Study examines how Lorentz transformations affect foliations in spacetime.
problem Understanding foliations and their relation to observers under Lorentz transformations.
method Exploration of kinematic characteristics of timelike congruences (observers) and their relation to foliations.
result Local Lorentz transformations impact the existence of transversal foliations.
We consider the kinematics of specific fluid spacetimes admitting timelike congruences of Ricci Solitons. These fluids includes string cloud, string fluid, perfect fluid, radially symmetric fluid, anisotropic fluid and relativistic magneto-fluid. Results are obtained and important physical aspects are discussed.
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…
We develop a new approach on the (1+3) threading of spacetime (M,g) with respect to a congruence of curves defined by an arbitrary timelike vector field. The study is based on spatial tensor fields and on the Riemannian spatial connection ∇⋆, which behave as 3D geometric objects. We obtain new formul…
Study on braid group quotients by congruence subgroups.
problem Understanding the image of congruence subgroups in GL(n,Z).
method Characterization through symplectic congruence subgroups.
result Open problem solved: image of congruence subgroups in GL(n,Z).
The paper describes the geometric properties of line congruences' singularities.
problem Understanding the singularities of generic line congruences.
method Use of an equiaffine pair to define generic line congruences.
result Geometric description of folds, cusps, and swallowtails as singularities of generic line congruences.
The paper classifies singularities of line congruences in 4D space.
problem Classifying singularities of line congruences in 4D space.
method Generic classification approach for 3-parameter line congruences and Blaschke normal congruences.
result Generic classification of singularities of 3-parameter line congruences in R4. Study on braid groups' congruence subgroups and their crystallographic quotients.
problem Understanding congruence subgroups and crystallographic quotients of braid groups.
method Investigation of lower central series of congruence braid groups related to B3. result Quotients of congruence braid groups are almost crystallographic.
The paper classifies singularities of plane congruences and affine distance functions.
problem Classifying singularities of plane congruences and affine distance functions.
method Classification through 2-parameter plane congruences in \(\mathbb{R^4}\) and affine normal plane congruences.
result Generic singularities of plane congruences and affine distance functions are classified.
Let f be an integer greater than one. We study three progressively finer equivalence relations on closed 3-manifolds generated by Dehn surgery with denominator f: weak f-congruence, f-congruence, and strong f-congruence. If f is odd, weak f-congruence preserves the ring structure on cohomology with Z_f-coefficients. We…
This paper explores geometric insights into discrete R-congruences and their envelopes.
problem Understanding the ambiguity in discrete R-congruences and their envelopes.
method Analyzes discrete R-congruences that are enveloped by specific types of surfaces and maps.
result Discovers a 2-parameter family of discrete enveloping surfaces for discrete R-congruences.
We enumerate all the principal congruence link complements in S3, there by answering a question of W. Thurston. Related articles: "Technical Report: All Principal Congruence Link Groups" (arXiv:1902.04722), "All Known Principal Congruence Links" (arXiv:1902.04426).
Paper explores relations between braid groups and their quotients.
problem Understanding relations between braid groups and their quotients.
method Recalling and introducing elements of congruence braid groups, establishing isomorphisms between crystallographic and congruence braid groups.
result Established isomorphisms between crystallographic braid groups and quotients of congruence braid groups.
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.
New BDEs reveal singular surfaces from line congruences.
problem Understanding binary differential equations associated with line congruences.
method Applied pointwise to quadratic differential forms, studying quotients of quadratic forms and associated polar lines.
result Introduced a new singular surface in Euclidean 3-space.
The study explores congruence subgroups of braid groups and their quotients.
problem Understanding the structure of congruence subgroups of braid groups.
method Utilizing the integral Burau representation and results from integral matrices, the study examines quotients of these subgroups.
result Findings of quotients that are not isomorphic to symmetric groups.
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…
Criterion for congruence RFRS towers in hyperbolic lattices.
problem Criteria for hyperbolic lattices to have congruence RFRS towers.
method Criterion based on congruence subgroups.
result Virtually fibered Bianchi groups and new RFRS Kähler groups.
Study of line congruences for Appell's rank-4 hypergeometric functions.
problem Understanding line congruences for Appell's rank-4 hypergeometric functions.
method Derived original formulae for Laplace transform of rank-4 system, applied to geometry of surfaces defined by these functions.
result Natural line congruences for Laplace transforms of Appell's rank-4 functions form a W-congruence.
