Closed Lorentz 4-manifolds have finite isometry groups with a bounded abelian subgroup.
problem Understanding the structure of isometry groups of closed Lorentz 4-manifolds.
method Proving that any finite subgroup of the isometry group of a closed Lorentz 4-manifold has a bounded abelian subgroup.
result Finite isometry groups of closed Lorentz 4-manifolds have a bounded abelian subgroup.
It is observed that on many 4-manifolds there is a unique smooth structure underlying a globally hyperbolic Lorentz metric. For instance, every contractible smooth 4-manifold admitting a globally hyperbolic Lorentz metric is diffeomorphic to the standard R4. Similarly, a smooth 4-manifold homeomorphic to the produc…
The fundamental equations of Gauss, Codazzi and Ricci provide the conditions for local isometric embeddability. In general, the three fundamental equations are independent for surfaces in Riemannian 4-manifolds. In contrast, we prove in this article that for arbitrary Lorentz surfaces in Lorentzian Kaehler surfaces the…
Locally symplectic structure found on Kerr space-time.
problem Understanding Kerr space-time using geodesics.
method Identifying locally conformally symplectic structure using characteristic classes and Kerr-Schild coordinates.
result Definition of cobordism category of contact 3-manifolds and locally conformally symplectic cobordisms.
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.
problem Existence of closed geodesics in compact Lorentz manifolds.
method Constructed compact Lorentz manifolds without closed geodesics.
result Compact Lorentz manifolds can't have closed geodesics.
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 study introduces canonical coordinates for Lorentz surfaces and proves a Bonnet-type theorem.
problem Characterizing Lorentz surfaces in R13. method Introduces canonical isotropic coordinates and a natural equation for the surfaces.
result Proves a Bonnet-type theorem for Lorentz surfaces of general type.
LLoCa makes any network Lorentz-equivariant, achieving high accuracy and efficiency.
problem Limitations of specialized layers in Lorentz-equivariant neural networks.
method LLoCa framework using local reference frames and geometric message passing.
result Models achieve competitive and state-of-the-art accuracy on particle physics tasks.
Geodesic completeness proven for all compact locally symmetric Lorentz manifolds.
problem Geodesic completeness of compact locally symmetric Lorentz manifolds.
method Proof in all remaining cases using completeness result.
result All compact, locally symmetric Lorentz manifolds are geodesically complete.
A world sheet in Lorentz-Minkowski space is a timelike submanifold consisting of a one-parameter family of spacelike submanifolds in Lorentz-Minkowski space. In this paper we investigate differential geometry of world sheets in Lorentz-Minkowski space as an application of the theory of big wave fronts.
Lecture notes on Lorentz geometry, focusing on curves and surfaces.
problem Diagonalization of the Weingarten map for timelike surfaces and classification of surfaces with constant Gaussian curvature.
method Analysis of linear algebra in pseudo-Euclidean space, application of the Fundamental Theorem of Curves, and use of split-complex algebra.
result Local classification of surfaces with constant Gaussian curvature and Weierstrass' representation formula.
Study on maximal surfaces with high genus in Lorentz-Minkowski space.
problem Existence of nonorientable maximal surfaces with high genus.
method Existence results for nonorientable maximal surfaces with high genus and one end.
result Existence of maximal surfaces with high genus in Lorentz-Minkowski space.
This paper considers aspects of 4-manifold topology from the point of view of the null cone of a neutral metric, a point of view we call neutral causal topology. In particular, we construct and investigate neutral 4-manifolds with null boundaries that arise from canonical 3- and 4-dimensional settings. A null hypersurf…
Defines and computes a generalized spectral action for Lorentz warped products.
problem Computing spectral actions for Lorentz warped products.
method Defines and computes the bimetric spectral Einstein-Hilbert action for Lorentz warped products.
result Derives a Kastler-Kalau-Walze type theorem for Lorentz warped products.
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 the totally null surfaces of the neutral Kaehler metric on certain 4-manifolds. The tangent spaces of totally null surfaces are either self-dual (α-planes) or anti-self-dual (β-planes) and so we consider α-surfaces and β-surfaces. The metric of the examples we study, which include the spaces of oriente…
The three-dimensional Heisenberg group H3 has three left-invariant Lorentz metrics g1, g2 and g3. They are not isometric each other. In this paper, we characterize the left-invariant Lorentzian metric g1 as a Lorentz Ricci soliton. This Ricci soliton g1 is a shrinking non-gradient Ricci soliton. Likew…
Classified spaces in low dimensions.
problem Irreducible homogeneous almost Hermite-Lorentz spaces in low dimensions.
method Classification through complex dimension 3.
result Classification of spaces in low dimensions.
