arXiv research

This is Part II of a series on noncompact isometry groups of Lorentz manifolds. We have introduced in Part I, a compactification of these isometry groups, and called ``bi-polarized'' those Lorentz manifolds having a ``trivial '' compactification. Here we show a geometric rigidity of non-bi-polarized Lorentz manifolds; …

The three-dimensional Heisenberg group $H_3$ has three left-invariant Lorentz metrics $g_1$, $g_2$ and $g_3$. They are not isometric each other. In this paper, we characterize the left-invariant Lorentzian metric $g_1$ as a Lorentz Ricci soliton. This Ricci soliton $g_1$ is a shrinking non-gradient Ricci soliton. Likew…

We classify isometries of compact Lorentz manifolds.

The paper examines Lorentz Ricci solitons on specific Lie groups.

Study of Gödel Universe as Lie group with specific metric.

This is the first part of a series on non-compact groups acting isometrically on compact Lorentz manifolds. This subject was recently investigated by many authors. In the present part we investigate the dynamics of affine, and especially Lorentz transformations. In particular we show how this is related to geodesic fol…

Study models of Gödel Universe using Lie groups and Iwasawa decomposition.

We show a geometric rigidity of isometric actions of non compact (semisimple) Lie groups on Lorentz manifolds. Namely, we show that the manifold has a warped product structure of a Lorentz manifold with constant curvature by a Riemannian manifold.

We give an explicit classification of translation-invariant, Lorentz-invariant continuous valuations on convex sets. We also classify the Lorentz-invariant even generalized valuations.

We study 3-dimensional non-Riemannian Lorentz geometries, i.e. compact locally homogeneous Lorentz 3-manifolds with non-compact (local) isotropy group. One result is that, up to a finite cover, all such manifolds admit Lorentz metrics of (non-positive) constant sectionnal curvature. If the geometry is maximal, then the…

We give a classification, up to local isomorphisms, of semi-simple Lie groups without compact factors that can act faithfully and conformally on a compact Lorentz manifold of dimension greater than or equal to $3$.

Here, we classify Lie groups acting isometrically on compact Lorentz manifolds, and in particular we describe the geometric structure of compact homogeneous Lorentz manifolds.

Equivariant neural network simplifies particle physics models.

In this paper, we give a complete topological, as well as geometrical classification of closed 3-dimensional Lorentz manifolds admitting a noncompact isometry group.

LLoCa makes any network Lorentz-equivariant, achieving high accuracy and efficiency.

We define naturally Hermite-Lorentz metrics on almost-complex manifolds as special case of pseudo-Riemannian metrics compatible with the almost complex structure. We study their isometry groups.

We prove that for any closed Lorentz $4$-manifold $(M,g)$ the isometry group $Isom(M,g)$ is Jordan. Namely, there exists a constant $C$ (depending on $M$ and $g$) such that any finite subgroup $Γ\leq Isom(M,g)$ has an abelian subgroup $A\leqΓ$ satisfying $[Γ:A]\leq C$.

We prove results toward classifying compact Lorentz manifolds on which Heisenberg groups act isometrically. We give a general construction, leading to a new example, of codimension-one actions--those for which the dimension of the Heisenberg group is one less than the dimension of the manifold. The main result is a cla…

While conformal transformations of the plane preserve Laplace's equation, Lorentz-conformal mappings preserve the wave equation. We discover how simple geometric objects, such as quadrilaterals and pairs of crossing curves, are transformed under nonlinear Lorentz-conformal mappings. Squares are transformed into curvili…

This paper gives a new proof of a result of Geoff Mess that the linear holonomy group of a complete flat Lorentz 3-manifold cannot be cocompact in SO(2,1). The proof uses a signed marked Lorentzian length-spectrum invariant developed by G.Margulis, reinterpreted in terms of deformations of hyperbolic surfaces.

We study the isometry group of a globally hyperbolic spatially compact Lorentz surface. Such a group acts on the circle, and we show that when the isometry group acts non properly, the subgroups of $\mathrm{Diff}(\mathbb{S}^1)$ obtained are semi conjugate to subgroups of finite covers of $\mathrm{PSL}(2,\mathbb{R})$ by…

Study timelike minimal surfaces in Heisenberg group using harmonic maps.

Constructs a topological cover of real line's multiplicative group.

