Proves nonemptyness of domains for specific group actions.
problem Nonemptyness of domains of proper discontinuity for Anosov groups of affine Lorentzian transformations.
method Proof of nonemptyness of domains of proper discontinuity.
result Proves nonemptyness of domains for Anosov groups of affine Lorentzian transformations.
Study explores solvable Lie groups' actions on closed Lorentzian manifolds.
problem Understanding conformal actions of solvable Lie groups on closed Lorentzian manifolds.
method Analyzes the identity component of the conformal group and uses algebraic hypotheses.
result Establishes conditions for conformal flatness and local embeddings.
In the present paper, we prove that no infinite group acts isometrically, effectively, and properly discontinuously on a certain class of Lorentzian manifolds that are not necessarily homogeneous.
We consider conformal actions of simple Lie groups on compact Lorentzian manifolds. Mainly motivated by the Lorentzian version of a conjecture of Lichnerowicz, we establish the alternative: Either the group acts isometrically for some metric in the conformal class, or the manifold is conformally flat - that is, everywh…
The paper explores conformal groups on plane waves and proves a conjecture in locally homogeneous settings.
problem Proving the Lorentzian conformal Lichnerowicz conjecture in locally homogeneous settings.
method Analyzing conformal groups on plane waves and proving the conjecture in a specific setting.
result The Lorentzian conformal Lichnerowicz conjecture is proven in a locally homogeneous setting.
Formula for index in Lorentzian spacetimes.
problem Developing an index formula for spacetimes with boundary.
method Reduction from equivariant to non-equivariant, Lorentzian spectral flow.
result Equivalence of equivariant index and spectral flow in Lorentzian spacetimes.
The paper classifies metrics on specific Lie groups and finds unique properties of these metrics.
problem Classifying left-invariant Lorentzian metrics on specific Lie groups.
method Analyzing the three-dimensional Heisenberg group and its direct product with Euclidean space.
result There are exactly six left-invariant Lorentzian metrics on the Lie group, one of which is flat and the others are Ricci solitons but not Einstein.
In this paper we prove that the space of differential invariants for curves with arc-length parameter in the light cone of Lorentzian R4, invariants under the centro-affine action of the Lorentzian group, is Poisson equivalent to the space of conformal differential invariants for curves in the Möbius sphere…
Embeds Lorentzian manifolds in \(\mathbb{R}^{n+2}\) with SO(2,n) compatibility.
problem Embedding Lorentzian manifolds in \(\mathbb{R}^{n+2}\) with specific metric properties.
method Embedding using SO(2,n) compatible metrics.
result Conformal transformations on submanifolds inherited from ambient space.
We study the mechanisms of the non properness of the action of the group of diffeomorphisms on the space of Lorentzian metrics of a compact manifold. In particular, we prove that nonproperness entails the presence of lightlike geodesic foliations of codimension 1. On the 2-torus, we prove that a metric with constant cu…
Study generalizes submetry concept for spacetimes, linking to curvature and foliations.
problem Understanding submetry in non-positive signature spacetimes.
method Developed Lorentzian submetries and established their equivalence to semi-Riemannian submersions.
result Lorentzian submetries correspond to locally C^{1,1} semi-Riemannian submersions under completeness assumptions.
In this work we investigate families of compact Lorentzian manifolds in dimension four. We show that every lightlike geodesic on such spaces is periodic, while there are closed and non-closed spacelike and timelike geodesics. Their isometry groups are computed. We also show that there is a non trivial action by isometr…
Study of recurrent Lorentzian Weyl spaces with detailed local and global structures.
problem Characterizing and classifying non-closed Lorentzian Weyl manifolds.
method Analyzing curvature tensors, Lie group actions, and differential invariants.
result Locally homogeneous non-closed recurrent Lorentzian Weyl manifolds are classified.
Compact group found for special Lorentzian manifolds.
problem Understanding the conformal group of specific Lorentzian manifolds.
method Analyzing real analytic Lorentzian manifolds with finite fundamental group.
result The conformal group is compact for these manifolds.
The paper classifies 3D Lorentzian Ein(2) Lie groups.
problem Classifying 3D Lorentzian Ein(2) Lie groups. method Complete classification through mathematical analysis.
result Three-dimensional Lorentzian Ein(2) Lie groups have been completely classified. Study left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups.
problem Characterize left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups.
method Analyze left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups, classify Lie algebras and groups.
result New results on left-invariant Lorentzian metrics with harmonic curvature and non-parallel Ricci operator.
