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.
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.
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.
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.
Derives spacetime regularity under specific curvature conditions.
problem Ensuring smoothness in spacetime models with given curvature constraints.
method General regularity estimate for 4-d spacetimes, using Ricci curvature and Lie derivatives.
result Establishes conditions for smoothness in spacetime models.
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.
We discuss contact invariant structures on the space of solutions of a third-order ordinary differential equation. Associated to any third-order differential equation modulo contact transformations, Chern introduced a degenerate conformal Lorentzian metric on the space of 2-jets of functions of one variable. When the W…
The notion of Γ-symmetric space is a natural generalization of the classical notion of symmetric space based on $\z_2$-grading of Lie algebras. In our case, we consider homogeneous spaces G/H such that the Lie algebra $\g$ of G admits a Γ-grading where Γ is a finite abelian group. In this work we study Rieman…
Survey on warped products and their curvature properties.
problem Understanding warped products and their curvature bounds.
method Construction and analysis of warped products between manifolds and metric spaces.
result Warp products have nice curvature properties, especially sectional and Ricci bounds.
The paper extends Hawking's singularity theorem to metrics with Hölder continuity and bounded curvature.
problem Proving singularity theorems for metrics with low regularity.
method Combining elliptic RT-equations for metric regularisation and manifold convolution for curvature refinement.
result Establishes globally hyperbolic and timelike incompleteness for metrics with Hölder continuity and bounded curvature.
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.
Researchers generalize space forms in Riemannian geometry using specific vector fields.
problem Generalizing space forms in Riemannian geometry with a distinguished vector field.
method Proposing and studying pairs (g,T) of Riemannian metrics and vector fields, leading to Lorentzian metrics with constant curvature.
result Only flat, product manifolds with universal covering as a product of R and N are the only pairs (g,T) whose corresponding Lorentzian metric is a space form in the compact setting.
The notion of Γ-symmetric space is a natural generalization of the classical notion of symmetric space based on Z2-grading of Lie algebras. In our case, we consider homogeneous spaces G/H such that the Lie algebra $\g$ of G admits a Γ-grading where Γ is a finite abelian group. In this work we study Rieman…
Continuous Lorentzian metrics yield infinitesimal Minkowskian spacetimes.
problem Understanding spacetime properties from continuous Lorentzian metrics.
method Proving infinitesimal Minkowskianity for causally simple metric measure spacetimes.
result Continuous Lorentzian metrics result in spacetimes that are infinitesimally Minkowskian.
The paper classifies tensors on specific Lorentzian metrics.
problem Classification of tensors on homogeneous plane waves.
method Framework of BGG operators to derive explicit formulae.
result Explicit formulae for irreducible Killing and conformal Killing 2-tensors identified.
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…
Proposes a weaker version of Strong Cosmic Censorship with curvature bounds.
problem The original Strong Cosmic Censorship conjecture.
method Weakens the conjecture to allow manifolds with bounded curvature and Lipschitz continuity of metrics.
result Proves the conjecture with bounded curvature for sufficiently large p (p>4 with uniform bounds, p>2 without uniform bounds).
We describe the structure of d-dimensional homogeneous Lorentzian G-manifolds M=G/H of a semisimple Lie group G. Due to a result by N. Kowalsky, it is sufficient to consider the case when the group G acts properly, that is the stabilizer H is compact. Then any homogeneous space G/Hˉ with a smaller gro…
We use a construction which we call generalized cylinders to give a new proof of the fundamental theorem of hypersurface theory. It has the advantage of being very simple and the result directly extends to semi-Riemannian manifolds and to embeddings into spaces of constant curvature. We also give a new way to identify …
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.
We determine all Ricci flat left invariant Lorentzian metrics on simply connected 2-step nilpotent Lie groups. We show that the 2k+1-dimensional Heisenberg Lie group H2k+1 carries a Ricci flat left invariant Lorentzian metric if and only if k=1. We show also that for any 2≤q≤k, H2k+1 carries a R…
Maximal causal curves for Lipschitz metrics are either lightlike or timelike.
problem Characterizing maximal causal curves for Lipschitz metrics.
method Analyzing the parametrization and geodesic equation for maximal causal curves in terms of Filippov solutions.
result Maximal causal curves for Lipschitz metrics are either everywhere lightlike or everywhere timelike.
Formula derived for curvature on smooth manifolds.
problem Calculating curvature on smooth manifolds.
method Derived a formula for sectional curvature.
result Formula for sectional curvature on smooth manifolds.
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.
We consider three-dimensional Lorentzian metrics that locally admit four independent Killing vectors. Their classification is summarized, and conditions for characterizing them are found. These consist of algebraic classification of the traceless Ricci tensor, and other conditions satisfied by the curvature and its der…
The study finds obstructions for certain Weyl curvature tensors on manifolds.
problem Can manifolds admit metrics with purely electric or magnetic Weyl tensors?
method Analyzes algebraic curvature tensors and their Pontryagin classes on scalar product spaces.
result Obstructions to the existence of metrics with PE or PM Weyl tensors in top-degree cohomology.