The paper explores unique properties of Darboux transformations of spacelike curves in the Lorentz-Minkowski plane.
problem Exploring Darboux transformations of spacelike curves in the Lorentz-Minkowski plane.
method Using the Penrose diagram for conformal compactification, the paper investigates unique properties of Darboux transformations of spacelike curves.
result The paper identifies unique properties of Darboux transformations of spacelike curves in the Lorentz-Minkowski plane, especially regarding singularities and blowup.
Study connects Riemann-Finsler geometry to Lorentz-violating scalar fields.
problem Exploring the connection between Riemann-Finsler geometries and Lorentz-violating scalar fields.
method Deriving quadratic actions and classical relativistic point-particle lagrangians in various spacetime dimensions.
result Support for open conjectures about Riemann-Finsler geometries in Lorentz-violating field theories.
A condition of Osserman type, called φ-null Osserman condition, is introduced and studied in the context of Lorentz globally framed f-manifolds. An explicit example shows the naturalness of this condition in the setting of Lorentz S-manifolds. We prove that a Lorentz S-manifold with constant…
We classify isometries of compact Lorentz manifolds.
problem Understanding the isometry group of compact Lorentz manifolds.
method Proving structure theorems and applying the Tits alternative.
result Classification of lattices acting on compact Lorentz manifolds.
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…
Lorentz-Finsler geometry reveals new and old inequalities.
problem Finding new inequalities using Lorentz-Finsler geometry.
method Applying reverse Cauchy-Schwarz and reverse triangle inequalities in Lorentz-Finsler geometry.
result Proved new and refined inequalities, including refinements of Aczél's inequality.
The paper examines Lorentz Ricci solitons on specific Lie groups.
problem Identifying Lorentz Ricci solitons on 4D non-Abelian nilpotent Lie groups.
method Investigates left-invariant Lorentz metrics on H3imesR and G4. result Only specific metrics on H3imesR and G4 satisfy the Ricci Soliton equation. Maximal surfaces in Lorentz-Minkowski space have conjugate graphs.
problem Characterizing maximal surfaces in Lorentz-Minkowski space.
method Three proofs showing correspondence to minimal surfaces in Euclidean space.
result Conjugate surface of a maximal graph over a convex domain is also a graph.
Investigates differences in solving mean curvature problems in Euclidean and Lorentz-Minkowski spaces.
problem Solving the Dirichlet problem for mean curvature equations in different spacetimes.
method Compares techniques and conditions for solvability in Euclidean and Lorentz-Minkowski spaces.
result Lorentz-Minkowski spacelike condition allows dropping mean convexity hypothesis.
The abstract develops weighted Ricci curvature in Lorentz-Finsler geometry and extends singularity theorems.
problem Extending singularity theorems in weighted Lorentz-Finsler geometry.
method Generalizing Jacobi, Riccati, and Raychaudhuri equations; applying generalized Bishop inequality.
result Weighted Lorentz-Finsler singularity theorems extended.
Unified estimates for mean curvature in Lorentz-Minkowski space.
problem Estimating mean curvature for space-like and time-like graphs.
method Using gradient bounds to derive Heinz-type estimates.
result Unified vanishing theorem for mean curvature of constant mean curvature graphs.
Paper constructs two series of Lorentz bi-quotients from polyhedra.
problem Creating fundamental domains for Lorentz bi-quotients.
method Explicit construction of polyhedral fundamental domains.
result Two infinite series of Lorentz bi-quotients constructed.
We study minimal Lorentz surfaces in the pseudo-Euclidean 4-space with neutral metric whose first normal space is two-dimensional and whose Gauss curvature K and normal curvature ϰ satisfy the inequality K2−ϰ2>0. Such surfaces we call minimal Lorentz surfaces of general type. On any surface of …
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.
Survey Bernstein-type theorems for graphical surfaces in Euclidean and Lorentz-Minkowski spaces.
problem Proving theorems for minimal and constant mean curvature graphs in Euclidean and Lorentz-Minkowski spaces.
method Explains several proofs and provides mean curvature estimates for graphs in Euclidean and Lorentz-Minkowski spaces.
result Bernstein-type theorems for constant mean curvature graphs in Euclidean 3-space and space-like graphs in Lorentz-Minkowski 3-space.