Study classifies Riemann solitons on specific 3D Lorentzian groups.
problem Classifying Riemann solitons on three-dimensional Lorentzian Lie groups.
method Complete classification through analysis of left-invariant structures.
result Comprehensive classification of Riemann solitons on these groups.
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…
Timelike minimal surfaces in Lorentzian Heisenberg group have singular points.
problem Characterizing singularities on timelike minimal surfaces.
method Constructing timelike minimal surfaces as Lorentzian harmonic maps and analyzing their singularities.
result Criteria for cuspidal edges, swallowtails, and cuspidal cross caps are provided.
A Margulis spacetime is a complete flat affine Lorentzian 3-manifold with free fundamental group. Associated to M is a noncompact complete hyperbolic surface Σ. We study double extensions of π1(M)≅π1(Σ) when Σ is homeomorphic to a projective plane minus two discs. We classify proper actions of this do…
The paper classifies Ricci solitons on specific Lorentzian Lie groups.
problem Classifying Ricci solitons on Lorentzian Lie groups.
method Computed Bott connections and their curvature; classified Ricci solitons.
result Classification of Ricci solitons on three-dimensional Lorentzian Lie groups.
Study of special Lorentzian Lie groups with 4D isometry group, finding all are non-gradient expanding Ricci solitons.
problem Characterizing homogeneous Lorentzian three-manifolds with a 4D isometry group.
method Explicit global coordinate description and proof of Ricci soliton properties.
result All special examples are non-gradient expanding Ricci solitons.
We study the geometry and the periodic geodesics of a compact Lorentzian manifold that has a Killing vector field which is timelike somewhere. Using a compactness argument for subgroups of the isometry group, we prove the existence of one timelike non self-intersecting periodic geodesic. If the Killing vector field is …
Classifies Ricci collineations on specific 3D Lorentzian Lie groups.
problem Classifying Ricci collineations on specific Lie groups.
method Classifying based on canonical and Kobayashi-Nomizu connections.
result Results in classification of Ricci collineations.
Study finds all conformal Ricci collineations on specific 3D Lorentzian groups.
problem Identifying conformal Ricci collineations on three-dimensional Lorentzian Lie groups.
method Analysis of Levi-Civita connection on specific Lie groups.
result Determined all conformal Ricci collineations associated with the Levi-Civita connection.
New conformally Einstein metrics on Heisenberg group found.
problem Finding new conformally Einstein metrics on specific Lie groups.
method Described Lorentzian semi-direct extensions of the Heisenberg group.
result Classified Bach-flat left-invariant Lorentzian metrics.
Classifies Ricci solitons on specific Lorentzian Lie groups.
problem Classifying Ricci solitons on three-dimensional Lorentzian Lie groups.
method Examined canonical and perturbed canonical connections, as well as Kobayashi-Nomizu connections and perturbed Kobayashi-Nomizu connections.
result Classified affine Ricci solitons on three-dimensional Lorentzian Lie groups with product structure.
Study on SU(2) group's Lorentzian problem, focusing on controllability and extremals.
problem Left-invariant Lorentzian problem on SU(2) group.
method Addressing complete controllability, existence of length maximizers, extremals.
result Results on controllability and extremals for the Lorentzian problem.
The paper classifies Ricci solitons on specific Lorentzian Lie groups.
problem Classifying algebraic Ricci solitons on three-dimensional Lorentzian Lie groups.
method Computed canonical and Kobayashi-Nomizu connections and their curvatures; defined algebraic Ricci solitons.
result Classified algebraic Ricci solitons on specific Lorentzian Lie groups.
In this paper, a survey of the recent results about the classification of the connected holonomy groups of the Lorentzian manifolds is given. A simplification of the construction of the Lorentzian metrics with all possible connected holonomy groups is obtained. As the applications, the Einstein equation, Lorentzian man…
The paper classifies Lorentzian Lie groups based on Codazzi tensors and quasi-statistical structures.
problem Classifying Lorentzian Lie groups based on specific tensor properties.
method Classification of three-dimensional Lorentzian Lie groups based on Ricci tensors and quasi-statistical structures associated with different affine connections.
result The paper classifies three-dimensional Lorentzian Lie groups based on Codazzi tensors and quasi-statistical structures associated with Bott, canonical, and Kobayashi-Nomizu connections.
