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…
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48 results for “Lorentzian”
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 of a series of Lorentzian structures on SL(2,R) with SO(1,1) symmetry.
problem Global optimality of extremal trajectories in a series of Lorentzian structures.
method Analysis of a one-parametric series of left-invariant Lorentzian structures on SL(2,R) with SO(1,1) symmetry.
result Properties of the Lorentzian structures deform to those of the sub-Lorentzian structure in a limit case.
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.
It is shown how one can apply the classification of the holonomy algebras of Lorentzian manifolds to solve some problems. In particular, a new proof to the classification of Lorentzian manifolds with recurrent curvature tensor is given; the classification of two-symmetric Lorentzian manifolds is explained; conformally …
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 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.
Research explores Lorentzian distances on a specific geometric plane.
problem Investigating Lorentzian structures on a 2D geometric plane.
method Analyzes sectional curvature, attainable sets, and Lorentzian length maximizers.
result Describes distance properties and spheres in the context of Lorentzian geometry.
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.
The paper classifies 3D Lorentzian Lie groups.
problem Classifying 3D Lorentzian Lie groups.
method Complete classification through mathematical analysis.
result Three-dimensional Lorentzian Lie groups have been completely classified.
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.
Study of 2D Lorentzian anti-de Sitter plane using geometric control theory.
problem Understanding extremal trajectories and reachable set on anti-de Sitter plane.
method Geometric control theory and differential geometry.
result Construction of optimal synthesis and description of Lorentzian distance.
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.
Diagonalizes metrics of 3D Lorentzian manifolds.
problem Diagonalizing metrics of 3D Lorentzian manifolds.
method Applying the technique of moving frames.
result Every smooth Lorentzian 3-manifold admits an atlas with a diagonal metric.
Characterizes Lorentzian manifolds embeddable in Minkowski spacetime.
problem Identifying Lorentzian manifolds embeddable in Minkowski spacetime.
method Characterization and proof of embeddability conditions.
result Lorentzian manifolds embeddable in Minkowski spacetime coincide with globally hyperbolic spacetimes.
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.
Topology change and selection rules for high-dimensional $\Spin(1, n)_0$-Lorentzian cobordismsmath.GT
We study necessary and sufficient conditions for the existence of Lorentzian and weak Lorentzian cobordisms between closed smooth manifolds of arbitrary dimension such that the structure group of the frame bundle of the cobordism is $\Spin(1, n)_0$. This extends a result of Gibbons-Hawking on $\Sl(2, \C)$-Lorentzian co…
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.
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.
In this paper, we study the invariant and noninvariant hypersurfaces of (1,1,1) almost contact manifolds, Lorentzian almost paracontact manifolds and Lorentzian para-Sasakian manifolds, respectively. We show that a noninvariant hypersurface of an (1,1,1) almost contact manifold admits an almost product structure. We in…
If an -manifold 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 is said to be a \emph{Lorentzian similarity manifold}. A Lorentzian similarity manifold is also a conformally flat Lorentzia…
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.
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.
The Lorentzian length, which is one of the most significant functions in Lorentzian geometry, is a complex-valued function. Its square gives a real-valued non-degenerate quadratic function. In this paper, we define naturally extended mappings of Lorentzian distance-squared functions, wherein each component is a Lorentz…
Holonomy of Weyl connections in Lorentzian space classified.
problem Classifying holonomy algebras of Weyl connections in Lorentzian signature.
method Classification through construction of examples of Weyl connections with all possible holonomy algebras.
result Examples of Weyl connections with all possible holonomy algebras constructed.
A non-lorentzian primerhep-th
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.
Gromov's reconstruction theorem and measured Gromov-Hausdorff convergence in Lorentzian geometrymath.DG
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.
Two flat sub-Lorentzian problems on Martinet distribution differ in attainable set intersections.
problem Flat sub-Lorentzian structures on Martinet distribution.
method Analysis of attainable sets, optimal trajectories, sub-Lorentzian distances and spheres.
result The attainable set for the first problem intersects with the Martinet plane, while for the second it does not.
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…
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.
In this paper we define and study the slant and semi-slant submanifolds of a Lorentzian almost paracontact manifold. Some necessary and sufficient conditions for a submanifold of a Lorentzian almost paracontact manifold to be slant are obtained. We give some examples for the slant and semi-slant submanifolds of a Loren…
Affine Ricci Solitons associated to the Bott connection on three-dimensional Lorentzian Lie groupsmath.DG
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.
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…
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.
Formula for sectional curvature on 2D Lorentzian manifolds derived.
problem Calculating sectional curvature on 2D Lorentzian manifolds.
method Obtained a formula for sectional curvature.
result Formula for sectional curvature on 2D Lorentzian manifolds.
Existence of the longest arcs for left-invariant three-dimensional contact sub-Lorentzian structuresmath.DG
Researchers found the longest arcs for specific sub-Lorentzian structures.
problem Finding the longest arcs for sub-Lorentzian structures.
method Optimal control problem with unbounded control set and concave cost functional. Sufficient conditions for existence of longest arcs proposed.
result Existence of the longest arcs for left-invariant three-dimensional contact sub-Lorentzian structures proved.
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.
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.
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 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.
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.
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 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.
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.
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.
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.
The classification of all possible holonomy algebras of Einstein and vacuum Einstein Lorentzian manifolds is obtained. It is shown that each such algebra appears as the holonomy algebra of an Einstein (resp., vacuum Einstein) Lorentzian manifold, the direct constructions are given. Also the holonomy algebras of totally…
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.