properness and cocompactness of the action on homogeneous spaces due to T. Kobayashi.

Semisimple subalgebras in simple Lie algebras and a computational approach to the compact Clifford–Klein forms problem

In this article, we develop the algorithm for 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.

Compactification of certain Clifford-Klein forms of reductive homogeneous spaces

We describe smooth compactifications of certain families of reductive homogeneous spaces such as group manifolds for classical Lie groups, or pseudo-Riemannian analogues of real hyperbolic spaces and their complex and quaternionic counterparts. We deduce compactifications of Clifford-Klein forms of these homogeneous spaces, namely quotients by discrete groups Gamma acting properly discontinuously, in the case that Gamma is word hyperbolic and acts via an Anosov representation. In particular, these Clifford-Klein forms are topologically tame.

A cohomological obstruction to the existence of compact Clifford–Klein forms

In this paper, we continue the study of the existence problem of compact Clifford-Klein forms from a cohomological point of view, which was initiated by Kobayashi-Ono and extended by Benoist-Labourie and the author. We give an obstruction to the existence of compact Clifford-Klein forms by relating a natural homomorphism from relative Lie algebra cohomology to de Rham cohomology with an upper-bound estimate for cohomological dimensions of discontinuous groups. From this obstruction, we derive some examples, e.g. $\mathrm{SO}_0(p+r, q)/(\mathrm{SO}_0(p,q) \times \mathrm{SO}(r))$ $(p,q,r \geq 1, \ q:\text{odd})$ and $\mathrm{SL}(p+q, \mathbb{C})/\mathrm{SU}(p,q)$ $(p,q \geq 1)$, of a homogeneous space that does not admit a compact Clifford-Klein form. To construct these examples, we apply H. Cartan's theorem on relative Lie algebra cohomology of reductive pairs and the theory of $\epsilon$-families of semisimple symmetric pairs.

On solvable compact Clifford-Klein forms

In this article we prove that under certain assumptions, a reductive homogeneous space G / H 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 H into G G .

On the Calabi–Markus phenomenon and a rigidity theorem for Euclidean motion groups

In this article, we study the rigidity properties of deformation parameters of the natural action of a discontinuous subgroup Γ⊂G, on a homogeneous space G/H, where H stands for a closed subgroup of a Euclidean motion group G:=On(R)⋉Rn. That is, we prove the following local (and global) rigidity theorem: the parameter space admits a rigid (equivalently a locally rigid) point if and only if Γ is finite. Remarkably, it turns out that H is compact whenever Γ is infinite, which makes accessible the study of the corresponding parameter and deformation spaces and their topological and local geometrical features. This shows that the Calabi–Markus phenomenon occurs in this setting. That is, if H is a closed noncompact subgroup of G, then G/H does not admit a compact Clifford–Klein form, unless G/H itself is compact. We also answer a question posed by T. Kobayashi. That is, no homogeneous space G/H admits a noncommutative free group as a discontinuous group.

A topological necessary condition for the existence of compace Clifford-Klein forms

We provide a necessary condition for the existence of a compact Clifford-Klein form of a given homogeneous space of reductive type. The key to the proof is to combine a result of Kobayashi-Ono with an elementary fact that certain two different Clifford-Klein forms have the same cohomology ring. We give some examples, SL(p+q, R)/SO(p, q) (p, q: odd) for instance, of homogeneous spaces which do not admit compact Clifford-Klein forms.

A restriction on proper group actions on homogeneous spaces of reductive type

Let $$L$$ be a reductive subgroup of a reductive Lie group $$G$$ . Let $$G/H$$ be a homogeneous space of reductive type. We provide a necessary condition for the properness of the action of $$L$$ on $$G/H$$ . As an application we give examples of spaces that do not admit standard compact Clifford-Klein forms.