arXiv subjects

Blandine Galiay

Publications and source records attributed to Blandine Galiay.

6 recordsLinked to original sources

arXiv2026

Metric properties of domains in real-type Nagano spaces

Nagano spaces are compact symmetric spaces that admit large transformation groups. They include for instance all the Grassmannians and the Einstein Universes. In this paper, we study a Kobayashi-type pseudometric on domains in real-type Nagano spaces. When the Nagano space is real projective space, this metric coincides with the classical Kobayashi pseudometric. For a dually convex domain of a general real-type Nagano space, we prove that this pseudometric is a genuine metric if and only if the domain does not contain a photon minus a point. We compute this metric on the proper symmetric domains and prove that it is obtained by integrating the $L^1$-norm along flats. We prove that in higher rank, the Kobayashi metric of a strongly $\mathcal{R}$-proper dually convex divisible domain is never Gromov hyperbolic. This contrasts with the rank-one case corresponding to real projective space, where a classical result of Benoist shows that this metric is Gromov hyperbolic if and only if the domain is strictly convex.

math.GR↗

arXiv2026

Completeness of closed Kleinian flat Pseudo-Riemannian Manifolds of Signature (2,2)

Let $\mathbb{R}^{2,2}$ denote the model space of flat pseudo-Riemannian manifolds of signature $(2,2)$. We prove that the only domain divisible by a discrete subgroup of the isometry group of $\mathbb{R}^{2,2}$ is $\mathbb{R}^{2,2}$ itself. In the Kleinian setting, this provides the first completeness theorem of closed flat pseudo-Riemannian manifolds beyond the Euclidean and Lorentzian cases. Along the proof, we show two results of independent interest. The first is a geometric reduction for certain divisible domains of affine space. The second concerns the existence of syndetic hulls in semidirect products $R \ltimes G$, where $G$ is a homothety Lie group. This construction generalizes earlier constructions in affine geometry due to Carrière and Dal'bo.

math.DG↗

arXiv2025

Proper Almost-Homogeneous Domains of the Einstein Universe

The Einstein universe $\mathbf{Ein}^{p,q}$ of signature $(p,q)$ is a pseudo-Riemannian analogue of the conformal sphere; it is the conformal compactification of the pseudo-Riemannian Minkowski space. For $p,q \geq 1$, we show that, up to a conformal transformation, there is only one almost-homogeneous domain in $\mathbf{Ein}^{p,q}$ that is bounded in a suitable stereographic projection. This domain, which we call a diamond, is a model for the symmetric space of $\operatorname{PO}(p,1) \times \operatorname{PO}(1,q)$. We deduce a classification of closed conformally flat manifolds with proper development.

math.DG↗

arXiv2025

Rigidity of proper almost-homogeneous domains in positive flag manifolds

We show that, inside the Shilov boundary of any given Hermitian symmetric space of tube type, there is, up to isomorphism, only one proper domain such that every point on its boundary belongs to the closure of an orbit under its automorphism group. This gives a classification of all closed proper manifolds locally modelled on such Shilov boundaries, and provides a positive answer, in the case of flag manifolds admitting a $Θ$-positive structure, to a rigidity question of Limbeek and Zimmer.

math.GR↗

arXiv2025

Transverse groups preserving proper domains in flag manifolds

Given a semisimple Lie group $G$ and a self-opposite flag manifold $\mathcal{F}$ of $G$, we establish a necessary condition for an infinite subgroup $H$ of $G$ to preserve a proper domain in $\mathcal{F}$. In the case where $G$ is a Hermitian Lie group of tube type, we introduce and study a notion of causal convexity in the Shilov boundary $\mathbf{Sb}(G)$ of the symmetric space of $G$, inspired by the one already existing in conformal Lorentzian geometry. We show that subgroups $H$ of $G$ that are transverse with respect to a parabolic subgroup of $G$ defining $\mathbf{Sb}(G)$ and that preserve a proper domain in $\mathbf{Sb}(G)$ satisfy a geometric property with respect to this causal convexity, close to the strong projective convex cocompactness defined by Danciger--Guéritaud--Kassel. This result highlights the spatial nature of the dynamics of $H$. We construct Zariski-dense examples of such transverse subgroups.

math.RT↗

arXiv2021

Lattices in the four-dimensional split oscillator group

Besides the oscillator group, there is another four-dimensional non-abelian solvable Lie group that admits a bi-invariant pseudo-Riemannian metric. It is called split oscillator group (sometimes also hyperbolic oscillator group or Boidol's group). We parametrise the set of lattices in this group and develop a method to classify these lattices up automorphisms of the ambient group. We show that their commensurability classes are in bijection with the set of real quadratic fields.

math.DG↗