Study left invariant Lorentzian metrics on 3D non-unimodular Lie groups.
problem Classify and analyze left invariant Lorentzian metrics on 3D non-unimodular Lie groups.
method Classify metrics up to automorphism, study curvature functions.
result Obtain Ricci operator, scalar curvature, and sectional curvatures as functions of metrics.
We define a concept of Lorentzian angle that works even when one or both of the directions involved is null (lightlike). Such angles play a role in Regge-Calculus, in the boundary- and corner- terms for the gravitational action, and in the Lorentzian Gauss-Bonnet theorem (for which we provide a proof).
Study finds all Ricci collineations for specific connections on 3D Lorentzian groups.
problem Identifying Ricci collineations for specific connections on 3D Lorentzian Lie groups.
method Examined left-invariant Ricci collineations associated with Bott connections on three-dimensional Lorentzian Lie groups.
result Determined all left-invariant Ricci collineations associated with the Bott connection.
We prove that each special Lorentzian holonomy group (with the exception of those including the isotropy groups of Kähler symmetric spaces with rank greater than one) can be realized as the holonomy group of a globally hyperbolic Lorentzian manifold.
Study on Heisenberg group's Lorentzian problems using Pontryagin's principle.
problem Lorentzian problems on the Heisenberg group.
method Applied Pontryagin's maximum principle to obtain extremal trajectories.
result Parameterization of abnormal and normal extremal trajectories, investigation of reachability sets and existence of optimal trajectories.
The paper classifies solitons on specific Lie groups.
problem Classifying solitons on three-dimensional Lorentzian Lie groups.
method Computing Wanas tensor and defining algebraic Wanas solitons.
result Classification of algebraic Wanas solitons on specific Lie groups.
Complete description of flat Lorentzian Lie groups solved.
problem Long-standing open problem in pseudo-Riemannian Lie groups.
method Refined analysis of double extension process and explicit classification of Lie algebras.
result All flat Lorentzian Lie algebras arise from flat Euclidean ones.
Optimal transport explored on a specific geometric space.
problem Optimal transport problem in sub-Lorentzian Heisenberg group.
method Synthetic metric spacetime structure analysis and sub-Lorentzian version of Brenier's theorem.
result Established sub-Lorentzian version of Brenier's theorem and derived Monge-Ampère equation.
Study on special Lie groups with Lorentzian metrics.
problem Characterize structure of 2-step nilpotent Lorentzian naturally reductive Lie groups. method Develop framework for naturally reductive Lie groups, extend to Lorentzian context, analyze degenerate and non-degenerate cases.
result Complete structural description of naturally reductive 2-step Lorentzian nilpotent Lie groups. The study classifies special flows on specific geometric groups.
problem Classifying special flows on specific geometric groups.
method Classification of Left-invariant Ricci collineations associated to Yano connections on three-dimensional Lorentzian Lie groups.
result Results in classifying these flows.
Classifies 1-connected Lorentzian manifolds with essential conformal groups.
problem Classifying Lorentzian manifolds with specific conformal groups.
method Proves existence of a metric making the manifold homogeneous and plane wave.
result Completes the classification of 1-connected Lorentzian manifolds with transitive conformal groups.
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…
Explains non-lorentzian theories and their dynamics.
problem Understanding non-lorentzian kinematics and dynamics.
method Review of kinematical spacetimes, construction of particle dynamics actions, discussion of gravity theories and field theories.
result Introduction and analysis of non-lorentzian gravity and field theories.
Classifies a specific type of Lie groups related to Einstein geometry.
problem Classifying Einstein Lorentzian 3-nilpotent Lie groups with 1-dimensional nondegenerate center.
method Complete classification through mathematical analysis.
result A full classification of the specified Lie groups.
Study classifies helix surfaces in Lorentzian Heisenberg group.
problem Classifying helix surfaces in Lorentzian Heisenberg group.
method Complete description of ambient space geometry, classification of minimal and CMC helix surfaces, investigation of constant angle surfaces.
result Explicit parametrizations of minimal and CMC helix surfaces in $\htt$.
Mathematical treatment of plane waves, proving their inextendibility and completeness.
problem Completeness and inextendibility of homogeneous plane waves.
method Cohomogeneity one Heisenberg actions, isometry group analysis.
result Proof of C2-inextendibility and geodesic completeness of non-flat homogeneous plane waves. Given a simple Lie group G of rank 1, we consider compact pseudo-Riemannian manifolds (M,g) of signature (p,q) on which G can act conformally. Precisely, we determine the smallest possible value for the index min(p,q) of the metric. When the index is optimal and G non-exceptional, we prove that the metric must be confo…