Timelike geometry of spherical simplices is shown to be isometric to vector spaces.
problem Characterizing the geometry of spherical simplices.
method Proved isometry to vector spaces with a timelike norm.
result Timelike spherical Hilbert geometry of simplices is isometric to a union of six copies of vector spaces.
A timelike space is a Hausdorff topological space equipped with a partial order relation < and a distance function ρ satisfying a collection of axioms including a set of compatibility conditions between the partial order relation and the distance function. The distance function is defined only on a subset of the pr…
We study the conformal geometry of timelike curves in the (1+2)-Einstein universe, the conformal compactification of Minkowski 3-space defined as the quotient of the null cone of R2,3 by the action by positive scalar multiplications. The purpose is to describe local and global conformal invariants of time…
New method finds closed timelike geodesics on Lorentzian manifolds.
problem Existence of closed timelike geodesics in Lorentzian geometry.
method Introducing timelike geodesic homotopy and combining with a local length argument.
result Provides new results on the existence of closed timelike geodesics.
Equivalence found between smooth and synthetic timelike curvature bounds.
problem Understanding timelike sectional curvature bounds in spacetime geometry.
method Established equivalence between sectional curvature bounds on timelike planes and synthetic timelike bounds.
result Equivalence proved for sectional curvature bounds on timelike planes and synthetic timelike bounds on strongly causal spacetimes.
This survey introduces synthetic timelike Ricci curvature bounds in Lorentzian spaces.
problem Synthetic timelike Ricci curvature bounds in non-smooth Lorentzian spaces.
method Optimal transport and entropy tools.
result Synthetic version of Hawking's singularity theorem and synthetic characterisation of Einstein's vacuum equations.
The paper proves properties of Lipschitz spacetimes with bounded Ricci curvature.
problem Properties of Lipschitz spacetimes with bounded Ricci curvature.
method Globally hyperbolic spacetimes with locally Lipschitz metrics and timelike Ricci curvature.
result New comparison theorems for Lipschitz spacetimes.
The paper studies quasi-umbilical timelike surfaces in a specific geometric setting.
problem Characterizing and understanding quasi-umbilical timelike surfaces in the (1+2)-Einstein universe. method Analyzes the conformal Lorentz geometry of these surfaces, proving their isothermic nature and constructing them from null curves.
result Quasi-umbilical surfaces are isothermic and their conformal deformations are determined by one arbitrary function.
The paper examines timelike conformal fields on 3-manifolds and finds they are rigidly tied to specific geometric structures.
problem Investigating timelike conformal vector fields on closed Lorentzian 3-manifolds.
method Performing conformal changes to unit vectors and analyzing the resulting flows through stable Hamiltonian structures and cohomology.
result Timelike conformal vector fields on 3-manifolds are either Reeb vector fields of Sasakian or co-Kähler structures.
The paper explores conjugate points in Lorentzian spaces, comparing different definitions and proving related theorems.
problem Understanding conjugate points in Lorentzian geometry.
method Introducing and comparing different definitions of conjugate points in synthetic Lorentzian length spaces.
result All defined notions of conjugate points are compatible with the smooth spacetime setting.
This study examines geometric properties and offsets of slant timelike-ruled surfaces.
problem Geometric properties and offsets of slant timelike-ruled surfaces in Minkowski 3-space.
method Derivation of parametric formulation, conditions for coaxial alignment, examination through Blaschke and Darboux frames.
result Conditions ensuring the coaxial alignment of the central normal with the ruling direction of the offset surface.
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.
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 …
The study introduces hyperbolic angles in Lorentzian spaces and characterizes curvature bounds.
problem Characterizing timelike curvature bounds in Lorentzian spaces.
method Synthetic geometric framework of Lorentzian (pre-)length spaces, introduction of hyperbolic angles, and angle monotonicity condition.
result Characterization of timelike curvature bounds with an angle monotonicity condition.
Synthetic splitting theorem for Lorentzian spaces with non-negative curvature.
problem Proving a splitting theorem for globally hyperbolic Lorentzian length spaces with non-negative timelike curvature.
method Synthetic approach using triangle comparison and parallelity of timelike lines.
result Establishes a splitting of a neighborhood of a complete timelike line, leading to global inextendibility.
Paper introduces a new time separation function for C0 spacetimes.
problem Lower semicontinuity of time separation function for C0 spacetimes. method Introduced nearly timelike curves to ensure lower semicontinuity.
result Lower semicontinuous time separation function for C0 spacetimes. The paper extends completeness notions to low-regularity spacetimes.
problem Defining completeness conditions for spacetimes with low-regularity metrics.
method Extending Beem's completeness notions to Lorentzian length spaces and proving relationships between them.
result Equivalence of completeness conditions for globally hyperbolic C1-spacetimes under certain conditions. The Sagnac effect is re-examined using Finslerian geometry.
problem Understanding the Sagnac effect in general relativity.
method Reviewing the geometry of the Sagnac effect with a focus on Finslerian metrics.
result An asymmetry in Finslerian metrics affects the Sagnac effect for both future-pointing null and timelike geodesics.
The basic theory on the conformal geometry of timelike surfaces in pseudo-Riemannian space forms is introduced, which is parallel to the classical framework of Burstall etc. for spacelike surfaces. Then we provide a discussion on the transforms of timelike (±)−isothermic surfaces (or real isothermic, complex isothe…
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 of timelike surfaces with time-minimizing rulings in Newtonian and relativistic spacetimes.
problem Understanding time-minimizing paths in spacetime geometries.
method Constructing timelike surfaces ruled by geodesics of Finsler or Jacobi metrics.
result Explicit examples of brachistochrone-ruled timelike surfaces in Minkowski and Schwarzschild spacetimes.
