Defines metrics for Lorentzian spaces and explores maximal developments.
problem Defining metrics and maximal developments in Lorentzian spaces.
method Defining Lorentzian spaces and using functorial properties.
result Explicit non-spacetime example of maximal globally hyperbolic Lorentzian space.
Study gluing of Lorentzian length spaces and their causal ladder properties.
problem Compatibility of Lorentzian amalgamation with length space properties.
method Conditions for gluing Lorentzian length spaces and criteria for causal ladder preservation.
result Gluing of Lorentzian length spaces yields again a Lorentzian length space under certain conditions.
Defines new metrics for Lorentzian spaces and their convergence.
problem Defining metrics for Lorentzian spaces and their convergence.
method Abstract approach to Lorentzian Gromov-Hausdorff distance and convergence, defining bounded Lorentzian-metric spaces, and proving stability under GH limits.
result GH limits of Lorentzian-metric spaces are isometric and homeomorphic.
Study on Lorentzian spaces with curvature bounds, proving comparison theorems.
problem Understanding curvature bounds in Lorentzian spaces.
method Introduced normalized angle for Lorentzian pre-length spaces, proving comparison theorems.
result Established local Lorentzian Toponogov theorem and Alexandrov convexity property.
In this article we study minimal flat Lorentzian surfaces in Lorentzian complex space forms. First we prove that, for minimal flat Lorentzian surfaces in a Lorentzian complex form, the equation of Ricci is a consequence of the equations of Gauss and Codazzi. Then we classify minimal flat Lorentzian surfaces in the Lore…
We investigate the twistor space and the Grassmannian fibre bundle of a Lorentzian 4-space with natural almost optical structures and its induced CR-structures. The twistor spaces of the Lorentzian space forms $\R^4_1, \Di{S}^4_1$ and $\Di{H}^4_1$ are explicitly discussed. The given twistor construction is applied to s…
The study proves a transverse diameter theorem for Lorentzian foliations.
problem Understanding the geometry of foliations in Lorentzian spacetimes.
method Developed a novel causality structure on leaf spaces via transverse Lorentzian geometry.
result Derived a transverse diameter theorem for Lorentzian foliations and orbifolds.
Study extends null distance concept to Lorentzian length spaces for spacetime analysis.
problem Understanding spacetime convergence and topology in Lorentzian geometry.
method Extend null distance concept to Lorentzian length spaces, study Gromov-Hausdorff convergence.
result First results on compatibility of null distance with synthetic curvature bounds in warped product Lorentzian length spaces.
Study spacelike loxodromes on helicoidal surfaces in Lorentzian n-space.
problem Characterize spacelike loxodromes on helicoidal surfaces in Lorentzian n-space.
method Find parametrizations of spacelike loxodromes on both spacelike and timelike helicoidal surfaces.
result Classification of spacelike loxodromes on helicoidal surfaces in Lorentzian n-space.
Causal properties of Lorentzian symmetric spaces are investigated in the paper. The global hyperbolicity of the Cahen--Wallach Lorentzian symmetric spaces is proved.
New method glues Lorentzian spaces, preserving curvature bounds.
problem Creating new spaces from existing ones in Lorentzian geometry.
method Introducing an amalgamation process for Lorentzian pre-length spaces.
result Gluing preserves upper curvature bounds in spacetimes.
Study how algebraic conditions on isotropy group affect Lorentzian homogeneous space geometry.
problem Understand how algebraic conditions on isotropy group affect the geometry and curvature of Lorentzian homogeneous spaces.
method Prove that a Lorentzian locally homogeneous space is locally isometric to a plane wave if it admits an Ambrose--Singer connection with indecomposable, non-irreducible holonomy.
result Generalize existing results about Lorentzian homogeneous spaces with irreducible isotropy and prove results about Lorentzian connections with parallel torsion and 2-symmetric connections.
Study properties of semi-symmetric Lorentzian spaces, foliated manifolds.
problem Properties of semi-symmetric pseudo-Riemannian manifolds.
method Investigate foliated manifolds with Lorentzian metrics and analyze Ricci operator eigenvalues.
result Ricci operator has only real eigenvalues for Lorentzian metrics.
Establishes a version of Bartnik's conjecture for Lorentzian length spaces.
problem Proving Bartnik's conjecture for Lorentzian length spaces.
method Using timelike completeness and non-negative timelike curvature bounds, the causal boundary is shown to be a single point.
result A globally hyperbolic Lorentzian length space splits as a metric Lorentzian product.
Splitting theorem for non-positively curved Lorentzian spaces.
problem Understanding curvature in Lorentzian spaces.
method Proving a splitting theorem with global non-positive timelike curvature and extending first variation formula.
result Splitting theorem for Lorentzian pre-length spaces with global non-positive timelike curvature.