Researchers found differential invariants for Kundt spacetimes.
problem Equivalence problem for Lorentzian metrics in Kundt spacetimes.
method Found generators for rational differential invariants for Kundt spacetimes.
result Relating findings to other approaches to the equivalence problem.
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.
We construct normal forms for Lorentzian metrics on Engel distributions under the assumption that abnormal curves are timelike future directed Hamiltonian geodesics. Then we indicate some cases in which the abnormal timelike future directed curve initiating at the origin is geometrically optimal. We also give certain e…
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. Study on completeness of metrics on specific Lie groups.
problem Completeness of left-invariant Lorentzian metrics on 3D non-unimodular Lie groups.
method Analyzing metrics with Lie algebra of the form R⋉AR2 for various A. result Determine all geodesically complete and incomplete metrics for different cases of A. We consider Lie groups equipped with a left-invariant cyclic Lorentzian metric. As in the Riemannian case, in terms of homogeneous structures, such metrics can be considered as different as possible from bi-invariant metrics. We show that several results concerning cyclic Riemannian metrics do not extend to their Loren…
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.
Study geometric isomorphisms between spacetime solutions using paracausal metrics.
problem Geometric isomorphisms between solutions of normally hyperbolic operators over different spacetimes.
method Introduce paracausal relation to define isomorphisms between spacetime metrics and use Møller operators.
result Møller operators preserve causal propagators and natural symplectic forms on initial data.
Generic singularities found in spacetimes with weakly trapped submanifolds.
problem Existence of singularities in spacetimes with weakly trapped submanifolds.
method Using Whitney topologies and singularity theorems, the study shows genericity of singularities in spacetimes with weakly trapped submanifolds.
result Generic singularities are found in spacetimes with weakly trapped submanifolds.
We prove that the "generic condition" used in singularity theorems of general relativity is generic in the space of Lorentzian metrics on a given manifold, in the sense that it is satisfied for all metrics in a residual set in the Whitney Ck-topology, for k depending on the dimension of the manifold.
New metrics constructed dual to specific wave-like geometries.
problem Constructing metrics dual to general plane-fronted wave Lorentzian metrics.
method Explains construction of extremal and non-Kähler almost-Kähler metrics.
result Constructs canonical almost-Kähler metrics dual to general plane-fronted wave Lorentzian metrics.
Study on recovering Lorentzian metrics from scattering data.
problem Recovering Lorentzian metrics from scattering data on a boundary.
method Analyzing the role of boundary distance functions and linearizing the light ray transform.
result Scattering rigidity can be reduced to boundary rigidity of magnetic systems.
We consider a class of S1-bundles whose total space admits a nowhere vanishing recurrent lightlike vector field with respect to a Lorentzian metric. This metric can be modified such that its restricted holonomy group is indecomposable and reducible. We apply Hodge theory to construct examples with Hermitian screen…
We investigate connections between pairs of (pseudo-)Riemannian metrics whose sum is a (tensor) product of a covector field with itself. A bijective mapping between the classes of Euclidean and Lorentzian metrics is constructed as a special result. The existence of such maps on a differentiable manifold is discussed. S…
We derive all possible causality conditions for conformally flat Lorentzian metrics on the two-dimensional cylinder.
We work on a 4-manifold equipped with Lorentzian metric g and consider a volume-preserving diffeomorphism which is the unknown quantity of our mathematical model. The diffeomorphism defines a second Lorentzian metric h, the pullback of g. Motivated by elasticity theory, we introduce a Lagrangian expressed algebra…
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. We show that the Hawking--Penrose singularity theorem, and the generalisation of this theorem due to Galloway and Senovilla, continue to hold for Lorentzian metrics that are of C1,1-regularity. We formulate appropriate weak versions of the strong energy condition and genericity condition for C1,1-metrics, an…
Novel Hodge theory developed for Carrollian bundles, addressing black hole event horizons.
problem Developing Hodge theory on Carrollian bundles with degenerate metrics.
method Constructing Hodge star and Laplacian operators on Carrollian Rimes-bundles. result Hodge--de Rham Laplacian defined on event horizons of Schwarzschild black holes.
Study recovers Lorentzian metrics from boundary data, proving local rigidity.
problem Recovering a Lorentzian metric from scattering data on a boundary.
method Analyzes jet and real analyticity of metrics near lightlike points.
result Metric can be recovered up to gauge transformations near lightlike strictly convex points.
Extends singularity theorems to low regularity metrics.
problem Proving singularity theorems for metrics with low regularity.
method Careful estimates of curvature and stability properties of geodesics.
result Complete proof of Hawking and Penrose singularity theorems for C1-Lorentzian metrics. Researchers find counterexamples to inverse problems for wave equations.
problem Inverse problems for wave equations on domains and Lorentzian manifolds.
method Constructing non-isometric Lorentzian metrics leading to same partial data measurements.
result Non-isometric Lorentzian metrics can produce identical partial data measurements.