Rigidity results for hypersurfaces in warped spacetimes.
problem Characterizing hypersurfaces in specific spacetime geometries.
method Analyzing mean curvature and conformal vectors in warped spacetimes.
result Compact hypersurfaces in certain spacetimes are slices.
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.
Unified approach to various energy conditions in spacetime geometry.
problem Synthetic quantification of energy conditions in spacetime.
method Introducing entropic timelike curvature dimension condition with variable Ricci curvature bounds.
result Unified approach to various energy conditions including strong, weak, and null energy conditions.
Lorentzian Ptolemy inequality linked to curvature bounds.
problem Global timelike sectional curvature bounds in Lorentzian geometry.
method Investigation of Ptolemy inequality in globally hyperbolic spacetimes.
result Equivalence of Lorentzian Ptolemy inequality to curvature bound.
In this paper, we establish equiform differential geometry of space and timelike curves in 4-dimensional Minkowski space. We obtain some conditions for these curves. Also, general helices with respect to their equiform curvatures are characterized.
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.
In this paper, we generalize the defining equation for de Sitter space by replacing the de Sitter radius with a function f satisfying certain conditions; each resulting hypersurface is diffeomorphic to de Sitter space, and has a geometry (and causal character) which is controlled by the choice of f. Necessary and s…
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.
Authors create déjà vu links in Legendrian geometry.
problem Understanding déjà vu moments in spacetime.
method Constructing Legendrian links with specific properties.
result Found examples of déjà vu links in various geometric settings.
Stability of timelike Ricci bounds in low-regularity spacetimes.
problem Stability of synthetic timelike Ricci curvature bounds under C0-limits. method Constructing smooth approximations and analyzing limiting behavior via Lorentzian optimal transport.
result Impulsive gravitational waves satisfy synthetic timelike Ricci curvature lower bounds.
Two different spacetimes can mimic each other's boundary measurements.
problem Lorentzian Calderón problem
method Counterexample construction
result Non-isometric spacetimes can have identical boundary measurements
We study when the Jacobi operator associated to the Weyl conformal curvature tensor has constant eigenvalues on the bundle of unit spacelike or timelike tangent vectors. This leads to questions in the conformal geometry of pseudo-Riemannian manifolds which generalize the Osserman conjecture to this setting. We also stu…
Study of timelike surfaces in Minkowski space with specific geometric properties.
problem Characterizing geometric properties of timelike surfaces in Minkowski space.
method Analytical study of two types of timelike general rotational surfaces.
result Explicit descriptions of minimal and surfaces with specific curvature properties.
Study curves of constant breadth in a specific 3D manifold.
problem Differential geometry of curves in Walker 3-manifolds.
method Investigate curves of constant breadth using Darboux frame.
result Properties of curves of constant breadth in Walker 3-manifolds.
There are three types of hypersurfaces in a pseudoconformal space C^n_1 of Lorentzian signature: spacelike, timelike, and lightlike. These three types of hypersurfaces are considered in parallel. Spacelike hypersurfaces are endowed with a proper conformal structure, and timelike hypersurfaces are endowed with a conform…
This paper is devoted to the study of the singularity phenomenon of timelike extremal hypersurfaces in Minkowski spacetime R1+3. We find that there are two explicit lightlike self-similar solutions to a graph representation of timelike extremal hypersurfaces in Minkowski spacetime R1+3, the …
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.
In this paper, using the classifications of timelike and spacelike ruled surfaces, we study the Mannheim offsets of timelike ruled surfaces in Minkowski 3-space. Firstly, we define the Mannheim offsets of a timelike ruled surface by considering the Lorentzian casual character of the offset surface. We obtain that the M…
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.
In this paper, we introduce the dual geodesic trihedron (dual Darboux frame) of a timelike ruled surface. By the aid of the E. Study Mapping, we consider timelike ruled surfaces as dual hyperbolic spherical curves and define the Mannheim offsets of timelike ruled surfaces by means of dual Darboux frame. We obtain the r…
The study proves timelike Ricci bounds for low regularity spacetimes using optimal transport.
problem Proving timelike Ricci bounds for spacetimes with low regularity.
method Using optimal transport to prove timelike measure-contraction property.
result Timelike curvature-dimension condition holds for C1,1 metrics. Timelike curvature and Brunn-Minkowski inequality linked in non-smooth spacetimes.
problem Equivalence between timelike Ricci curvature and Brunn-Minkowski inequality in synthetic Lorentzian spaces.
method Introducing strong q-timelike Brunn-Minkowski condition and proving equivalence to curvature conditions. result Timelike curvature dimension condition equivalent to timelike Brunn-Minkowski inequality in specific settings.
The study classifies timelike meridian surfaces in Minkowski 4-space.
problem Characterizing timelike meridian surfaces in Minkowski space.
method Analyzing surfaces as one-parameter systems of meridians of rotational hypersurfaces.
result Classification of timelike meridian surfaces with various curvature properties.
Note shows equivalence of recent NEC reformulation to classical NEC for C2-metrics.
problem Consistency of null energy condition in Lorentzian length spaces.
method Shows equivalence of recent reformulation of null energy condition to classical formulation for C2-metrics. result Equivalence of recent reformulation of null energy condition to classical formulation for C2-metrics. 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.
Paper proves timelike minimal surfaces with any number of ends exist.
problem Existence of timelike minimal surfaces with arbitrary ends.
method Analyzes asymptotic behavior and topology of constructed surfaces.
result Timelike minimal surfaces with arbitrary ends exist.
We study the geometry of compact Lorentzian manifolds that admit a somewhere timelike Killing vector field, and whose isometry group has infinitely many connected components. Up to a finite cover, such manifolds are products (or amalgamated products) of a flat Lorentzian torus and a compact Riemannian (resp., lightlike…