Study defines new products for Lorentzian spaces and analyzes causal diamonds.
problem Understanding causal diamonds in Lorentzian spaces.
method Introduced taxicab and uniform products for Lorentzian pre-length spaces. Defined D(RimesTX) space and analyzed its properties. result The space D(RimesTX) is geodesic and globally hyperbolic for complete X. Geodesic completeness proven for certain Lorentzian spaces.
problem Geodesic completeness of compact Lorentzian locally symmetric spaces.
method Using recent and earlier results on Lorentzian and de Rham-Wu decompositions.
result Geodesically complete if Lorentzian factor is of Cahen-Wallach type or maximal flat factor is one-dimensional and time-like.
Defines doubling conditions for Lorentzian spaces linked to curvature bounds.
problem Relating doubling conditions to curvature bounds in Lorentzian geometry.
method Defines doubling conditions using chronological diamonds and proves implications with timelike curvature bounds.
result Relates doubling to curvature bounds in Lorentzian geometry.
The paper reconstructs Lorentzian spacetimes from causal sets.
problem Reconstructing Lorentzian spacetimes from causal sets.
method Introduced a concept of isomorphy and three types of convergence.
result Established Gromov's reconstruction theorem in Lorentzian geometry.
Third-order symmetric Lorentzian manifolds, i.e. Lorentzian manifold with zero third derivative of the curvature tensor, are classified. These manifolds are exhausted by a special type of pp-waves, they generalize Cahen-Wallach spaces and second-order symmetric Lorentzian spaces.
Study finds formulas for special curves in complex spaces.
problem Understanding special curves in complex spaces.
method Obtained explicit formulas for Killing magnetic curves.
result Explicit formulas for Killing magnetic curves in non-flat Lorentzian-Heisenberg spaces.
New Alexandrov-Patchwork construction for Lorentzian spaces with curvature bounds.
problem Understanding finite diameter constraints in Lorentzian geometry.
method Constructing Alexandrov-Patchwork and proving Bonnet-Myers theorem for Lorentzian spaces.
result Lorentzian spaces with curvature bounds have finite diameter.
Study finds holonomy algebras for Lorentzian Weyl spin manifolds with specific spinors.
problem Characterizing Lorentzian Weyl spin manifolds with weighted parallel spinors.
method Analyzing holonomy algebras and introducing special coordinates.
result Local forms and examples of Lorentzian Weyl spin manifolds with weighted parallel spinors.
Three functors link Lorentzian geometry concepts.
problem Finiteness results, singularity theorems, boundary constructions.
method Review of three functors from Lorentzian categories.
result Novel functor from ordered measure spaces to Lorentzian pre-length spaces.
Researchers found explicit formulae for special solitons in 3D spaces.
problem Understanding Ricci solitons in specific 3D Lorentzian spaces.
method Analyzing homogeneous Ricci solitons on Bianchi-Cartan-Vranceanu spaces.
result Explicit formulae for Ricci solitons were derived.
We determine, for all three-dimensional non-unimodular Lie groups equipped with a Lorentzian metric, the set of homogeneous geodesics through a point. Together with the results of [C] and [CM2], this leads to the full classification of three-dimensional Lorentzian g.o. spaces and naturally reductive spaces.
Researchers extend the concept of metric spaces to Lorentzian spaces and prove the feasibility of their c-completion.
problem Extending the concept of metric spaces to Lorentzian spaces and proving their c-completion.
method Revisiting Lorentzian metric spaces, constructing c-completion, proving feasibility and endowing with Lorentzian metric space structure.
result The c-completion of Lorentzian metric spaces is feasible and well-suited, completing the original space in a precise sense.
Develops optimal transport in Lorentzian spaces with synthetic curvature bounds.
problem Synthetic curvature bounds for Lorentzian spaces.
method Optimal transport, convexity analysis of entropy functionals.
result Synthetic notion of timelike Ricci curvature lower bounds.
Proves a synthetic Lorentzian Cartan-Hadamard theorem.
problem Formulates and proves a theorem for Lorentzian geometry.
method Uses an appropriate notion of local concavity for Lorentzian (pre-)length spaces.
result Establishes existence and uniqueness of timelike geodesics.
In this paper, we address the problem of determining a function in terms of its orbital integrals on Lorentzian symmetric spaces. It has been solved by S. Helgason for even-dimensional isotropic Lorentzian symmetric spaces via a limit formula involving the Laplace-Beltrami operator. The result has been extended by J. O…
Globalisation theorem for Lorentzian spaces with curvature bounds.
problem Synthetic geometric analysis of Lorentzian length spaces.
method Cat's cradle construction and synthetic geometry.
result An analogue of Toponogov's Globalisation Theorem for Lorentzian length spaces.