Unified geometry for relativity and beyond.
problem Unified geometric framework for relativity and beyond.
method Unified Lorentz-Finsler geometry.
result Unified framework for relativity and beyond.
Study geodesic orbit Lorentz nilmanifolds, proving structural properties.
problem Understanding the structure of geodesic orbit Lorentz nilmanifolds.
method Analyzing geodesic orbit Lorentz nilmanifolds with reductive decompositions.
result Proves properties of nilpotent subgroups and their nilpotency steps.
Since J. L. Lagrange initiated in 1760 the study of minimal surfaces of Euclidean 3-space, minimal surfaces in real space forms have been studied extensively by many mathematicians during the last two and half centuries. In contrast, so far very few results on minimal Lorentz surfaces in indefinite space forms are know…
New inequality shows all special submanifolds in light cone are totally umbilical spheres.
problem Characterizing submanifolds with parallel mean curvature in Lorentz-Minkowski spacetime.
method Established an integral inequality and used it to derive a rigidity result.
result All compact submanifolds with parallel mean curvature in light cone are totally umbilical spheres.
Paper connects surfaces in 4D and 3D spacetime.
problem Finding relations between Lorentz surfaces in different spacetime dimensions.
method Weierstrass-type representations for null curves and surfaces.
result Relation between minimal Lorentz surfaces in R24 and R13. Study of Gödel Universe as Lie group with specific metric.
problem Characterize geodesics in the Gödel Universe.
method Geometric theory of optimal control applied to Lie groups with left-invariant Lorentz metrics.
result No closed timelike or isotropic geodesics in the Gödel Universe.
Paper solves Hessian quotient equations in Lorentz-Minkowski space with Dirichlet boundary conditions.
problem Existence and uniqueness of solutions to Hessian quotient equations in Lorentz-Minkowski space.
method Suitable settings to prove existence and uniqueness of solutions.
result Existence and uniqueness of solutions to the class of Hessian quotient equations.
L-GATr transforms high-energy physics data using geometric algebra and Lorentz symmetry.
problem Extracting scientific understanding from particle-physics experiments with high precision and efficiency.
method L-GATr, a geometric algebra Transformer, representing data in 4D space-time and being equivariant under Lorentz transformations.
result L-GATr achieves performance comparable to or better than domain-specific baselines on regression, classification, and generative tasks.
A new model uses Lorentz-Finsler geometry to predict wave propagation.
problem Modeling wave propagation in anisotropic and rheonomic media.
method Identifying wave trajectories as lightlike pregeodesics of a specific Lorentz-Finsler metric, solving ODE systems.
result Wave trajectories can be easily computed in real time.
Navigation in Lorentz Finsler geometry induces isoparametric hypersurfaces.
problem Defining and analyzing isoparametric hypersurfaces in Lorentz Finsler geometry.
method Using a navigation process with a Finsler metric and a tangent vector field, isoparametric functions and hypersurfaces are defined and analyzed.
result Local correspondences between isoparametric functions and hypersurfaces are established.
The paper studies pseudo-torsion functions of spacelike curves in Lorentz-Minkowski space.
problem Understanding the behavior of pseudo-torsion functions along spacelike curves with isolated lightlike points.
method Introducing pseudo-torsion functions and proving the fundamental theorem.
result A necessary and sufficient condition for real analytic spacelike curves to be planar.
A Lorentz surface in the four-dimensional pseudo-Euclidean space with neutral metric is called quasi-minimal if its mean curvature vector is lightlike at each point. In the present paper we obtain the complete classification of quasi-minimal Lorentz surfaces with pointwise 1-type Gauss map.
Study geometric properties of spacelike foliations on Lorentz manifolds.
problem Investigate conditions for spacelike foliations to be totally umbilical or geodesic.
method Develop an equation relating foliation to ambient manifold, apply Maximum principle.
result Obtain an obstruction for totally geodesic foliations in spacetimes with positive Ricci curvature.
Paper classifies ruled surfaces in Lorentz-Minkowski space for a specific flow.
problem Classifying ruled surfaces in Lorentz-Minkowski space.
method Examining homothetic self-similar solutions of the inverse mean curvature flow.
result Existence of two classes of non-cylindrical homothetic solitons.