In 1995, S. Adams and G. Stuck as well as A. Zeghib independently provided a classification of non-compact Lie groups which can act isometrically and locally effectively on compact Lorentzian manifolds. In the case that the corresponding Lie algebra contains a direct summand isomorphic to the two-dimensional special li…

Based on the work of Adams and Stuck as well as on the work of Zeghib, we classify the Lie groups which can act isometrically and locally effectively on Lorentzian manifolds of finite volume. In the case that the corresponding Lie algebra contains a direct summand isomorphic to the two-dimensional special linear algebr…

The main result of this paper is a construction of fundamental domains for certain group actions on Lorentz manifolds of constant curvature. We consider the simply connected Lie group G~, the universal cover of the group SU(1,1) of orientation-preserving isometries of the hyperbolic plane. The Killing form on the Lie g…

The main result of this paper is the conformal flatness of real-analytic compact Lorentz manifolds of dimension at least $3$ admitting a conformal essential (i.e. conformal, but not isometric) action of a Lie group locally isomorphic to PSL(2,R). It is established by using a general result of M. Gromov on local isometr…

L-GATr transforms high-energy physics data using geometric algebra and Lorentz symmetry.

The main aim of this paper is to give two infinite series of examples of Lorentz space forms that can be obtained from Lorentz polyhedra by identification of faces. These Lorentz space forms are bi-quotients of the form $Γ_1\backslash G/Γ_2$, where $G=\widetilde{\operatorname{SU}(1,1)}\cong\widetilde{\operatorname{SL}(…

We develop the classification of weakly symmetric pseudo--riemannian manifolds $G/H$ where $G$ is a semisimple Lie group and $H$ is a reductive subgroup. We derive the classification from the cases where $G$ is compact, and then we discuss the (isotropy) representation of $H$ on the tangent space of $G/H$ and the signa…

Let H be a closed, noncompact subgroup of a simple Lie group G, such that G/H admits an invariant Lorentz metric. We show that if G = SO(2,n), with n > 2, then the identity component of H is conjugate to the identity component of SO(1,n). Also, if G = SO(1,n), with n > 2, then the identity component of H is conjugate t…

A study is made of 4-dimensional Lorentz manifolds which are projectively related, that is, whose Levi-Civita connections give rise to the same (unparameterised) geodesics. A brief review of some relevant recent work is provided and a list of new results connecting projective relatedness and the holonomy type of the Lo…

We prove a structure theorem for compact aspherical Lorentz manifolds with abundant local symmetry. If M is a compact, aspherical, real-analytic, complete Lorentz manifold such that the isometry group of the universal cover has semisimple identity component, then the local isometry orbits in M are roughly fibers of a f…

The above title is the same, but with "semisimple" instead of "simple," as that of a notice by N. Kowalsky. There, she announced many theorems on the subject of actions of simple Lie groups preserving a Lorentz structure. Unfortunately, she published proofs for essentially only half of the announced results before her …

Geodesic orbit property studied for Lorentz manifolds.

Study Lorentz harmonic maps and spacelike surfaces in anti-de Sitter space.

New findings on plane waves in 3D spacetimes, showing non-unimodular elliptic plane waves are unique.

If an $m+2$-manifold $M$ is locally modeled on $\RR^{m+2}$ with coordinate changes lying in the subgroup $G=\RR^{m+2}\rtimes ({\rO}(m+1,1)\times \RR^+)$ of the affine group ${\rA}(m+2)$, then $M$ is said to be a \emph{Lorentzian similarity manifold}. A Lorentzian similarity manifold is also a conformally flat Lorentzia…

This study explores magnetic trajectories on the Heisenberg group, finding symmetries and solutions.

The paper classifies and proves properties of symmetry breaking operators for specific groups.

The paper extends invariant theory to non-compact and non-reductive actions, classifying four regimes.

Building on the universal covering group of the general linear group, we introduce the composite spinor bundle whose subbundles are Lorentz spin structures associated with different gravitational fields. General covariant transformations of this composite spinor bundle are canonically defined.

We study conformal Fefferman-Lorentz manifolds introduced by Fefferman. To do so, we introduce Fefferman-Lorentz structure on (2n+2)-dimensional manifolds. By using causal conformal vector fields preserving that structure, we shall establish two theorems on compact Fefferman-Lorentz manifolds: One is the coincidence of…

Study shows compact Lorentz manifolds can't have closed geodesics.

Using the Morse-theoretic methods introduced by Hitchin, we prove that the moduli space of $\SO_0(1,n)$-Higgs bundles when $n$ is odd has two connected components.

The study introduces canonical coordinates for Lorentz surfaces and proves a Bonnet-type theorem.

Geodesic completeness proven for all compact locally symmetric Lorentz manifolds.

Develops neural networks for reductive Lie groups, enhancing symmetry respect.