The abstract discusses a new type of space and its properties.
problem The abstract tackles the concept of non-Hilbertian (Lorentzian) length spaces.
method The abstract introduces a new type of space and analyzes its properties.
result The abstract finds that normed spaces without inner products have no sectional curvature bounds.
Paper classifies timelike Bonnet surfaces in Lorentzian 3-manifolds.
problem Classifying timelike Bonnet surfaces in Lorentzian 3-manifolds.
method Provided a necessary and sufficient condition for timelike Bonnet surfaces in Lorentzian 3-spaces.
result Theorem classifying timelike Bonnet surfaces in a Lorentzian 3-space.
The paper explores Lorentzian connections with parallel skew torsion.
problem Understanding metric connections with parallel skew-symmetric torsion in Lorentzian signature.
method Analyzing holonomy algebras, torsion, and curvature; constructing examples; classifying homogeneous spaces.
result Complete classification of Lorentzian naturally reductive homogeneous spaces in low dimensions.
Researchers found a spinorial representation for surfaces in 3D Lorentzian spaces.
problem Representation of surfaces in Lorentzian spaces.
method Spinorial representation approach.
result New correspondences between different types of surfaces in 3D spaces.
Proves existence of longest paths in sub-Lorentzian problems.
problem Existence of longest paths in sub-Lorentzian problems.
method Generalizes classical theorem for Lorentzian manifolds to sub-Lorentzian problems.
result Longest paths exist for any left-invariant sub-Lorentzian structures on Carnot groups.
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.
Synthetic approach to conformal transformations in metric and Lorentzian spaces.
problem Defining consistent conformal transformations in spaces of low regularity.
method Introducing conformal transformations in metric and Lorentzian spaces, focusing on Lorentzian pre-length spaces.
result Established a consistent notion of conformal length and proved its properties.
Study of longest arcs and cut loci in deformed anti de-Sitter spaces.
problem Existence and properties of time-like cycles in deformed Lorentzian manifolds.
method Analysis of universal covering, admissible curves, and Lorentzian geodesics.
result Identification of cut time and cut locus in deformed anti de-Sitter spaces.
Sharp inequality for Lorentzian spaces with timelike Ricci bounds.
problem Establishing bounds on achronal hypersurfaces in Lorentzian spaces.
method Optimal transport and synthetic TCDpe(K,N) spaces. result Sharp isoperimetric-type inequality for Lorentzian spaces.
Constructs Lorentzian harmonic maps and associated timelike surfaces.
problem Lorentzian harmonic maps and timelike surfaces properties.
method Constructs framed null curves and solves eigenvalue equation.
result Characterizes singularities on timelike minimal surfaces.
Proves Lorentzian manifold properties for analytic 3D spaces.
problem Analyzing properties of Lorentzian manifolds.
method Analyzes compact, real-analytic, three-dimensional Lorentzian manifolds.
result Identity conformal group preserves metric or manifold is flat.
We give local, explicit representation formulas for n-dimensional spacelike submanifolds which are marginally trapped in the Minkowski space, the de Sitter and anti de Sitter spaces and the Lorentzian products of the sphere and the hyperbolic space by the real line.
The paper studies timelike loxodromes on specific Lorentzian helicoidal surfaces.
problem Analyzing timelike loxodromes on Lorentzian helicoidal surfaces.
method First-order differential equations, general solutions, explicit parametrizations.
result Explicit parametrizations of timelike loxodromes on Lorentzian helicoidal surfaces.
The study constructs a Lorentzian length space and explores its properties and relationships with metric and causal geometry.
problem Understanding the relationship between metric and causal geometry in Lorentzian spaces.
method Constructing a Lorentzian length space with an orthogonal splitting on a product of an interval and a metric space, and using synthetic time-like Ricci curvature bounds.
result Established sufficient conditions for global hyperbolicity and formulated time-like Ricci curvature bounds without push-up and regularity assumptions.
We solve the twistor equation on all indecomposable Lorentzian symmetric spaces explicity.
The paper generalizes rectifying and normal curves in Lorentzian n-space.
problem Characterizing and classifying g−rectifying and g−normal curves in Lorentzian n-space. method Introducing a g−position vector field and defining g−rectifying and g−normal curves based on this field. result Comprehensive characterization and classification of g−rectifying and g−normal curves. The paper constructs Gromov-Hausdorff metrics for Lorentzian spaces and calculates dimensions.
problem Understanding the geometry and dimensions of Lorentzian spaces.
method Construction of Gromov-Hausdorff metrics, calculation of dimensions, and analysis of Lorentzian spaces.
result Dushnik-Miller dimension of Minkowski spaces is countably infinite.