We give a sufficient condition for isometric actions to have the congruency of orbits, that is, all orbits are isometrically congruent to each other. As applications, we give simple and unified proofs for some known congruence results, and also provide new examples of isometric actions on symmetric spaces of noncompact…
Characterizes W-congruences to study their stable umbilical points.
problem Understanding singularities of W-congruences.
method Characterization of W-congruences to study umbilical points.
result Examples of isolated stable umbilical points of type Am. 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. We continue the investigation of the correspondence between systems of conservation laws and congruences of lines in projective space. Relationship between "additional" conservation laws and hypersurfaces conjugate to a congruence is established. This construction allows us to introduce, in a purely geometric way, the …
Proves congruence subgroup property for mapping class groups of hyperbolic surfaces.
problem Residual finiteness of hyperbolic groups and congruence subgroup property for mapping class groups.
method Assumption of residual finiteness of hyperbolic groups leads to proof of congruence subgroup property.
result Congruence subgroup property for mapping class groups of hyperbolic surfaces.
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.
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.
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.
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.
Extends Kummer's theory to singular surfaces for line congruences.
problem Applying Kummer's theory to singular surfaces for line congruences.
method Analyzing the equation of principal surfaces and developable surfaces for normal congruences.
result The multiplicative factor for the principal surfaces is associated with the singular set of ξ. 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.
Timelike Thomsen surfaces are timelike minimal surfaces that are also affine minimal. In this paper, we make use of both the Lorentz conformal coordinates and the null coordinates, and their respective representation theorems of timelike minimal surfaces, to obtain a complete global classification of these surfaces and…
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.
Timelike curves in Lorentzian length spaces have a total curvature notion that agrees with smooth curves.
problem Estimating the length of timelike curves in Lorentzian length spaces.
method Introducing a synthetic timelike total curvature notion.
result Proving timelike curves of finite total curvature are rectifiable.
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.
We study "how far away" a finite index subgroup G of SL(2,Z) is from being a congruence group. For this we define its deficiency of being a congruence group. We show that the index of the image of G in SL(2,Z/nZ) is biggest, if n is the general Wohlfahrt level. We furthermore show that the Veech groups of origamis (or …
We introduce a new approach to the study of timelike minimal surfaces in the Lorentz-Minkowski space through a split-complex representation formula for this kind of surface. As applications, we solve the Björling problem for timelike surfaces and obtain interesting examples and related results. Using the Björling repre…
Let S be a smooth affine algebraic curve, and let S' be the Riemann surface obtained by removing a point from S. We provide evidence for the congruence subgroup property of the mapping class group Mod(S') by showing that its congruence kernel lies in the centralizer of every braid in Mod(S'). As a corollary, we obtain …
Study timelike meridian surfaces in Minkowski 4-space with specific properties.
problem Characterize timelike meridian surfaces with special properties in Minkowski 4-space.
method Analyze different classes of timelike meridian surfaces with constant Gauss curvature, mean curvature, and parallel normalized mean curvature vector field.
result Explicit solutions to PDEs describing timelike surfaces with parallel normalized mean curvature vector field.
Study timelike minimal surfaces in Heisenberg group using harmonic maps.
problem Characterize timelike minimal surfaces in the Heisenberg group.
method Characterization through non-conformal Lorentz harmonic maps and generalized Weierstrass type representation.
result Established generalized Weierstrass type representation for timelike minimal surfaces.
Study minimal timelike surfaces in 3D Lorentz-Minkowski space using holomorphic functions.
problem Characterize minimal timelike surfaces in R13. method Use a Weierstrass-type formula with holomorphic functions in split-complex numbers to find canonical parameters and corresponding holomorphic functions.
result Enneper surfaces are the only minimal timelike surfaces with polynomial parametrization of degree 3 in isothermal parameters.
The paper explores affine geometry of line congruences using singularity theory.
problem Understanding the affine geometry of line congruences and their focal sets.
method Use of singularity theory to describe generic phenomena and singularities.
result Identification of a key projective quadric in the tangent space.
Defines timelike ideal boundary for non-positively curved Lorentzian spaces.
problem Understanding the geometry of non-positively curved Lorentzian spaces.
method Introduces timelike ideal boundary as asymptotic classes of geodesic rays, endows with topology and metric, and studies upper curvature bounds.
result Established upper curvature bounds for the resulting metric space.
The paper characterizes timelike rectifying curves in De Sitter 3-space.
problem Characterizing timelike rectifying curves in De Sitter 3-space.
method Defining timelike rectifying curves and conical surfaces, providing characterizations and results.
result Characterizations and results of timelike rectifying curves in De Sitter 3-space.
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.
It is known that the level 2 principal congruence subgroup of GL(n;Z) has a finite generating set. In this paper, we give a finite presentation of the level 2 principal congruence subgroup of GL(n;Z).